arXiv subjects

Jeremy Gunawardena

Publications and source records attributed to Jeremy Gunawardena.

10 recordsLinked to original sources

arXiv2026

Algebraic formulas for first-passage times of Markov processes in the linear framework: generalising the work of Hill and Kac

In a preceding paper, we used the graph-theoretic linear framework to show how transient properties of continuous-time Markov processes -- splitting probabilities and the moments of first-passage time (FPT) distributions -- could be expressed as rational algebraic functions of the transition rates, by using spanning forests of the underlying graph. This contrasts with the related rational formulas for steady-state (s.s.) probabilities, which use only spanning trees. The biophysicist Terrell Hill sketched a procedure for calculating mean FPTs and splitting probabilities in terms of the s.s. probabilities of a modified Markov process, thereby converting calculations using ensembles of trajectories to those using a single trajectory. Similarly, Mark Kac showed that the mean recurrence time to a state of a Markov process could be expressed in terms of the s.s. probability of that state. Here, we explore further the relationships between transient and s.s. properties, forests and trees, and ensemble and single-trajectory calculations. We formalise Hill's procedure by introducing a Hill operator, $H_u[G]$, on a graph, $G$, and use it to calculate all moments of the conditional and unconditional FPTs from $u$ as rational functions of the s.s. probabilities of $H_u[G]$, which arise from trees and "exchange factors" which arise from forests. We then combine this with an unravelling operator, $U_v[G]$, to calculate all moments of the recurrence time distribution as rational functions of the s.s. probabilities of $G$ and related exchange factors. Surprisingly, the Hill operator turns out to be a left-inverse to the unravelling operator, suggesting that the algebra of operators on linear framework graphs may be of broader interest. Our results integrate and generalise previously disparate findings into a common repertoire of rational algebraic formulas for FPTs of Markov processes.

math.PR↗

arXiv2023

Universal Bounds on Information-Processing Capabilities of Markov Processes

We consider a finite-state, continuous-time Markov process, represented in the "linear framework" by a directed graph with labelled edges which specifies the infinitesimal generator of the process. If the graph is strongly connected, the process has a unique steady-state probability distribution, $p$, which may not be one of thermodynamic equilibrium. If the label (rate) of any edge (transition) is perturbed, to reach the new steady-state probability distribution $p'$, we find that the Kullback-Leibler (KL) divergence between these distributions is bounded by the change in the thermodynamic affinity, $ΔA(C)$, of any cycle, $C$, that includes the altered transition, D$_{KL}$$(p'||p) \leq |ΔA(C)|$, irrespective of the structure of the graph. It follows that, if an equilibrium distribution is shifted away from equilibrium by perturbing a single rate, then the free energy difference between these distributions is similarly bounded $F^{neq}-F^{eq}\leq |ΔA(C)|$. Our analysis reveals universal, energy-induced bounds on the information-processing capabilities of Markov systems operating arbitrarily far from thermodynamic equilibrium.

physics.bio-ph↗

arXiv2022

Thermodynamic bounds on ultrasensitivity in covalent switching

Switch-like motifs are among the basic building blocks of biochemical networks. A common motif that can serve as an ultrasensitive switch consists of two enzymes acting antagonistically on a substrate, one making and the other removing a covalent modification. To work as a switch, such covalent modification cycles must be held out of thermodynamic equilibrium by continuous expenditure of energy. Here, we exploit the linear framework for timescale separation to establish tight bounds on the performance of any covalent-modification switch, in terms of the chemical potential difference driving the cycle. The bounds apply to arbitrary enzyme mechanisms, not just Michaelis-Menten, with arbitrary rate constants, and thereby reflect fundamental physical constraints on covalent switching.

q-bio.MN↗

arXiv2021

Reformulating non-equilibrium steady-states and generalised Hopfield discrimination

Despite substantial progress in non-equilibrium physics, steady-state (s.s.) probabilities remain intractable to analysis. For a Markov process, s.s. probabilities can be expressed in terms of transition rates using the Matrix-Tree theorem (MTT) in the graph-based linear framework. The MTT reveals that, away from equilibrium, s.s. probabilities become globally dependent on all rates, with expressions growing exponentially in the system size. This overwhelming complexity and lack of thermodynamic interpretation have greatly impeded analysis. Here, we show that s.s. probabilities are proportional to the average of $\exp(-S(P))$, where $S(P)$ is the entropy generated along minimal paths, $P$, in the graph, and the average is taken over a probability distribution on spanning trees. Assuming Arrhenius rates, this "arboreal" distribution becomes Boltzmann-like, with the energy of a tree being its total edge barrier energy. This reformulation offers a thermodynamic interpretation that smoothly generalises equilibrium statistical mechanics and reorganises the expression complexity: the number of distinct minimal-path entropies depends on the entropy production index, a new graph-theoretic measure of non-equilibrium complexity, not on graph size. We demonstrate the power of this reformulation by extending Hopfield's analysis of discrimination by kinetic proofreading to any graph with index 1. We derive a general formula for the error ratio and use it to show that local energy dissipation can yield optimal discrimination through global synergy.

cond-mat.stat-mech↗

arXiv2020

Reversal symmetries for cyclic paths away from thermodynamic equilibrium

If a system is at thermodynamic equilibrium, an observer cannot tell whether a film of it is being played forward or in reverse: any transition will occur with the same frequency in the forward as in the reverse direction. However, if expenditure of energy changes the rate of even a single transition to yield a non-equilibrium steady state, such time-reversal symmetry undergoes a widespread breakdown, far beyond the point at which the energy is expended. An explosion of interdependency also arises, with steady-state probabilities of system states depending in a complicated manner on the rate of every transition in the system. Nevertheless, in the midst of this global non-equilibrium complexity, we find that cyclic paths have reversibility properties that remain local, and which can exhibit symmetry, no matter how far the system is from thermodynamic equilibrium. Specifically, given any cycle of reversible transitions, the ratio of the frequencies with which the cycle is traversed in one direction versus the other is determined, in the long-time limit, only by the thermodynamic force on the cycle itself, without requiring knowledge of transition rates elsewhere in the system. In particular, if there is no net energy expenditure on the cycle, then, over long times, the cycle traversal frequencies are the same in either direction.

cond-mat.stat-mech↗

arXiv2017

An energy-speed-accuracy relation in complex networks for biological discrimination

Discriminating between correct and incorrect substrates is a core process in biology but how is energy apportioned between the conflicting demands of accuracy ($μ$), speed ($σ$) and total entropy production rate ($P$)? Previous studies have focussed on biochemical networks with simple structure or relied on simplifying kinetic assumptions. Here, we use the linear framework for timescale separation to analytically examine steady-state probabilities away from thermodynamic equilibrium for networks of arbitrary complexity. We also introduce a method of scaling parameters that is inspired by Hopfield's treatment of kinetic proofreading. Scaling allows asymptotic exploration of high-dimensional parameter spaces. We identify in this way a broad class of complex networks and scalings for which the quantity $σ\ln(μ)/P$ remains asymptotically finite whenever accuracy improves from equilibrium, so that $μ_{eq}/μ\to 0$. Scalings exist, however, even for Hopfield's original network, for which $σ\ln(μ)/P$ is asymptotically infinite, illustrating the parametric complexity. Outside the asymptotic regime, numerical calculations suggest that, under more restrictive parametric assumptions, networks satisfy the bound, $σ\ln(μ/μ_{eq})/P < 1$, and we discuss the biological implications for discrimination by ribosomes and DNA polymerase. The methods introduced here may be more broadly useful for analysing complex networks that implement other forms of cellular information processing.

q-bio.MN↗

arXiv2011

A linear elimination framework

Key insights in molecular biology, such as enzyme kinetics, protein allostery and gene regulation emerged from quantitative analysis based on time-scale separation, allowing internal complexity to be eliminated and resulting in the well-known formulas of Michaelis-Menten, Monod-Wyman-Changeux and Ackers-Johnson-Shea. In systems biology, steady-state analysis has yielded eliminations that reveal emergent properties of multi-component networks. Here we show that these analyses of nonlinear biochemical systems are consequences of the same linear framework, consisting of a labelled, directed graph on which a Laplacian dynamics is defined, whose steady states can be algorithmically calculated. Analyses previously considered distinct are revealed as identical, while new methods of analysis become feasible.

q-bio.MN↗

arXiv2007

Modular model building

Mathematical models are increasingly used in both academia and the pharmaceutical industry to understand how phenotypes emerge from systems of molecular interactions. However, their current construction as monolithic sets of equations presents a fundamental barrier to progress. Overcoming this requires modularity, enabling sub-systems to be specified independently and combined incrementally, and abstraction, enabling general properties to be specified independently of specific instances. These in turn require models to be represented as programs rather than as datatypes. Programmable modularity and abstraction enables libraries of modules to be created for generic biological processes, which can be instantiated and re-used repeatedly in different contexts with different components. We have developed a computational infrastructure to support this. We show here why these capabilities are needed, what is required to implement them and what can be accomplished with them that could not be done previously.

q-bio.MN↗

arXiv2007

Multi-bit information storage by multisite phosphorylation

Cells store information in DNA and in stable programs of gene expression, which thereby implement forms of long-term cellular memory. Cells must also possess short-term forms of information storage, implemented post-translationally, to transduce and interpret external signals. CaMKII, for instance, is thought to implement a one-bit (bistable) short-term memory required for learning at post-synaptic densities. Here we show by mathematical analysis that multisite protein phosphorylation, which is ubiquitous in all eukaryotic signalling pathways, exhibits multistability for which the maximal number of steady states increases with the number of sites. If there are n sites, the maximal information storage capacity is at least log_2 (n+2)/2 bits when n is even and log_2 (n+1)/2 bits when n is odd. Furthermore, when substrate is in excess, enzyme saturation together with an alternating low/high pattern in the site-specific relative catalytic efficiencies, enriches for multistability. That is, within physiologically plausible ranges for parameters, multistability becomes more likely than monostability. We discuss the experimental challenges in pursuing these predictions and in determining the biological role of short-term information storage.

q-bio.MN↗

arXiv2003

The Perron-Frobenius Theorem for Homogeneous, Monotone Functions

If A is a nonnegative matrix whose associated directed graph is strongly connected, the Perron-Frobenius theorem asserts that A has an eigenvector in the positive cone, (R^+)^n. We associate a directed graph to any homogeneous, monotone function, f: (R^+)^n -> (R^+)^n, and show that if the graph is strongly connected then f has a (nonlinear) eigenvector in (R^+)^n. Several results in the literature emerge as corollaries. Our methods show that the Perron-Frobenius theorem is ``really'' about the boundedness of invariant subsets in the Hilbert projective metric. They lead to further existence results and open problems.

math.FA↗