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The maintenance of tissue and organ structures during dynamic homeostasis is often not well understood. In order for a system to be stable, cell renewal, cell migration and cell death must be finely balanced. Moreover, a tissue's shape must remain relatively unchanged. Simple epithelial tissues occur in various structures throughout the body, such as the endothelium, mesothelium, linings of the lungs, saliva and thyroid glands, and gastrointestinal tract. Despite the prevalence of simple epithelial tissues, there are few models which accurately describe how these tissues maintain a stable structure. Here, we present a novel, 3D, deformable, multilayer, cell-centre model of a simple epithelium. Cell movement is governed by the minimisation of a bending potential across the epithelium, cell-cell adhesion, and viscous effects. We show that the model is capable of maintaining a consistent tissue structure while undergoing self renewal. We also demonstrate the model's robustness under tissue renewal, cell migration and cell removal. The model presented here is a valuable advancement towards the modelling of tissues and organs with complex and generalised structures.

The question of whether a population will persist or go extinct is of key interest throughout ecology and biology. Various mathematical techniques allow us to generate knowledge regarding individual behaviour, which can be analysed to obtain predictions about the ultimate survival or extinction of the population. A common model employed to describe population dynamics is the lattice-based random walk model with crowding (exclusion). This model can incorporate behaviour such as birth, death and movement, while including natural phenomena such as finite size effects. Performing sufficiently many realisations of the random walk model to extract representative population behaviour is computationally intensive. Therefore, continuum approximations of random walk models are routinely employed. However, standard continuum approximations are notoriously incapable of making accurate predictions about population extinction. Here, we develop a new continuum approximation, the state space diffusion approximation, which explicitly accounts for population extinction. Predictions from our approximation faithfully capture the behaviour in the random walk model, and provides additional information compared to standard approximations. We examine the influence of the number of lattice sites and initial number of individuals on the long-term population behaviour, and demonstrate the reduction in computation time between the random walk model and our approximation.

BondGraphTools is a Python library for scripted modelling of complex multi-physics systems. In contrast to existing modelling solutions, BondGraphTools is based upon the well established bond graph methodology, provides a programming interface for symbolic model composition, and is intended to be used in conjunction with the existing scientific Python toolchain. Here we discuss the design, implementation and use of BondGraphTools, demonstrate how it can be used to accelerate systems modelling with an example from optomechanics, and comment on current and future applications in cross-domain modelling, particularly in systems biology.

Systems biology and whole-cell modelling are demanding increasingly comprehensive mathematical models of cellular biochemistry. These models require the development of simplified models of specific processes which capture essential biophysical features but without unnecessarily complexity. Recently there has been renewed interest in thermodynamically-based modelling of cellular processes. Here we present an approach to developing of simplified yet thermodynamically consistent (hence physically plausible) models which can readily be incorporated into large scale biochemical descriptions but which do not require full mechanistic detail of the underlying processes. We illustrate the approach through development of a simplified, physically plausible model of the mitochondrial electron transport chain and show that the simplified model behaves like the full system.

Calcium (Ca2+) plays a central role in mediating both contractile function and hypertrophic signalling in ventricular cardiomyocytes. L-type Ca2+ channels trigger release of Ca2+ from ryanodine receptors (RyRs) for cellular contraction, while signalling downstream of Gq coupled receptors stimulates Ca2+ release via inositol 1,4,5-trisphosphate receptors (IP3Rs), engaging hypertrophic signalling pathways. Modulation of the amplitude, duration, and duty cycle of the cytosolic Ca2+ contraction signal, and spatial localisation, have all been proposed to encode this hypertrophic signal. Given current knowledge of IP3Rs, we develop a model describing the effect of functional interaction (cross-talk) between RyR and IP3R channels on the Ca2+ transient, and examine the sensitivity of the Ca2+ transient shape to properties of IP3R activation. A key result of our study is that IP3R activation increases Ca2+ transient duration for a broad range of IP3R properties, but the effect of IP3R activation on Ca2+ transient amplitude is dependent on IP3 concentration. Furthermore we demonstrate that IP3-mediated Ca2+ release in the cytosol increases the duty cycle of the Ca2+ transient, the fraction of the cycle for which [Ca2+] is elevated, across a broad range of parameter values and IP3 concentrations. When coupled to a model of downstream transcription factor (NFAT) activation, we demonstrate that there is a high correspondence between the Ca transient duty cycle and the proportion of activated NFAT in the nucleus. These findings suggest increased cytosolic Ca2+ duty cycle as a plausible mechanism for IP3-dependent hypertrophic signalling via Ca2+-sensitive transcription factors such as NFAT in ventricular cardiomyocytes., Comment: Biophysical Journal, in press

Environments with immobile obstacles or void regions that inhibit and alter the motion of individuals within that environment are ubiquitous. Correlation in the location of individuals within such environments arises as a combination of the mechanisms governing individual behavior and the heterogeneous structure of the environment. Measures of spatial structure and correlation have been successfully implemented to elucidate the roles of the mechanisms underpinning the behavior of individuals. In particular, the pair correlation function has been used across biology, ecology and physics to obtain quantitative insight into a variety of processes. However, naively applying standard pair correlation functions in the presence of obstacles may fail to detect correlation, or suggest false correlations, due to a reliance on a distance metric that does not account for obstacles. To overcome this problem, here we present an analytic expression for calculating a corrected pair correlation function for lattice-based domains containing obstacles. We demonstrate that this corrected pair correlation function is necessary for isolating the correlation associated with the behavior of individuals, rather than the structure of the environment. Using simulations that mimic cell migration and proliferation we demonstrate that the corrected pair correlation function recovers the short-range correlation known to be present in this process, independent of the heterogeneous structure of the environment. Further, we show that the analytic calculation of the corrected pair correlation derived here is significantly faster to implement than the corresponding numerical approach.

The drive for reproducibility in the computational sciences has provoked discussion and effort across a broad range of perspectives: technological, legislative/policy, education, and publishing. Discussion on these topics is not new, but the need to adopt standards for reproducibility of claims made based on computational results is now clear to researchers, publishers and policymakers alike. Many technologies exist to support and promote reproduction of computational results: containerisation tools like Docker, literate programming approaches such as Sweave, knitr, iPython or cloud environments like Amazon Web Services. But these technologies are tied to specific programming languages (e.g. Sweave/knitr to R; iPython to Python) or to platforms (e.g. Docker for 64-bit Linux environments only). To date, no single approach is able to span the broad range of technologies and platforms represented in computational biology and biotechnology. To enable reproducibility across computational biology, we demonstrate an approach and provide a set of tools that is suitable for all computational work and is not tied to a particular programming language or platform. We present published examples from a series of papers in different areas of computational biology, spanning the major languages and technologies in the field (Python/R/MATLAB/Fortran/C/Java). Our approach produces a transparent and flexible process for replication and recomputation of results. Ultimately, its most valuable aspect is the decoupling of methods in computational biology from their implementation. Separating the 'how' (method) of a publication from the 'where' (implementation) promotes genuinely open science and benefits the scientific community as a whole.

The Bond Graph approach and the Chemical Reaction Network approach to modelling biomolecular systems developed independently. This paper brings together the two approaches by providing a bond graph interpretation of the chemical reaction network concept of complexes. Both closed and open systems are discussed. The method is illustrated using a simple enzyme-catalysed reaction and a trans-membrane transporter.

Membrane transporters contribute to the regulation of the internal environment of cells by translocating substrates across cell membranes. Like all physical systems, the behaviour of membrane transporters is constrained by the laws of thermodynamics. However, many mathematical models of transporters, especially those incorporated into whole-cell models, are not thermodynamically consistent, leading to unrealistic behaviour. In this paper we use a physics-based modelling framework, in which the transfer of energy is explicitly accounted for, to develop thermodynamically consistent models of transporters. We then apply this methodology to model two specific transporters: the cardiac sarcoplasmic/endoplasmic Ca$^{2+}$ ATPase (SERCA) and the cardiac Na$^+$/K$^+$ ATPase.

Efficient energy transduction is one driver of evolution; and thus understanding biomolecular energy transduction is crucial to understanding living organisms. As an energy-orientated modelling methodology, bond graphs provide a useful approach to describing and modelling the efficiency of living systems. This paper gives some new results on the efficiency of metabolism based on bond graph models of the key metabolic processes: glycolysis., Comment: Accepted for 13th International Conference on Bond Graph Modeling (ICBGM 18), July 9-12, 2018, Bordeaux, France

Mathematical models of cardiac action potentials have become increasingly important in the study of heart disease and pharmacology, but concerns linger over their robustness during long periods of simulation, in particular due to issues such as model drift and non-unique steady states. Previous studies have linked these to violation of conservation laws, but only explored those issues with respect to charge conservation in specific models. Here, we propose a general and systematic method of identifying conservation laws hidden in models of cardiac electrophysiology by using bond graphs, and develop a bond graph model of the cardiac action potential to study long-term behaviour. Bond graphs provide an explicit energy-based framework for modelling physical systems, which makes them well-suited for examining conservation within electrophysiological models. We find that the charge conservation laws derived in previous studies are examples of the more general concept of a "conserved moiety". Conserved moieties explain model drift and non-unique steady states, generalising the results from previous studies. The bond graph approach provides a rigorous method to check for drift and non-unique steady states in a wide range of cardiac action potential models, and can be extended to examine behaviours of other excitable systems.

The Na$^+$/K$^+$ ATPase is an essential component of cardiac electrophysiology, maintaining physiological Na$^+$ and K$^+$ concentrations over successive heart beats. Terkildsen et al. (2007) developed a model of the ventricular myocyte Na$^+$/K$^+$ ATPase to study extracellular potassium accumulation during ischaemia, demonstrating the ability to recapitulate a wide range of experimental data, but unfortunately there was no archived code associated with the original manuscript. Here we detail an updated version of the model and provide CellML and MATLAB code to ensure reproducibility and reusability. We note some errors within the original formulation which have been corrected to ensure that the model is thermodynamically consistent, and although this required some reparameterisation, the resulting model still provides a good fit to experimental measurements that demonstrate the dependence of Na$^+$/K$^+$ ATPase pumping rate upon membrane voltage and metabolite concentrations. To demonstrate thermodynamic consistency we also developed a bond graph version of the model. We hope that these models will be useful for community efforts to assemble a whole-cell cardiomyocyte model which facilitates the investigation of cellular energetics.

Decomposition of biomolecular reaction networks into pathways is a powerful approach to the analysis of metabolic and signalling networks. Current approaches based on analysis of the stoichiometric matrix reveal information about steady-state mass flows (reaction rates) through the network. In this work we show how pathway analysis of biomolecular networks can be extended using an energy-based approach to provide information about energy flows through the network. This energy-based approach is developed using the engineering-inspired bond graph methodology to represent biomolecular reaction networks. The approach is introduced using glycolysis as an exemplar; and is then applied to analyse the efficiency of free energy transduction in a biomolecular cycle model of a transporter protein (Sodium-Glucose Transport Protein 1, SGLT1). The overall aim of our work is to present a framework for modelling and analysis of biomolecular reactions and processes which considers energy flows and losses as well as mass transport.

Motivation: Clustering techniques are routinely applied to identify patterns of co-expression in gene expression data. Co-regulation, and involvement of genes in similar cellular function, is subsequently inferred from the clusters which are obtained. Increasingly sophisticated algorithms have been applied to microarray data, however, less attention has been given to the statistical significance of the results of clustering studies. We present a technique for the analysis of commonly used hierarchical linkage-based clustering called Significance Analysis of Linkage Trees (SALT). Results: The statistical significance of pairwise similarity levels between gene expression profiles, a measure of co-expression, is established using a surrogate data analysis method. We find that a modified version of the standard linkage technique, complete-linkage, must be used to generate hierarchical linkage trees with the appropriate properties. The approach is illustrated using synthetic data generated from a novel model of gene expression profiles and is then applied to previously analysed microarray data on the transcriptional response of human fibroblasts to serum stimulation.

Many ion channels spontaneously switch between different levels of activity. Although this behaviour known as modal gating has been observed for a long time it is currently not well understood. Despite the fact that appropriately representing activity changes is essential for accurately capturing time course data from ion channels, systematic approaches for modelling modal gating are currently not available. In this paper, we develop a modular approach for building such a model in an iterative process. First, stochastic switching between modes and stochastic opening and closing within modes are represented in separate aggregated Markov models. Second, the continuous-time hierarchical Markov model, a new modelling framework proposed here, then enables us to combine these components so that in the integrated model both mode switching as well as the kinetics within modes are appropriately represented. A mathematical analysis reveals that the behaviour of the hierarchical Markov model naturally depends on the properties of its components. We also demonstrate how a hierarchical Markov model can be parameterised using experimental data and show that it provides a better representation than a previous model of the same data set. Because evidence is increasing that modal gating reflects underlying molecular properties of the channel protein, it is likely that biophysical processes are better captured by our new approach than in earlier models., Comment: 28 pages, 8 figures, 3 tables

Energy-based bond graph modelling of biomolecular systems is extended to include chemoelectrical trans- duction thus enabling integrated thermodynamically-compliant modelling of chemoelectrical systems in general and excitable membranes in particular. Our general approach is illustrated by recreating a well-known model of an excitable membrane. This model is used to investigate the energy consumed during a membrane action potential thus contributing to the current debate on the trade-off between the speed of an action potential event and energy consumption. The influx of Na+ is often taken as a proxy for energy consumption; in contrast, this paper presents an energy based model of action potentials. As the energy based approach avoids the assumptions underlying the proxy approach it can be directly used to compute energy consumption in both healthy and diseased neurons. These results are illustrated by comparing the energy consumption of healthy and degenerative retinal ganglion cells using both simulated and in vitro data.

We give a review of the current state of the art of data-driven modelling of the inositol trisphosphate receptor (IPR). After explaining that the IPR plays a crucial role as a central regulator in calcium dynamics, several sources of relevant experimental data are introduced. Single ion channels are best studied by recording single-channel currents under different ligand concentrations via the patch-clamp technique. The particular relevance of modal gating, the spontaneous switching between different levels of channel activity that occur even at constant ligand concentrations, is highlighted. In order to investigate the interactions of IPRs, calcium release from small clusters of channels, so-called calcium puffs, can be used. We then present the mathematical framework common to all models based on single-channel data, aggregated continuous-time Markov models, and give a short review of statistical approaches for parameterising these models with experimental data. The process of building a Markov model that integrates various sources of experimental data is illustrated using two recent examples, the model by Ullah et al. and the "Park-Drive" model by Siekmann et al., the only models that account for all sources of data currently available. Finally, it is demonstrated that the essential features of the Park-Drive model in different models of calcium dynamics are preserved after reducing it to a two-state model that only accounts for the switching between the inactive "park" and the active "drive" mode. This highlights the fact that modal gating is the most important mechanism of ligand regulation in the IPR. It also emphasises that data-driven models of ion channels do not necessarily have to lead to detailed models but can be constructed so that relevant data is selected to represent ion channels at the appropriate level of complexity for a given application., Comment: 23 pages, 7 figures

With the wealth of high-throughput sequencing data generated by recent large-scale consortia, predictive gene expression modelling has become an important tool for integrative analysis of transcriptomic and epigenetic data. However, sequencing data-sets are characteristically large, and previously modelling frameworks are typically inefficient and unable to leverage multi-core or distributed processing architectures. In this study, we detail an efficient and parallelised MapReduce implementation of gene expression modelling. We leverage the computational efficiency of this framework to provide an integrative analysis of over fifty histone modification data-sets across a variety of cancerous and non-cancerous cell-lines. Our results demonstrate that the genome-wide relationships between histone modifications and mRNA transcription are lineage, tissue and karyotype-invariant, and that models trained on matched epigenetic/transcriptomic data from non-cancerous cell-lines are able to predict cancerous expression with equivalent genome-wide fidelity., Comment: 10 pages, 3 figures

The bond graph approach to modelling biochemical networks is extended to allow hierarchical construction of complex models from simpler components. This is made possible by representing the simpler components as thermodynamically open systems exchanging mass and energy via ports. A key feature of this approach is that the resultant models are robustly thermodynamically compliant: the thermodynamic compliance is not dependent on precise numerical values of parameters. Moreover, the models are reusable due to the well-defined interface provided by the energy ports. To extract bond graph model parameters from parameters found in the literature, general and compact formulae are developed to relate free-energy constants and equilibrium constants. The existence and uniqueness of solutions is considered in terms of fundamental properties of stoichiometric matrices. The approach is illustrated by building a hierarchical bond graph model of glycogenolysis in skeletal muscle., Comment: 35 pages, 12 figures

Motivation: Predictive modelling of gene expression is a powerful framework for the in silico exploration of transcriptional regulatory interactions through the integration of high-throughput -omics data. A major limitation of previous approaches is their inability to handle conditional and synergistic interactions that emerge when collectively analysing genes subject to different regulatory mechanisms. This limitation reduces overall predictive power and thus the reliability of downstream biological inference. Results: We introduce an analytical modelling framework (TREEOME: tree of models of expression) that integrates epigenetic and transcriptomic data by separating genes into putative regulatory classes. Current predictive modelling approaches have found both DNA methylation and histone modification epigenetic data to provide little or no improvement in accuracy of prediction of transcript abundance despite, for example, distinct anti-correlation between mRNA levels and promoter-localised DNA methylation. To improve on this, in TREEOME we evaluate four possible methods of formulating gene-level DNA methylation metrics, which provide a foundation for identifying gene-level methylation events and subsequent differential analysis, whereas most previous techniques operate at the level of individual CpG dinucleotides. We demonstrate TREEOME by integrating gene-level DNA methylation (bisulfite-seq) and histone modification (ChIP-seq) data to accurately predict genome-wide mRNA transcript abundance (RNA-seq) for H1-hESC and GM12878 cell lines. Availability: TREEOME is implemented using open-source software and made available as a pre-configured bootable reference environment. All scripts and data presented in this study are available online at http://sourceforge.net/projects/budden2015treeome/., Comment: 14 pages, 6 figures

Thermodynamic aspects of chemical reactions have a long history in the Physical Chemistry literature. In particular, biochemical cycles - the building-blocks of biochemical systems - require a source of energy to function. However, although fundamental, the role of chemical potential and Gibb's free energy in the analysis of biochemical systems is often overlooked leading to models which are physically impossible. The bond graph approach was developed for modelling engineering systems where energy generation, storage and transmission are fundamental. The method focuses on how power flows between components and how energy is stored, transmitted or dissipated within components. Based on early ideas of network thermodynamics, we have applied this approach to biochemical systems to generate models which automatically obey the laws of thermodynamics. We illustrate the method with examples of biochemical cycles. We have found that thermodynamically compliant models of simple biochemical cycles can easily be developed using this approach. In particular, both stoichiometric information and simulation models can be developed directly from the bond graph. Furthermore, model reduction and approximation while retaining structural and thermodynamic properties is facilitated. Because the bond graph approach is also modular and scaleable, we believe that it provides a secure foundation for building thermodynamically compliant models of large biochemical networks.