Mutual Linearity is a Generic Property of Steady-State Markov Networks

Abstract

Understanding and predicting how complex systems respond to external perturbations is a central challenge in nonequilibrium statistical physics. Here we consider continuous-time Markov networks, which we subject to perturbations along a single edge. We find that in steady state the probabilities of any two states are linearly related to one another. We show that this mutual linearity of probabilities extends to a broad class of observables, including currents but also generic counting and state-dependent observables. Moreover, we derive an exact relation between the relative response of any state's probability and the ratio of two steady-state probabilities. Leveraging the Markov chain tree theorem, we further show that probabilities and the considered observables are constrained by the topological and kinetic properties of the network and provide analytical expressions in terms of spanning tree polynomials. Our results are general, holding for arbitrary rate parameterizations and extending far from equilibrium.

Figures (6)

• Figure 1: (a) Markov network consisting of four states. The controlled input edge $I$ (orange) is spanned by states $i = 1$ and $j = 2$. (b) Occupation probabilities as function of rate $k_{+I}$ while $k_{-I}$ is fixed. Probabilities are colored according to the labels in (a). (c) Linear relation [Eq. \ref{['eq:linearity']}] between probabilities. All probabilities are linear in the reference probability $p_2$. Dashed lines depict the limits of values accessible to $p_2$ [Eq. \ref{['eq:bounds']}]. (d) Probability $p_4$ over $p_2$ together with the rectangular phase space region spanned by their respective bounds and markers indicate the corners of the diagonal that contains the possible solutions. The affine coefficient $p_{4,2}^{(0)}$ and susceptibility $\chi_{4,2}$ entering Eq. \ref{['eq:linearity']} are indicated in orange.
• Figure 2: (a) Adaptation model for a chemoreceptor with a total of $|\mathcal{N}|=2(M+1)$ states comprising the activity state $a\in\lbrace 0,1\rbrace$ and the methylation level $m \in \lbrace 0, \ldots, M \rbrace$ (purple triangles). Occupation probabilities are denoted as $p(m,a)$. (b) Mutual linearity between the average methylation $\langle m \rangle$ [Eq. \ref{['app_eq:avgm']}], average activity $\langle a \rangle$ [Eq. \ref{['app_eq:ptumble']}], mesoscopic edge traffic $t^{(1)}_{0\leftrightarrow M}$ [Eq. \ref{['app_eq:traf']}], and current through the input edge $j_{m^\ast}^{(1)\to (0)}$ [Eq. \ref{['app_eq:cur']}], and probabilities $p(4,0)$ and $p(4,1)$. (c) Comparison between the ratio of probabilities connected by the input edge $p(2,1)/p(2,0)$ (colors) and the relative response $-\partial_{k_{+I}} p(0,1)/[\partial_{k_{-I}} p(0,1)]$ (wireframe). For the dashed green line, edge $I$ obeys detailed balance and the probability and response ratios equal $k_{-I}/k_{+I}$. The solid orange line indicates the protocol we adopt for input rates in panels (b) and (d). In panels (b) and (c) we consider a sensory network with $M=4$ methylation levels, $\Delta \mu=1.3$, and the input edge connects activity states at methylation level $m^\ast=2$. (d) Normalized susceptibilities of the adaptation model with $M=20$ methylation levels as we vary the driving strength $\Delta \mu$. Each circle corresponds to a state of the active $a=1$ (top) or passive $a=0$ branch (bottom) and is colored according to its susceptibility obtained from normalizing the results of Eq. \ref{['eq:suscs']}, with reference probability $p(10,1)$. The input edge is located at $m^\ast = 10$ and indicated in orange.
• Figure 3: (a) Calmodulin folding network stigler11 with an added unidirectional resetting transition with rate $r$ (orange) from an intermediate "$3$" to the unfolded state "$1$". The unidirectional rate is subject to external control. (b) Linear relation between steady-state probabilities and $p_1$. The black dashed lines and grey shaded areas indicate the bounds on $p_1$ given by Eqs. \ref{['eq:bounds']} and \ref{['eq:tight_bounds']} respectively. The grey dotted line shows the bound $\mathcal{B}_1^-$ [Eq. \ref{['eq:bounds']}]. (c) Currents through the two different folding pathways $\mathcal{J}_1 \equiv j_{15} + j_{56}$ and $\mathcal{J}_2\equiv j_{12}+j_{26}$, as well as the resetting flux $\mathcal{J}_r \equiv p_3r$ (see inset) over the resetting rate $r$. (d) The same observables as in (c) plotted over the entropy production rate of all microscopically reversible transitions $\sigma_\mathrm{rev} \equiv \sum_{n < m\in \mathcal{N}\backslash \lbrace I \rbrace} j_{nm} \ln(k_{nm}/k_{mn})$.
• Figure 4: Steady-state probabilities of a chemotaxis ladder with $M=20$ methylation levels for various driving strengths $\Delta \mu$ (in steps of $0.5$). Each circle corresponds to a state of the active $a=1$ (top) and passive $a=0$ (bottom) branch. Colors indicate the occupation probabilities, where we cap the colorbar at a value of $0.08$ to improve readability. The input edge located at $m^\ast = 10$ is highlighted in orange. Here, we use $S=0$, such that cycles are centered around $m^\ast$sm.
• Figure S1: We revisit the 4-state model from Fig. 1(a) of the main text. (a) Spanning trees $\mathcal{T^\mu}$ of the network. (b) Pictorial calculation of $K_{ij}^\mu$ for an example tree. Taking the ratio of rate polynomials leaves an unique reaction pathway (blue). (c) Upper (orange) and lower (red) bounds on the relative response of steady-state probabilities (black). The pathways that correspond to the respective bound are indicated in the inset.