Modular Bond-graph Modelling and Analysis of Biomolecular Systems

Bond graphs can be used to build thermodynamically-compliant hierarchical models of biomolecular systems. As bond graphs have been widely used to model, analyse and synthesise engineering systems, this paper suggests that they can play the same role in the modelling, analysis and synthesis of biomolecular systems. The particular structure of bond graphs arising from biomolecular systems is established and used to elucidate the relation between thermodynamically closed and open systems. Block diagram representations of the dynamics implied by these bond graphs are used to reveal implicit feedback structures and are linearised to allow the application of control-theoretical methods. Two concepts of modularity are examined: computational modularity where physical correctness is retained and behavioural modularity where module behaviour (such as ultrasensitivity) is retained. As well as providing computational modularity, bond graphs provide a natural formulation of behavioural modularity and reveal the sources of retroactivity. A bond graph approach to reducing retroactivity, and thus inter-module interaction, is shown to require a power supply such as that provided by the ATP = ADP + Pi reaction. The MAPK cascade (Raf-MEK-ERK pathway) is used as an illustrative example.

💡 Research Summary

The paper presents a comprehensive framework for modeling, analyzing, and synthesizing biomolecular systems using bond graphs, a graphical language originally developed for engineering systems that captures energy flow through effort–flow pairs. By treating chemical potential (µ) as effort and molar flow rate (v) as flow, the authors map biochemical reactions onto bond‑graph elements: C‑components store species amounts, Re‑components represent reactions as two‑port nonlinear resistors, and TF‑components encode stoichiometric coefficients.

A key contribution is the systematic distinction between thermodynamically closed and open systems. Closed systems are modeled solely with C, Re, and 0/1 junctions, ensuring internal conservation of mass and energy. To represent interaction with the environment, the authors introduce “chemostats” (fixed chemical potentials) and “flowstats” (fixed molar flows). Adding these external ports converts a closed bond graph into an open one, making explicit how material and energy exchange with the surroundings occur.

The paper then explores two notions of modularity. Computational modularity refers to the ability to interconnect modules while preserving physical correctness; this is guaranteed by the bond‑graph connection rules. Behavioral modularity concerns whether a module’s intrinsic dynamic properties—such as ultrasensitivity, bistability, or switch‑like responses—remain unchanged when the module is embedded in a larger network. Biological networks typically lack unidirectional buffers, leading to retroactivity, where downstream loads feed back and alter upstream behavior. The bond‑graph formalism naturally captures retroactivity as a mismatch in power flow at module interfaces.

To mitigate retroactivity, the authors demonstrate that a dedicated power supply—most naturally the ATP ↔ ADP + Pi reaction—must be present. By providing a high‑energy phosphate reservoir, each signaling module can behave like an independent voltage source, reducing the impact of downstream loads and restoring behavioral modularity.

Linearization is employed to bridge the gap between the inherently nonlinear bond‑graph equations and classical control‑theoretic tools. Around a steady‑state operating point, the authors perform a first‑order Taylor expansion, yielding a linear state‑space model that can be represented as a block diagram. This representation makes explicit the feedback loops inherent in the biochemical network, allowing the application of sensitivity analysis, stability criteria, and controller design methods familiar to control engineers.

The theoretical developments are illustrated with the MAPK cascade (Raf‑MEK‑ERK pathway). Instead of using Michaelis–Menten approximations, which ignore reverse reactions and lead to non‑thermodynamic models, the authors construct a fully thermodynamically compliant bond‑graph model that includes enzyme‑substrate complexes, reversible steps, and ATP consumption. Each cascade layer is treated as a separate module; ATP hydrolysis is introduced as a power source to each layer, thereby reducing retroactivity between layers. After linearization, the resulting block diagram reveals the cascade’s intrinsic feedback structure and confirms that ultrasensitivity is preserved when the modules are interconnected.

Overall, the paper makes four major contributions: (1) a rigorous mapping of biochemical reaction networks onto bond‑graph formalism, preserving thermodynamic consistency; (2) a clear methodology for converting closed systems to open systems via chemostats and flowstats; (3) a nuanced analysis of computational versus behavioral modularity, with retroactivity quantified in energetic terms; and (4) a demonstration that control‑theoretic linearization and block‑diagram techniques can be directly applied to biomolecular systems, opening avenues for systematic design and analysis of synthetic biological circuits. The work suggests future research directions including automated generation of bond‑graph models from omics data, development of nonlinear control strategies for biochemical networks, and the formulation of design principles for modular synthetic biology based on energetic isolation.