Bond Graph Modeling of Chemiosmotic Biomolecular Energy Transduction

Sensitivity analysis of biochemical systems using bond graphs

The sensitivity of systems biology models to parameter variation can give insights into which parameters are most important for physiological function, and also direct efforts to estimate parameters. However, in general, kinetic models of biochemical systems do not remain thermodynamically consistent after perturbing parameters. To address this issue, we analyse the sensitivity of biological reaction networks in the context of a bond graph representation. We find that the parameter sensitivities can themselves be represented as bond graph components, mirroring potential mechanisms for controlling biochemistry. In particular, a sensitivity system is derived which re-expresses parameter variation as additional system inputs. The sensitivity system is then linearized with respect to these new inputs to derive a linear system which can be used to give local sensitivity to parameters in terms of linear system properties such as gain and time constant. This linear system can also be used to find so-called sloppy parameters in biological models. We verify our approach using a model of the Pentose Phosphate Pathway, confirming the reactions and metabolites most essential to maintaining the function of the pathway.

SBML to bond graphs: From conversion to composition

The Systems Biology Markup Language (SBML) is a popular software-independent XML-based format for describing models of biological phenomena. The BioModels Database is the largest online repository of SBML models. Several tools and platforms are available to support the reuse and composition of SBML models. However, these tools do not explicitly assess whether models are physically plausible or thermodynamically consistent. This often leads to ill-posed models that are physically impossible, impeding the development of realistic complex models in biology. Here, we present a framework that can automatically convert SBML models into bond graphs, which imposes energy conservation laws on these models. The new bond graph models are easily mergeable, resulting in physically plausible coupled models. We illustrate this by automatically converting and coupling a model of pyruvate distribution to a model of the pentose phosphate pathway.

Network thermodynamics of biological systems: A bond graph approach

Edmund Crampin (1973-2021) was at the forefront of Systems Biology research and his work will influence the field for years to come. This paper brings together and summarises the seminal work of his group in applying energy-based bond graph methods to biological systems. In particular, this paper: (a) motivates the need to consider energy in modelling biology; (b) introduces bond graphs as a methodology for achieving this; (c) describes extensions to modelling electrochemical transduction; (d) outlines how bond graph models can be constructed in a modular manner and (e) describes stoichiometric approaches to deriving fundamental properties of reaction networks. These concepts are illustrated using a new bond graph model of photosynthesis in chloroplasts.

The Cell Physiome: What Do We Need in a Computational Physiology Framework for Predicting Single-Cell Biology?

Modern biology and biomedicine are undergoing a big data explosion, needing advanced computational algorithms to extract mechanistic insights on the physiological state of living cells. We present the motivation for the Cell Physiome project: a framework and approach for creating, sharing, and using biophysics-based computational models of single-cell physiology. Using examples in calcium signaling, bioenergetics, and endosomal trafficking, we highlight the need for spatially detailed, biophysics-based computational models to uncover new mechanisms underlying cell biology. We review progress and challenges to date toward creating cell physiome models. We then introduce bond graphs as an efficient way to create cell physiome models that integrate chemical, mechanical, electromagnetic, and thermal processes while maintaining mass and energy balance. Bond graphs enhance modularization and reusability of computational models of cells at scale. We conclude with a look forward at steps that will help fully realize this exciting new field of mechanistic biomedical data science. Expected final online publication date for the Annual Review of Biomedical Data Science, Volume 5 is August 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Analysing and simulating energy-based models in biology using BondGraphTools

Like all physical systems, biological systems are constrained by the laws of physics. However, mathematical models of biochemistry frequently neglect the conservation of energy, leading to unrealistic behaviour. Energy-based models that are consistent with conservation of mass, charge and energy have the potential to aid the understanding of complex interactions between biological components, and are becoming easier to develop with recent advances in experimental measurements and databases. In this paper, we motivate the use of bond graphs (a modelling tool from engineering) for energy-based modelling and introduce, BondGraphTools, a Python library for constructing and analysing bond graph models. We use examples from biochemistry to illustrate how BondGraphTools can be used to automate model construction in systems biology while maintaining consistency with the laws of physics.

Modular assembly of dynamic models in systems biology

It is widely acknowledged that the construction of large-scale dynamic models in systems biology requires complex modelling problems to be broken up into more manageable pieces. To this end, both modelling and software frameworks are required to enable modular modelling. While there has been consistent progress in the development of software tools to enhance model reusability, there has been a relative lack of consideration for how underlying biophysical principles can be applied to this space. Bond graphs combine the aspects of both modularity and physics-based modelling. In this paper, we argue that bond graphs are compatible with recent developments in modularity and abstraction in systems biology, and are thus a desirable framework for constructing large-scale models. We use two examples to illustrate the utility of bond graphs in this context: a model of a mitogen-activated protein kinase (MAPK) cascade to illustrate the reusability of modules and a model of glycolysis to illustrate the ability to modify the model granularity.

Modular dynamic biomolecular modelling with bond graphs: the unification of stoichiometry, thermodynamics, kinetics and data

Renewed interest in dynamic simulation models of biomolecular systems has arisen from advances in genome-wide measurement and applications of such models in biotechnology and synthetic biology. In particular, genome-scale models of cellular metabolism beyond the steady state are required in order to represent transient and dynamic regulatory properties of the system. Development of such whole-cell models requires new modelling approaches. Here, we propose the energy-based bond graph methodology, which integrates stoichiometric models with thermodynamic principles and kinetic modelling. We demonstrate how the bond graph approach intrinsically enforces thermodynamic constraints, provides a modular approach to modelling, and gives a basis for estimation of model parameters leading to dynamic models of biomolecular systems. The approach is illustrated using a well-established stoichiometric model of Escherichia coli and published experimental data.

Hierarchical semantic composition of biosimulation models using bond graphs

Simulating complex biological and physiological systems and predicting their behaviours under different conditions remains challenging. Breaking systems into smaller and more manageable modules can address this challenge, assisting both model development and simulation. Nevertheless, existing computational models in biology and physiology are often not modular and therefore difficult to assemble into larger models. Even when this is possible, the resulting model may not be useful due to inconsistencies either with the laws of physics or the physiological behaviour of the system. Here, we propose a general methodology for composing models, combining the energy-based bond graph approach with semantics-based annotations. This approach improves model composition and ensures that a composite model is physically plausible. As an example, we demonstrate this approach to automated model composition using a model of human arterial circulation. The major benefit is that modellers can spend more time on understanding the behaviour of complex biological and physiological systems and less time wrangling with model composition.

Energy-Based Modeling of the Feedback Control of Biomolecular Systems With Cyclic Flow Modulation

Energy-based modelling brings engineering insight to the understanding of biomolecular systems. It is shown how well-established control engineering concepts, such as loop-gain, arise from energy feedback loops and are therefore amenable to control engineering insight. In particular, a novel method is introduced to allow the transfer function based approach of classical linear control to be utilised in the analysis of feedback systems modelled by network thermodynamics and thus amalgamate energy-based modelling with control systems analysis. The approach is illustrated using a class of metabolic cycles with activation and inhibition leading to the concept of Cyclic Flow Modulation.

Network Thermodynamical Modeling of Bioelectrical Systems: A Bond Graph Approach

Interactions among biomolecules, electrons, and protons are essential to many fundamental processes sustaining life. It is therefore of interest to build mathematical models of these bioelectrical processes not only to enhance understanding but also to enable computer models to complement in vitro and in vivo experiments. Such models can never be entirely accurate; it is nevertheless important that the models are compatible with physical principles. Network Thermodynamics, as implemented with bond graphs, provide one approach to creating physically compatible mathematical models of bioelectrical systems. This is illustrated using simple models of ion channels, redox reactions, proton pumps, and electrogenic membrane transporters thus demonstrating that the approach can be used to build mathematical and computer models of a wide range of bioelectrical systems.

Physically-plausible modelling of biomolecular systems: A simplified, energy-based model of the mitochondrial electron transport chain

Advances in systems biology and whole-cell modelling demand increasingly comprehensive mathematical models of cellular biochemistry. Such models require the development of simplified representations 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.

A thermodynamic framework for modelling membrane transporters

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 Ca²⁺ ATPase (SERCA) and the cardiac Na⁺/K⁺ ATPase.

Bond graph modelling of the cardiac action potential: implications for drift and non-unique steady states

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, generalizing 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.

Computing Biomolecular System Steady-States

A new approach to compute the equilibria and the steady-states of biomolecular systems modeled by bond graphs is presented. The approach is illustrated using a model of a biomolecular cycle representing a membrane transporter and a model of the mitochondrial electron transport chain.

Bond graph modelling of chemoelectrical energy transduction

Energy‐based bond graph modelling of biomolecular systems is extended to include chemoelectrical transduction 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 study 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.