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journal.pcbi.1006637

Dynamical anchoring of distant arrhythmia sources by fibrotic regions via restructuring of the activation pattern

Many clinically relevant cardiac arrhythmias are conjectured to be organized by rotors ., A rotor is an extension of the concept of a reentrant source of excitation into two or three dimensions with an area of functional block in its center , referred to as the core ., Rapid and complex reentry arrhythmias such as atrial fibrillation ( AF ) and ventricular fibrillation ( VF ) are thought to be driven by single or multiple rotors ., A clinical study by Narayan et al . 1 indicated that localized rotors were present in 68% of cases of sustained AF ., Rotors ( phase singularities ) were also found in VF induced by burst pacing in patients undergoing cardiac surgery 2 , 3 and in VF induced in patients undergoing ablation procedures for ventricular arrhythmias 4 ., Intramural rotors were also reported in early phase of VF in the human Langendorff perfused hearts 5 , 6 ., It was also demonstrated that in most cases rotors originate and stabilize in specific locations 4–8 ., A main mechanism of rotor stabilization at a particular site in cardiac tissue was proposed in the seminal paper from the group of Jalife 9 ., It was observed that rotors can anchor and exhibit a stable rotation around small arteries or bands of connective tissue ., Later , it was experimentally demonstrated that rotors in atrial fibrillation in a sheep heart can anchor in regions of large spatial gradients in wall thickness 10 ., A recent study of AF in the right atrium of the explanted human heart 11 revealed that rotors were anchored by 3D micro-anatomic tracks formed by atrial pectinate muscles and characterized by increased interstitial fibrosis ., The relation of fibrosis and anchoring in atrial fibrillation was also demonstrated in several other experimental and numerical studies 8 , 11–14 ., Initiation and anchoring of rotors in regions with increased intramural fibrosis and fibrotic scars was also observed in ventricles 5 , 7 , 15 ., One of the reasons for rotors to be present at the fibrotic scar locations is that the rotors can be initiated at the scars ( see e . g . 7 , 15 ) and therefore they can easily anchor at the surrounding scar tissue ., However , rotors can also be generated due to different mechanisms , such as triggered activity 16 , heterogeneity in the refractory period 16 , 17 , local neurotransmitter release 18 , 19 etc ., What will be the effect of the presence of the scar on rotors in that situation , do fibrotic areas ( scars ) actively affect rotor dynamics even if they are initially located at some distance from them ?, In view of the multiple observations on correlation of anchoring sites of the rotors with fibrotic tissue this question translates to the following: is this anchoring just a passive probabilistic process , or do fibrotic areas ( scars ) actively affect the rotor dynamics leading to this anchoring ?, Answering these questions in experimental and clinical research is challenging as it requires systematic reproducible studies of rotors in a controlled environment with various types of anchoring sites ., Therefore alternative methods , such as realistic computer modeling of the anchoring phenomenon , which has been extremely helpful in prior studies , are of great interest ., The aim of this study is therefore to investigate the processes leading to anchoring of rotors to fibrotic areas ., Our hypothesis is that a fibrotic scar actively affects the rotor dynamics leading to its anchoring ., To show that , we first performed a generic in-silico study on rotor dynamics in conditions where the rotor was initiated at different distances from fibrotic scars with different properties ., We found that in most cases , scars actively affect the rotor dynamics via a dynamical reorganization of the excitation pattern leading to the anchoring of rotors ., This turned out to be a robust process working for rotors located even at distances more than 10 cm from the scar region ., We then confirmed this phenomenon in a patient-specific model of the left ventricle from a patient with remote myocardial infarction ( MI ) and compared the properties of this process with clinical ECG recordings obtained during induction of a ventricular arrhythmia ., Our anatomical model is based on an individual heart of a post-MI patient reconstructed from late gadolinium enhanced ( LGE ) magnetic resonance imaging ( MRI ) was described in detail previously 20 ., Briefly , a 1 . 5T Gyroscan ACS-NT/Intera MR system ( Philips Medical Systems , Best , the Netherlands ) system was used with standardized cardiac MR imaging protocol ., The contrast –gadolinium ( Magnevist , Schering , Berlin , Germany ) ( 0 . 15 mmol/kg ) – was injected 15 min before acquisition of the LGE sequences ., Images were acquired with 24 levels in short-axis view after 600–700 ms of the R-wave on the ECG within 1 or 2 breath holds ., The in-plane image resolution is 1 mm and through-plane image resolution is 5 mm ., Segmentation of the contours for the endocardium and the epicardium was performed semi-automatically on the short-axis views using the MASS software ( Research version 2014 , Leiden University Medical Centre , Leiden , the Netherlands ) ., The myocardial scar was identified based on signal intensity ( SI ) values using a validated algorithm as described by Roes et al . 21 ., In accordance with the algorithm , the core necrotic scar is defined as a region with SI >41% of the maximal SI ., Regions with lower SI values were considered as border zone areas ., In these regions , we assigned the fibrosis percentage as normalized values of the SI as in Vigmond et al . 22 ., In the current paper , fibrosis was introduced by generating a random number between 0 and 1 for each grid point and if the random number was less than the normalized SI at the corresponding pixel the grid point was considered as fibroblast ., Currently there is no consensus on how the SI values should be used for clinical assessment of myocardial fibrosis and various methods have been reported to produce significantly different results 23 ., However , the method from Vigmond et al . properly describes the location of the necrotic scar region in our model as for the fibrosis percentage of more than 41% we observe a complete block of propagation inside the scar ., This means that all tissue which has a fibrotic level higher than 41% behaves like necrotic scar ., The approach and the 2D model was described in detail in previous work 24–26 ., Briefly , for ventricular cardiomyocyte we used the ten Tusscher and Panfilov ( TP06 ) model 27 , 28 , and the cardiac tissue was modeled as a rectangular grid of 1024 × 512 nodes ., Each node represented a cell that occupied an area of 250 × 250 μm2 ., The equations for the transmembrane voltage are given by, C m d V i k d t = ∑ α , β ∈ { - 1 , + 1 } η i k α β g gap ( V i + α , k + β - V i k ) - I ion ( V i k , … ) , ( 1 ), where Vik is the transmembrane voltage at the ( i , k ) computational node , Cm is membrane capacitance , ggap is the conductance of the gap junctions connecting two neighboring myocytes , Iion is the sum of all ionic currents and η i k α β is the connectivity tensor whose elements are either one or zero depending on whether neighboring cells are coupled or not ., Conductance of the gap junctions ggap was taken to be 103 . 6 nS , which results in a maximum velocity planar wave propagation in the absence of fibrotic tissue of 72 cm/s at a stimulation frequency of 1 Hz ., ggap was not modified in the fibrotic areas ., A similar system of differential equations was used for the 3D computations where instead of the 2D connectivity tensor η i k α β we used a 3D weights tensor w i j k α β γ whose elements were in between 0 and 1 , depending both on coupling of the neighbor cells and anisotropy due to fiber orientation ., Each node in the 3D model represented a cell of the size of 250 × 250 × 250 μm3 ., 20s of simulation in 3D took about 3 hours ., Fibrosis was modeled by the introduction of electrically uncoupled unexcitable nodes 29 ., The local percentage of fibrosis determined the probability for a node of the computational grid to become an unexcitable obstacle , meaning that for high percentages of fibrosis , there is a high chance for a node to be unexcitable ., As previous research has demonstrated that LGE-MRI enhancement correlates with regions of fibrosis identified by histological examination 30 , we linearly interpolated the SI into the percentage of fibrosis for the 3D human models ., In addition , the effect of ionic remodeling in fibrotic regions was taken into account for several results of the paper 31 , 32 ., To describe ionic remodeling we decreased the conductance of INa , IKr , and IKs and depending on local fibrosis level as:, G Na = ( 1 - 1 . 55 f 100 % ) G Na 0 , ( 2 ) G Kr = ( 1 - 1 . 75 f 100 % ) G Kr 0 , ( 3 ) G Ks = ( 1 - 2 f 100 % ) G Ks 0 , ( 4 ), where GX is the peak conductance of IX ionic current , G X 0 is the peak conductance of the current in the absence of remodeling , and f is the local fibrosis level in percent ., These formulas yield a reduction of 62% for INa , of 70% for IKr , and of 80% for IKs if the local fibrosis f is 40% ., These values of reduction are , therefore , in agreement with the values published in 33 , 34 ., The normal conduction velocity at CL 1000 ms is 72 cm/s ( CL 1000 ms ) ., However , as the compact scar is surrounded by fibrotic tissue , the velocity of propagation in that region gradually decreases with the increase in the fibrosis percentage ., For example for fibrosis of 30% , the velocity decreases to 48 cm/s ( CL 1000 ms ) ., We refer to Figure 1 in Ten Tusscher et al 25 for the planar conduction velocity as a function of the percentage fibrosis in 2D tissue and 3D tissue ., The geometry and extent of fibrosis in the human left ventricles were determined using the LGE MRI data ., The normalized signal intensity was used to determine the density of local fibrosis ., The fiber orientation is presented in detail in the supplementary S1 Appendix ., The model for cardiac tissue was solved by the forward Euler integration scheme with a time step of 0 . 02 ms . The numerical solver was implemented using the CUDA toolkit for performing the computations on graphics processing units ., Simulations were performed on a GeForce GTX Titan Black graphics card using single precision calculations ., The eikonal equations for anisotropy generation were solved by the fast marching Sethian’s method 35 ., The eikonal solver and the 3D model generation pipeline were implemented in the OCaml programming language ., Rotors were initiated by an S1S2 protocol , as shown in the supplementary S1 Fig . Similarly , in the whole heart simulations , spiral waves ( or scroll waves ) were created by an S1S2 protocol ., For the compact scar geometry used in our simulations the rotation of the spiral wave was stationary , the period of rotation of the anchored rotor was always more than 280 ms , while the period of the spiral wave was close to 220 msec ., Therefore , we determined anchoring as follows: if the period of the excitation pattern was larger than 280 ms over a measuring time interval of 320 ms we classified the excitation as anchored ., When the type of anchoring pattern was important ( single or multi-armed spiral wave ) we determined it visually ., If in all points of the tissue , the voltage was below -20 mV , the pattern was classified as terminated ., We applied the classification algorithm at t = 40 s in the simulation ., In the whole heart , the pseudo ECGs were calculated by assuming an infinite volume conductor and calculating the dipole source density of the membrane potential Vm in all voxel points of the ventricular myocardium , using the following equation 36, E C G ( t ) = ∫ ( r → , D ( r → ) ∇ → V ( t ) ) | r → | 3 d 3 r ( 5 ), whereby D is the diffusion tensor , V is the voltage , and r → is the vector from each point of the tissue to the recording electrode ., The recording electrode was placed 10 cm from the center of the ventricles in the transverse plane ., Twelve-lead ECGs of all induced ventricular tachycardia ( VT ) of patients with prior myocardial infarction who underwent radiofrequency catheter ablation ( RFCA ) for monomorphic VT at LUMC were reviewed ., All patients provided informed consent and were treated according to the clinical protocol ., Programmed electrical stimulation ( PES ) is routinely performed before RFCA to determine inducibility of the clinical/presumed clinical VT ., All the patients underwent PES and ablation according to the standard clinical protocol , therefore no ethical approval was required ., Ablation typically targets the substrate for scar-related reentry VT ., After ablation PES is repeated to test for re-inducibility and evaluate morphology and cycle length of remaining VTs ., The significance of non-clinical , fast VTs is unclear and these VTs are often not targeted by RFCA ., PES consisted of three drive cycle lengths ( 600 , 500 and 400 ms ) , one to three ventricular extrastimuli ( ≥200 ms ) and burst pacing ( CL ≥200 ms ) from at least two right ventricular ( RV ) sites and one LV site ., A positive endpoint for stimulation is the induction of any sustained monomorphic VT lasting 30 s or requiring termination ., ECG and intracardiac electrograms ( EG ) during PES were displayed and recorded simultaneously on a 48-channel acquisition system ( Prucka CardioLab EP system , GE Healthcare , USA ) for off-line analysis ., Fibrotic scars can not only anchor the rotors but can dynamically anchor them from a large distance ., In the first experiments we studied spiral wave dynamics with and without a fibrotic scar in a generic study ., The diameter of the fibrotic region was 6 . 4 cm , based on the similar size of the scars from patients with documented and induced VT ( see the Methods section , Magnetic Resonance Imaging ) ., The percentage of fibrosis changed linearly from 50% at the center of the scar to 0% at the scar boundary ., We initiated a rotor at a distance of 15 . 5 cm from the scar ( Fig 1 , panel A ) which had a period of 222 ms and studied its dynamics ., First , after several seconds the activation pattern became less regular and a few secondary wave breaks appeared at the fibrotic region ( Fig 1 , panel B ) ., These irregularities started to propagate towards the tip of the initial rotor ( Fig 1 , panel C-D ) creating a complex activation picture in between the scar and the initial rotor ., Next , one of the secondary sources reached the tip of the original rotor ( Fig 1 , panel E ) ., Then , this secondary source merged with the initial rotor ( Fig 1 , panel F ) , which resulted in a deceleration of the activation pattern and promoted a chain reaction of annihilation of all the secondary wavebreaks in the vicinity of the original rotor ., At this moment , a secondary source located more closely to the scar dominated the simulation ( Fig 1 , panel G ) ., The whole process now started again ( Fig 1 , panels H-K ) , until finally only one source became the primary source anchored to the scar ( Fig 1L ) with a rotation period of 307 ms . For clarity , a movie of this process is provided as supplementary S1 Movie ., Note that this process occurs only if a scar with surrounding fibrotic zone was present ., In the simulation entitled as ‘No scar’ in Fig 1 , we show a control experiment when the same initial conditions were used in tissue without a scar ., In the panel entitled as ‘Necrotic scar’ in Fig 1 , a simulation with only a compact region without the surrounding fibrotic tissue is shown ., In both cases the rotor was stable and located at its initial position during the whole period of simulation ., The important difference here from the processes shown in Fig 1 ( Fibrotic scar ) is that in cases of ‘No scar’ and ‘Necrotic scar’ no new wavebreaks occur and thus we do not have a complex dynamical process of re-arrangement of the excitation patterns ., We refer to this complex dynamical process leading to anchoring of a distant rotor as dynamical anchoring ., Although this process contains a phase of complex behaviour , overall it is extremely robust and reproducible in a very wide range of conditions ., In the second series of simulations , the initial rotor was placed at different distances from the scar border , ranging from 1 . 8 to 14 . 3 cm , to define the possible outcomes , see Fig 2 . Here , in addition to a single anchored rotor shown in Fig 1H we could also obtain other final outcomes of dynamical anchoring: we obtained rotors rotating in the opposite direction ( Fig 2A , top ) , double armed anchored rotors which had 2 wavefronts rotating around the fibrotic regions ( Fig 2A , middle ) or annihilation of the rotors ( Fig 2A , bottom , which show shows no wave around the scar ) , which normally occurred as a result of annihilation of a figure-eight-reentrant pattern ., To summarize , we therefore had the following possible outcomes:, Termination of activity A rotor rotating either clockwise or counter-clockwise A two- or three-armed rotor rotating either clockwise or counter-clockwise, Fig 2 , panel B presents the relative chance of the mentioned activation patterns to occur depending on the distance between the rotor and the border of the scar ., We see , indeed , that for smaller initial distances the resulting activation pattern is always a single rotor rotating in the same direction ., With increasing distance , other anchoring patterns are possible ., If the distance was larger than about 9 cm , there is at least a 50% chance to obtain either a multi-armed rotor or termination of activity ., Also note that such dynamical anchoring occurred from huge distances: we studied rotors located up to 14 cm from the scar ., However , we observed that even for very large distances such as 25 cm or more such dynamical anchoring ( or termination of the activation pattern ) was always possible , provided enough time was given ., We measured the time required for the anchoring of rotors as a function of the distance from the scar ., For each distance , we performed about 60 computations using different seed values of the random number generator , both with and without taking ionic remodeling into account ., The results of these simulations are shown in Fig 3 . We see that the time needed for dynamical anchoring depends linearly on the distance between the border of the scar and the initial rotor ., The blue and yellow lines correspond to the scar model with and without ionic remodeling , respectively ( ionic remodeling was modelled by decreasing the conductance of INa , IKr , and IKs as explained in the Methods Section ) ., We interpret these results as follows; The anchoring time is mainly determined by the propagation of the chaotic regime towards the core of the original rotor and this process has a clear linear dependency ., For distant rotors , propagation of this chaotic regime mainly occurs outside the region of ionic remodelling , and thus both curves in Fig 3 have the same slope ., However , in the presence of ionic remodelling , the APD in the scar region is prolonged ., This creates a heterogeneity and as a consequence the initial breaks in the scar region are formed about 3 . 5 s earlier in the scar model with remodeling compared with the scar model without remodeling ., To identify some properties of the substrate necessary for the dynamical anchoring we varied the size and the level of fibrosis within the scar and studied if the dynamical anchoring was present ., Due to the stochastic nature of the fibrosis layout we performed about 300 computations with different textures of the fibrosis for each given combination of the scar size and the fibrosis level ., The results of this experiment are shown in Fig 4 . Dynamical anchoring does not occur when the scar diameter was below 2 . 6 cm , see Fig 4 . For scars of such small size we observed the absence of both the breakup and dynamical anchoring ., We explain this by the fact that if the initial separation of wavebreaks formed at the scar is small , the two secondary sources merge immediately , repairing the wavefront shape and preventing formation of secondary sources 37 ., Also , we see that this effect requires an intermediate level of fibrosis density ., For small fibrosis levels no secondary breaks are formed ( close to the boundary of the fibrotic tissue ) ., Also , no breaks could be formed if the fibrosis level is larger than 41% in our 2D model ( i . e . closer to the core ) , as the tissue behaves like an inexcitable scar ., For a fibrosis > 41% the scar effectively becomes a large obstacle that is incapable of breaking the waves of the original rotor 37 ., Close to the threshold of 41% we have also observed another interesting pattern when the breaks are formed inside the core of the scar ( inside the > 41% region ) only and cannot exit to the surrounding tissue , see the supplementary S1 Movie ., Finally , note that Fig 4 illustrates only a few factors important for the dynamical anchoring in a simple setup in an isotropic model of cardiac tissue ., The particular values of the fibrosis level and the size of the scar can also depend on anisotropy , the texture of the fibrosis and its possible heterogeneous distribution ., To verify that the dynamical anchoring takes place in a more realistic geometry , we developed and investigated this effect in a patient-specific model of the human left ventricle , see the Method section for details ., The scar in this dataset has a complex geometry with several compact regions with size around 5-7 cm in which the percentage of fibrosis changes gradually from 0% to 41% at the core of the scar based on the imaging data , see Methods section ., The remodeling of ionic channels at the whole scar region was also included to the model ( including borderzone as described the Fibrosis Model in the method section ) ., We studied the phenomenon of dynamical anchoring for 16 different locations of cores of the rotor randomly distributed in a slice of the heart at about 4 cm from the apex ( see Fig 5 ) ., Cardiac anisotropy was generated by a rule-based approach described in details in the Methods section ( Model of the Human Left Ventricle ) ., Of the 16 initial locations , shown in Fig 5 , there was dynamical anchoring to the fibrotic tissue in all cases , with and without ionic remodeling ., After the anchoring , in 4 cases the rotor annihilated ., The effect of the attraction was augmented by the electrophysiolical remodelling , similar as in 2D ., A representative example of our 3D simulations is shown in Fig 5 . We followed the same protocol as for the 2D simulations ., The top 2 rows the modified anterior view and the modified posterior view in the case the scar was present ., In column A , we see the original location of the spiral core ( 5 cm from the scar ) indicated with the black arrow in anterior view ., In column B , breaks are formed due to the scar tissue , and the secondary source started to appear ., After 3 . 7 s , the spiral is anchored around the scar , indicated with the black arrow in the posterior view , and persistently rotated around it ., In the bottom row , we show the same simulation but the scar was not taken into account ., In this case , the spiral does not change its original location ( only a slight movement , see the black arrows ) ., To evaluate if this effect can potentially be registered in clinical practice we computed the ECG for our 3D simulations ., The ECG that corresponds to the example in Fig 5 is shown in Fig 6 . During the first three seconds , the ECG shows QRS complexes varying in amplitude and shape and then more uniform beat-to-beat QRS morphology with a larger amplitude ., This change in morphology is associated with anchoring of the rotor which occurs around three seconds after the start of the simulation ., The initial irregularity is due to the presence of the secondary sources that have a slightly higher period than the original rotor ., After the rotor is anchored , the pattern becomes relatively stable which corresponds to a regular saw-tooth ECG morphology ., Additional ECGs for the cases of termination of the arrhythmia and anchoring are shown in supplementary S2 Fig . For the anchoring dynamics we see similar changes in the ECG morphology as in Fig 6 . The dynamical anchoring is accompanied by an increase of the cycle length ( 247 ± 16 ms versus 295 ± 30 ms ) ., The reason for this effect is that the rotation of the rotor around an obstacle –anatomical reentry– is usually slower than the rotation of the rotor around its own tip—functional reentry , which is typically at the limit of cycle length permitted by the ERP ., In the previous section , we showed that the described results on dynamical anchoring in an anatomical model of the LV of patients with post infarct scars correspond to the observations on ECGs during initiation of a ventricular arrhythmia ., After initiation , in 18 out of 30 patients ( 60% ) a time dependent change of QRS morphology was observed ., Precordial ECG leads V2 , V3 and V4 from two patients are depicted in Fig 7 . For both patients the QRS morphology following the extra stimuli gradually changed , but the degree of changes here was different ., In patient A , this morphological change is small and both parts of the ECG may be interpreted as a transition from one to another monomorhpic ventricular tachycardia ( MVT ) morphology ., However , for patient B the transition from polymorphic ventricular tachycardia ( PVT ) to MVT is more apparent ., In the other 16 cases we observed different variations between the 2 cases presented in Fig 7 . Supplementary S3 Fig shows examples of ECGs of 4 other patients ., Here , in patients 1 and 2 , we see substantial variations in the QRS complexes after the arrhythmia initiation and subsequently a transformation to MVT ., The recording in patient 3 is less polymorphic and in patient 4 we observe an apparent shift of the ECG from one morphology to another ., It may occur , for example , if due to underlying tissue heterogeneity additional sources of excitation are formed by the initial source ., Overall , the morphology with clear change from PVT to MVT was observed in 5/18 or 29% of the cases ., These different degrees of variation in QRS morphology may be due to many reasons , namely the proximity of the created source of arrhythmia to the anchoring region , the underlying degree of heterogeneity and fibrosis at the place of rotor initiation , complex shape of scar , etc ., Although this finding is not a proof , it supports that the anchoring phenomenon may occur in clinical settings and serve as a possible mechanism of fast VT induced by programmed stimulation ., In this study , we investigated the dynamics of arrhythmia sources –rotors– in the presence of fibrotic regions using mathematical modeling ., We showed that fibrotic scars not only anchor but also induce secondary sources and dynamical competition of these sources normally results their annihilation ., As a result , if one just compares the initial excitation pattern in Fig 1A and final excitation pattern in Fig 1L , it may appear as if a distant spiral wave was attracted and anchored to the scar ., However , this is not the case and the anchored spiral here is a result of normal anchoring and competition of secondary sources which we call dynamical anchoring ., This process is different from the usual drift or meandering of rotors where the rotor gradually changes its spatial position ., In dynamical anchoring , the break formation happens in the fibrotic scar region , then it spreads to the original rotor and merges with this rotor tip and reorganizes the excitation pattern ., This process repeats itself until a rotor is anchored around the fibrotic scar region ., Dynamical anchoring may explain the organization from fast polymorphic to monomorphic VT , also accompanied by prolongation in CL , observed in some patients during re-induction after radio frequency catheter ablation of post-infarct scar related VT ., In our simulations the dynamics of rotors in 2D tissue were stable and for given parameter values they do not drift or meander ., This type of dynamics was frequently observed in cardiac monolayers 38 , 39 which can be considered as a simplified experimental model for cardiac tissue ., We expect that more complex rotor dynamics would not affect our main 2D results , as drift or meandering will potentate the disappearance of the initial rotor and thus promote anchoring of the secondary wavebreaks ., In our 3D simulations in an anatomical model of the heart , the dynamics of rotors is not stationary and shows the ECG of a polymorphic VT ( Fig 6 ) ., The dynamical anchoring combines several processes: generation of new breaks at the scar , spread of breaks toward the original rotor , rotor disappearance and anchoring or one of the wavebreaks at the scar ., The mechanisms of the formation of new wavebreaks at the scar has been studied in several papers 15 , 37 , 40 and can occur due to ionic heterogeneity in the scar region or due to electrotonic effects 40 ., However the process of spread of breaks toward the original rotors is a new type of dynamics and the mechanism of this phenomenon remains to be studied ., To some extent it is similar to the global alternans instability reported in Vandersickel et al . 41 ., Indeed in Vandersickel et al . 41 it was shown that an area of 1:2 propagation block can extend itself towards the original spiral wave and is related to the restitution properties of cardiac tissue ., Although in our case we do not have a clear 1:2 block , wave propagation in the presence of breaks is disturbed resulting in spatially heterogeneous change of diastolic interval which via the restitution effects can result in breakup extension ., This phenomenon needs to be further studied as it may provide new ways for controlling rotor anchoring processes and therefore can affect the dynamics of a cardiac arrhythmia ., In this paper , we used the standard method of representing fibrosis by placement of electrically uncoupled unexcitable nodes with no-flux boundary conditions ., Although such representation is a simplification based on the absence of detailed 3D data , it does reproduce the main physiological effects observed in fibrotic tissue , such as formation of wavebreaks , fractionated electrograms , etc 22 ., The dynamical anchoring reported in this paper occurs as a result of the restructuring of the activation pattern and relies only on these basic properties of the fibrotic scar , i . e . the ability to generate wavebreaks and the ability to anchor rotors , which is reproduced by this representation ., In addition , for each data point , we performed simulations with at least 60 different textures ., Therefore , we expect that the effect observed in our paper is general and should exist for any possible representation of the fibrosis ., The specific conditions , e . g . the size and degree of fibrosis necessary for dynamical anchoring may depend on the detailed fibrosis structure and it would be useful to perform simulations with detailed experimentally based 3D structures of the fibrotic scars , when they become available ., Similar processes can not only occur at fibrotic scars , but also at ionic heterogeneities ., In Defauw et al . 42 , it has been shown that rotors can be attracted by ionic heterogeneities of realistic size and shape , similar to those measured in the ventricles of the human heart 43 ., These ionic heterogeneities had a prolonged APD and also caused wavebreaks , creating a similar dynamical process as described in Fig 1 ., In this study however , we demonstrated that structural heterogeneity is sufficient to trigger this type of dynamical anchoring ., It is important to note that in this study fibrosis was modeled as regions with many small inexcitable obstacles ., However , the outcome can depend on how the cellular electrophysiology and regions of fibrosis have been represented ., In modeling studies , regions of fibrosis can also be represented

Introduction, Materials and methods, Results, Discussion

Rotors are functional reentry sources identified in clinically relevant cardiac arrhythmias , such as ventricular and atrial fibrillation ., Ablation targeting rotor sites has resulted in arrhythmia termination ., Recent clinical , experimental and modelling studies demonstrate that rotors are often anchored around fibrotic scars or regions with increased fibrosis ., However , the mechanisms leading to abundance of rotors at these locations are not clear ., The current study explores the hypothesis whether fibrotic scars just serve as anchoring sites for the rotors or whether there are other active processes which drive the rotors to these fibrotic regions ., Rotors were induced at different distances from fibrotic scars of various sizes and degree of fibrosis ., Simulations were performed in a 2D model of human ventricular tissue and in a patient-specific model of the left ventricle of a patient with remote myocardial infarction ., In both the 2D and the patient-specific model we found that without fibrotic scars , the rotors were stable at the site of their initiation ., However , in the presence of a scar , rotors were eventually dynamically anchored from large distances by the fibrotic scar via a process of dynamical reorganization of the excitation pattern ., This process coalesces with a change from polymorphic to monomorphic ventricular tachycardia .

Rotors are waves of cardiac excitation like a tornado causing cardiac arrhythmia ., Recent research shows that they are found in ventricular and atrial fibrillation ., Burning ( via ablation ) the site of a rotor can result in the termination of the arrhythmia ., Recent studies showed that rotors are often anchored to regions surrounding scar tissue , where part of the tissue still survived called fibrotic tissue ., However , it is unclear why these rotors anchor to these locations ., Therefore , in this work , we investigated why rotors are so abundant in fibrotic tissue with the help of computer simulations ., We performed simulations in a 2D model of human ventricular tissue and in a patient-specific model of a patient with an infarction ., We found that even when rotors are initially at large distances from the fibrotic region , they are attracted by this region , to finally end up at the fibrotic tissue ., We called this process dynamical anchoring and explained how the process works .

dermatology, medicine and health sciences, diagnostic radiology, engineering and technology, cardiovascular anatomy, cardiac ventricles, fibrosis, magnetic resonance imaging, developmental biology, electrocardiography, bioassays and physiological analysis, cardiology, research and analysis methods, scars, arrhythmia, imaging techniques, atrial fibrillation, electrophysiological techniques, rotors, mechanical engineering, radiology and imaging, diagnostic medicine, cardiac electrophysiology, anatomy, biology and life sciences, heart

journal.pcbi.1002283

Chemotaxis when Bacteria Remember: Drift versus Diffusion

The bacterium E . coli moves by switching between two types of motions , termed ‘run’ and ‘tumble’ 1 ., Each results from a distinct movement of the flagella ., During a run , flagella motors rotate counter-clockwise ( when looking at the bacteria from the back ) , inducing an almost constant forward velocity of about , along a near-straight line ., In an environment with uniform nutrient concentration , run durations are distributed exponentially with a mean value of about 2 ., When motors turn clockwise , the bacterium undergoes a tumble , during which , to a good approximation , it does not translate but instead changes its direction randomly ., In a uniform nutrient-concentration profile , the tumble duration is also distributed exponentially but with a much shorter mean value of about 3 ., When the nutrient ( or , more generally , chemoattractant ) concentration varies in space , bacteria tend to accumulate in regions of high concentration ( or , equivalently , the bacteria can also be repelled by chemorepellants and tend to accumulate in low chemical concentration ) 4 ., This is achieved through a modulation of the run durations ., The biochemical pathway that controls flagella dynamics is well understood 1 , 5–7 and the stochastic ‘algorithm’ which governs the behavior of a single motor is experimentally measured ., The latter is routinely used as a model for the motion of a bacteria with many motors 1 , 8–11 ., This algorithm represents the motion of the bacterium as a non-Markovian random walker whose stochastic run durations are modulated via a memory kernel , shown in Fig . 1 ., Loosely speaking , the kernel compares the nutrient concentration experienced in the recent past with that experienced in the more distant past ., If the difference is positive , the run duration is extended; if it is negative , the run duration is shortened ., In a complex medium bacterial navigation involves further complications; for example , interactions among the bacteria , and degradations or other dynamical variations in the chemical environment ., These often give rise to interesting collective behavior such as pattern formation 12 , 13 ., However , in an attempt to understand collective behavior , it is imperative to first have at hand a clear picture of the behavior of a single bacterium in an inhomogeneous chemical environment ., We are concerned with this narrower question in the present work ., Recent theoretical studies of single-bacterium behavior have shown that a simple connection between the stochastic algorithm of motion and the average chemotactic response is far from obvious 8–11 ., In particular , it appeared that favorable chemotactic drift could not be reconciled with favorable accumulation at long times , and chemotaxis was viewed as resulting from a compromise between the two 11 ., The optimal nature of this compromise in bacterial chemotaxis was examined in Ref ., 10 ., In various approximations , while the negative part of the response kernel was key to favorable accumulation in the steady state , it suppressed the drift velocity ., Conversely , the positive part of the response kernel enhanced the drift velocity but reduced the magnitude of the chemotactic response in the steady state ., Here , we carry out a detailed study of the chemotactic behavior of a single bacterium in one dimension ., We find that , for an ‘adaptive’ response kernel ( i . e . , when the positive and negative parts of the response kernel have equal weight such that the total area under the curve vanishes ) , there is no incompatibility between a strong steady-state chemotaxis and a large drift velocity ., A strong steady-state chemotaxis occurs when the positive peak of the response kernel occurs at a time much smaller than and the negative peak at a time much larger than , in line with experimental observation ., Moreover , we obtain that the drift velocity is also large in this case ., For a general ‘non-adaptive’ response kernel ( i . e . , when the area under the response kernel curve is non-vanishing ) , however , we find that a large drift velocity indeed opposes chemotaxis ., Our calculations show that , in this case , a position-dependent diffusivity is responsible for chemotactic accumulation ., In order to explain our numerical results , we propose a simple coarse-grained model which describes the bacterium as a biased random walker with a drift velocity and diffusivity , both of which are , in general , position-dependent ., This simple model yields good agreement with results of detailed simulations ., We emphasize that our model is distinct from existing coarse-grained descriptions of E . coli chemotaxis 13–16 ., In these , coarse-graining was performed over left- and right-moving bacteria separately , after which the two resulting coarse-grained quantities were then added to obtain an equation for the total coarse-grained density ., We point out why such approaches can fail and discuss the differences between earlier models and the present coarse-grained model ., Following earlier studies of chemotaxis 9 , 17 , we model the navigational behavior of a bacterium by a stochastic law of motion with Poissonian run durations ., A switch from run to tumble occurs during the small time interval between and with a probability ( 1 ) Here , and is a functional of the chemical concentration , , experienced by the bacterium at times ., In shallow nutrient gradients , the functional can be written as ( 2 ) The response kernel , , encodes the action of the biochemical machinery that processes input signals from the environment ., Measurements of the change in the rotational bias of a flagellar motor in wild-type bacteria , in response to instantaneous chemoattractant pulses were reported in Refs ., 17 , 18; experiments were carried out with a tethering assay ., The response kernel obtained from these measurements has a bimodal shape , with a positive peak around and a negative peak around ( see Fig . 1 ) ., The negative lobe is shallower than the positive one and extends up to , beyond which it vanishes ., The total area under the response curve is close to zero ., As in other studies of E . coli chemotaxis , we take this response kernel to describe the modulation of run duration of swimming bacteria 8–11 ., Recent experiments suggest that tumble durations are not modulated by the chemical environment and that as long as tumbles last long enough to allow for the reorientation of the cell , bacteria can perform chemotaxis successfully 19 , 20 ., The model defined by Eqs ., 1 and 2 is linear ., Early experiments pointed to a non-linear , in effect a threshold-linear , behavior of a bacterium in response to chemotactic inputs 17 , 18 ., In these studies , a bacterium modulated its motion in response to a positive chemoattractant gradient , but not to a negative one ., In the language of present model , such a threshold-linear response entails replacing the functional defined in Eq ., 2 by zero whenever the integral is negative ., More recent experiments suggest a different picture , in which a non-linear response is expected only for a strong input signal whereas the response to weak chemoattractant gradient is well described by a linear relation 21 ., Here , we present an analysis of the linear model ., For the sake of completeness , in Text S1 , we present a discussion of models which include tumble modulations and a non-linear response kernel ., Although recent experiments have ruled out the existence of both these effects in E . coli chemotaxis , in general such effects can be relevant to other systems with similar forms of the response function ., The shape of the response function hints to a simple mechanism for the bacterium to reach regions with high nutrient concentration ., The bilobe kernel measures a temporal gradient of the nutrient concentration ., According to Eq ., 1 , if the gradient is positive , runs are extended; if it is negative , runs are unmodulated ., However , recent literature 8 , 9 , 11 has pointed out that the connection between this simple picture and a detailed quantitative analysis is tenuous ., For example , de Gennes used Eqs ., 1 to calculate the chemotactic drift velocity of bacteria 8 ., He found that a singular kernel , , where is a Dirac function and a positive constant , lead to a mean velocity in the direction of increasing nutrient concentration even when bacteria are memoryless ( ) ., Moreover , any addition of a negative contribution to the response kernel , as seen in experiments ( see Fig . 1 ) , lowered the drift velocity ., Other studies considered the steady-state density profile of bacteria in a container with closed walls , both in an approximation in which correlations between run durations and probability density were ignored 11 and in an approximation in which the memory of the bacterium was reset at run-to-tumble switches 9 ., Both these studies found that , in the steady state , a negative contribution to the response function was mandatory for bacteria to accumulate in regions of high nutrient concentration ., These results seem to imply that the joint requirement of favorable transient drift and steady-state accumulation is problematic ., The paradox was further complicated by the observation 9 that the steady-state single-bacterium probability density was sensitive to the precise shape of the kernel: when the negative part of the kernel was located far beyond it had little influence on the steady-state distribution 11 ., In fact , for kernels similar to the experimental one , model bacteria accumulated in regions with low nutrient concentration in the steady state 9 ., In order to resolve these paradoxes and to better understand the mechanism that leads to favorable accumulation of bacteria , we perform careful numerical studies of bacterial motion in one dimension ., In conformity with experimental observations 17 , 18 , we do not make any assumption of memory reset at run-to-tumble switches ., We model a bacterium as a one-dimensional non-Markovian random walker ., The walker can move either to the left or to the right with a fixed speed , , or it can tumble at a given position before initiating a new run ., In the main paper , we present results only for the case of instantaneous tumbling with , while results for non-vanishing are discussed in Text S1 ., There , we verify that for an adaptive response kernel does not have any effect on the steady-state density profile ., For a non-adaptive response kernel , the correction in the steady-state slope due to finite is small and proportional to ., The run durations are Poissonian and the tumble probability is given by Eq ., 1 ., The probability to change the run direction after a tumble is assumed to have a fixed value , , which we treat as a parameter ., The specific choice of the value of does not affect our broad conclusions ., We find that , as long as , only certain detailed quantitative aspects of our numerical results depend on ., ( See Text S1 for details on this point . ), We assume that bacteria are in a box of size with reflecting walls and that they do not interact among each other ., We focus on the steady-state behavior of a population ., Reflecting boundary conditions are a simplification of the actual behavior 22 , 23; as long as the total ‘probability current’ ( see discussion below ) in the steady state vanishes , our results remain valid even if the walls are not reflecting ., As a way to probe chemotactic accumulation , we consider a linear concentration profile of nutrient: ., We work in a weak gradient limit , i . e . , the value of is chosen to be sufficiently small to allow for a linear response ., Throughout , we use in our numerics ., From the linearity of the problem , results for a different attractant gradient , , can be obtained from our results through a scaling factor ., In the linear reigme , we obtain a spatially linear steady-state distribution of individual bacterium positions , or , equivalently , a linear density profile of a bacterial population ., Its slope , which we denote by , is a measure of the strength of chemotaxis ., A large slope indicates strong bacterial preference for regions with higher nutrient concentration ., Conversely , a vanishing slope implies that bacteria are insensitive to the gradient of nutrient concentration and are equally likely to be anywhere along the line ., We would like to understand the way in which the slope depends on the different time scales present in the system ., In order to gain insight into our numerical results , we developed a simple coarse-grained model of chemotaxis ., For the sake of simplicity , we first present the model for a non-adaptive , singular response kernel , , and , subsequently , we generalize the model to adaptive response kernels by making use of linear superposition ., The memory trace embodied by the response kernel induces temporal correlations in the trajectory of the bacterium ., However , if we consider the coarse-grained motion of the bacterium over a spatial scale that exceeds the typical run stretch and a temporal scale that exceeds the typical run duration , then we can assume that it behaves as a Markovian random walker with drift velocity and diffusivity ., Since the steady-state probability distribution , , is flat for , for small we can write ( 4 ) ( 5 ) ( 6 ) Here , and ., Since we are neglecting all higher order corrections in , our analysis is valid only when is sufficiently small ., In particular , even when , we assume that the inequality is still satisfied ., The chemotactic drift velocity , , vanishes if ; it is defined as the mean displacement per unit time of a bacterium starting a new run at a given location ., Clearly , even in the steady state when the current , defined through , vanishes , may be non-vanishing ( see Eq . 8 below ) ., In general , the non-Markovian dynamics make dependent on the initial conditions ., However , in the steady state this dependence is lost and can be calculated , for example , by performing a weighted average over the probability of histories of a bacterium ., This is the quantity that is of interest to us ., An earlier calculation by de Gennes showed that , if the memory preceding the last tumble is ignored , then for a linear profile of nutrient concentration the drift velocity is independent of position and takes the form 8 ., While the calculation applies strictly in a regime with ( because of memory erasure ) , in fact its result captures the behavior well over a wide range of parameters ( see Fig . 4 ) ., To measure in our simulations , we compute the average displacement of the bacterium between two successive tumbles in the steady state , and we extract therefrom the drift velocity ., ( For details of the derivation , see Text S1 . ), We find that is negative for and that its magnitude falls off with increasing values of ( Fig . 4 ) ., We also verify that indeed does not show any spatial dependence ( data shown in Fig . of Text S1 ) ., We recall that , in our numerical analysis , we have used a small value of ; this results in a low value of ., We show below that for an experimentally measured bilobe response kernel , obtained by superposition of singular response kernels , the magnitude of becomes larger and comparable with experimental values ., To obtain the diffusivity , , we first calculate the effective mean free path in the coarse-grained model ., The tumbling frequency of a bacterium is and depends on the details of its past trajectory ., In the coarse-grained model , we replace the quantity by an average over all the trajectories within the spatial resolution of the coarse-graining ., Equivalently , in a population of non-interacting bacteria , the average is taken over all the bacteria contained inside a blob , and , hence , denotes the position of the center of mass of the blob at a time in the past ., As mentioned above , the drift velocity is proportional to , so that ., The average tumbling frequency then becomes and , consequently , the mean free path becomes ., As a result , the diffusivity is expressed as ., We checked this form against our numerical results ( Fig . 5 ) ., Having evaluated the drift velocity , , and the diffusivity , , we now proceed to write down the continuity equation ( for a more rigorous but less intuitive approach , see 10 ) ., For a biased random walker on a lattice , with position-dependent hopping rates and towards the right and the left , respectively , one has and , where is the lattice constant ., In the continuum limit , the temporal evolution of the probability density is given by a probability current , as ( 7 ) where the current takes the form ( 8 ) For reflecting boundary condition , in the steady state ., This constraint yields a steady-state slope ( 9 ) for small ., We use our measured values for and ( Figs . 4 and 5 ) , and compute the slope using Eq ., 9 ., ( For details of the measurement of , see Text S1 . ), We compare our analytical and numerical results in Fig . 2 , which exhibits close agreement ., According to Eq ., 9 , steady-state chemotaxis results from a competition between drift motion and diffusion ., For , the drift motion is directed toward regions with a lower nutrient concentration and hence opposes chemotaxis ., Diffusion is spatially dependent and becomes small for large nutrient concentrations ( again for ) , thus increasing the effective residence time of the bacteria in favorable regions ., For large values of , the drift velocity vanishes and one has a strong chemotaxis as increases ( Fig . 2 ) ., Finally , for , the calculation by de Gennes yields which exactly cancels the spatial gradient of ( to linear order in ) , and there is no accumulation 8 , 11 ., These conclusions are easily generalized to adaptive response functions ., For , within the linear response regime , the effective drift velocity and diffusivity can be constructed by simple linear superposition: The drift velocity reads ., Interestingly , the spatial dependence of cancels out and ., The resulting slope then depends on the drift only and is calculated as ( 10 ) In this case , the coarse-grained model is a simple biased random walker with constant diffusivity ., For and , the net velocity , proportional to , is positive and gives rise to a favorable chemotactic response , according to which bacteria accumulate in regions with high food concentration ., Moreover , the slope increases as the separation between and grows ., We emphasize that there is no incompatibility between strong steady-state chemotaxis and large drift velocity ., In fact , in the case of an adaptive response function , strong chemotaxis occurs only when the drift velocity is large ., For a bilobe response kernel , approximated by a superposition of many delta functions ( Fig . 1 ) , the slope , , can be calculated similarly and in Fig . 3 we compare our calculation to the simulation results ., We find close agreement in the case of a linear model with a bilobe response kernel and , in fact , also in the case of a non-linear model ( see Text S1 ) ., The experimental bilobe response kernel is a smooth function , rather than a finite sum of singular kernels over a set of discrete values ( as in Fig . 1 ) ., Formally , we integrate singular kernels over a continuous range of to obtain a smooth response kernel ., If we then integrate the expression for the drift velocity obtained by de Gennes , according to this procedure , we find an overall drift velocity , for the concentration gradient considered ( ) ., By scaling up the concentration gradient by a factor of , the value of can also be scaled up by and can easily account for the experimentally measured velocity range ., We carried out a detailed analysis of steady-state bacterial chemotaxis in one dimension ., The chemotactic performance in the case of a linear concentration profile of the chemoattractant , , was measured as the slope of the bacterium probability density profile in the steady state ., For a singular impulse response kernel , , the slope was a scaling function of , which vanished at the origin , increased monotonically , and saturated at large argument ., To understand these results we proposed a simple coarse-grained model in which bacterial motion was described as a biased random walk with drift velocity , , and diffusivity , ., We found that for small enough values of , was independent of and varied linearly with nutrient concentration ., By contrast , was spatially uniform and its value decreased monotonically with and vanished for ., We presented a simple formula for the steady-state slope in terms of and ., The prediction of our coarse-grained model agreed closely with our numerical results ., Our description is valid when is small enough , and all our results are derived to linear order in ., We assume is always satisfied ., Our results for an impulse response kernel can be easily generalized to the case of response kernels with arbitrary shapes in the linear model ., For an adaptive response kernel , the spatial dependence of the diffusivity , , cancels out but a positive drift velocity , , ensures bacterial accumulation in regions with high nutrient concentration , in the steady state ., In this case , the slope is directly proportional to the drift velocity ., As the delay between the positive and negative peaks of the response kernel grows , the velocity increases , with consequent stronger chemotaxis ., Earlier studies of chemotaxis 13–16 put forth a coarse-grained model different from ours ., In the model first proposed by Schnitzer for a single chemotactic bacterium 14 , he argued that , in order to obtain favorable bacterial accumulation , tumbling rate and ballistic speed of a bacterium must both depend on the direction of its motion ., In his case , the continuity equation reads ( 11 ) where is the ballistic speed and is the tumbling frequency of a bacterium moving toward the left ( right ) ., For E . coli , as discussed above , , a constant independent of the location ., In that case , Eq ., 11 predicts that in order to have a chemotactic response in the steady state , one must have a non-vanishing drift velocity , i . e . , ., This contradicts our findings for non-adaptive response kernels , according to which a drift velocity only hinders the chemotactic response ., The spatial variation of the diffusivity , instead , causes the chemotactic accumulation ., This is not captured by Eq ., 11 ., In the case of adaptive response kernels , the diffusivity becomes uniform while the drift velocity is positive , favoring chemotaxis ., Comparing the expression of the flux , , obtained from Eqs ., 7 and 8 with that from Eq ., 11 , and matching the respective coefficients of and , we find and ., As we argued above in discussing the coarse-grained model for adaptive response kernels , both and are spatially independent ., This puts strict restrictions on the spatial dependence of and ., For example , as in E . coli chemotaxis , our coarse-grained description is recovered only if and are also independent of ., We comment on a possible origin of the discrepancy between our work and earlier treatments ., In Ref ., 14 , a continuity equation was derived for the coarse-grained probability density of a bacterium , starting from a pair of approximate master equations for the probability density of a right-mover and a left-mover , respectively ., As the original process is non-Markovian , one can expect a master equation approach to be valid only at scales that exceed the scale over which spatiotemporal correlations in the behavior of the bacterium are significant ., In particular , a biased diffusion model can be viewed as legitimate only if the ( coarse-grained ) temporal resolution allows for multiple runs and tumbles ., If so , at the resolution of the coarse-grained model , left- and right-movers become entangled , and it is not possible to perform a coarse-graining procedure on the two species separately ., Thus one cannot define probability densities for a left- and a right-mover that evolves in a Markovian fashion ., In our case , left- and right-movers are coarse-grained simultaneously , and the total probability density is Markovian ., Thus , our diffusion model differs from that of Ref ., 14 because it results from a different coarse-graining procedure ., The model proposed in Ref ., 14 has been used extensively to investigate collective behaviors of E . coli bacteria such as pattern formation 13 , 15 , 16 ., It would be worth asking whether the new coarse-grained description can shed new light on bacterial collective behavior .

Introduction, Models, Results, Discussion

Escherichia coli ( E . coli ) bacteria govern their trajectories by switching between running and tumbling modes as a function of the nutrient concentration they experienced in the past ., At short time one observes a drift of the bacterial population , while at long time one observes accumulation in high-nutrient regions ., Recent work has viewed chemotaxis as a compromise between drift toward favorable regions and accumulation in favorable regions ., A number of earlier studies assume that a bacterium resets its memory at tumbles – a fact not borne out by experiment – and make use of approximate coarse-grained descriptions ., Here , we revisit the problem of chemotaxis without resorting to any memory resets ., We find that when bacteria respond to the environment in a non-adaptive manner , chemotaxis is generally dominated by diffusion , whereas when bacteria respond in an adaptive manner , chemotaxis is dominated by a bias in the motion ., In the adaptive case , favorable drift occurs together with favorable accumulation ., We derive our results from detailed simulations and a variety of analytical arguments ., In particular , we introduce a new coarse-grained description of chemotaxis as biased diffusion , and we discuss the way it departs from older coarse-grained descriptions .

The chemotaxis of Escherichia coli is a prototypical model of navigational strategy ., The bacterium maneuvers by switching between near-straight motion , termed runs , and tumbles which reorient its direction ., To reach regions of high nutrient concentration , the run-durations are modulated according to the nutrient concentration experienced in recent past ., This navigational strategy is quite general , in that the mathematical description of these modulations also accounts for the active motility of C . elegans and for thermotaxis in Escherichia coli ., Recent studies have pointed to a possible incompatibility between reaching regions of high nutrient concentration quickly and staying there at long times ., We use numerical investigations and analytical arguments to reexamine navigational strategy in bacteria ., We show that , by accounting properly for the full memory of the bacterium , this paradox is resolved ., Our work clarifies the mechanism that underlies chemotaxis and indicates that chemotactic navigation in wild-type bacteria is controlled by drift while in some mutant bacteria it is controlled by a modulation of the diffusion ., We also propose a new set of effective , large-scale equations which describe bacterial chemotactic navigation ., Our description is significantly different from previous ones , as it results from a conceptually different coarse-graining procedure .

physics, statistical mechanics, theoretical biology, biophysics theory, biology, computational biology, biophysics simulations, biophysics

journal.pcbi.1005644

A phase transition induces chaos in a predator-prey ecosystem with a dynamic fitness landscape

In many natural ecosystems , at least one constituent species evolves quickly enough relative to its population growth that the two effects become interdependent ., This phenomenon can occur when selection forces are tied to such sudden environmental effects as algal blooms or flooding 1 , or it can arise from more subtle , population-level effects such as overcrowding or resource depletion 2 ., Analysis of such interactions within a unified theory of “eco-evolutionary dynamics” has been applied to a wide range of systems—from bacteria-phage interactions to bighorn sheep 3—by describing population fluctuations in terms of the feedback between demographic change and natural selection 4 ., The resulting theoretical models relate the fitness landscape ( or fitness function ) to population-level observables such as the population growth rate and the mean value of an adapting phenotypic trait ( such as horn length , cell wall thickness , etc ) ., The fitness landscape may have an arbitrarily complex topology , as it can depend on myriad factors ranging from environmental variability 5 , 6 , to inter- and intraspecific competition 7 , 8 , to resource depletion 9 ., However , these complex landscapes can be broadly classified according to whether they result in stabilizing or disruptive selection ., In the former , the landscape may possess a single , global maximum that causes the population of individuals to evolve towards a state in which most individuals have trait values at or near this maximum 10 ., Conversely , in disruptive selection , the fitness landscape may contain multiple local maxima , in which case the population could have a wide distribution of trait values and occupy multiple distinct niches 11 ., In eco-evolutionary models , the shape of the fitness landscape may itself depend on the population densities of the interacting species it describes ., Specifically , the concept that the presence of competition can lead a single-peaked fitness landscape to spontaneously develop additional peaks originates in the context of “competitive speciation” first proposed by Rosenzweig 12 ., This is formalized in genetic models in which sympatric speciation is driven by competitive pressures rather than geographic isolation 13 ., Competition-induced disruptive selection has been observed in natural populations of stickleback fish 14 , microbial communities 15 , and fruit flies 16 , 17 ., Here , we model eco-evolutionary dynamics of a predator-prey system based on first-order “gradient dynamics” 10 , 18 , a class of models that explicitly define the fitness in terms of the population growth rate r , which is taken to depend only on the mean value of the trait across the entire population , c ¯ 19 ., Despite this simplification , gradient dynamics models display rich behavior that can account for a wide range of effects observed in experimental systems—in particular , recent work by Cortez and colleagues has shown that these models can result in irregular cycles and dynamical bifurcations that depend on the standing genetic variation present in a population 20 , 21 ., In our model , gradient dynamics cause the prey fitness landscape to change as a result of predation , and we find that the resulting dynamical system exhibits chaotic dynamics ., Chaos is only possible in systems in which three or more dependent dynamical variables vary in time 22 , and previously it has been observed in predator-prey systems comprising three or more mutually interdependent species , or in which an external environmental variable ( such as seasonal variation or generic noise ) is included in the dynamics 23 , 24 ., Here we show that evolution of just one species in a two-species ecosystem is sufficient to drive the ecosystem into chaos ., Moreover , we find that chaos is driven by a density-dependent change of the fitness landscape from a stabilizing to disruptive state , and that this transition has hysteretic behavior with mathematical properties that are strongly reminiscent of a first-order phase transition in a thermodynamical system ., The resulting dynamics display intermittent properties typically associated with ecosystems poised at the “edge of chaos , ” which we suggest has implications for the study of ecological stability and speciation ., Adapting the notation and formulation used by Cortez ( 2016 ) 21 , we use a two-species competition model with an additional dynamical variable introduced to account for a prey trait on which natural selection may act ., The most general fitness function for the prey , r , accounts for density-dependent selection on a prey trait c ,, r ( x , y , c ¯ , c ) ≡ G ( x , c , c ¯ ) - D ( c , c ¯ ) - f ( x , y ) , ( 1 ), where x = x ( t ) is the time-dependent prey density , y = y ( t ) is the time-dependent predator density , c is a trait value for an individual in the prey population , and c ¯ = c ¯ ( t ) is the mean value of the trait across the entire prey population at time t ., r comprises a density-dependent birth rate G , a density-independent death rate D , and a predator-prey interaction term f , which for simplicity is assumed to depend on neither c nor c ¯ ., Thus the trait under selection in our model is not an explicit predator avoidance trait such as camouflage , but rather an endogenous advancement ( i . e . , improved fecundity , faster development , or reduced mortality ) that affects the prey’s ability to exploit resources in its environment , even in the absence of predation ., The continuous-time “gradient dynamics” model that we study interprets the fitness r as the growth rate of the prey: 19 , 25, x ˙ = x r ( x , y , c ¯ , c ) | c → c ¯ ( 2 ), y ˙ = y ( f ( x , y ) - D ˜ ( y ) ) ( 3 ), c ¯ ˙ = V ∂ r ( x , y , c ¯ , c ) ∂ c | c → c ¯ ., ( 4 ) Eq ( 2 ) is evaluated with all individual trait values c set to the mean value c ¯ because the total prey population density is assumed to change based on the fitness function , which in turn depends on the population-averaged value of the prey trait c ¯ 21 ., The timescale of the dynamics in c ¯ are set by V , which is interpreted as the additive genetic variance of the trait 10 ., While Eq ( 2 ) depends only on the mean trait value c ¯ , the full distribution of individual trait values c present in a real-world population may change over time as the relative frequencies of various phenotypes change ., In principle , additional differential equations of the form of Eq ( 4 ) could be added to account for higher moments of the distribution of c across an ensemble of individuals , allowing the gradient dynamics model to be straightforwardly extended to model a trait’s full distribution rather than just the population mean ., However , here we focus on the case where the prey density dynamics x ˙ depend only on the mean trait value to first order , and we do not include differential equations for higher-order moments of the prey trait value distribution ., The use of a single Eq ( 4 ) to describe the full dynamics of the trait distribution represents an approximation that is exact only when the phenotypic trait distribution stays nearly symmetric and the prey population maintains a constant standing genetic variation V 10 ., However , V may remain fixed even if the phenotypic variance changes , a property that is observed phenomenologically in experimental systems , and which may be explained by time-dependent heritability , breeding effects , mutation , or other transmission effects not explicitly modeled here 26–29 ., More broadly , this assumption may imply that gene selection is weak compared to phenotype selection 30 , 31 ., S1D Appendix further describes the circumstances under which V remains fixed , and also provides a first-order estimate of the magnitude of error introduced by ignoring higher-order effects ( such as skewness ) in the trait distribution ., The results suggest that these effects are small for the parameter values ( and resulting range of x and y values ) used here , due in part to limitations on the maximum skewness that a hypothetical trait distribution can achieve on the fitness landscapes studied here ., In S1D Appendix , we also compare the results presented below to an equivalent model in which a full trait distribution is present , in which case Eq ( 2 ) becomes a full integro-differential equation involving averages of the trait value over the entire prey population ., Detailed numerical study of this integro-differential equation is computationally prohibitive for the long timescales studied here , but direct comparison of the contributions of various terms in the velocity field suggests general accuracy of the gradient dynamics model for the fitness landscapes and conditions we study here ., However , in general the appropriateness of the gradient dynamics model should be checked whenever using Eq ( 4 ) with an arbitrary fitness function ., Fig 1A shows a schematic summarizing the gradient dynamics model , and noting the primary assumptions underlying this formulation ., Next , we choose functional forms for f , G , D , and D ˜ in Eqs ( 2 ) and ( 3 ) ., We start with the assumption that , for fixed values of the trait c an d its mean c ¯ , the population dynamics should have the form of a typical predator-prey system in the absence of evolutionary effects ., Because the predator dynamics are not directly affected by evolutionary dynamics , we choose a simple form for predator growth consisting of a fixed death rate and a standard Holling Type II birth rate , 32, f ( x , y ) = a 2 x y 1 + b 2 x ( 5 ), D ˜ ( y ) = d 2 ( 6 ), The predator birth rate f saturates at large values of the prey density , which is more realistic than the standard Lotka-Volterra competition term xy in cases where the prey density is large or fluctuating 22 ., A saturating interaction term ensures that solutions of the system remain bounded for a wider range of parameter values , a necessity for realistic models of long-term interactions 33 ., For the prey net growth rate ( Eq ( 1 ) , the fitness ) in the absence of the predator , we use the following functional forms ,, G ( x , c ¯ , c ) = a 1 c ¯ 1 + b 1 c ¯ ( 1 - k 1 x ( c - c ¯ ) ) ( 7 ), D ( c , c ¯ ) = d 1 ( 1 - k 2 ( c 2 - c ¯ 2 ) + k 4 ( c 4 - c ¯ 4 ) ) ., ( 8 ), The first term in Eq ( 7 ) specifies that the prey population density growth rate r | c → c ¯ depends only on a primary saturating contribution of the mean trait to the birth rate G . In other models a similar effect is achieved by modifying the mean trait evolution Eq ( 4 ) , such that extremal values of the trait are disadvantaged 21; alternative coupling methods based on exponential saturation would be expected to yield similar qualitative results 19 ., However , the additional series terms in Eqs ( 7 ) and ( 8 ) ensure that the any individual’s fitness r may differ from the rest of the population depending on the difference between its trait value c and the population mean c ¯ ., Because the functional form of this difference is unknown , its contribution expressed as second-order truncation of the series difference of the form r ( c , c ¯ ) = r ˜ | c → 0 + ( r ˜ ( c ) - r ˜ ( c ) | c → c ¯ ) ( where r ˜ represents an unscaled fitness function ) ., This ensures that when c ˙ = 0 or c = c ¯ , the system reduces to a standard prey model with a Holling Type II increase in birth rate in response to increasing mean trait value 25 ., In the results reported below , we observe that all dynamical variables remain bounded as long as parameter values are chosen such that the predator density does not equilibrate to zero ., This is a direct consequence of our use of saturating Holling Type II functional forms in Eqs ( 7 ) and ( 8 ) , which prevent the fitness landscape from increasing without bound at large c , c ¯ and also ensure that the predator and prey densities do not jointly diverge ., That the dynamics should stay bounded due to saturating terms is justified by empirical studies of predator-prey systems 34 , 35; moreover , other saturating functional forms are expected to yield similar results if equivalent parameter values are chosen 33 , 36 ., The nonlinear dependence of the mortality rate Eq ( 8 ) on the trait is based on mechanistic models of mortality with individual variation 19 , 37 , 38 ., The specific choice of a quartic in Eq ( 8 ) allows the fitness function r to have a varying number of real roots and local maxima in the domain c , c ¯ > 0 , affording the system dynamical freedom not typically possible in predator prey models with constant or linear prey fitness—in particular , for different values of k2 , k4 the fitness landscape can have a single optimal phenotype , multiple optimal phenotypes , or no optimal intermediate values ., Because any even , continuous form for the fitness landscape can be approximated using a finite number of terms in its Taylor series around c = 0 , our choice of a quartic form simply constitutes truncation of this expansion at the quartic order in order to include the simplest case in which the fitness function admits multiple local maxima—for this reason , a quartic will always represent the leading-order series expansion of a fitness landscape with multiple local maxima ., Below , we observe numerically that ∣ c - c ¯ ∣ < 1 , ex post facto justifying truncation of the higher order terms in this series expansion ., However , if the trait value c was strictly bounded to only take non-zero values on a finite interval ( as opposed to the entire real line ) , then a second-order , quadratic fitness landscape would be sufficient to admit multiple local maxima ( at the edges of the interval ) 14 ., However , the choice here of an unbounded trait value c avoids creating boundary effects , and it has little consequence due to the steep decay of the quartic function at large values of |c| , which effectively confines the possible values of c ¯ accessible by the system ., In physics , similar reasons—unbounded domains , multiple local optima , and continuity—typically justify the use of quartic free energy functions in minimal models of systems exhibiting multiple energetic optima , such as the Ginzberg-Landau free energy used in models of superconducting phase transitions 39 ., We note that the birth rate Eq ( 7 ) contributes a density-dependent term to the fitness function even in the absence of predation ( y = 0 ) 21 ., Unlike the death rate function , the effect of the individual trait value on this term is directional: the sign of c - c ¯ determines whether birth rates increase or decrease ., As the population density x increases , the effect of these directional effects is amplified , consistent with the observed effect of intraspecific competition and crowding in experimental studies of evolution 40 , 41 ., The chaotic dynamics reported below arise from this density-dependent term because the term prevents the Jacobian of the system ( 2 ) , ( 3 ) and ( 4 ) from having a row and column with all zeros away from the diagonal; in this case , the prey trait ( and thus evolutionary dynamics ) would be uncoupled from the rest of the system , and would thus relax to a stable equilibrium ( as is necessary for a first-order single-variable equation ) ., In that case , c ¯ would essentially remain fixed and the predator-prey dynamics would become two-dimensional in x and y , precluding chaos ., For similar reasons , density-dependent selection has been found to be necessary for chaos in some discrete-time evolutionary models , for which chaotic dynamics require a certain minimum degree of association between the fitness and the trait frequencies 42 ., Inserting Eqs ( 5 ) , ( 7 ) and ( 8 ) , into Eq ( 1 ) results in a final fitness function of the form, r ( x , y , c ¯ , c ) = a 1 c ¯ 1 + b 1 c ¯ ( 1 - k 1 x ( c - c ¯ ) ) - d 1 ( 1 - k 2 ( c 2 - c ¯ 2 ) + k 4 ( c 4 - c ¯ 4 ) ) - a 2 x y 1 + b 2 x ., ( 9 ), This fitness landscape is shown in Fig 1B , for typical parameter values and predator and prey densities used in the numerical results below ., Depending on the current predator and prey densities , the local maximum of the system can appear in two different locations , which directly affects the dynamics described in the next section ., Inserting Eq ( 9 ) into Eqs ( 2 ) , ( 3 ) and ( 4 ) results in a final form for the dynamical equations ,, x ˙ = x ( a 1 c ¯ 1 + b 1 c ¯ - a 2 y 1 + b 2 x - d 1 ) ( 10 ), y ˙ = y ( y a a 2 x 1 + b 2 x - d 2 ) ( 11 ), c ¯ ˙ = c ¯ V ( ( 2 k 2 d 1 ) - ( 4 k 4 d 1 ) c ¯ 2 - ( a 1 k 1 ) x 1 + b 1 c ¯ ) ., ( 12 ), Due to the Holling coupling terms , the form of these equations qualitatively resembles models of vertical , tritrophic food webs—the mean trait value c ¯ affects the growth rate of the prey , which in turn affects the growth rate of the predator 24 , 32 , 43 ., The coupling parameter ya introduces asymmetry into the competition when ya ≠ 1; however , it essentially acts as a scale factor that only affects the amplitude of the y cycles and equilibria rather than the dynamics ., Additionally , because the predator-prey interaction term Eq ( 5 ) is unaffected by the trait , our model contains no triple-product x y c ¯ interaction terms , which typically stabilize the dynamics ., For our analysis of the system ( 10 ) , ( 11 ) and ( 12 ) , we first consider the case where evolution proceeds very slowly relative to population dynamics ., In the case of both no evolution ( V = 0 ) and no predation ( y = 0 ) , the prey growth Eq ( 10 ) advances along the one-dimensional nullcline y ˙ , c ¯ ˙ = 0 , y = 0 ., Depending on whether the fixed mean trait value c ¯ exceeds a critical value ( c ¯ † ≡ d 1 / ( a 1 - b 1 d 1 ) ) , the prey density will either grow exponentially ( c ¯ > c ¯ † ) or collapse exponentially ( c ¯ < c ¯ † ) because the constant c ¯ remains too low to sustain the prey population in the absence of evolutionary adaptation ., The requirement that c ¯ > c ¯ † carries over to the case where a predator is added to the system but evolutionary dynamics remain fixed , corresponding to a two dimensional system advancing along the two-dimensional nullcline c ¯ ˙ = 0 ., In this case , as long as c ¯ > c ¯ † , the prey density can exhibit continuous growth or cycling depending in the relative magnitudes of the various parameters in Eqs ( 10 ) and ( 11 ) ., The appearance and disappearance of these cycles is determined by a series of bifurcations that depends on the values of c ¯ and b1 , b2 relative to the remaining parameters a1 , a2 , d1 , d2 ( S1A Appendix ) ., In the full three-variable system ( 10 ) , ( 11 ) and ( 12 ) , c ¯ passes through a range of values as time progresses , resulting in more complex dynamics than those observed in the two-dimensional case ., For very small values of V , the evolutionary dynamics c ¯ ˙ are slow enough that the system approaches the equilibrium predicted by the two-variable model with c ¯ constant ., The predator and prey densities initially grow , but the prey trait value does not change fast enough for the prey population growth to sustain—eventually resulting in extinction of both the predator and prey ., However , if V takes a slightly larger value , so that the mean trait value can gradually change with a growing prey population density ( due to the density-dependent term in Eq ( 10 ) ) , then the population dynamics begin to display regular cycling with fixed frequencies and amplitudes ( Fig 2A , top ) ., This corresponds to a case where the evolutionary dynamics are slow compared to the ecological dynamics , but not so slow as to be completely negligible ., Finally , when V is the same order of magnitude as the parameters governing the ecological dynamics , the irregular cycles become fully chaotic , with both amplitudes and frequencies that vary widely over even relatively short time intervals ( Fig 2A , bottom ) ., Typically , the large V case would correspond to circumstances in which the prey population develops a large standing genetic variation 10 , 44 ., That the dynamics are chaotic , rather than quasi-periodic , is suggested by the presence of multiple broad , unevenly-spaced peaks in the power spectrum 45 ( Figure A in S1E Appendix ) , as well as by numerical tabulation of the Lyapunov spectrum ( described further below ) ., Due to the hierarchical coupling of Eqs ( 10 ) , ( 11 ) and ( 12 ) , when plotted in three-dimensions the chaotic dynamics settle onto a strange attractor that resembles the “teacup” attractor found in models of tritrophic food webs 24 , 46 ( Fig 2B ) ., Poincare sections though various planes of the teacup all appear linear , suggesting that the strange attractor is effectively two-dimensional—consistent with pairings of timescales associated with different dynamical variables at different points in the process ( Figure B in S1E Appendix ) ., In the “rim” of the teacup , the predator density changes slowly relative to the prey density and mean trait value ., This is visible in a projection of the attractor into the x - c ¯ plane ( Fig 2B , bottom inset ) ., However , in the “handle” of the teacup , the mean trait value varies slowly relative to the ecological dynamics ( c ¯ ˙ ≈ 0 ) , resulting in dynamics that qualitatively resemble the two-dimensional “reduced” system described above for various fixed values of c ¯ ( Fig 2B , top inset ) ., The structure of the attractor suggests that the prey alternately enters periods of evolutionary change and periods of competition with the predator ., A closer inspection of a typical transition reveals that this “two timescale” dynamical separation is responsible for the appearance of chaos in the system ( Fig 3A ) ., As the system explores configuration space , it reaches a metastable configuration corresponding to a high mean trait value c ¯ , which causes the prey density to nearly equilibrate to a low density due to the negative density-dependent term in Eq ( 10 ) ., During this period ( the “rim” of the teacup ) , the predator density gradually declines due to the lack of prey ., However , once the predator density becomes sufficiently small , the prey population undergoes a sudden population increase , which triggers a period of rapid cycling in the system ( the “handle” of the teacup attractor ) ., During this time , the predator density continuously increases , causing an equivalent decrease in the prey density that resets the cycle to the metastable state ., The sudden increase in the prey population at low predator densities can be understood from how the fitness function r ( from Eq ( 9 ) ) changes over time ., Fig 3B shows a kymograph of the log-scaled fitness Eq ( 9 ) as a function of individual trait values c , across each timepoint and corresponding set of ( x , y , c ¯ ) values given in panel A . Overlaid on this time-dependent fitness landscape are curves indicating the instantaneous location of the local maximum ( black ) and minimum ( white ) ., By comparing panels A and B , it is apparent that the mean trait value during the “metastable” period of the dynamics stays near the local maximum of the fitness function , which barely varies as the predator density y changes ., However , when y ( t ) ≈ 0 . 25 , the fitness function changes so that the local minimum and local maximum merge and disappear from the system , leading to a new maximum spontaneously appearing at c = 0 ., Because V is large enough ( for these parameters ) that the gradient dynamics occur over timescales comparable to the competition dynamics , the system tends to move rapidly towards this new maximum in the fitness landscape , resulting in rapidly-changing dynamics in x and c ¯ ., Importantly , because of the symmetric coupling of the prey fitness landscape r to the prey density x , this rapid motion resets the fitness landscape so that the maximum once again occurs at the original value , resulting in a period of rapid cycling ., The fitness landscape at two representative timepoints in the dynamics is shown in Fig 3C ., That the maxima in the fitness Function ( 9 ) suddenly change locations with continuous variation in x , y is a direct consequence of the use of a high-order ( here , quartic ) polynomial in c to describe the fitness landscape ., The quartic represents the simplest analytic function that admits more than one local maxima in its domain , and the number of local maxima is governed by the relative signs of the coefficients of the ( c 2 - c ¯ 2 ) and ( c 4 - c ¯ 4 ) terms in Eq ( 9 ) , which change when the system enters the rapid cycling portion of the chaotic dynamics at t = 500 in Fig 3A ., This transition marks the mean prey trait switching from being drawn ( via the gradient dynamics ) to a single fitness peak at an intermediate value of the trait ceq ≈ 0 . 707 to being drawn instead to one of two peaks: the existing peak , or a new peak at the origin ., Thus the metastable period of the dynamics corresponds to a period of stabilizing selection: if the fitness landscape were frozen in time during this period , then an ensemble of prey would all evolve to a single intermediate trait value corresponding to the location of the global maximum ., Conversely , if the fitness landscape were held fixed in the multipeaked form it develops during a period of rapid cycling , given sufficient time an ensemble of prey would evolve towards subpopulations with trait values at the location of each local fitness maximum—representing disruptive selection ., That the fitness landscape does not remain fixed for extended durations in either a stabilizing or disruptive state—but rather switches between the two states due to the prey density-dependent term in Eq ( 9 ) — underlies the onset of chaotic cycling in the model ., Density-dependent feedback similarly served to induce chaos in many early discrete-time ecosystem models 23 ., However , the “two timescale” form of the chaotic dynamics and strange attractor here is a direct result of reversible transitions between stabilizing and disruptive selection ., If the assumptions underlying the gradient dynamics model do not strictly hold—if the additive genetic variance V slowly varies via an additional dynamical equation , or if the initial conditions are such that significant skewness would be expected to persist in the phenotypic distribution , then the chaotic dynamics studied here would be transient rather than indefinite ., While the general stability analysis shown above ( and in the S1 Appendix ) would still hold , additional dynamical equations for V or for high-order moments of the trait distribution would introduce additional constraints on the values of the parameters , which would ( in general ) increase the opportunities for the dynamics to become unstable and lead to diverging predator or prey densities ., However , in some cases these additional effects may actually serve to stabilize the system against both chaos and divergence ., For example , if additional series terms were included in Eq ( 8 ) such that the dependence of mortality rate on c ¯ and c had an upper asymptote 25 , then c ¯ ˙ = 0 would be true for a larger range of parameter values—resulting in the dynamical system remaining planar for a larger range of initial conditions and parameter values , precluding chaos ., The transition between stabilizing and disruptive selection that occurs when the system enters a period of chaotic cycling is strongly reminiscent of a first-order phase transition ., Many physical systems can be described in terms of a free energy landscape , the negative gradient of which determines the forces acting on the system ., Minima of the free energy landscape correspond to equilibrium points of the system , which the dynamical variables will approach with first-order dynamics in an overdamped limit ., When a physical system undergoes a phase transition—a qualitative change in its properties as a single “control” parameter , an externally-manipulable variable such as temperature , is smoothly varied—the transition can be understood in terms of how the control parameter changes the shape of the free energy landscape ., The Landau free energy model represents the simplest mathematical framework for studying such phase transitions: a one-dimensional free energy landscape is defined as a function of the control parameter and an additional independent variable , the “order parameter , ” a derived quantity ( such as particle density or net magnetization ) with respect to which the free energy can have local minima or maxima ., In a first-order phase transition in the Landau model , as the control parameter monotonically changes the relative depth of a local minimum at the origin decreases , until a new local minimum spontaneously appears at a fixed nonzero value of the order parameter—resulting in dynamics that suddenly move towards the new minimum , creating discontinuities in thermodynamic properties of the system such as the entropy 47 ., First-order phase transitions are universal physical models , which have been used to describe a broad range of processes spanning from superconductor breakdown 48 to primordial black hole formation in the early universe 49 ., In the predator-prey model with prey evolution , the fitness function is analogous to the free energy , with the individual trait value c serving as the “order parameter” for the system ., The control parameter for the transition is the prey density , x , which directly couples into the dynamics via the density-dependent term in Eq ( 7 ) ., Because the fitness consists of a linear combination of this term in Eq ( 7 ) and a quartic landscape Eq ( 8 ) , the changing prey density “tilts” the landscape and provokes the appearance of the additional , disruptive peak visible in Fig 3C ., The appearance and disappearance of local maxima as the system switches between stabilizing and disruptive selection is thus analogous to a first-order phase transition , with chaotic dynamics being a consequence of repeated increases and decreases of the control parameter x above and below the critical prey densities x* , x** at which the phase transition occurs ., Similar chaotic dynamics emerge from repeated first-order phase transitions in networks of coupled oscillators , which may alternate between synchronized and incoherent states that resemble the “metastable” and “rapid cycling” portions of the predator-prey dynamics 50 ., The analogy between a first-order phase transition and the onset of disruptive selection can be used to study the chaotic dynamics in terms of dynamical hysteresis , a defining feature of such phase transitions 47 ., For different values of x , the three equilibria corresponding to the locations of the local minima and maxima of the fitness landscape , ceq , can be calculated from the roots of the cubic in Eq ( 12 ) ., The resulting plots of ceq vs x in Fig 4 are generated by solving for the roots in the limit of fast prey equilibration , c ¯ → c e q , which holds in the vicinity of the equilibria ( S1B Appendix ) ., The entry into the transient chaotic cycling occurs when x increases gradually and shifts ceq with it; x eventually attains a critical value x* ( x* ≈ 0 . 45 for the parameters used in the figures ) , causing ceq to jump from its first critical value c* to the origin ( the red “forward” branch in Fig 4 ) ., This jump causes rapid re-equilibration of c ¯ ( t ) , resulting in the rapid entry into cycling observable in Fig 3A ., However , x cannot increase indefinitely due to predation; rather , it decreases until it reaches a second critical value x** , at which point ceq jumps back from the origin to a positive value ( the blue “return” branch in Fig 4; x** = 0 . 192 for these parameter values ) ., This second critical point marks the return to the metastable dynamics in Fig 3A ., This asymmetry in the forward and backwards dynamics of x lead to dynamical time-irreversibility ( hysteresis ) and the jagged , sawtooth-like cycles visible in the dynamics of the full system ., Because the second jump in ceq is steeper , the parts of the trajectories associated with the “return” transition in Fig 3A appear steeper ., Additionally , the maximum value obtained by c ¯ ( t ) anywhere on the attractor , c e q m a x , is determined by the limiting value o

Introduction, Model, Results, Discussion

In many ecosystems , natural selection can occur quickly enough to influence the population dynamics and thus future selection ., This suggests the importance of extending classical population dynamics models to include such eco-evolutionary processes ., Here , we describe a predator-prey model in which the prey population growth depends on a prey density-dependent fitness landscape ., We show that this two-species ecosystem is capable of exhibiting chaos even in the absence of external environmental variation or noise , and that the onset of chaotic dynamics is the result of the fitness landscape reversibly alternating between epochs of stabilizing and disruptive selection ., We draw an analogy between the fitness function and the free energy in statistical mechanics , allowing us to use the physical theory of first-order phase transitions to understand the onset of rapid cycling in the chaotic predator-prey dynamics ., We use quantitative techniques to study the relevance of our model to observational studies of complex ecosystems , finding that the evolution-driven chaotic dynamics confer community stability at the “edge of chaos” while creating a wide distribution of opportunities for speciation during epochs of disruptive selection—a potential observable signature of chaotic eco-evolutionary dynamics in experimental studies .

Evolution is usually thought to occur very gradually , taking millennia or longer in order to appreciably affect a species survival mechanisms ., Conversely , demographic shifts due to predator invasion or environmental change can occur relatively quickly , creating abrupt and lasting effects on a species survival ., However , recent studies of ecosystems ranging from the microbiome to oceanic predators have suggested that evolutionary and ecological processes can often occur over comparable timescales—necessitating that the two be addressed within a single , unified theoretical framework ., Here , we show that when evolutionary effects are added to a minimal model of two competing species , the resulting ecosystem displays erratic and chaotic dynamics not typically observed in such systems ., We then show that these chaotic dynamics arise from a subtle analogy between the evolutionary concept of fitness , and the concept of the free energy in thermodynamical systems ., This analogy proves useful for understanding quantitatively how the concept of a changing fitness landscape can confer robustness to an ecosystem , as well as how unusual effects such as history-dependence can be important in complex real-world ecosystems ., Our results predict a potential signature of a chaotic past in the distribution of timescales over which new species can emerge during the competitive dynamics , a potential waypoint for future experimental work in closed ecosystems with controlled fitness landscapes .

ecology and environmental sciences, predator-prey dynamics, population dynamics, systems science, mathematics, population biology, thermodynamics, computer and information sciences, ecosystems, dynamical systems, free energy, community ecology, physics, population metrics, ecology, predation, natural selection, trophic interactions, biology and life sciences, physical sciences, population density, evolutionary biology, evolutionary processes

journal.pgen.1000098

Evaluating Statistical Methods Using Plasmode Data Sets in the Age of Massive Public Databases: An Illustration Using False Discovery Rates

“Omic” technologies ( genomic , proteomic , etc . ) have led to high dimensional experiments ( HDEs ) that simultaneously test thousands of hypotheses ., Often these omic experiments are exploratory , and promising discoveries demand follow-up laboratory research ., Data from such experiments require new ways of thinking about statistical inference and present new challenges ., For example , in microarray experiments an investigator may test thousands of genes aiming to produce a list of promising candidates for differential genetic expression across two or more treatment conditions ., The larger the list , the more likely some genes will prove to be false discoveries , i . e . genes not actually affected by the treatment ., Statistical methods often estimate both the proportion of tested genes that are differentially expressed due to a treatment condition and the proportion of false discoveries in a list of genes selected for follow-up research ., Because keeping the proportion of false discoveries small ensures that costly follow-on research will yield more fruitful results , investigators should use some statistical method to estimate or control this proportion ., However , there is no consensus on which of the many available methods to use 1 ., How should an investigator choose ?, Although the performance of some statistical methods for analyzing HDE data has been evaluated analytically , many methods are commonly evaluated using computer simulations ., An analytical evaluation ( i . e . , one using mathematical derivations to assess the accuracy of estimates ) may require either difficult-to-verify assumptions about a statistical model that generated the data or a resort to asymptotic properties of a method ., Moreover , for some methods an analytical evaluation may be mathematically intractable ., Although evaluations using computer simulations may overcome the challenge of intractability , most simulation methods still rely on the assumptions inherent in the statistical models that generated the data ., Whether these models accurately reflect reality is an open question , as is how to determine appropriate parameters for the model , what realistic “effect sizes” to incorporate in selected tests , as well as if and how to incorporate correlation structure among the many thousands of observations per unit 2 ., Plasmode data sets may help overcome the methodological challenges inherent in generating realistic simulated data sets ., Catell and Jaspers 3 made early use of the term when they defined a plasmode as “a set of numerical values fitting a mathematico-theoretical model . That it fits the model may be known either because simulated data is produced mathematically to fit the functions , or because we have a real—usually mechanical—situation which we know with certainty must produce data of that kind . ”, Mehta et al . ( p . 946 ) 2 more concisely refer to a plasmode as “a real data set whose true structure is known . ”, The plasmodes can accommodate unknown correlation structures among genes , unknown distributions of effects among differentially expressed genes , an unknown null distribution of gene expression data , and other aspects that are difficult to model using theoretical distributions ., Not surprisingly , the use of plasmode data sets is gaining traction as a technique of simulating reality-based data from HDEs 4 ., A plasmode data set can be constructed by spiking specific mRNAs into a real microarray data set 5 ., Evaluating whether a particular method correctly detects the spiked mRNAs provides information about the methods ability to detect gene expression ., A plasmode data set can also be constructed by using a current data set as a template for simulating new data sets for which some truth is known ., Although in early microarray experiments , sample sizes were too small ( often only 2 or 3 arrays per treatment condition ) to use as a basis for a population model for simulating data sets , larger HDE data sets have recently become publicly available , making their use feasible for simulation experiments ., In this paper , we propose a technique to simulate plasmode data sets from previously produced data ., The source-data experiment was conducted at the Center for Nutrient–Gene Interaction ( CNGI , www . uab . edu/cngi ) , at the University of Alabama at Birmingham ., We use a data set from this experiment as a template for producing a plasmode null data set , and we use the distribution of effect sizes from the experiment to select expression levels for differentially expressed genes ., The technique is intuitively appealing , relatively straightforward to implement , and can be adapted to HDEs in contexts other than microarray experiments ., We illustrate the value of plasmodes by comparing 15 different statistical methods for estimating quantities of interest in a microarray experiment , namely the proportion of true nulls ( hereafter denoted π0 ) , the false discovery rate ( FDR ) 6 and a local version of FDR ( LFDR ) 7 ., This type of analysis enables us , for the first time , to compare key omics research tools according to their performance in data that , by definition , are realistic exemplars of the types of data biologists will encounter ., The illustrations given here provide some insight into the relative performance characteristics of the 15 methods in some circumstances , but definitive claims regarding uniform superiority of one method over another would require more extensive evaluations over multiple types of data sets ., Steps for plasmode creation that are described herein are relatively straightforward ., First , an HDE data set is obtained that reflects the type of experiment for which statistical methods will be used to estimate quantities of interest ., Data from a rat microarray experiment at CNGI were used here ., Other organisms might produce data with different structural characteristics and methods may perform differently on such data ., The CNGI data were obtained from an experiment that used rats to test the pathways and mechanisms of action of certain phytoestrogens 8 , 9 ., In brief , rats were divided into two large groups , the first sacrificed at day 21 ( typically the day of weaning for rats ) , the second sacrificed at day 50 ( the day , corresponding to late human puberty , when rats are most susceptible to chemically induced breast cancer ) ., Each of these groups was subdivided into smaller groups according to diet ., At 21 and 50 days , respectively , the relevant tissues from these rat groups were appropriately processed , and gene expression levels were extracted using GCOS ( GeneChip Operating Software ) ., We exported the microarray image ( * . CEL ) files from GCOS and analyzed them with the Affymetrix Package of Bioconductor/R to extract the MAS 5 . 0 processed expression intensities ., The arrays and data were investigated for outliers using Pearsons correlation , spatial artifacts 10 and a deleted residuals approach 11 ., It is important to note that only one normalization method was considered , but the methods could be compared on RMA normalized data as well ., In fact , comparisons of methods performances on data from different normalization techniques could be done using the plasmode technique ., Second , an HDE data set that compares effect of a treatment ( s ) is analyzed and the vector of effect sizes is saved ., The effect size used here was a simple standardized mean difference ( i . e . , a two sample t-statistics ) but any meaningful metric could be used ., Plasmodes , in fact , could be used to compare the performance of statistical methods when different statistical tests were used to produce the P-values ., We chose two sets of HDE data as templates to represent two distributions of effect sizes and two different null distributions ., We refer to the 21-day experiment using the control group ( 8 arrays ) and the treatment group ( EGCG supplementation , 10 arrays ) as data set 1 , and the 50-day experiment using the control group ( 10 arrays ) and the treatment group ( Resveratrol supplementation , 10 arrays ) as data set, 2 . There were 31 , 042 genes on each array , and two sample pooled variance t-tests for differential expression were used to create a distribution of P-values ., Histograms of the distributions for both data sets are shown in Figure, 1 . The distribution of P-values for data set 1 shows a stronger signal ( i . e . , a larger collection of very small P-values ) than that for data set 2 , suggesting either that more genes are differentially expressed or that those that are expressed have a larger magnitude treatment effect ., This second step provided a distribution of effects sizes from each data set ., Next , create the plasmode null data set ., For each of the HDE data sets , we created a random division of the control group of microarrays into two sets of equal size ., One consideration in doing so is that if some arrays in the control group are ‘different’ from others due to some artifact in the experiment , then the null data set can be sensitive to how the arrays are divided into two sets ., Such artifacts can be present in data from actual HDEs , so this issue is not a limitation of plasmode use but rather an attribute of it , that is , plasmodes are designed to reflect actual structure ( including artifacts ) in a real data set ., We obtained the plasmode null data set from data set 1 by dividing the day 21 control group of 8 arrays into two sets of 4 , and for data set 2 by dividing the control group of 10 arrays into two sets of 5 arrays ., Figure 2 shows the two null distributions of P-values obtained using the two sample t-test on the plasmode null data sets ., Both null distributions are , as expected , approximately uniform , but sampling variability allows for some deviation from uniformity ., A proportion 1−π0 of effect sizes were then sampled from their respective distributions using a weighted probability sampling technique described in the Methods section ., What sampling probabilities are chosen can be a tuning parameter in the plasmode creation procedure ., The selected effects were incorporated into the associated null distribution for a randomly selected proportion 1−π0 of genes in a manner also described in the Methods section ., What proportion of genes is selected may depend upon how many genes in an HDE are expected to be differentially expressed ., This may determine whether a proportion equal to 0 . 01 or 0 . 5 is chosen to construct a plasmode ., Proportions between 0 . 05 and 0 . 2 were used here as they are in the range of estimated proportions of differentially expressed genes that we have seen from the many data sets we have analyzed ., Finally , the plasmode data set was analyzed using a selected statistical method ., We used two sample t-tests to obtain a plasmode distribution of P-values for each plasmode data set because the methods compared herein all analyze a distribution of P-values from an HDE ., P-values were declared statistically significant if smaller than a threshold τ ., Box 1 summarizes symbol definitions ., When comparing the 15 statistical methods , we used three values of π0 ( 0 . 8 , 0 . 9 , and 0 . 95 ) and two thresholds ( τ\u200a=\u200a0 . 01 and 0 . 001 ) ., For each choice of π0 and threshold τ , we ran B\u200a=\u200a100 simulations ., All 15 methods provided estimates of π0 , 14 provided estimates of FDR , and 7 provided estimates of LFDR ., Because the true values of π0 and FDR are known for each plasmode data set , we can compare the accuracy of estimates from the different methods ., There are two basic strategies for estimating FDR , both predicated on an estimated value for π0 , the first using equation ( 1 ) below , the second using a mixture model approach ., Let PK\u200a=\u200aM/K be the proportion of tests that were declared significant at a given threshold , where M and K were defined with respect to quantities in Table, 1 . Then one estimate for FDR at this threshold is , ( 1 ) The mixture model ( usually a two-component mixture ) approach uses a model of the form , ( 2 ) where f is a density , p represents a P-value , f0 a density of a P-value under the null hypothesis , f1 a density of a P-value under the alternative hypothesis , π0 is interpreted as before , and θ a ( possibly vector ) parameter of the distribution ., Since valid P-values are assumed , f0 is a uniform density ., LFDR is defined with respect to this mixture model as , ( 3 ) FDR is defined similarly except that the densities in ( 3 ) are replaced by the corresponding cumulative distribution functions ( CDF ) , that is , ( 4 ) where F1 ( τ ) is the CDF under the alternative hypothesis , evaluated at a chosen threshold τ ., ( There are different definitions of FDR and the definition in ( 4 ) is , under some conditions , the definition of a positive false discovery rate 12 ., However , in cases with a large number of genes many of the variants of FDR are very close 13 ) ., The methods are listed for quick reference in Table, 2 . Methods 1–8 use different estimates for π0 and , as implemented herein , proceed to estimate FDR using equation ( 1 ) ., Method 9 uses a unique algorithm to estimate LFDR and does not supply an estimate of FDR ., Methods 10–15 are based on a mixture model framework and estimate FDR and LFDR using equations ( 3 ) and ( 4 ) where the model components are estimated using different techniques ., All methods were implemented using tuning parameter settings from the respective paper or ones supplied as default values with the code in cases where the code was published online ., First , to compare their differences , we used the 15 methods to analyze the original two data sets , with data set 1 having a “stronger signal” ( i . e . , lower estimates of π0 and FDR ) ., Estimates of π0 from methods 3 through 15 ranged from 0 . 742 to 0 . 837 for data set 1 and 0 . 852 to 0 . 933 for data set, 2 . ( Methods 1 and 2 are designed to control for rather than estimate FDR and are designed to be conservative; hence , their estimates were much closer to, 1 . ) Results of these analyses can be seen in the Supplementary Tables S1 and S2 ., Next , using the two template data sets we constructed plasmode data sets in order to compare the performance of the 15 methods for estimating π0 ( all methods ) , FDR ( all methods except method 9 ) , and LFDR ( methods 9–15 ) ., Figures 3 and 4 show some results based on data set, 2 . More results are available in the Figures S1 , S2 , S3 , S4 , S5 , and S6 ., Figure 3 shows the distribution of 100 estimates for π0 using data set 2 when the true value of π0 is equal to 0 . 8 and 0 . 9 ., Methods 1 and 2 are designed to be conservative ( i . e . , true values are overestimated ) ., With a few exceptions , the other methods tend to be conservative when π0\u200a=\u200a0 . 8 and liberal ( the true value is underestimated ) when π0\u200a=\u200a0 . 9 ., The variability of estimates for π0 is similar across methods , but some plots show a slightly larger variability for methods 12 and 15 when π0\u200a=\u200a0 . 9 ., Figure 4 shows the distribution of estimates for FDR and LFDR at the two thresholds ., The horizontal lines in the plots show the mean ( solid line ) and the minimum and maximum ( dashed lines ) of the true FDR value for the 100 simulations ., A true value for LFDR is not known in the simulation procedure ., The methods tend to be conservative ( overestimate FDR ) when the threshold τ\u200a=\u200a0 . 01 and are more accurate at the lower threshold ., Estimates of FDR are more variable for methods 11 , 13 , and 14 and estimates for LFDR more variable for methods 13 and 14 , with the exception of a few unusual estimates obtained from method 9 ., The high variability of FDR estimates from method 11 may be due to a “less than optimal” choice of the spanning parameter in a numerical smoother ( see also Pounds and Cheng 27 ) ., We did not attempt to tune any of the methods for enhanced performance ., Researchers have been evaluating the performance of the burgeoning number of statistical methods for the analysis of high dimensional omic data , relying on a mixture of mathematical derivations , computer simulations , and sadly , often single dataset illustrations or mere ipse dixit assertions ., Recognizing that the latter two approaches are simply unacceptable approaches to method validation 2 and that the first two suffer from limitations described earlier , an increasing number of investigators are turning to plasmode datasets for method evaluation 28 ., An excellent example is the Affycomp website ( http://affycomp . biostat . jhsph . edu/ ) that allows investigators to compare different microarray normalization methods on datasets of known structure ., Other investigators have also recently used plasmode-like approaches which they refer to as ‘data perturbation’ 29 , 30 , yet it is not clear that these ‘perturbed datasets’ can distinguish true from false positives , suggesting greater need for articulation of principles or standards of plasmode generation ., As more high dimensional experiments with larger sample sizes become available , researchers can use a new kind of simulation experiment to evaluate the performance of statistical analysis methods , relying on actual data from previous experiments as a template for generating new data sets , referred to herein as plasmodes ., In theory , the plasmode method outlined here will enable investigators to choose on an empirical basis the most appropriate statistical method for their HDEs ., Our results also suggest that large , searchable databases of plasmode data sets would help investigators find existing data sets relevant to their planned experiments ., ( We have already implemented a similar idea for planning sample size requirements in HDEs 31 , 32 . ), Investigators could then use those data sets to compare and evaluate several analytical methods to determine which best identifies genes affected by the treatment condition ., Or , investigators could use the plasmode approach on their own data sets to glean some understanding of how well a statistical method works on their type of data ., Our results compare the performance of 15 statistical methods as they process the specific plasmode data sets constructed from the CNGI data ., Although identifying one uniformly superior method ( if there is one ) is difficult within the limitations of this one comparison , our results suggest that certain methods could be sensitive to tuning parameters or different types of data sets ., A comparison over multiple types of source data sets with different distributions of effects sizes could add the detail necessary to clearly recommend certain methods over others 1 ., Other papers have used simulation studies to compare the performance of methods for estimating π0 and FDR ( e . g . , Hsueh et al . 33; Nguyen 34; Nettleton et al . 35 ) ., We compared methods that use the distribution of P-values as was done in Broberg 36 and Yang and Yang 37 ., Unlike our plasmode approach , most earlier comparison studies used normal distributions to simulate gene expression data and incorporated dependence using a block diagonal correlation structure as in Allison et al 26 ., A key implication and recommendation of our paper is that , as data from the growing number of HDEs is made publicly available , researchers may identify a previous HDE similar to one they are planning or have recently conducted and use data from these experiments to construct plasmode data sets with which to evaluate candidate statistical methods ., This will enable investigators to choose the most appropriate method ( s ) for analyzing their own data and thus to increase the reliability of their research results ., In this manner , statistical science ( as a discipline that studies the methods of statistics ) becomes as much an empirical science as a theoretical one ., The quantities in Table 1 are those for a typical microarray experiment ., Let N\u200a=\u200aA+B and M\u200a=\u200aC+D and note that both N and M will be known and K\u200a=\u200aN+M ., However , the number of false discoveries is equal to an unknown number C . The proportion of false discoveries for this experiment is C/M ., Benjamini and Hochberg 6 defined FDR as , P ( M>0 ) where I{M>0} is an indicator function equal to 1 if M>0 and zero otherwise ., Storey 12 defined the positive FDR as ., Since P ( M>0 ) ≥1− ( 1−τ ) K , and since K is usually very large , FDR≈pFDR , so we do not distinguish between FDR and pFDR as the parameter being estimated and simply refer to it as FDR with estimates denoted ( and ) ., Suppose we identify a template data set corresponding to a two treatment comparison for differential gene expression for K genes ., Obtain a vector , δ , of effect sizes ., One suggestion is the usual t-statistic , where the ith component of δ , is given by ( 5 ) where ntrt , nctrl are number of biological replicates in the treatment and control group , respectively , X̅i , trt , X̅i , ctrl are the mean gene expression levels for gene i in treatment and control groups , and , is the usual pooled sample variance for the ith gene , where the two sample variances are given by , ., In what follows , we will use this choice for δi since it allows for effects to be described by a unitless quantity , i . e . , it is scaled by the standard error of the observed mean difference X̅i , trt−X̅i , ctrl for each gene ., For convenience , assume that nctrl is an even number and divide the control group into two sets of equal size ., Requiring nctrl≥4 allows for at least two arrays in each set , thus allowing estimates of variance within each of the two sets ., This will be the basis for the plasmode “null” data set ., There are ways of making this division ., Without loss of generality , assume that the first nctrl/2 arrays after the division are the plasmode control group and the second nctrl/2 are the plasmode treatment group ., Specify a value of π0 and specify a threshold , τ , such that a P-value ≤τ is declared evidence of differential expression ., Execute the following steps ., One can then obtain another data set and repeat the entire process to evaluate a method on a different type of data , perhaps from a different organism having a different null distribution , or a different treatment type giving a different distribution of effect sizes , δ ., Alternatively , one might choose to randomly divide the control group again and repeat the entire process ., This would help assess how differences in arrays within a group or possible correlation structure might affect results from a method ., If some of the arrays in the control group have systematic differences among them ( e . g . , differences arising from variations in experimental conditions—day , operator , technology , etc . ) , then the null distribution can be sensitive to the random division of the original control group into the two plasmode groups , particularly if nctrl is small .

Introduction, Results, Discussion, Methods

Plasmode is a term coined several years ago to describe data sets that are derived from real data but for which some truth is known ., Omic techniques , most especially microarray and genomewide association studies , have catalyzed a new zeitgeist of data sharing that is making data and data sets publicly available on an unprecedented scale ., Coupling such data resources with a science of plasmode use would allow statistical methodologists to vet proposed techniques empirically ( as opposed to only theoretically ) and with data that are by definition realistic and representative ., We illustrate the technique of empirical statistics by consideration of a common task when analyzing high dimensional data: the simultaneous testing of hundreds or thousands of hypotheses to determine which , if any , show statistical significance warranting follow-on research ., The now-common practice of multiple testing in high dimensional experiment ( HDE ) settings has generated new methods for detecting statistically significant results ., Although such methods have heretofore been subject to comparative performance analysis using simulated data , simulating data that realistically reflect data from an actual HDE remains a challenge ., We describe a simulation procedure using actual data from an HDE where some truth regarding parameters of interest is known ., We use the procedure to compare estimates for the proportion of true null hypotheses , the false discovery rate ( FDR ) , and a local version of FDR obtained from 15 different statistical methods .

Plasmode is a term used to describe a data set that has been derived from real data but for which some truth is known ., Statistical methods that analyze data from high dimensional experiments ( HDEs ) seek to estimate quantities that are of interest to scientists , such as mean differences in gene expression levels and false discovery rates ., The ability of statistical methods to accurately estimate these quantities depends on theoretical derivations or computer simulations ., In computer simulations , data for which the true value of a quantity is known are often simulated from statistical models , and the ability of a statistical method to estimate this quantity is evaluated on the simulated data ., However , in HDEs there are many possible statistical models to use , and which models appropriately produce data that reflect properties of real data is an open question ., We propose the use of plasmodes as one answer to this question ., If done carefully , plasmodes can produce data that reflect reality while maintaining the benefits of simulated data ., We show one method of generating plasmodes and illustrate their use by comparing the performance of 15 statistical methods for estimating the false discovery rate in data from an HDE .

biotechnology, mathematics, science policy, computational biology, molecular biology, genetics and genomics

journal.pcbi.1006166

Variability in pulmonary vein electrophysiology and fibrosis determines arrhythmia susceptibility and dynamics

Success rates for catheter ablation of persistent atrial fibrillation ( AF ) patients are currently low; however , there is a subset of patients for whom pulmonary vein isolation ( PVI ) alone is a successful treatment strategy 1 ., PVI ablation may work by preventing triggered beats from entering the left atrial body , or by converting rotors or functional reentry around the left atrial/pulmonary vein ( LA/PV ) junction to anatomical reentry around a larger circuit , potentially converting AF to a simpler tachycardia 2 ., It is difficult to predict whether PVI represents a sufficient treatment strategy for a given patient with persistent AF 1 , and it is unclear what to do for the majority of patients for whom it is not effective ., Patients with AF exhibit distinct properties in effective refractory period ( ERP ) and conduction velocity ( CV ) in the PVs ., For example , paroxysmal AF patients have shorter ERP and longer conduction delays compared to control patients 3 ., AF patients show a number of other differences to control patients: PVs are larger 4; PV fibrosis is increased; and fiber direction may be more disorganised , particularly at the PV ostium 5 ., There are also differences within patient groups; for example , patients for whom persistent AF is likely to terminate after PVI have a larger ERP gradient compared to those who require further ablation 1 , 3 ., Electrical driver location changes as AF progresses; drivers ( rotors or focal sources ) are typically located close to the PVs in early AF , but are also located elsewhere in the atria with longer AF duration 6 ., Atrial fibrosis is a major factor associated with AF and modifies conduction ., However , there is conflicting evidence on the relationship between fibrosis distribution and driver location 7 , 8 ., It is difficult to clinically separate the individual effects of these factors on arrhythmia susceptibility and maintenance ., We hypothesise that the combination of PV properties and atrial body fibrosis determines driver location and , thus , the likely effectiveness of PVI ., In this study , we tested this hypothesis by using computational modelling to gain mechanistic insight into the individual contribution of PV ERP , CV , fiber direction , fibrosis and anatomy on arrhythmia susceptibility and dynamics ., We incorporated data on APD ( action potential duration , as a surrogate for ERP ) and CV for the PVs to determine mechanisms underlying arrhythmia susceptibility , by testing inducibility from PV ectopic beats ., We also predicted driver location , and PVI outcome ., All simulations were performed using the CARPentry simulator ( available at https://carp . medunigraz . at/carputils/ ) ., We used a previously published bi-atrial bilayer model 9 , which consists of resistively coupled endocardial and epicardial surfaces ., This model incorporates detailed atrial structure and includes transmural heterogeneity at a similar computational cost to surface models ., We chose to use a bilayer model rather than a volumetric model incorporating thickness for this study because of the large numbers of parameters investigated , which was feasible with the reduced computational cost of the bilayer model ., As previously described , the bilayer model was constructed from computed tomography scans of a patient with paroxysmal AF , which were segmented and meshed to create a finite element mesh suitable for electrophysiology simulations ., Fiber information was included in the model using a semi-automatic rule based method that matches histological descriptions of atrial fiber orientation 10 ., The left atrium of the bilayer model consists of linearly coupled endocardial and epicardial layers , while the right atrium is an epicardial layer , with endocardial atrial structures including the pectinate muscles and crista terminalis ., The left and right atrium of the model are electrically connected through three pathways: Bachmann’s bundle , the coronary sinus and the fossa ovalis ., Tissue conductivities were tuned to human activation mapping data from Lemery et al . 9 , 11 ., The Courtemanche-Ramirez-Nattel human atrial ionic model was used with changes representing electrical remodelling during persistent AF 12 , together with a doubling of sodium conductance to produce realistic action potential upstroke velocities 9 , and a decrease in IK1 by 20% to match clinical restitution data 13 ., Regional heterogeneity in repolarisation was included by modifying ionic conductances of the cellular model , as described in Bayer et al . 14 , which follows Aslanidi et al . and Seemann et al . 15 , 16 ., Parameters for the baseline PV model were taken from Krueger et al . 17 ., The following PV properties were varied as shown in schematic Fig 1: APD , CV , fiber direction , the inclusion of fibrosis in the PVs and the atrial geometry ., These are described in the following sections ., To investigate the effects of PV length and diameter on arrhythmia inducibility and arrhythmia dynamics , bi-atrial bilayer meshes were constructed from MRI data for twelve patients ., All patients gave written informed consent; this study is in accordance with the Declaration of Helsinki , and approved by the Institutional Ethics Committee at the University of Bordeaux ., Patient-specific models with electrophysiological heterogeneity and fiber direction were constructed using our modelling pipeline , which uses a universal atrial coordinate system to map scalar and vector data from the original bilayer model to a new patient specific mesh ., Late gadolinium enhancement MRI ( average resolution 0 . 625mm x 0 . 625mm x 2 . 5mm ) was performed using a 1 . 5T system ( Avanto , Siemens Medical Solutions , Erlangen , Germany ) ., These LGE-MRI data were manually segmented using the software MUSIC ( Electrophysiology and Heart Modeling Institute , University of Bordeaux , Bordeaux France , and Inria , Sophia Antipolis , France , http://med . inria . fr ) ., The resulting endocardial surfaces were meshed ( using the Medical Imaging Registration Toolkit mcubes algorithm 18 ) and cut to create open surfaces at the mitral valve , the four pulmonary veins , the tricuspid valve , and each of the superior vena cava , the inferior vena cava and the coronary sinus using ParaView software ( Kitware , Clifton Park , NY , USA ) ., The meshes were then remeshed using mmgtools meshing software ( http://www . mmgtools . org/ ) , with parameters chosen to produce meshes with an average edge length of 0 . 34mm to match the resolution of the previously published bilayer model 9 ., Two atrial coordinates were defined for each of the LA and RA , which allow automatic transfer of atrial structures to the model , such as the pectinate muscles and Bachmann’s bundle ., These coordinates were also used to map fiber directions to the bilayer model ., To investigate the effects of PV electrophysiology on arrhythmia inducibility and dynamics , we varied PV APD and CV by modifying the value of the inward rectifier current ( IK1 ) conductance and tissue level conductivity respectively ., IK1 conductance was chosen in this case to investigate macroscopic differences in APD 19 , although several ionic conductances are known to change with AF 20 ., Modifications were either applied homogeneously or following a ostial-distal gradient ., This gradient was implemented by calculating geodesic distances from the rim of mesh nodes at the distal PV boundary to all nodes in the PV and from the rim of nodes at the LA/PV junction to all nodes in the PV ., The ratio of these two distances was then used as a distance parameter from the LA/PV junction to the distal end of the PV ( see Fig 1 ) ., IK1 conductance was multiplied by a value in the range 0 . 5–2 . 5 , resulting in PV APDs in the clinical range of 100–190ms 3 , 21 , 22 ., This rescaling was either a homogeneous change or followed a gradient along the PV length ., Gradients of IK1 conductance varied from the baseline value at the LA/PV junction , to a maximum scaling factor at the distal boundary ., PV APDs are reported at 90% repolarisation for a pacing cycle length of 1000ms ., LA APD is 185ms , measured at a LA pacing cycle length of 200ms ., To cover the clinically observed range of PV CVs , longitudinal and transverse tissue conductivities were divided by 1 , 2 , 3 or 5 , resulting in CVs , measured along the PV axis , in the range: 0 . 28–0 . 67m/s 3 , 21–24 ., To model heterogeneous conduction slowing , conductivities were varied as a function of distance from the LA/PV junction , ranging from baseline at the junction to a maximum rescaling ( minimum conductivity ) at the distal boundary ., The direction of this gradient was also reversed to model conduction slowing at the LA/PV junction 5 ., Motivated by the findings of Hocini et al . 5 , interstitial fibrosis was modelled for the PVs with a density varying along the vein , increasing from the LA/PV junction to the distal boundary ., This was implemented by randomly selecting edges of elements of the mesh with probability scaled by the distance parameter and the angle of the edge compared to the element fiber direction , where edges in the longitudinal fiber direction were four times more likely to be selected than those in the transverse direction , following our previous methodology 25 ., To model microstructural discontinuities , no flux boundary conditions were applied along the connected edge networks , following Costa et al . 26 ., An example of modelled PV interstitial fibrosis is shown in S1A Fig . For a subset of simulations , interstitial fibrosis was incorporated in the biatrial model based on late gadolinium enhancement ( LGE ) -MRI data , using our previously published methodology 25 ., In brief , likelihood of interstitial fibrosis depended on both LGE intensity and the angle of the edge compared to the element fiber direction ( see S1B Fig ) ., LGE intensity distributions were either averaged over a population of patients 27 , or for an individual patient ., The averaged distributions were for patients with paroxysmal AF ( averaged over 34 patients ) , or persistent AF ( averaged over 26 patients ) ., For patient-specific simulations , the model arrhythmia dynamics were compared to AF recordings from a commercially available non-invasive ECGi mapping technology ( CardioInsight Technologies Inc . , Cleveland , OH ) for which phase mapping analysis was performed as previously described 28 ., PV fiber direction shows significant inter-patient variability ., Endocardial and epicardial fiber direction in the four PVs was modified according to fiber arrangements described in the literature 5 , 29 , 30 ., Six arrangements were considered , as follows:, 1 . circular arrangement on both the endocardium and epicardium;, 2 . spiralling arrangement on both the endocardium and epicardium;, 3 . circular arrangement on the endocardium , with longitudinal epicardial fibers;, 4 . fibers progress from longitudinal at the distal vein to circumferential at the ostium , with identical endocardial and epicardial fibers;, 5 . epicardial layer fibers as per case 4 , with circumferential endocardial fibers;, 6 . as per case 4 , but with a chaotic fiber arrangement at the LA/PV junction ., These fiber distributions are shown in S2 Fig . Cases 4–6 were implemented by setting the fiber angle to be a function of the distance along the vein , measured from the LA/PV junction to the distal boundary , varying from circumferential at the junction to longitudinal at the distal end ( representing a change of 90 degrees ) ., The disorder in fiber direction at the LA/PV junction for case 6 was implemented by taking the fibers of case 4 and adding independent standard Gaussian distributions scaled by the distance from the distal boundary , resulting in the largest perturbations at the ostium ., Arrhythmia inducibility was tested by extrastimulus pacing from each of the four PVs individually using a clinically motivated protocol 31 , to simulate the occurrence of PV ectopics ., Simulations were performed for each of the PVs , to determine the effects of ectopic beat location on inducibility ., Sinus rhythm was simulated by stimulating the sinoatrial node region of the model at a cycle length of 700ms throughout the simulation ., Each PV was paced individually with five beats at a cycle length of 160ms , and coupling intervals between the first PV beat and a sinus rhythm beat in the range 200–500 ms . Thirty-two pacing protocols were applied for each model set up: eight coupling intervals ( coupling interval = 200 , 240 , 280 , 320 , 360 , 400 , 440 , 480ms ) , for each of the four PVs ., Inducibility is reported as the proportion of cases resulting in reentry; termed the inducibility ratio ., The effects of PVI were determined for model set-ups that used the original bilayer geometry and in which the arrhythmia lasted for greater than two seconds ., PVI was applied two seconds post AF initiation in each case by setting the tissue conductivity close to zero ( 0 . 001 S/m ) in the regions shown in S3 Fig . For each case , ten seconds of arrhythmia data were analysed , starting from two seconds post AF initiation , to identify re-entrant waves and wavefront break-up using phase ., The phase of the transmembrane voltage was calculated for each node of the mesh using the Hilbert transform , following subtraction of the mean 32 ., Phase singularities ( PSs ) for the transmembrane potential data were identified by calculating the topological charge of each element in the mesh 33 , and PS spatial density maps were calculated using previously published methods 14 ., PS density maps were then partitioned into the LA body , PV regions , and the RA to assess where drivers were located in relation to the PVs ( see S3 Fig ) ., The PV region was defined as the areas enclosed by , and including , the PVI lines; the LA region was then the rest of the LA and left atrial appendage ., The PV PS density ratio was then defined as the total PV PS count divided by the total model PS count over both atria ., A difference in APD between the model LA and PVs was required for AF induction ., Modelling the PVs using LA cellular properties resulted in non-inducibility , whereas , modelling the LA using PV cellular properties resulted in either non-inducibility or macroreentry ., The effects of modifying PV APD homogeneously or following a gradient are shown in Table 1 ., Simulations in which PV APD was longer than LA APD were non-inducible ( PV APD 191ms ) ., As APD was decreased below the baseline value ( 181ms ) , inducibility initially increased and then fluctuated ., Comparing cases with equal distal APD , arrhythmia inducibility was significantly higher for APD following a ostial-distal gradient than for homogeneous APD ( p = 0 . 03 from McNemar’s test ) ., PS location was also affected by PV APD ., PV PS density was low in cases of short APD , an example of which is shown in Fig 2 where reentry is no longer seen around the LA/PV junction in the case of short APD ( 120ms ) ., This change was more noticeable for cases with homogeneous PV APD than for a gradient in APD; PV reentry was observed for the baseline case and a heterogeneous APD case , but not for a homogeneous decrease in APD ., Arrhythmia inducibility decreased with homogeneous CV slowing ( from 0 . 38 i . e . 12/32 at 0 . 67m/s to 0 . 03 i . e . 1/32 at 0 . 28m/s ) ., In the baseline model , reentry occurs close to the LA/PV junction due to conduction block when the paced PV beat encounters a change in fiber direction at the base of the PVs , together with a longer LA APD compared to the PV APD ., In this case , the wavefront encounters a region of refractory tissue due to the longer APD in the LA ., However , when PV CV is slowed homogeneously , the wavefront takes longer to reach the LA tissue , giving the tissue enough time to recover , such that conduction block and reentry no longer occurs ., Modifying conductivity following a gradient means that , unlike the homogeneous case , the time taken for the extrastimulus wavefront to reach the LA tissue is similar to the baseline case , so the LA tissue might still be refractory and conduction block might occur ., In the case that conduction was slowest at the distal vein , the inducibility was similar to the baseline case ( see Table 2 , GA , inducibility is 0 . 38 at baseline and 0 . 34 for the cases with CV slowing ) ., Cases with greatest conduction slowing at the LA/PV junction ( see Table 2 , GB ) exhibit an increase in inducibility ( from 0 . 38 to 0 . 53 ) when CV is decreased because of the discontinuity in conductivity at the junction ., Fig 2 shows that reentry is seen around the LA/PV junction in cases with both baseline and slow CV , indicating that the presence of reentry at the LA/PV junction is independent of PV CV ., PV conduction properties are also affected by PV fiber direction ., Modifications in fiber direction increased inducibility compared to the baseline fiber direction ( baseline case: 0 . 38; modified fiber direction cases 1-6: 0 . 53-0 . 63 ) ., The highest inducibility occurred with circular fibers at the ostium ( cases 1 and 4 , 0 . 63 ) , independent of fiber direction at the distal PV end ., This inducibility was reduced if the epicardial fibers were not circular at the ostium ( case 3 , 0 . 56 ) , or if fibers were spiralling ( case 2 , 0 . 56 ) instead of circular ., Next we investigated the interplay between PV properties and atrial fibrosis ., LA fibrosis properties were varied to represent interstitial fibrosis in paroxysmal and persistent AF patients , incorporating average LGE-MRI distributions 27 into the model ., These control , paroxysmal and persistent AF levels of fibrosis were then combined with PV properties varied as follows: baseline CV and APD ( 0 . 67m/s , 181ms ) , slow CV ( 0 . 51m/s ) , short APD ( 120ms ) , slow CV and short APD ., PS distributions in Fig 2 show that reentry occurred around the LA/PV junction in the case of baseline PV APD for control or paroxysmal levels of fibrosis , but not for shorter PV APD ., Modifying PV CV did not affect whether LA/PV reentry is observed ., Rotors were found to stabilise to regions of high fibrosis density in the persistent AF case ., Models with PV fibrosis had a higher inducibility compared to the baseline case ( 0 . 47 vs . 0 . 38 ) and a higher PV PS density since reentry localised there ., Fig 3 shows an example with moderate PV fibrosis ( A ) in which reentry changed from around the RIPV to the LIPV later in the simulation; adding a higher level of PV fibrosis resulted in a more stable reentry around the right PVs ( B ) ., The relationship between LA fibrosis and PV properties on driver location was investigated on an individual patient basis for four patients ., For patients for whom rotors were located away from the PVs ( Fig 4 LA1 ) , increasing model fibrosis from low to high increased the model agreement with clinical PS density 2 . 3 ± 1 . 0 fold ( comparing the sensitivity of identifying clinical regions of high PS density using model PS density between the two simulations ) ., For other patients , lower levels of fibrosis were more appropriate ( 2 . 1 fold increase in agreement for lower fibrosis , Fig 4 LA2 ) , and PV isolation converted fibrillation to macroreentry in the model ., Arrhythmia inducibility showed a large variation between patient geometries ( 0 . 16–0 . 47 ) ., Increasing PV area increased inducibility to a different degree for each vein: right superior PV ( RSPV ) inducibility was generally high ( > 0 . 75 for all but one geometry ) independent of PV area; left superior PV ( LSPV ) inducibility increased with PV area ( Spearman’s rank correlation coefficient of 0 . 36 indicating positive correlation; line of best fit gradient 0 . 27 , R2 = 0 . 3 ) ; left inferior PV ( LIPV ) and right inferior PV ( RIPV ) inducibility exhibited a threshold effect , in which veins were only inducible above a threshold area ( Fig 5A ) ., There is no clear relationship between PV length and inducibility ., PV PS density ratio increased as PV area increased ( Fig 5B , Spearman’s rank correlation coefficient of 0 . 41 indicating positive correlation ) ., Fig 5C shows that rotor and wavefront trajectories depend on patient geometry , exhibiting varied importance of the PVs compared to other atrial regions ., PVI outcome was assessed for cases with varied PV APD ( both with a homogeneous change or following a gradient ) , with the inclusion of PV fibrosis and with varied PV fiber direction because these factors were found to affect the PV PS density ratio ., PVI outcome was classified into three classes depending on the activity 1 second after PVI was applied in the model: termination , meaning there was no activity; macroreentry , meaning that there was a macroreentry around the LA/PV junctions; AF sustained by LA rotors , meaning there were drivers in the LA body ., These classes accounted for different proportions of the outcomes: termination ( 27 . 3% of cases ) , macroreentry ( 39 . 4% ) , or AF sustained by LA rotors ( 33 . 3% ) ., Calculating the PV PS density ratio before PVI for each of these classes shows that cases in which the arrhythmia either terminated or changed to a macroreentry are characterised by a statistically higher PV PS density ratio pre-PVI than cases sustained by LA rotors post-PVI ( see Fig 6 , t-test comparing termination and LA rotors shows they are significantly different , p<0 . 001; comparing macroreentry and LA rotors p = 0 . 01 ) ., High PV PS density ratio may indicate likelihood of PVI success ., In this computational modelling study , we demonstrated that the PVs can play a large role in arrhythmia maintenance and initiation , beyond being simply sources of ectopic beats ., We separated the effects of PV properties and atrial fibrosis on arrhythmia inducibility , maintenance mechanisms and the outcome of PVI , based on population or individual patient data ., PV properties affect arrhythmia susceptibility from ectopic beats; short PV APD increased arrhythmia susceptibility , while longer PV APD was found to be protective ., Arrhythmia inducibility increased with slower CV at the LA/PV junction , but not for cases with homogeneous CV changes or slower CV at the distal PV ., The effectiveness of PVI is usually attributed to PV ectopy , but our study demonstrates that the PVs affect reentry in other ways and this may , in part , also account for success or failure of PVI ., Both PV properties and fibrosis distribution affect arrhythmia dynamics , which varies from meandering rotors to PV reentry ( in cases with baseline or long APD ) , and then to stable rotors at regions of high fibrosis density ., PS density in the PV region was high for cases with PV fibrosis ., The measurement of fibrosis and PV properties may indicate patient specific susceptibility to AF initiation and maintenance ., PV PS density before PVI was higher in cases in which AF terminated or converted to a macroreentry; thus , high PV PS density may indicate likelihood of AF termination by PVI alone ., PV repolarisation is heterogeneous in the PVs 23 , and exhibits distinct properties in AF patients , with Rostock et al . reporting a greater decrease in PV ERP than LA ERP in patients with AF , termed AF begets AF in the PVs 21 ., Jais et al . found that PV ERP is greater than LA ERP in AF patients , but this gradient is reversed in AF patients 3 ., ERP measured at the distal PV is shorter than at the LA/PV junction during AF 5 , 22 ., Motivated by these clinical and experimental studies , we modelled a decrease in PV APD , which was applied either homogeneously , or as a gradient of decreasing APD along the length of the PV , with the shortest APD at the distal PV rim ., An initial decrease in APD increased inducibility ( Table 1 ) , which agrees with clinical findings of increased inducibility for AF patients ., Applying this change following a gradient , as observed in previous studies , led to an increased inducibility compared to a homogeneous change in APD ., Similar to Calvo et al . 34 we found that rotor location depends on PV APD ( Fig 2 ) ., Thus PV APD affects PVI outcome in two ways; on the one hand , decreasing APD increases inducibility , emphasising the importance of PVI in the case of ectopic beats; on the other hand , PV PS density decreases for cases with short PV APD , and PVI was less likely to terminate AF ., Multiple studies have measured conduction slowing in the PVs 3 , 5 , 21–24 ., We modelled changes in tissue conductivity either homogeneously , or as a function of distance along the PV ., Simply decreasing conductivity and thus decreasing CV , decreased inducibility ( Table 2 ) ., Kumagai et al . reported that conduction delay was longer for the distal to ostial direction 22 ., We found that modifying conductivity following a gradient , with CV decreasing towards the LA/PV junction , resulted in an increase in inducibility in the model ., This agrees with the clinical observations of Pascale et al . 1 ., This suggests that PVI should be performed in cases in which CV decreases towards the LA/PV junction as these cases have high inducibility ., Changes in CV may also be due to other factors , including gap junction remodelling , modified sodium conductance or changes in fiber direction 5 , 29 ., A variety of PV fiber patterns have been described in the literature and there is variability between patients ., Interestingly , all of the PV fiber directions considered in our study showed an increased inducibility compared to the baseline model ., Verheule et al . 29 documented circumferential strands that spiral around the lumen of the veins , motivating the arrangements for cases 1 and 4 in our study; Aslanidi et al . 15 reported that fibers run in a spiralling arrangement ( case 2 ) ; Ho et al . 30 measured mainly circular or spiral bundles , with longitudinal bundles ( cases 3 and 5 ) ; Hocini et al . 5 reported longitudinal fibers at the distal PV , with circumferential and a mixed chaotic fiber direction at the PV ostium ( case 6 ) ., Using current imaging technologies , PV fiber direction cannot be reliably measured in vivo ., In our study , fiber direction at the PV ostium was found to be more important than at the distal PV; the greatest inducibility was for cases with circular fibers at the ostium on both endocardial and epicardial surfaces , independent of fiber direction at the distal PV end ., Similar to modelling studies by both Coleman 35 and Aslanidi 15 , inducibility increased due to conduction block near the PVs ., PVs may be larger in AF patients compared to controls 4 , 36 , and this difference may vary between veins; Lin et al . found dilatation of the superior PVs in patients with focal AF originating from the PVs , but no difference in the dimensions of inferior PVs compared to control or to patients with focal AF from the superior vena cava or crista terminalis 37 ., We found that inducibility increased with PV area for the LSPV , LIPV and RIPV , but not for the RSPV ( see Fig 5 ) ., In addition , PV PS density ratio increased with total PV area , suggesting that PVI alone is more likely to be a successful treatment strategy in the case of larger veins ., However , Den Uijl et al . found no relation between PV dimensions and the outcome of PVI 38 ., Rotors were commonly found in areas of high surface curvature , including the LA/PV junction and left atrial appendage ostia , which agrees with findings of Tzortzis et al . 39 ., However , there were differences in PS density between geometries , with varying importance of the LA/PV junction ( Fig 5 ) , demonstrating the importance of modelling the geometry of an individual patient ., Myocardial tissue within the PVs is significantly fibrotic , which may lead to slow conduction and reentry 5 , 30 , 40 ., More fibrosis is found in the distal PV , with increased connective tissue deposition between myocardial cells 41 ., We modelled interstitial PV fibrosis with increasing density distally , and found that the inclusion of PV fibrosis increased PS density in the PV region of the model due to increased reentry around the LA/PV junction and wave break in the areas of fibrosis ., This , together with the results in Fig 6 , suggests that PVI alone is more likely to be a successful in cases of high PV fibrosis ., There are multiple methodologies for modelling atrial fibrosis 25 , 42 , 43 , and the choice of method may affect this localisation ., Population based distributions of atrial fibrosis were modelled for paroxysmal and persistent patients , together with varied PV properties ., The presence of LA/PV reentry depends on both PV properties and the presence of fibrosis; reentry is seen at the LA/PV junction for cases with baseline PV APD , but not for short PV APD , and stabilised to areas of high fibrosis in persistent AF , for which LA/PV reentry no longer occurred ., This suggests that rotor location depends on both fibrosis and PV properties ., This finding may explain the clinical findings of Lim et al . in which drivers are primarily located in the PV region in early AF , but AF complexity increased with increased AF duration , and drivers are also located at sites away from the PVs 6 ., During early AF , PV properties may be more important , while with increasing AF duration , there is increased atrial fibrosis in the atrial body that affects driver location ., This suggests that in cases with increased atrial fibrosis in the atrial body , ablation in addition to PVI is likely to be required ., Simulations of models with patient-specific atrial fibrosis together with varied PV properties performed in this study offer a proof of concept for using this approach in future studies ., The level of atrial fibrosis and PV properties that gave the best fit of the model PS density to the clinical PS density varied between patients ., Measurement of PV ERP and conduction properties using a lasso catheter before PVI could be used to tune the model properties , together with LGE-MRI or an electro-anatomic voltage map ., It is difficult to predict whether PVI alone is likely to be a successful treatment strategy for a patient with persistent AF 44 ., This will depend on both the susceptibility to AF from ectopic beats , together with electrical driver location , and electrical size ., Our study describes multiple factors that affect the susceptibility to AF from ectopic beats ., Measurement of PV APD , PV CV and PV size will allow prediction of the susceptibility to AF from ectopic beats ., Arrhythmia susceptibility increased in cases with short PV APD , slower CV at the LA/PV junction and larger veins , suggesting the importance of PVI in these cases ., The likelihood that PVI terminates AF was also found to depend on driver location , assessed using PS density ., Our simulation studies suggest that high PV PS density indicates likelihood of PVI success ., Thus either measuring this clinically using non-invasive ECGi recordings , or running patient-specific simulations to estimate this value may suggest whether ablation in addition to PVI should be performed ., In a recent clinical study , Navara et al . observed AF termination during ablation near the PVs , before complete isolation , in cases where rotational and focal activity were identified close to these ablation sites 45 ., These data may support the PV PS density metric suggested in our study ., Our simulations show that PV PS density depends on PV APD , the degree of PV fibrosis and to a lesser extent on PV fiber direction ., To the best of the authors’ knowledge , there are no previous studies on the relationship between fibrosis in the PVs , or PV fiber direction , and the success rate of PVI ., Measuring atrial electrogram properties , including AF cycle length , before and after ablation may indicate changes in local tissue refractoriness 46 ., PV APD can be estimated clinically by pacing to find the PV ERP; and PV fibrosis may be estimated using LGE-MRI , although this is challenging , as the tissue is thin ., PV

Introduction, Materials and methods, Results, Discussion

Success rates for catheter ablation of persistent atrial fibrillation patients are currently low; however , there is a subset of patients for whom electrical isolation of the pulmonary veins alone is a successful treatment strategy ., It is difficult to identify these patients because there are a multitude of factors affecting arrhythmia susceptibility and maintenance , and the individual contributions of these factors are difficult to determine clinically ., We hypothesised that the combination of pulmonary vein ( PV ) electrophysiology and atrial body fibrosis determine driver location and effectiveness of pulmonary vein isolation ( PVI ) ., We used bilayer biatrial computer models based on patient geometries to investigate the effects of PV properties and atrial fibrosis on arrhythmia inducibility , maintenance mechanisms , and the outcome of PVI ., Short PV action potential duration ( APD ) increased arrhythmia susceptibility , while longer PV APD was found to be protective ., Arrhythmia inducibility increased with slower conduction velocity ( CV ) at the LA/PV junction , but not for cases with homogeneous CV changes or slower CV at the distal PV ., Phase singularity ( PS ) density in the PV region for cases with PV fibrosis was increased ., Arrhythmia dynamics depend on both PV properties and fibrosis distribution , varying from meandering rotors to PV reentry ( in cases with baseline or long APD ) , to stable rotors at regions of high fibrosis density ., Measurement of fibrosis and PV properties may indicate patient specific susceptibility to AF initiation and maintenance ., PV PS density before PVI was higher for cases in which AF terminated or converted to a macroreentry; thus , high PV PS density may indicate likelihood of PVI success .

Atrial fibrillation is the most commonly encountered cardiac arrhythmia , affecting a significant portion of the population ., Currently , ablation is the most effective treatment but success rates are less than optimal , being 70% one-year post-treatment ., There is a large effort to find better ablation strategies to permanently cure the condition ., Pulmonary vein isolation by ablation is more or less the standard of care , but many questions remain since pulmonary vein ectopy by itself does not explain all of the clinical successes or failures ., We used computer simulations to investigate how electrophysiological properties of the pulmonary veins can affect rotor formation and maintenance in patients suffering from atrial fibrillation ., We used complex , biophysical representations of cellular electrophysiology in highly detailed geometries constructed from patient scans ., We heterogeneously varied electrophysiological and structural properties to see their effects on rotor initiation and maintenance ., Our study suggests a metric for indicating the likelihood of success of pulmonary vein isolation ., Thus either measuring this clinically , or running patient-specific simulations to estimate this metric may suggest whether ablation in addition to pulmonary vein isolation should be performed ., Our study provides motivation for a retrospective clinical study or experimental study into this metric .

medicine and health sciences, engineering and technology, cardiovascular anatomy, fibrosis, electrophysiology, endocardium, simulation and modeling, developmental biology, epicardium, research and analysis methods, cardiology, arrhythmia, atrial fibrillation, rotors, mechanical engineering, anatomy, physiology, biology and life sciences, heart

journal.pntd.0006075

Development and preliminary evaluation of a multiplexed amplification and next generation sequencing method for viral hemorrhagic fever diagnostics

Outbreaks of viral hemorrhagic fever ( VHF ) occur in many parts of the world 1 , 2 ., VHFs are caused by various single-stranded RNA viruses , the majority of which are classified in Arenaviridae , Filoviridae , and Flaviviridae families and Bunyavirales order 3 ., Human infections show high morbidity and mortality rates , can spread easily , and require rapid responses based on comprehensive pathogen identification 1 , 3 , 4 ., However , routine diagnostic approaches are challenged when fast and simultaneous screening for different viral pathogens in higher numbers of individuals is necessary 5 ., Even PCR as a widely used diagnostic method , usually providing specific virus identification , requires intense hands-on time for parallel screening of larger quantity of specimens and provides limited genetic information about the target virus ., Multiplexing of different specific PCR assays aims at dealing with these drawbacks; however , until recently , it was limited to a few primer pairs in one reaction due to a lack of amplicon identification approaches for more than five targets 6 , 7 ., Next Generation Sequencing ( NGS ) has provided novel options for the identification of viruses , including simultaneous and unbiased screening for different pathogens and multiplexing of various samples in a single sequencing run 8 ., Furthermore , the development of real-time sequencing platforms has enabled processing and analysis of individual specimens within reasonable timeframes 9 ., However , virus identification with NGS is also accompanied by major drawbacks , such as diminished sensitivity when viral genome numbers in the sample are insufficient and masked by unbiased sequencing of all nucleic acids present in the specimen , including the host genome 10 , 11 ., Attempts to increase the sensitivity of NGS-based diagnostics have focused on enrichment of virus material and libraries before sequencing , including amplicon sequencing , PCR-generated baits , and solution-based capture techniques 12–14 ., The strategy of ultrahigh-multiplex PCR with subsequent NGS has previously been employed for human single nucleotide polymorphism typing , genetic variations in human cardiomyopathies , and bacterial biothreat agents 15–17 ., In this study , we describe the development and initial evaluation of a novel method for targeted amplification and NGS-identification of viral febrile disease and hemorrhagic fever agents and assess the feasibility of this approach in diagnostics ., The human specimens , used for the evaluation of the developed panel were obtained from adults after written informed consent and in full compliance of the local ethics board approval ( Ankara Research and Training Hospital , 13 . 07 . 11/0426 ) ., Viruses reported to cause VHF as well as related strains , associated with febrile disease accompanied by arthritis , respiratory symptoms , or meningoencephalitis , were included in the design to enable differential diagnosis ( Table 1 ) ., For each virus strain , all genetic variants with complete or near-complete genomes deposited in GenBank ( https://www . ncbi . nlm . nih . gov/genbank/ ) were assembled into groups of >90% nucleotide sequence identity via the Geneious software ( version 9 . 1 . 3 ) 18 ., The consensus sequence of each group was included in the design ., The primer sequences were deduced using the Ion AmpliSeq Designer online tool ( https://ampliseq . com/browse . action ) which provides a custom multiplex primer pool design for NGS ( Thermo Fisher Scientific , Waltham , MA ) ., For initial evaluation of the approach and as internal controls , human-pathogenic viruses belonging in identical and/or distinct families/genera but not associated with hemorrhagic fever or febrile disease were included in the design ( Table 1 ) ., The designed primers were tested in silico for specific binding to the target virus strains , including all known genotypes and genetic variants ., The primer sets were aligned to their specific target reference sequences and relative primer orientation , amplicon size and overlap , and total mismatches for each primer were evaluated using the Geneious software 18 ., Pairs targeting a specific virus with less than two mismatches in sense and antisense primers were defined as a hit and employed for sensitivity calculations ., Unspecific binding of each primer to non-viral targets was investigated via the BLASTn algorithm , implemented within the National Center for Biotechnology Information website ( https://blast . ncbi . nlm . nih . gov/Blast . cgi ) 19 ., The sensitivity and specificity of the primer panel for each virus were determined via standard methods as described previously 20 ., The performance of the novel panel for the detection of major VHF agents was evaluated via selected virus strains ., For this purpose , nucleic acids from Yellow fever virus ( YFV ) strain 17D , Rift Valley fever virus ( RVFV ) strain MP-12 , Crimean-Congo hemorrhagic fever virus ( CCHFV ) strain UCCR4401 , Zaire Ebola virus ( EBOV ) strain Makona-G367 , Chikungunya virus ( CHIKV ) strain LR2006-OPY1 and Junin mammarenavirus ( JUNV ) strain P3766 were extracted with the QIAamp Viral RNA Mini Kit ( Qiagen , Hilden , Germany ) with subsequent cDNA synthesis according to the SuperScript IV Reverse Transcriptase protocol ( Thermo Fisher Scientific ) ., Genome concentration of all strains was determined by specific quantitative real-time PCRs using plasmid-derived virus standards , as described previously ( protocols are available upon request ) ., Genome equivalents ( ge ) of 100–103 for each virus were prepared and mixed with 10 ng of human genetic material recovered from HeLa cells ., In order to compare the efficiency of amplification with the novel panel versus direct NGS , all virus cDNAs were further subjected to second strand cDNA synthesis using the NEBNext RNA Second Strand Synthesis Module ( New England BioLabs GmbH , Frankfurt , Germany ) according to the manufacturer’s instructions ., Reagent-only mixes and HeLa cell extracts were employed as negative controls in the experiments ., The performance of the panel was further tested on clinical specimens from individuals with a clinical and laboratory diagnosis of VHF 21 ., For this purpose , previously stored sera with quantifiable CCHFV RNA and lacking IgM or IgG antibodies were employed and processed via High Pure Viral Nucleic Acid Kit ( Roche , Mannheim , Germany ) and the SuperScript IV Reverse Transcriptase ( Thermo Fisher Scientific ) protocols , as suggested by the manufacturer ., Two human sera , without detectable nucleic acids of the targeted viral strains were tested in parallel as negative controls ., The specimens were amplified using the custom primer panels designed for HFVs with the following PCR conditions for each pool: 2 μl of viral cDNA mixed with human genetic material , 5 μl of primer pool , 0 . 5 mM dNTP ( Invitrogen , Karlsruhe , Germany ) , 5 μl of 10 x Platinum Taq buffer , 4 mM MgCl2 , and 10 U Platinum Taq polymerase ( Invitrogen ) with added water to a final volume of 25 μl ., Cycling conditions were 94°C for 7 minutes , 45 amplification cycles at 94°C for 20 seconds , 60°C for 1 minute , and 72°C for 20 seconds , and a final extension step for 6 minutes ( at 72°C ) ., Thermal cycling was performed in an Eppendorf Mastercycler Pro ( Eppendorf Vertrieb Deutschland , Wesseling-Berzdorf , Germany ) with a total runtime of 90 minutes ., The amplicons obtained from the virus strains were subjected to the Ion Torrent Personal Genome Machine ( PGM ) System for NGS analysis ( Thermo Fisher Scientific Inc . ) ., Initially , the specimens were purified with an equal volume of Agencourt AMPure XP Reagent ( Beckman Coulter , Krefeld , Germany ) ., PGM libraries were prepared according to the Ion Xpress Plus gDNA Fragment Library Kit , using the “Amplicon Libraries without Fragmentation” protocol ( Thermo Fisher Scientific ) ., For direct NGS , specimens were fragmented with the Ion Shear Plus Reagents Kit ( Thermo Fisher Scientific ) with a reaction time of 8 minutes ., Subsequently , libraries were prepared using the Ion Xpress Plus gDNA Fragment Library Preparation kit and associated protocol ( Thermo Fisher Scientific ) ., All libraries were quality checked using the Agilent Bioanalyzer ( Agilent Technologies , Frankfurt , Germany ) , quantitated with the Ion Library Quantitation Kit ( Thermo Fisher Scientific ) , and pooled equimolarly ., Enriched , template-positive Ion PGM Hi-Q Ion Sphere Particles were prepared using the Ion PGM Hi-Q Template protocol with the Ion PGM Hi-Q OT2 400 Kit ( Thermo Fisher Scientific ) ., Sequencing was performed with the Ion PGM Hi-Q Sequencing protocol , using a 318 chip ., Amplicons obtained from CCHFV-infected individuals and controls were processed for nanopore sequencing via MinION ( Oxford Nanopore Technologies , Oxford , United Kingdom ) ., The libraries were prepared using the ligation sequencing kit 1D , SQK-LSK108 , R9 . 4 ( Oxford Nanopore Technologies ) ., Subsequently , the libraries were loaded on Oxford Nanopore MinION SpotON Flow Cells Mk I , R9 . 4 ( Oxford Nanopore Technologies ) using the library loading beads and run until initial viral reads were detected ., The sequences generated by PGM sequencing were trimmed to remove adaptors from each end using Trimmomatic 22 , and reads shorter than 50 base pairs were discarded ., All remaining reads were mapped against the viral reference database prepared during the design process via Geneious 9 . 1 . 3 software 18 ., During and after MinION sequencing , all basecalled reads in fast5 format were extracted in fasta format using Poretools software 23 ., The BLASTn algorithm was employed for sequence similarity searches in the public databases when required ., The AmpliSeq design for the custom multiplex primer panel resulted in two pools of 285 and 256 primer pairs for the identification of 46 virus species causing hemorrhagic fevers , encompassing 6 , 130 genetic variants of the strains involved ., All amplicons were designed to be within a range of 125–375 base pairs ., Melting temperature values of the primers ranged from 55 . 3°C to 65 . 0°C ., No amplicons <1 , 000 base pairs with primer pairs in relative orientation and distance to each other could be identified , leading to an overall specificity of 100% for all virus species ., The primer sequences in the panels are provided in S1 Table ., The overall sensitivity of the panel reached 97 . 9% , with the primer pairs targeting 6 , 007 out of 6 , 130 genetic variants ( 1 mismatch in one or both of each primers of a primer pair accepted , as described above ) ( Fig 1 ) ., Impaired sensitivity was noted for Hantaan virus ( 0 . 05 ) ., Evaluation of all Hantaan virus variants in GenBank revealed that newly added virus sequences were divergent by up to 17% from sequences included in the panel design , leading to diminished primer binding ., These sequences could be fully covered by two sets of additional primers ., Amplification of viral targets with the multiplex PCR panel prior to NGS resulted in a significant increase of viral read numbers compared to direct NGS ( Figs 2 and 3 , S2 Table ) ., In specimens with 103 ge of the target strain , the ratio of viral reads to unspecific background increased from 1×10−3 to 0 . 25 ( CCHFV ) , 3×10−5 to 0 . 34 ( RVFV ) , 1×10−4 to 0 . 27 ( EBOV ) , and 2×10−5 to 0 . 64 ( CHIKV ) with fold-changes of 247 , 10 , 297 , 1 , 633 , and 25 , 398 , respectively ., In direct NGS , no viral reads could be detected for CCHFV and CHIKV genomic concentrations lower than 103 , and this approach failed to identify YFV and JUNV regardless of the initial virus count ., In targeted NGS , the limit of detection was noted as 100 ge for YFV , CCHFV , RVFV , EBOV , and CHIKV and 101 ge for JUNV ., For the viruses detectable via direct NGS , amplification provided significant increases in specific viral reads over total reads ratios , from 10−4 to 0 . 19 ( CCHFV , 1 , 900-fold change ) , 2×10−5 to 0 . 19 ( RVFV , 9 , 500-fold change ) , and 3×10−4 to 0 . 56 ( EBOV , 1 , 866-fold change ) ., The average duration of the workflow of direct and targeted NGS via PGM was 19 and 20 . 5 hours , respectively ., In all patient sera evaluated via nanopore sequencing following amplification , the causative agent could be detected after 1 to 9 minutes of the NGS run ( Table 2 ) ., The characterized sequences were 89–99% identical to the CCHFV strain Kelkit L segment ( GenBank accession: GQ337055 ) known to be in circulation in Turkey 24 , 25 ., No targeted viral sequence could be observed in human sera used as negative controls during 1 hour of sequencing ., The preparation , amplification , and sequencing steps of the clinical specimens could be completed with a total sample-to-result time of less than 3 . 5 hours ., In this study , we report the development and evaluation of an ultrahigh-multiplex PCR for the enrichment of viral targets before NGS , which aims to provide a robust molecular diagnosis in VHFs ., The panel was observed to be highly specific and sensitive and to have the capacity to detect over 97% of all known genetic variants of the targeted 46 viral species in silico ., The sensitivity of the primer panel was impaired by virus sequences not included in the original design , as noted for Hantaan virus in this study ., As 36 out of a total of 59 isolates have been published after panel design was completed , these genetic variants of Hantaan virus could not be detected with a comparable sensitivity or not at all with the current panel ., This indicates that the panel has to be adapted to newly-available sequences in public databases ., We have evaluated how the panel could be updated to accommodate these recently-added sequences and observed that two additional primer pairs could sufficiently cover all divergent entries ., Although the approach for the panel design as well as the actual design with the AmpliSeq pipeline was successful for all genetic variants included , the amplification of viral sequences significantly diverging from the panel could not be guaranteed , which may also apply for novel viruses ., Unlike other pathogenic microorganisms , viruses can be highly variable in their genome ., Only rarely do they share genes among all viruses or virus species that could be targeted as a virus-generic marker by amplification ., Our strategy for primer design and the AmpliSeq pipeline do not permit the generation of degenerated primers or the targeting of very specific consensus sequences ., However , the design of the primer panel is relatively flexible , and additional primer pairs can be appended in response to recently published virus genomes ., Moreover , an updated panel will also encompass non-viral pathogens relevant for differential diagnosis , and syndrome-specific panels targeting only VHF agents or virally induced febrile diseases such as West Nile fever and Chikungunya can be developed ., We have further tested the panel using quantitated nucleic acids of six well-characterized viruses responsible for VHF or severe febrile disease , with a background of human genetic material to simulate specimens likely to be submitted for diagnosis , using the semiconductor PGM sequencing platform ., The impact of amplification was evaluated with a comparison of direct and amplicon-based NGS runs ., Overall , targeted amplification prior to NGS ensured viral read detection in specimens with the lowest virus concentration ( 1 ge ) in five of the six viruses evaluated and 10 ge in the remaining strain , which is within the range of the established real-time PCR assays ., Furthermore , this approach enabled significant increases in specific viral reads over background in all of the viruses , with varying fold changes in different strains and concentrations ( Figs 2 and 3 ) ., The increased sensitivity and specificity provided with the targeted amplification suggest that it can be directly employed for the investigation of suspected VHF cases where viremia is usually short and the time point of maximum virus load is often missed 1 , 5 ., Finally , we evaluated the VHF panel by using serum specimens obtained during the acute phase of CCHFV-induced disease and employed an alternate NGS platform based on nanopore sequencing ., This approach enabled virus detection and characterization within 10 minutes of the NGS run and can be completed in less than 3 . 5 hours in total ( Table 2 ) ., The impact of the nanopore sequencing has been revealed previously , during the EBOV outbreak in West Africa where the system provided an efficient method for real-time genomic surveillance of the causative agent in a resource-limited setting 26 ., Field-forward protocols based on nanopore sequencing have also been developed recently for pathogen screening in arthropods 27 ., Specimen processing time is likely to be further reduced via the recently developed rapid library preparation options ., While the duration of the workflow is longer , the PGM and similar platforms are well-suited for the parallel investigation of higher specimen numbers ., Although we have demonstrated in this study that targeted amplification and NGS-based characterization of VHF and febrile disease agents is an applicable strategy for diagnosis and surveillance , there are also limitations of this approach ., In addition to the requirement of primer sequence updates , the majority of the workflow requires non-standard equipment and well-trained personnel , usually out of reach for the majority of laboratories in underprivileged geographical regions mainly affected by these diseases ., However , NGS technologies are becoming widely available with reduced total costs and can be swiftly transported and set up in temporary facilities in field conditions 26 , 27 ., During outbreak investigations , where it is impractical and expensive to test for several individual agents via specific PCRs , this approach can easily provide information on the causative agent , facilitating timely implementation of containment and control measures ., Additional validation of the approach will be provided with the evaluation of well-characterized clinical specimen panels and direct comparisons with established diagnostic assays ., In conclusion , virus enrichment via targeted amplification followed by NGS is an applicable method for the diagnosis of VHFs which can be adapted for high-throughput or nanopore sequencing platforms and employed for surveillance or outbreak monitoring .

Introduction, Methods, Results, Discussion

We describe the development and evaluation of a novel method for targeted amplification and Next Generation Sequencing ( NGS ) -based identification of viral hemorrhagic fever ( VHF ) agents and assess the feasibility of this approach in diagnostics ., An ultrahigh-multiplex panel was designed with primers to amplify all known variants of VHF-associated viruses and relevant controls ., The performance of the panel was evaluated via serially quantified nucleic acids from Yellow fever virus , Rift Valley fever virus , Crimean-Congo hemorrhagic fever ( CCHF ) virus , Ebola virus , Junin virus and Chikungunya virus in a semiconductor-based sequencing platform ., A comparison of direct NGS and targeted amplification-NGS was performed ., The panel was further tested via a real-time nanopore sequencing-based platform , using clinical specimens from CCHF patients ., The multiplex primer panel comprises two pools of 285 and 256 primer pairs for the identification of 46 virus species causing hemorrhagic fevers , encompassing 6 , 130 genetic variants of the strains involved ., In silico validation revealed that the panel detected over 97% of all known genetic variants of the targeted virus species ., High levels of specificity and sensitivity were observed for the tested virus strains ., Targeted amplification ensured viral read detection in specimens with the lowest virus concentration ( 1–10 genome equivalents ) and enabled significant increases in specific reads over background for all viruses investigated ., In clinical specimens , the panel enabled detection of the causative agent and its characterization within 10 minutes of sequencing , with sample-to-result time of less than 3 . 5 hours ., Virus enrichment via targeted amplification followed by NGS is an applicable strategy for the diagnosis of VHFs which can be adapted for high-throughput or nanopore sequencing platforms and employed for surveillance or outbreak monitoring .

Viral hemorrhagic fever is a severe and potentially lethal disease , characterized by fever , malaise , vomiting , mucosal and gastrointestinal bleeding , and hypotension , in which multiple organ systems are affected ., Due to modern transportation and global trade , outbreaks of viral hemorrhagic fevers have the potential to spread rapidly and affect a significant number of susceptible individuals ., Thus , urgent and robust diagnostics with an identification of the causative virus is crucial ., However , this is challenged by the number and diversity of the viruses associated with hemorrhagic fever ., Several viruses classified in Arenaviridae , Filoviridae , and Flaviviridae families and Bunyavirales order may cause symptoms of febrile disease with hemorrhagic symptoms ., We have developed and evaluated a novel method that can potentially identify all viruses and their genomic variants known to cause hemorrhagic fever in humans ., The method relies on selected amplification of the target viral nucleic acids and subsequent high throughput sequencing technology for strain identification ., Computer-based evaluations have revealed very high sensitivity and specificity , provided that the primer design is kept updated ., Laboratory tests using several standard hemorrhagic virus strains and patient specimens have demonstrated excellent suitability of the assay in various sequencing platforms , which can achieve a definitive diagnosis in less than 3 . 5 hours .

sequencing techniques, medicine and health sciences, rift valley fever virus, pathology and laboratory medicine, togaviruses, pathogens, tropical diseases, microbiology, alphaviruses, viruses, next-generation sequencing, chikungunya virus, rna viruses, genome analysis, neglected tropical diseases, molecular biology techniques, microbial genetics, bunyaviruses, microbial genomics, research and analysis methods, viral hemorrhagic fevers, infectious diseases, viral genomics, genomics, crimean-congo hemorrhagic fever virus, medical microbiology, microbial pathogens, molecular biology, virology, viral pathogens, transcriptome analysis, genetics, biology and life sciences, viral diseases, computational biology, dna sequencing, hemorrhagic fever viruses, organisms

journal.pcbi.1007284

Fast and near-optimal monitoring for healthcare acquired infection outbreaks

"Since the time of Hippocrates , the “father of western medicine” , a central tenet of medical c(...TRUNCATED)

Introduction, Materials and methods, Results, Discussion

"According to the Centers for Disease Control and Prevention ( CDC ) , one in twenty five hospital p(...TRUNCATED)

"Healthcare acquired infections ( HAIs ) lead to significant losses of lives and result in heavy eco(...TRUNCATED)

"medicine and health sciences, gut bacteria, medical personnel, sociology, social sciences, health c(...TRUNCATED)

journal.pcbi.1003705

Rethinking Transcriptional Activation in the Arabidopsis Circadian Clock

"The task of the circadian clock is to synchronize a multitude of biological processes to the daily (...TRUNCATED)

Introduction, Results, Methods, Discussion

"Circadian clocks are biological timekeepers that allow living cells to time their activity in antic(...TRUNCATED)

"Like most living organisms , plants are dependent on sunlight , and evolution has endowed them with(...TRUNCATED)

"systems biology, physiological processes, computer and information sciences, network analysis, phys(...TRUNCATED)

journal.pcbi.1006080

"Bamgineer: Introduction of simulated allele-specific copy number variants into exome and targeted s(...TRUNCATED)

"The emergence and maturation of next-generation sequencing technologies , including whole genome se(...TRUNCATED)

Introduction, Results, Discussion, Materials and methods

"Somatic copy number variations ( CNVs ) play a crucial role in development of many human cancers .,(...TRUNCATED)

"We present Bamgineer , a software program to introduce user-defined , haplotype-specific copy numbe(...TRUNCATED)

"sequencing techniques, alleles, genetic mapping, genome analysis, copy number variation, molecular (...TRUNCATED)

journal.pcbi.1006772

"A component overlapping attribute clustering (COAC) algorithm for single-cell RNA sequencing data a(...TRUNCATED)

"Single cell ribonucleic acid sequencing ( scRNA-seq ) offers advantages for characterization of cel(...TRUNCATED)

Introduction, Results, Discussion, Methods and materials

"Recent advances in next-generation sequencing and computational technologies have enabled routine a(...TRUNCATED)

"Single-cell RNA sequencing ( scRNA-seq ) can reveal complex and rare cell populations , uncover gen(...TRUNCATED)

"biotechnology, medicine and health sciences, clinical research design, engineering and technology, (...TRUNCATED)