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One of the early changes upon tuber induction is the switch from apoplastic to symplastic unloading. Whether and how this change in unloading mode contributes to sink strength has remained unclear. In addition, developing tubers also change from energy to storage-based sucrose metabolism. Here, we investigated the coordination between changes in unloading mode and sucrose metabolism and their relative role in tuber sink strength by looking into callose and sucrose metabolism gene expression combined with a model of apoplastic and symplastic unloading. Gene expression analysis suggests that callose deposition in tubers is decreased by lower callose synthase expression. Furthermore, changes in callose and sucrose metabolism are strongly correlated, indicating a well-coordinated developmental switch. Modelling indicates that symplastic unloading is not the most efficient unloading mode per se. Instead, it is the concurrent metabolic switch that provides the physiological conditions necessary to potentiate symplastic transport and thereby enhance tuber sink strength ., Competing Interests: The authors declare no competing interests., (© The Author(s) 2024.)

Quantitative approaches in plant biology have a long history that have led to several ground-breaking discoveries and given rise to new principles, new paradigms and new methodologies. We take a short historical trip into the past to explore some of the many great scientists and influences that have led to the development of quantitative plant biology. We have not been constrained by historical fact, although we have tried not to deviate too much. We end with a forward look, expressing our hopes and ambitions for this exciting interdisciplinary field., Competing Interests: The authors declare no conflicts of interest., (© The Author(s), 2021. Published by Cambridge University Press in association with The John Innes Centre 2021.)

After germination, the meristem of the embryonic plant root becomes activated, expands in size and subsequently stabilizes to support post-embryonic root growth. The plant hormones auxin and cytokinin, together with master transcription factors of the PLETHORA (PLT) family have been shown to form a regulatory network that governs the patterning of this root meristem. Still, which functional constraints contributed to shaping the dynamics and architecture of this network, has largely remained unanswered. Using a combination of modeling approaches we reveal how the interplay between auxin and PLTs enables meristem activation in response to above-threshold stimulation, while its embedding in a PIN-mediated auxin reflux loop ensures localized PLT transcription and thereby, a finite meristem size. We furthermore demonstrate how this constrained PLT transcriptional domain enables independent control of meristem size and division rates, further supporting a division of labor between auxin and PLT. We subsequently reveal how the weaker auxin antagonism of the earlier active Arabidopsis response regulator 12 (ARR12) may arise from the absence of a DELLA protein interaction domain. Our model indicates that this reduced strength is essential to prevent collapse in the early stages of meristem expansion while at later stages the enhanced strength of Arabidopsis response regulator 1 (ARR1) is required for sufficient meristem size control. Summarizing, our work indicates that functional constraints significantly contribute to shaping the auxin-cytokinin-PLT regulatory network., Competing Interests: The authors declare no conflict of interest.

A plants' fitness to a large extent depends on its capacity to adapt to spatio-temporally varying environmental conditions. One such environmental condition to which plants display extensive phenotypic plasticity is soil nitrate levels and patterns. In response to heterogeneous nitrate distribution, plants show a so-called preferential foraging response. Herein root growth is enhanced in high nitrate patches and repressed in low nitrate locations beyond a level that can be explained from local nitrate sensing. Although various molecular players involved in this preferential foraging behavior have been identified, how these together shape root system adaptation has remained unresolved. Here we use a simple modeling approach in which we incrementally incorporate the known molecular pathways to investigate the combination of regulatory mechanisms that underly preferential root nitrate foraging. Our model suggests that instead of involving a growth suppressing supply signal, growth reduction on the low nitrate side may arise from reduced root foraging and increased competition for carbon. Additionally, our work suggests that the long distance CK signaling involved in preferential root foraging may function as a supply signal modulating demand signaling strength. We illustrate how this integration of demand and supply signals prevents excessive preferential foraging under conditions in which demand is not met by sufficient supply and a more generic foraging in search of nitrate should be maintained., (Copyright © 2020 Boer, Santos Teixeira and Ten Tusscher.)

The root system is a major determinant of a plant's access to water and nutrients. The architecture of the root system to a large extent depends on the repeated formation of new lateral roots. In this review, we discuss lateral root development from a systems biology perspective. We focus on studies combining experiments with computational modeling that have advanced our understanding of how the auxin-centered regulatory modules involved in different stages of lateral root development exert their specific functions. Moreover, we discuss how these regulatory networks may enable robust transitions from one developmental stage to the next, a subject that thus far has received limited attention. In addition, we analyze how environmental factors impinge on these modules, and the different manners in which these environmental signals are being integrated to enable coordinated developmental decision making. Finally, we provide some suggestions for extending current models of lateral root development to incorporate multiple processes and stages. Only through more comprehensive models we can fully elucidate the cooperative effects of multiple processes on later root formation, and how one stage drives the transition to the next., (Copyright © 2019 The Author. Published by Elsevier Inc. All rights reserved.)

Auxin plays a major role in a variety of processes involved in plant developmental patterning and its adaptation to environmental conditions. Therefore, an important question is how specificity in auxin signalling is achieved, that is, how a single signalling molecule can carry so many different types of information. In recent years, many studies on auxin specificity have been published, unravelling increasingly more details on differential auxin sensitivity, expression domains and downstream partners of the auxin receptors (transport inhibitor response 1 (TIR1) and other auxin signaling F-box proteins (AFB)), transcriptional repressors that are degraded in response to auxin (AUX/IAA) and downstream auxin response factors (ARF) that together constitute the plant's major auxin response pathways. These data are critical to explain how, in the same cells, different auxin levels may trigger different responses, as well as how in different spatial or temporal contexts similar auxin signals converge to different responses. However, these insights do not yet answer more complex questions regarding auxin specificity. As an example, they leave open the question of how similar sized auxin changes at similar locations result in different responses depending on the duration and spatial extent of the fluctuation in auxin levels. Similarly, it leaves unanswered how, in the case of certain tropisms, small differences in signal strength at both sides of a plant organ are converted into an instructive auxin asymmetry that enables a robust tropic response. Finally, it does not explain how, in certain cases, substantially different auxin levels become translated into similar cellular responses, while in other cases similar auxin levels, even when combined with similar auxin response machinery, may trigger different responses. In this review, we illustrate how considering the regulatory networks and contexts in which auxin signalling takes place helps answer these types of fundamental questions., Competing Interests: The authors declare no conflict of interest.

Conditioning small groups of root pericycle cells for future lateral root formation has a major impact on overall plant root architecture. This priming of lateral roots occurs rhythmically, involving temporal oscillations in auxin response in the root tip. During growth, this process generates a spatial pattern of prebranch sites, an early stage in lateral root formation characterized by a stably maintained high auxin response. To date, the molecular mechanism behind this rhythmicity has remained elusive. Some data implicate a cell-autonomous oscillation in gene expression, while others strongly support the importance of tissue-level modulations in auxin fluxes. Here, we summarize the experimental data on periodic lateral root priming. We present a theoretical framework that distinguishes between a priming signal and its subsequent memorization and show how major roles for auxin fluxes and gene expression naturally emerge from this framework. We then discuss three mechanisms that could potentially induce oscillations of auxin response: cell-autonomous oscillations, Turing-type patterning, and tissue-level oscillations in auxin fluxes, along with specific properties of lateral root priming that may be used to discern which type of mechanism is most likely to drive lateral root patterning. We conclude with suggestions for future experiments and modeling studies., (© 2017 American Society of Plant Biologists. All rights reserved.)

A key characteristic of plant development is its plasticity in response to various and dynamically changing environmental conditions. Tropisms contribute to this flexibility by allowing plant organs to grow from or towards environmental cues. Halotropism is a recently described tropism in which plant roots bend away from salt. During halotropism, as in most other tropisms, directional growth is generated through an asymmetric auxin distribution that generates differences in growth rate and hence induces bending. Here, we develop a detailed model of auxin transport in the Arabidopsis root tip and combine this with experiments to investigate the processes generating auxin asymmetry during halotropism. Our model points to the key role of root tip architecture in allowing the decrease in PIN2 at the salt-exposed side of the root to result in a re-routing of auxin to the opposite side. In addition, our model demonstrates how feedback of auxin on the auxin transporter AUX1 amplifies this auxin asymmetry, while a salt-induced transient increase in PIN1 levels increases the speed at which this occurs. Using AUX1-GFP imaging and pin1 mutants, we experimentally confirmed these model predictions, thus expanding our knowledge of the cellular basis of halotropism., Competing Interests: The authors declare no competing or financial interests., (© 2016. Published by The Company of Biologists Ltd.)

Background: The evolution of animal segmentation is a major research focus within the field of evolutionary-developmental biology. Most studied segmented animals generate their segments in a repetitive, anterior-to-posterior fashion coordinated with the extension of the body axis from a posterior growth zone. In the current study we ask which selection pressures and ordering of evolutionary events may have contributed to the evolution of this specific segmentation mode., Results: To answer this question we extend a previous in silico simulation model of the evolution of segmentation by allowing the tissue growth pattern to freely evolve. We then determine the likelihood of evolving oscillatory sequential segmentation combined with posterior growth under various conditions, such as the presence or absence of a posterior morphogen gradient or selection for determinate growth. We find that posterior growth with sequential segmentation is the predominant outcome of our simulations only if a posterior morphogen gradient is assumed to have already evolved and selection for determinate growth occurs secondarily. Otherwise, an alternative segmentation mechanism dominates, in which divisions occur in large bursts through the entire tissue and all segments are created simultaneously., Conclusions: Our study suggests that the ancestry of a posterior signalling centre has played an important role in the evolution of sequential segmentation. In addition, it suggests that determinate growth evolved secondarily, after the evolution of posterior growth. More generally, we demonstrate the potential of evo-devo simulation models that allow us to vary conditions as well as the onset of selection pressures to infer a likely order of evolutionary innovations.

Myocardial fibrosis is an important risk factor for cardiac arrhythmias. Previous experimental and numerical studies have shown that the texture and spatial distribution of fibrosis may play an important role in arrhythmia onset. Here, we investigate how spatial heterogeneity of fibrosis affects arrhythmia onset using numerical methods. We generate various tissue textures that differ by the mean amount of fibrosis, the degree of heterogeneity and the characteristic size of heterogeneity. We study the onset of arrhythmias using a burst pacing protocol. We confirm that spatial heterogeneity of fibrosis increases the probability of arrhythmia induction. This effect is more pronounced with the increase of both the spatial size and the degree of heterogeneity. The induced arrhythmias have a regular structure with the period being mostly determined by the maximal local fibrosis level. We perform ablations of the induced fibrillatory patterns to classify their type. We show that in fibrotic tissue fibrillation is usually of the mother rotor type but becomes of the multiple wavelet type with increase in tissue size. Overall, we conclude that the most important factor determining the formation and dynamics of arrhythmia in heterogeneous fibrotic tissue is the value of maximal local fibrosis.

Convergent extension, the simultaneous extension and narrowing of tissues, is a crucial event in the formation of the main body axis during embryonic development. It involves processes on multiple scales: the sub-cellular, cellular and tissue level, which interact via explicit or intrinsic feedback mechanisms. Computational modelling studies play an important role in unravelling the multiscale feedbacks underlying convergent extension. Convergent extension usually operates in tissue which has been patterned or is currently being patterned into distinct domains of gene expression. How such tissue patterns are maintained during the large scale tissue movements of convergent extension has thus far not been investigated. Intriguingly, experimental data indicate that in certain cases these tissue patterns may drive convergent extension rather than requiring safeguarding against convergent extension. Here we use a 2D Cellular Potts Model (CPM) of a tissue prepatterned into segments, to show that convergent extension tends to disrupt this pre-existing segmental pattern. However, when cells preferentially adhere to cells of the same segment type, segment integrity is maintained without any reduction in tissue extension. Strikingly, we demonstrate that this segment-specific adhesion is by itself sufficient to drive convergent extension. Convergent extension is enhanced when we endow our in silico cells with persistence of motion, which in vivo would naturally follow from cytoskeletal dynamics. Finally, we extend our model to confirm the generality of our results. We demonstrate a similar effect of differential adhesion on convergent extension in tissues that can only extend in a single direction (as often occurs due to the inertia of the head region of the embryo), and in tissues prepatterned into a sequence of domains resulting in two opposing adhesive gradients, rather than alternating segments.

Segmentation of the major body axis into repeating units is arguably one of the major inventions in the evolution of animal body plan pattering. It is found in current day vertebrates, annelids and arthropods. Most segmented animals seem to use a clock-and-wavefront type mechanism in which oscillations emanating from a posterior growth zone become transformed into an anterior posterior sequence of segments. In contrast, few animals such as Drosophila use a complex gene regulatory hierarchy to simultaneously subdivide their entire body axis into segments. Here I discuss how in silico models simulating the evolution of developmental patterning can be used to investigate the forces and constraints that helped shape these two developmental modes. I perform an analysis of a series of previous simulation studies, exploiting the similarities and differences in their outcomes in relation to model characteristics to elucidate the circumstances and constraints likely to have been important for the evolution of sequential and simultaneous segmentation modes. The analysis suggests that constraints arising from the involved growth process and spatial patterning signal--posterior elongation producing a propagating wavefront versus a tissue wide morphogen gradient--and the evolutionary history--ancestral versus derived segmentation mode--strongly shaped both segmentation mechanisms. Furthermore, this implies that these patterning types are to be expected rather than random evolutionary outcomes and supports the likelihood of multiple parallel evolutionary origins.

A major goal of evolutionary developmental biology (evo-devo) is to understand how multicellular body plans of increasing complexity have evolved, and how the corresponding developmental programs are genetically encoded. It has been repeatedly argued that key to the evolution of increased body plan complexity is the modularity of the underlying developmental gene regulatory networks (GRNs). This modularity is considered essential for network robustness and evolvability. In our opinion, these ideas, appealing as they may sound, have not been sufficiently tested. Here we use computer simulations to study the evolution of GRNs' underlying body plan patterning. We select for body plan segmentation and differentiation, as these are considered to be major innovations in metazoan evolution. To allow modular networks to evolve, we independently select for segmentation and differentiation. We study both the occurrence and relation of robustness, evolvability and modularity of evolved networks. Interestingly, we observed two distinct evolutionary strategies to evolve a segmented, differentiated body plan. In the first strategy, first segments and then differentiation domains evolve (SF strategy). In the second scenario segments and domains evolve simultaneously (SS strategy). We demonstrate that under indirect selection for robustness the SF strategy becomes dominant. In addition, as a byproduct of this larger robustness, the SF strategy is also more evolvable. Finally, using a combined functional and architectural approach, we determine network modularity. We find that while SS networks generate segments and domains in an integrated manner, SF networks use largely independent modules to produce segments and domains. Surprisingly, we find that widely used, purely architectural methods for determining network modularity completely fail to establish this higher modularity of SF networks. Finally, we observe that, as a free side effect of evolving segmentation and differentiation in combination, we obtained in-silico developmental mechanisms resembling mechanisms used in vertebrate development.

Background: Sexual reproduction has classically been considered as a barrier to the buildup of discrete phenotypic differentiation. This notion has been confirmed by models of sympatric speciation in which a fixed genetic architecture and a linear genotype phenotype mapping were assumed. In this paper we study the influence of a flexible genetic architecture and non-linear genotype phenotype map on differentiation under sexual reproduction.We use an individual based model in which organisms have a genome containing genes and transcription factor binding sites. Mutations involve single genes or binding sites or stretches of genome. The genome codes for a regulatory network that determines the gene expression pattern and hence the phenotype of the organism, resulting in a non-linear genotype phenotype map. The organisms compete in a multi-niche environment, imposing selection for phenotypic differentiation., Results: We find as a generic outcome the evolution of discrete clusters of organisms adapted to different niches, despite random mating. Organisms from different clusters are distinct on the genotypic, the network and the phenotypic level. However, the genome and network differences are constrained to a subset of the genome locations, a process we call genotypic canalization. We demonstrate how this canalization leads to an increased robustness to recombination and increasing hybrid fitness. Finally, in case of assortative mating, we explain how this canalization increases the effectiveness of assortativeness., Conclusion: We conclude that in case of a flexible genetic architecture and a non-linear genotype phenotype mapping, sexual reproduction does not constrain phenotypic differentiation, but instead constrains the genotypic differences underlying it. We hypothesize that, as genotypic canalization enables differentiation despite random mating and increases the effectiveness of assortative mating, sympatric speciation is more likely than is commonly suggested.

Sudden cardiac death is a major health problem in the industrialized world. The lethal event is typically ventricular fibrillation (VF), during which the co-ordinated regular contraction of the heart is overthrown by a state of mechanical and electrical anarchy. Understanding the excitation patterns that sustain VF is important in order to identify potential therapeutic targets. In this paper, we studied the organization of human VF by combining clinical recordings of electrical excitation patterns on the epicardial surface during in vivo human VF with simulations of VF in an anatomically and electrophysiologically detailed computational model of the human ventricles. We find both in the computational studies and in the clinical recordings that epicardial surface excitation patterns during VF contain around six rotors. Based on results from the simulated three-dimensional excitation patterns during VF, which show that the total number of electrical sources is 1.4 +/- 0.12 times greater than the number of epicardial rotors, we estimate that the total number of sources present during clinically recorded VF is 9.0 +/- 2.6. This number is approximately fivefold fewer compared with that observed during VF in dog and pig hearts, which are of comparable size to human hearts. We explain this difference by considering differences in action potential duration dynamics across these species. The simpler spatial organization of human VF has important implications for treatment and prevention of this dangerous arrhythmia. Moreover, our findings underline the need for integrated research, in which human-based clinical and computational studies complement animal research.

Sudden cardiac death is one of the major causes of death in the industrialized world. It is most often caused by a cardiac arrhythmia called ventricular fibrillation (VF). Despite its large social and economical impact, the mechanisms for VF in the human heart yet remain to be identified. Two of the most frequently discussed mechanisms observed in experiments with animal hearts are the multiple wavelet and mother rotor hypotheses. Most recordings of VF in animal hearts are consistent with the multiple wavelet mechanism. However, in animal hearts, mother rotor fibrillation has also been observed. For both multiple wavelet and mother rotor VF, cardiac heterogeneity plays an important role. Clinical data of action potential restitution measured from the surface of human hearts have been recently published. These in vivo data show a substantial degree of spatial heterogeneity. Using these clinical restitution data, we studied the dynamics of VF in the human heart using a heterogeneous computational model of human ventricles. We hypothesized that this observed heterogeneity can serve as a substrate for mother rotor fibrillation. We found that, based on these data, mother rotor VF can occur in the human heart and that ablation of the mother rotor terminates VF. Furthermore, we found that both mother rotor and multiple wavelet VF can occur in the same heart depending on the initial conditions at the onset of VF. We studied the organization of these two types of VF in terms of filament numbers, excitation periods, and frequency domains. We conclude that mother rotor fibrillation is a possible mechanism in the human heart.

The onset of ventricular fibrillation (VF) has been associated with steep action potential duration restitution in both clinical and computational studies. Recently, detailed clinical restitution properties in cardiac patients were reported showing a substantial degree of heterogeneity in restitution slopes at the epicardium of the ventricles. The aim of the present study was to investigate the effect of heterogeneous restitution properties in a three-dimensional model of the ventricles using these clinically measured restitution data. We used a realistic model of the human ventricles, including detailed descriptions of cell electrophysiology, ventricular anatomy, and fiber direction anisotropy. We extended this model by mapping the clinically observed epicardial restitution data to our anatomic representation using a diffusion-based algorithm. Restitution properties were then fitted by regionally varying parameters of the electrophysiological model. We studied the effects of restitution heterogeneity on the organization of VF by analyzing filaments and the distributions of excitation periods. We found that the number of filaments and the excitation periods were both dependent on the extent of heterogeneity. An increased level of heterogeneity leads to a greater number of filaments and a broader distribution of excitation periods, thereby increasing the complexity and dynamics of VF. Restitution heterogeneity may play an important role in providing a substrate for cardiac arrhythmias.

The His-Purkinje conduction system initiates the normal excitation of the ventricles and is a major component of the specialized conduction system of the heart. Abnormalities and propagation blocks in the Purkinje system result in abnormal excitation of the heart. Experimental findings suggest that the Purkinje network plays an important role in ventricular tachycardia and fibrillation, which is the major cause of sudden cardiac death. Nowadays an important area in the study of cardiac arrhythmias is anatomically accurate modelling. The majority of current anatomical models have not included a description of the Purkinje network. As a consequence, these models cannot be used to study the important role of the Purkinje system in arrhythmia initiation and maintenance. In this article we provide an overview of previous work on modelling of the Purkinje system and report on the development of a His-Purkinje system for our human ventricular model. We use the model to simulate the normal activation pattern as well as abnormal activation patterns resulting from bundle branch block and bundle branch reentry.

Aims: During ageing, after infarction, in cardiomyopathies and other cardiac diseases, the percentage of fibrotic (connective) tissue may increase from 6% up to 10-35%. The presence of increased amounts of connective tissue is strongly correlated with the occurrence of arrhythmias and sudden cardiac death., Methods and Results: In this article, we investigate the role of diffuse fibrosis on wave propagation, arrhythmogenesis, and arrhythmia mechanism in human ventricular tissue using computer modelling. We show that diffuse fibrosis slows down wave propagation and increases tissue vulnerability to wave break and spiral wave formation. We also demonstrate that diffuse fibrosis increases the period of re-entrant arrhythmias and can suppress the restitution-induced transition from tachycardia to fibrillation., Conclusion: The latter suggests that mechanisms different from restitution-induced spiral break-up might be more likely to account for the onset of fibrillation in the presence of large amounts of diffuse fibrotic tissue.

Sudden cardiac death is a major cause of death in the industrialized world, claiming approximately 300,000 victims annually in the United States alone. In most cases, sudden cardiac death is caused by ventricular fibrillation (VF). Experimental studies in large animal hearts have shown that the uncoordinated contractions during VF are caused by large numbers of chaotically wandering reentrant waves of electrical activity. However, recent clinical data on VF in the human heart seem to suggest that human VF may have a markedly different organization. Here, we use a detailed model of the human ventricles, including a detailed description of cell electrophysiology, ventricular anatomy, and fiber direction anisotropy, to study the organization of human VF. We show that characteristics of our simulated VF are qualitatively similar to the clinical data. Furthermore, we find that human VF is driven by only approximately 10 reentrant sources and thus is much more organized than VF in animal hearts of comparable size, where VF is driven by approximately 50 sources. We investigate the influence of anisotropy ratio, tissue excitability, and restitution properties on the number of reentrant sources driving VF. We find that the number of rotors depends strongest on minimum action potential duration, a property that differs significantly between human and large animal hearts. Based on these findings, we suggest that the simpler spatial organization of human VF relative to VF in large animal hearts may be caused by differences in minimum action potential duration. Both the simpler spatial organization of human VF and its suggested cause may have important implications for treating and preventing this dangerous arrhythmia in humans.

In this paper, we formulate a model for human ventricular cells that is efficient enough for whole organ arrhythmia simulations yet detailed enough to capture the effects of cell level processes such as current blocks and channelopathies. The model is obtained from our detailed human ventricular cell model by using mathematical techniques to reduce the number of variables from 19 to nine. We carefully compare our full and reduced model at the single cell, cable and 2D tissue level and show that the reduced model has a very similar behaviour. Importantly, the new model correctly produces the effects of current blocks and channelopathies on AP and spiral wave behaviour, processes at the core of current day arrhythmia research. The new model is well over four times more efficient than the full model. We conclude that the new model can be used for efficient simulations of the effects of current changes on arrhythmias in the human heart.

Ventricular fibrillation (VF) is one of the main causes of death in the Western world. According to one hypothesis, the chaotic excitation dynamics during VF are the result of dynamical instabilities in action potential duration (APD) the occurrence of which requires that the slope of the APD restitution curve exceeds 1. Other factors such as electrotonic coupling and cardiac memory also determine whether these instabilities can develop. In this paper we study the conditions for alternans and spiral breakup in human cardiac tissue. Therefore, we develop a new version of our human ventricular cell model, which is based on recent experimental measurements of human APD restitution and includes a more extensive description of intracellular calcium dynamics. We apply this model to study the conditions for electrical instability in single cells, for reentrant waves in a ring of cells, and for reentry in two-dimensional sheets of ventricular tissue. We show that an important determinant for the onset of instability is the recovery dynamics of the fast sodium current. Slower sodium current recovery leads to longer periods of spiral wave rotation and more gradual conduction velocity restitution, both of which suppress restitution-mediated instability. As a result, maximum restitution slopes considerably exceeding 1 (up to 1.5) may be necessary for electrical instability to occur. Although slopes necessary for the onset of instabilities found in our study exceed 1, they are within the range of experimentally measured slopes. Therefore, we conclude that steep APD restitution-mediated instability is a potential mechanism for VF in the human heart.

In this paper we briefly review currently published models for human ventricular cells and tissues. We discuss the Priebe-Beuckelmann (PB) model and the reduced version of this model constructed by Bernus et al. (redPB), the Ten Tusscher-Noble-Noble-Panfilov (TNNP) model and the Iyer-Mazhari-Winslow (IMW) model. We compare several characteristics of these models such as: sources of experimental data the models are based on, action potential morphology, action potential duration (APD) and conduction velocity (CV) restitution and computational efficiency. Finally, we discuss the application of a subset of these models-the redPB and the TNNP model-to study simulated spiral wave dynamics in 2D tissue sheets and in the human ventricles. We discuss the suitability of the different models for particular research questions and their limitations.

Recently, Wellner et al. [Proc. Natl. Acad. Sci. U.S.A. 99, 8015 (2002)]] proposed a principle for predicting a stable scroll wave filament shape as a geodesic in a 3D space with a metric determined by the inverse diffusivity tensor of the medium. Using the Hamilton-Jacobi theory we show that this geodesic is the shortest path for a wave propagating through the medium. This allows the use of shortest path algorithms to predict filament shapes, which we confirm numerically for a medium with orthotropic anisotropy. Our method can be used in cardiac tissue experiments since it does not require knowledge of the tissue anisotropy.

The experimental and clinical possibilities for studying cardiac arrhythmias in human ventricular myocardium are very limited. Therefore, the use of alternative methods such as computer simulations is of great importance. In this article we introduce a mathematical model of the action potential of human ventricular cells that, while including a high level of electrophysiological detail, is computationally cost-effective enough to be applied in large-scale spatial simulations for the study of reentrant arrhythmias. The model is based on recent experimental data on most of the major ionic currents: the fast sodium, L-type calcium, transient outward, rapid and slow delayed rectifier, and inward rectifier currents. The model includes a basic calcium dynamics, allowing for the realistic modeling of calcium transients, calcium current inactivation, and the contraction staircase. We are able to reproduce human epicardial, endocardial, and M cell action potentials and show that differences can be explained by differences in the transient outward and slow delayed rectifier currents. Our model reproduces the experimentally observed data on action potential duration restitution, which is an important characteristic for reentrant arrhythmias. The conduction velocity restitution of our model is broader than in other models and agrees better with available data. Finally, we model the dynamics of spiral wave rotation in a two-dimensional sheet of human ventricular tissue and show that the spiral wave follows a complex meandering pattern and has a period of 265 ms. We conclude that the proposed model reproduces a variety of electrophysiological behaviors and provides a basis for studies of reentrant arrhythmias in human ventricular tissue.

We study spiral wave dynamics in the presence of nonexcitable cells in two-dimensional (2D) and three-dimensional (3D) excitable media, described by the Aliev-Panfilov model. We find that increasing the percentage of randomly distributed nonexcitable cells can prevent the breaking up of a spiral wave into a complex spatiotemporal pattern. We show that this effect is more pronounced in 2D than 3D excitable media. We explain the observed 2D vs 3D differences by a different dependence of the period and diastolic interval of the spiral wave on the percentage of nonexcitable cells in 2D and 3D excitable media.

Heterogeneity of cardiac tissue is an important factor determining the initiation and dynamics of cardiac arrhythmias. In this paper, we studied the effects of gradients of electrophysiological heterogeneity on reentrant excitation patterns using computer simulations. We investigated the dynamics of spiral waves in a two-dimensional sheet of cardiac tissue described by the Luo-Rudy phase 1 (LR1) ventricular action potential model. A gradient of action potential duration (APD) was imposed by gradually varying the local current density of K(+) current or inward rectifying K(+) current along one axis of the tissue sheet. We show that a gradient of APD resulted in spiral wave drift. This drift consisted of two components. The longitudinal (along the gradient) component was always directed toward regions of longer spiral wave period. The transverse (perpendicular to the gradient) component had a direction dependent on the direction of rotation of the spiral wave. We estimated the velocity of the drift as a function of the magnitude of the gradient and discuss its implications.