Laplacian Dynamics with Synthesis and Degradation

Abstract

Analyzing qualitative behaviors of biochemical reactions using its associated network structure has proven useful in diverse branches of biology. As an extension of our previous work, we introduce a graph-based framework to calculate steady state solutions of biochemical reaction networks with synthesis and degradation. Our approach is based on a labeled directed graph G and the associated system of linear non-homogeneous differential equations with first order degradation and zeroth order synthesis. We also present a theorem which provides necessary and sufficient conditions for the dynamics to engender a unique stable steady state. Although the dynamics are linear, one can apply this framework to nonlinear systems by encoding nonlinearity into the edge labels. We answer an open question from our previous work concerning the non-positiveness of the elements in the inverse of a perturbed Laplacian matrix. Moreover, we provide a graph theoretical framework for the computation of the inverse of such a matrix. This also completes our previous framework and makes it purely graph theoretical. Lastly, we demonstrate the utility of this framework by applying it to a mathematical model of insulin secretion through ion channels in pancreatic β-cells.

Reformulating non-equilibrium steady-states and generalised Hopfield discrimination

“…Such graph-theoretic approximations have been further developed within chemical physics, especially for analysing complex free-energy landscapes at thermodynamic equilibrium [34,4]. Independently of these developments, a graph-theoretic approach to analysing biochemical systems under timescale separation, the "linear framework", was introduced in systems biology [14,26,25,38,39]. This was applied to bulk populations of biochemical entities, such as posttranslational modification systems [8,28], but the same mathematics can be used to analyse individual stochastic entities, such as genes [9,3,36,29,37].…”

Allosteric conformational ensembles have unlimited capacity for integrating information

“…The linear framework was introduced in (Gunawardena, 2012), developed in Gunawardena, 2013, Mirzaev andBortz, 2015), applied to various biological problems in (Ahsendorf et al, 2014, Dasgupta et al, 2014, Estrada et al, 2016, Wong et al, 2018a,b, Yordanov and Stelling, 2018, Biddle et al, 2019, Yordanov and Stelling, 2020 and reviewed in (Gunawardena, 2014, Wong andGunawardena, 2020). Technical details and proofs of the ideas described here can be found in (Gunawardena, 2012, Mirzaev andGunawardena, 2013) as well as in the Supplementary Information of (Estrada et al, 2016, Wong et al, 2018b, Biddle et al, 2019.…”

Efficient manipulation and generation of Kirchhoff polynomials for the analysis of non-equilibrium biochemical reaction networks

“…Note that the Kirchhoff polynomial κ(G) in the denominator of the steady-state expression for closed systems acts as a non-equilibrium partition function [9]. For more details on LFMs, derivations, equilibrium steady states and steady states in non-strongly connected graphs see [7,8,23]. With change of variables, each expression tree from the forest is assigned a pointer counting as 1 to the size of the representation and pointing to the leaves of other expression trees where it should be substituted to obtain the expression tree of the complete Kirchhoff polynomial.…”