Derivation of stationary distributions of biochemical reaction networks via structure transformation

Hong, Hyukpyo; Kim, Jinsu; Al-Radhawi, M. Ali; Sontag, Eduardo D.; Kim, Jae Kyoung

doi:10.1038/s42003-021-02117-x

Cited by 9 publications

(3 citation statements)

References 58 publications

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“…A MATLAB computational package was built to facilitate the process. Importantly, network translation was applied to several networks [12][13][14][15][16][17].…”

Section: Resultsmentioning

confidence: 99%

A Decomposition-Based Approach for Deriving Positive Steady States of a Class of Chemical Reaction Networks with Non-Mass-Action Kinetics

Hernandez,

Lubenia

2024

MATCH

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Steady states are frequently used to investigate the long-term behaviors of (bio)chemical systems. Recently, there has been a growing interest in network-based approaches due to their efficiency in deriving parametrizations of positive steady states in systems with mass-action kinetics. In this study, we extend this approach to derive positive steady states in networks under non-mass-action kinetics, specifically mixed kinetics. In a system with mixed kinetics, some reactions {may follow} mass-action kinetics, while others in the same network follow different rate laws, such as quotient rate laws. An example of such complexity is evident in a mathematical model of the insulin signaling pathway in type 2 diabetes. To compute its positive {steady states}, we adapt our existing network decomposition approach, originally designed for mass-action kinetics, to handle networks with non-mass-action kinetics. This approach involves breaking down a given network into smaller, independent subnetworks to derive the positive steady states of each subnetwork separately. These individual steady states are then combined to obtain the positive steady states of the entire network. This strategy makes computations more manageable for complex and large networks. More importantly, this method could separate reactions with purely mass-action kinetics into certain subnetworks from those that follow different rate laws. We also present an illustrative example that provides insights into methods for transforming networks with mixed kinetics into their associated mass-action systems.

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“…A MATLAB computational package was built to facilitate the process. Importantly, network translation was applied to several networks [12][13][14][15][16][17].…”

Section: Resultsmentioning

confidence: 99%

A Decomposition-Based Approach for Deriving Positive Steady States of a Class of Chemical Reaction Networks with Non-Mass-Action Kinetics

Hernandez,

Lubenia

2024

MATCH

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“…Unlike the deterministic QSSA ( Eq (3) ), the stochastic QSSA, which is the stationary average number conditioned on the total numbers of the bound and unbound species, has a complex form [ 41 , 55 , 56 ]. For instance, the stochastic QSSA for the number of A (〈 A 〉) can be expressed in terms of the dimensionless variables and parameters, , where Ω is the volume of a system (e.g., , ) as follows (see Methods for details): where A 0 = max{0, A T − B T }.…”

Section: Resultsmentioning

confidence: 99%

Universally valid reduction of multiscale stochastic biochemical systems using simple non-elementary propensities

Hong

Kim

2021

PLoS Comput Biol

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Biochemical systems consist of numerous elementary reactions governed by the law of mass action. However, experimentally characterizing all the elementary reactions is nearly impossible. Thus, over a century, their deterministic models that typically contain rapid reversible bindings have been simplified with non-elementary reaction functions (e.g., Michaelis-Menten and Morrison equations). Although the non-elementary reaction functions are derived by applying the quasi-steady-state approximation (QSSA) to deterministic systems, they have also been widely used to derive propensities for stochastic simulations due to computational efficiency and simplicity. However, the validity condition for this heuristic approach has not been identified even for the reversible binding between molecules, such as protein-DNA, enzyme-substrate, and receptor-ligand, which is the basis for living cells. Here, we find that the non-elementary propensities based on the deterministic total QSSA can accurately capture the stochastic dynamics of the reversible binding in general. However, serious errors occur when reactant molecules with similar levels tightly bind, unlike deterministic systems. In that case, the non-elementary propensities distort the stochastic dynamics of a bistable switch in the cell cycle and an oscillator in the circadian clock. Accordingly, we derive alternative non-elementary propensities with the stochastic low-state QSSA, developed in this study. This provides a universally valid framework for simplifying multiscale stochastic biochemical systems with rapid reversible bindings, critical for efficient stochastic simulations of cell signaling and gene regulation. To facilitate the framework, we provide a user-friendly open-source computational package, ASSISTER, that automatically performs the present framework.

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“…While we focus on the steady states of deterministic systems, the stationary distributions of stochastic systems can also be derived analytically for WR and DZ biochemical reaction networks [23,[39][40][41]. It would be interesting in future work to investigate whether the combination of network decomposition and translation can be used to derive stationary distributions analytically for a large class of biochemical reaction networks.…”

Section: Plos Computational Biologymentioning

confidence: 99%

A framework for deriving analytic steady states of biochemical reaction networks

Hernandez

Lubenia²,

Johnston

et al. 2023

PLoS Comput Biol

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The long-term behaviors of biochemical systems are often described by their steady states. Deriving these states directly for complex networks arising from real-world applications, however, is often challenging. Recent work has consequently focused on network-based approaches. Specifically, biochemical reaction networks are transformed into weakly reversible and deficiency zero generalized networks, which allows the derivation of their analytic steady states. Identifying this transformation, however, can be challenging for large and complex networks. In this paper, we address this difficulty by breaking the complex network into smaller independent subnetworks and then transforming the subnetworks to derive the analytic steady states of each subnetwork. We show that stitching these solutions together leads to the the analytic steady states of the original network. To facilitate this process, we develop a user-friendly and publicly available package, COMPILES (COMPutIng anaLytic stEady States). With COMPILES, we can easily test the presence of bistability of a CRISPRi toggle switch model, which was previously investigated via tremendous number of numerical simulations and within a limited range of parameters. Furthermore, COMPILES can be used to identify absolute concentration robustness (ACR), the property of a system that maintains the concentration of particular species at a steady state regardless of any initial concentrations. Specifically, our approach completely identifies all the species with and without ACR in a complex insulin model. Our method provides an effective approach to analyzing and understanding complex biochemical systems.

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Derivation of stationary distributions of biochemical reaction networks via structure transformation

Cited by 9 publications

References 58 publications

A Decomposition-Based Approach for Deriving Positive Steady States of a Class of Chemical Reaction Networks with Non-Mass-Action Kinetics

A Decomposition-Based Approach for Deriving Positive Steady States of a Class of Chemical Reaction Networks with Non-Mass-Action Kinetics

Universally valid reduction of multiscale stochastic biochemical systems using simple non-elementary propensities

A framework for deriving analytic steady states of biochemical reaction networks

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