Properties of two human atrial cell models in tissue: restitution, memory, propagation, and reentry

To date, two detailed ionic models of human atrial cell electrophysiology have been developed, the Nygren et al. model (NM) and the Courtemanche et al. model (CM). Although both models draw from similar experimental data, they have vastly different properties. This paper provides the first systematic analysis and comparison of the dynamics of these models in spatially extended systems including one-dimensional cables and rings, two-dimensional sheets, and a realistic three-dimensional human atrial geometry. We observe that, as in single cells, the CM adapts to rate changes primarily by changes in action potential duration (APD) and morphology, while for the NM rate changes affect resting membrane potential (RMP) more than APD. The models also exhibit different memory properties as assessed through S1-S2 APD and conduction velocity (CV) restitution curves with different S1 cycle lengths. Reentrant wave dynamics also differ, with the NM exhibiting stable, non-breaking spirals and the CM exhibiting frequent transient wave breaks. The realistic atrial geometry modifies dynamics in some cases through drift, transient pinning, and breakup. Previously proposed modifications to represent atrial fibrillation-remodeled electrophysiology produce altered dynamics, including reduced rate adaptation and memory for both models and conversion to stable reentry for the CM. Furthermore, proposed variations to the NM to reproduce action potentials more closely resembling those of the CM do not substantially alter the underlying dynamics of the model, so that tissue simulations using these modifications still behave more like the unmodified NM. Finally, interchanging the transmembrane current formulations of the two models suggests that currents contribute more strongly to RMP and CV, intracellular calcium dynamics primarily determine reentrant wave dynamics, and both are important in APD restitution and memory in these models. This finding implies that the formulation of intracellular calcium processes is as important to producing realistic models as transmembrane currents.     

Effective dispersal rate is a function of habitat size and corridor shape: mechanistic formulation of a two‐patch compartment model for spatially continuous systems

Two‐patch compartment models have been explored to understand the spatial processes that promote species coexistence. However, a phenomenological definition of the inter‐patch ‘dispersal rate’ has limited the quantitative predictability of these models to community dynamics in spatially continuous habitats. Here, we mechanistically rederived a two‐patch Lotka–Volterra competition model for a spatially continuous reaction‐diffusion system where a narrow corridor connects two large habitats. We provide a mathematical formula of the dispersal rate appearing in the two‐patch compartment model as a function of habitat size, corridor shape (ratio of its width to its length), and organism diffusion coefficients. For most reasonable settings, the two‐patch compartment model successfully approximated not only the steady states, but also the transient dynamics of the reaction–diffusion model. Further numerical simulations indicated the general applicability of our formula to other types of community dynamics, e.g. driven by resource‐competition, in spatially homogeneous and heterogeneous environments. Our results suggest that the spatial configuration of habitats plays a central role in community dynamics in space. Furthermore, our new framework will help to improve experimental designs for quantitative test of metacommunity theories and reduce the gaps among modeling, empirical studies, and their application to landscape management.     

Spatial-temporal modelling of bacterial colony growth on solid media

The enormous diversity among bacterial colonies of different species, and among colonies of the same species under different environmental conditions, has long interested microbiologists. Yet it is only comparatively recently that quantitative, rather than merely conceptual, models have been developed to explain the dynamics of bacterial colony formation and growth. Understanding the fundamental processes that drive these dynamics is still at a rudimentary level, though a number of advances have been made. This review traces the history of bacterial colony growth modelling, from the pioneering work of Pirt in the late 1960s, through experimental investigations by Wimpenny and his colleagues in the 1970s, and further models extending from that work to understand complex bacterial colony formations. It concludes with recent results which find that both diameter and height of the colony follow simple power-law behaviour over the entire active growth period (from a few hours to several days old), and that the results of Pirt, Wimpenny and their contemporaries can be re-interpreted.     

Towards an artificial brain

Three components of a brain model operating on neuromolecular computing principles are described. The first component comprises neurons whose input-output behavior is controlled by significant internal dynamics. Models of discrete enzymatic neurons, reaction-diffusion neurons operating on the basis of the cyclic nucleotide cascade, and neurons controlled by cytoskeletal dynamics are described. The second component of the model is an evolutionary learning algorithm which is used to mold the behavior of enzyme-driven neurons or small networks of these neurons for specific function, usually pattern recognition or target seeking tasks. The evolutionary learning algorithm may be interpreted either as representing the mechanism of variation and natural selection acting on a phylogenetic time scale, or as a conceivable ontogenetic adaptation mechanism. The third component of the model is a memory manipulation scheme, called the reference neuron scheme. In principle it is capable of orchestrating a repertoire of enzyme-driven neurons for coherent function. The existing implementations, however, utilize simple neurons without internal dynamics. Spatial navigation and simple game playing (using tic-tac-toe) provide the task environments that have been used to study the properties of the reference neuron model. A memory-based evolutionary learning algorithm has been developed that can assign credit to the individual neurons in a network. It has been run on standard benchmark tasks, and appears to be quite effective both for conventional neural nets and for networks of discrete enzymatic neurons. The models have the character of artificial worlds in that they map the hierarchy of processes in the brain (at the molecular, neuronal, and network levels), provide a task environment, and use this relatively self-contained setup to develop and evaluate learning and adaptation algorithms.     

Elucidating the in vivo phosphorylation dynamics of the ERK MAP kinase using quantitative proteomics data and Bayesian model selection

Ever since reversible protein phosphorylation was discovered, it has been clear that it plays a key role in the regulation of cellular processes. Proteins often undergo double phosphorylation, which can occur through two possible mechanisms: distributive or processive. Which phosphorylation mechanism is chosen for a particular cellular regulation bears biological significance, and it is therefore in our interest to understand these mechanisms. In this paper we study dynamics of the MEK/ERK phosphorylation. We employ a model selection algorithm based on approximate Bayesian computation to elucidate phosphorylation dynamics from quantitative time course data obtained from PC12 cells in vivo. The algorithm infers the posterior distribution over four proposed models for phosphorylation and dephosphorylation dynamics, and this distribution indicates the amount of support given to each model. We evaluate the robustness of our inferential framework by systematically exploring different ways of parameterizing the models and for different prior specifications. The models with the highest inferred posterior probability are the two models employing distributive dephosphorylation, whereas we are unable to choose decisively between the processive and distributive phosphorylation mechanisms.     

Translating biochemical network models between different kinetic formats

Mechanistic biochemical network models describe the dynamics of intracellular metabolite pools in terms of substance concentrations, stoichiometry and reaction kinetics. Data from stimulus response experiments are currently the most informative source for in-vivo parameter estimation in such models. However, only a part of the parameters of classical enzyme kinetic models can usually be estimated from typical stimulus response data. For this reason, several alternative kinetic formats using different “languages” (e.g. linear, power laws, linlog, generic and convenience) have been proposed to reduce the model complexity. The present contribution takes a rigorous “multi-lingual” approach to data evaluation by translating biochemical network models from one kinetic format into another. For this purpose, a new high-performance algorithm has been developed and tested. Starting with a given model, it replaces as many kinetic terms as possible by alternative expressions while still reproducing the experimental data. Application of the algorithm to a published model for Escherichia coli's sugar metabolism demonstrates the power of the new method. It is shown that model translation is a powerful tool to investigate the information content of stimulus response data and the predictive power of models. Moreover, the local and global approximation capabilities of the models are elucidated and some pitfalls of traditional single model approaches to data evaluation are revealed.     

Benchmarking novel approaches for modelling species range dynamics

Increasing biodiversity loss due to climate change is one of the most vital challenges of the 21st century. To anticipate and mitigate biodiversity loss, models are needed that reliably project species’ range dynamics and extinction risks. Recently, several new approaches to model range dynamics have been developed to supplement correlative species distribution models (SDMs), but applications clearly lag behind model development. Indeed, no comparative analysis has been performed to evaluate their performance. Here, we build on process‐based, simulated data for benchmarking five range (dynamic) models of varying complexity including classical SDMs, SDMs coupled with simple dispersal or more complex population dynamic models (SDM hybrids), and a hierarchical Bayesian process‐based dynamic range model (DRM). We specifically test the effects of demographic and community processes on model predictive performance. Under current climate, DRMs performed best, although only marginally. Under climate change, predictive performance varied considerably, with no clear winners. Yet, all range dynamic models improved predictions under climate change substantially compared to purely correlative SDMs, and the population dynamic models also predicted reasonable extinction risks for most scenarios. When benchmarking data were simulated with more complex demographic and community processes, simple SDM hybrids including only dispersal often proved most reliable. Finally, we found that structural decisions during model building can have great impact on model accuracy, but prior system knowledge on important processes can reduce these uncertainties considerably. Our results reassure the clear merit in using dynamic approaches for modelling species’ response to climate change but also emphasize several needs for further model and data improvement. We propose and discuss perspectives for improving range projections through combination of multiple models and for making these approaches operational for large numbers of species.     

GeneMark.hmm: new solutions for gene finding.

The number of completely sequenced bacterial genomes has been growing fast. There are computer methods available for finding genes but yet there is a need for more accurate algorithms. The GeneMark. hmm algorithm presented here was designed to improve the gene prediction quality in terms of finding exact gene boundaries. The idea was to embed the GeneMark models into naturally derived hidden Markov model framework with gene boundaries modeled as transitions between hidden states. We also used the specially derived ribosome binding site pattern to refine predictions of translation initiation codons. The algorithm was evaluated on several test sets including 10 complete bacterial genomes. It was shown that the new algorithm is significantly more accurate than GeneMark in exact gene prediction. Interestingly, the high gene finding accuracy was observed even in the case when Markov models of order zero, one and two were used. We present the analysis of false positive and false negative predictions with the caution that these categories are not precisely defined if the public database annotation is used as a control.     

A single-rate context-dependent learning process underlies rapid adaptation to familiar object dynamics

Motor learning has been extensively studied using dynamic (force-field) perturbations. These induce movement errors that result in adaptive changes to the motor commands. Several state-space models have been developed to explain how trial-by-trial errors drive the progressive adaptation observed in such studies. These models have been applied to adaptation involving novel dynamics, which typically occurs over tens to hundreds of trials, and which appears to be mediated by a dual-rate adaptation process. In contrast, when manipulating objects with familiar dynamics, subjects adapt rapidly within a few trials. Here, we apply state-space models to familiar dynamics, asking whether adaptation is mediated by a single-rate or dual-rate process. Previously, we reported a task in which subjects rotate an object with known dynamics. By presenting the object at different visual orientations, adaptation was shown to be context-specific, with limited generalization to novel orientations. Here we show that a multiple-context state-space model, with a generalization function tuned to visual object orientation, can reproduce the time-course of adaptation and de-adaptation as well as the observed context-dependent behavior. In contrast to the dual-rate process associated with novel dynamics, we show that a single-rate process mediates adaptation to familiar object dynamics. The model predicts that during exposure to the object across multiple orientations, there will be a degree of independence for adaptation and de-adaptation within each context, and that the states associated with all contexts will slowly de-adapt during exposure in one particular context. We confirm these predictions in two new experiments. Results of the current study thus highlight similarities and differences in the processes engaged during exposure to novel versus familiar dynamics. In both cases, adaptation is mediated by multiple context-specific representations. In the case of familiar object dynamics, however, the representations can be engaged based on visual context, and are updated by a single-rate process.     

Top-down analysis of temporal hierarchy in biochemical reaction networks

The study of dynamic functions of large-scale biological networks has intensified in recent years. A critical component in developing an understanding of such dynamics involves the study of their hierarchical organization. We investigate the temporal hierarchy in biochemical reaction networks focusing on: (1) the elucidation of the existence of "pools" (i.e., aggregate variables) formed from component concentrations and (2) the determination of their composition and interactions over different time scales. To date the identification of such pools without prior knowledge of their composition has been a challenge. A new approach is developed for the algorithmic identification of pool formation using correlations between elements of the modal matrix that correspond to a pair of concentrations and how such correlations form over the hierarchy of time scales. The analysis elucidates a temporal hierarchy of events that range from chemical equilibration events to the formation of physiologically meaningful pools, culminating in a network-scale (dynamic) structure-(physiological) function relationship. This method is validated on a model of human red blood cell metabolism and further applied to kinetic models of yeast glycolysis and human folate metabolism, enabling the simplification of these models. The understanding of temporal hierarchy and the formation of dynamic aggregates on different time scales is foundational to the study of network dynamics and has relevance in multiple areas ranging from bacterial strain design and metabolic engineering to the understanding of disease processes in humans.     

A unifying framework for metapopulation dynamics

Many biologically important processes, such as genetic differentiation, the spread of disease, and population stability, are affected by the (natural or enforced) subdivision of populations into networks of smaller, partly isolated, subunits. Such "metapopulations" can have extremely complex dynamics. We present a new general model that uses only two functions to capture, at the metapopulation scale, the main behavior of metapopulations. We show how complex, structured metapopulation models can be translated into our generalized framework. The metapopulation dynamics arising from some important biological processes are illustrated: the rescue effect, the Allee effect, and what we term the "antirescue effect." The antirescue effect captures instances where high migration rates are deleterious to population persistence, a phenomenon that has been largely ignored in metapopulation conservation theory. Management regimes that ignore a significant antirescue effect will be inadequate and may actually increase extinction risk. Further, consequences of territoriality and conspecific attraction on metapopulation-level dynamics are investigated. The new, simplified framework can incorporate knowledge from epidemiology, genetics, and population biology in a phenomenological way. It opens up new possibilities to identify and analyze the factors that are important for the evolution and persistence of the many spatially subdivided species.     

Modelling Southern Ocean ecosystems: krill, the food-web, and the impacts of harvesting

The ecosystem approach to fisheries recognises the interdependence between harvested species and other ecosystem components. It aims to account for the propagation of the effects of harvesting through the food-web. The formulation and evaluation of ecosystem-based management strategies requires reliable models of ecosystem dynamics to predict these effects. The krill-based system in the Southern Ocean was the focus of some of the earliest models exploring such effects. It is also a suitable example for the development of models to support the ecosystem approach to fisheries because it has a relatively simple food-web structure and progress has been made in developing models of the key species and interactions, some of which has been motivated by the need to develop ecosystem-based management. Antarctic krill, Euphausia superba, is the main target species for the fishery and the main prey of many top predators. It is therefore critical to capture the processes affecting the dynamics and distribution of krill in ecosystem dynamics models. These processes include environmental influences on recruitment and the spatially variable influence of advection. Models must also capture the interactions between krill and its consumers, which are mediated by the spatial structure of the environment. Various models have explored predator-prey population dynamics with simplistic representations of these interactions, while others have focused on specific details of the interactions. There is now a pressing need to develop plausible and practical models of ecosystem dynamics that link processes occurring at these different scales. Many studies have highlighted uncertainties in our understanding of the system, which indicates future priorities in terms of both data collection and developing methods to evaluate the effects of these uncertainties on model predictions. We propose a modelling approach that focuses on harvested species and their monitored consumers and that evaluates model uncertainty by using alternative structures and functional forms in a Monte Carlo framework.     

Multi-scale computational modelling in biology and physiology

Recent advances in biotechnology and the availability of ever more powerful computers have led to the formulation of increasingly complex models at all levels of biology. One of the main aims of systems biology is to couple these together to produce integrated models across multiple spatial scales and physical processes. In this review, we formulate a definition of multi-scale in terms of levels of biological organisation and describe the types of model that are found at each level. Key issues that arise in trying to formulate and solve multi-scale and multi-physics models are considered and examples of how these issues have been addressed are given for two of the more mature fields in computational biology: the molecular dynamics of ion channels and cardiac modelling. As even more complex models are developed over the coming few years, it will be necessary to develop new methods to model them (in particular in coupling across the interface between stochastic and deterministic processes) and new techniques will be required to compute their solutions efficiently on massively parallel computers. We outline how we envisage these developments occurring.     

Analysis of logistic growth models

A variety of growth curves have been developed to model both unpredated, intraspecific population dynamics and more general biological growth. Most predictive models are shown to be based on variations of the classical Verhulst logistic growth equation. We review and compare several such models and analyse properties of interest for these. We also identify and detail several associated limitations and restrictions.A generalized form of the logistic growth curve is introduced which incorporates these models as special cases. Several properties of the generalized growth are also presented. We furthermore prove that the new growth form incorporates additional growth models which are markedly different from the logistic growth and its variants, at least in their mathematical representation. Finally, we give a brief outline of how the new curve could be used for curve-fitting.     

A numerical algorithm for investigating the role of the motor-cargo linkage in molecular motor-driven transport

We extend the numerical algorithm developed by Wang et al. (2003. J. Theor. Biol. 221, 491-511) for studying biomolecular transport processes to include the linkage that connects molecular motors to their cargo. The new algorithm is used to investigate how the stiffness of the linkage affects the average velocity, effective diffusion coefficient, and randomness parameter. Three different models for molecular motors are considered: (1) a discrete stepping motor (2) a motor moving in a tilted-periodic potential and (3) a motor driven by a flashing potential. We demonstrate that a flexible motor-cargo linkage can make inferences on motor behavior based on measurements of the cargo's position difficult. We also show that even for the case of a tilted-periodic potential there exists an optimal stiffness of the linkage at which transport is maximized. The MATLAB code used in this paper is available at: http://www.unc.edu/approximatelytelston/code/.     

A new method for inferring hidden markov models from noisy time sequences

We present a new method for inferring hidden Markov models from noisy time sequences without the necessity of assuming a model architecture, thus allowing for the detection of degenerate states. This is based on the statistical prediction techniques developed by Crutchfield et al. and generates so called causal state models, equivalent in structure to hidden Markov models. The new method is applicable to any continuous data which clusters around discrete values and exhibits multiple transitions between these values such as tethered particle motion data or Fluorescence Resonance Energy Transfer (FRET) spectra. The algorithms developed have been shown to perform well on simulated data, demonstrating the ability to recover the model used to generate the data under high noise, sparse data conditions and the ability to infer the existence of degenerate states. They have also been applied to new experimental FRET data of Holliday Junction dynamics, extracting the expected two state model and providing values for the transition rates in good agreement with previous results and with results obtained using existing maximum likelihood based methods. The method differs markedly from previous Markov-model reconstructions in being able to uncover truly hidden states.     

Regression modeling of ordinal data with nonzero baselines

This paper develops regression models for ordinal data with nonzero control response probabilities. The models are especially useful in dose-response studies where the spontaneous or natural response rate is nonnegligible and the dosage is logarithmic. These models generalize Abbott's formula, which has been commonly used to model binary data with nonzero background observations. We describe a biologically plausible latent structure and develop an EM algorithm for fitting the models. The EM algorithm can be implemented using standard software for ordinal regression. A toxicology data set where the proposed model fits the data but a more conventional model fails is used to illustrate the methodology.     

Computer simulation of the rough lipopolysaccharide membrane of Pseudomonas aeruginosa.

Lipopolysaccharides (LPSs) form the major constituent of the outer membrane of Gram-negative bacteria, and are believed to play a key role in processes that govern microbial metal binding, microbial adsorption to mineral surfaces, and microbe-mediated oxidation/reduction reactions at the bacterial exterior surface. A computational modeling capability is being developed for the study of geochemical reactions at the outer bacterial envelope of Gram-negative bacteria. A molecular model for the rough LPS of Pseudomonas aeruginosa has been designed based on experimentally determined structural information. An electrostatic model was developed based on Hartree-Fock SCF calculations of the complete LPS molecule to obtain partial atomic charges. The exterior of the bacterial membrane was assembled by replication of a single LPS molecule and a single phospholipid molecule. Molecular dynamics simulations of the rough LPS membrane of P. aeruginosa were carried out and trajectories were analyzed for the energetic and structural factors that determine the role of LPS in processes at the cell surface.