Topology of the simplest gene switch

TL;DR

The paper addresses why a minimal self-repressing gene switch exhibits multimodal expression despite deterministic expectations. It develops a topological framework based on spectral flow of a counting-field–deformed generator for a nonuniform Markov network, introducing adiabaticity $\omega = f/k$ and local/global winding numbers $X^{\mathrm{ad}}$ and $X^{\mathrm{eq}}$. It identifies three dynamical regimes—non-adiabatic, oscillatory, and adiabatic—derives domain-wall positions $n^*_1$ and $n^*_{\mathrm{tot}}$, and shows how oscillations arise via exceptional-point transitions; it also predicts steady-state peak locations and relative heights that match the topological analysis. This topology-based approach offers a robust, parameter-insensitive lens for stochastic gene regulation and lays groundwork for generalizing to larger, more complex biological networks.

Abstract

Complex gene regulatory networks often display emergent simple behavior. Sometimes this simplicity can be traced to a nearly equivalent energy landscape, but not always. Here, we show how a topological theory for stochastic and biochemical networks can predict phase transitions between dynamical regimes, where the simplest landscape paradigm would fail. We demonstrate the utility of this topological approach for a simple gene network, revealing a new oscillatory regime in addition to previously recognized multimodal stationary phases. We show how local winding numbers predict the steady-state locations in the single-mode and bimodal phases, and a flux analysis predicts the respective strengths of the steady-state peaks.

Figures (6)

• Figure 1: Self-repressing gene network. (a) A gene generates proteins at rate $g_1$. Each protein degrades at rate $k$ or binds to a transcription site at rate $h$, repressing generation when bound. The bound protein unbinds at rate $f$. (b) The network forms a ladder structure defined by protein number $n$ and transcription factor state (bound/unbound). Right inset: to compute spectral flow, transitions between states $(n^*,1)$ and $(n^*+1, 1)$ are multiplied by phase factors $e^{\pm i\chi}$.
• Figure 2: Spectral flow predicts three distinct phases in a self-repressing gene network. (a) The network spectrum as a function of $\chi$ shows spectral flow in three regimes of adiabaticity $\omega=f/k$, represented by $\omega=0.1$, $1$ and $5$. (b) Coherence as a function of adiabaticity $\omega$ reveals an oscillatory regime at intermediate $\omega\approx 1$. (c) Steady state has two peaks in the non-adiabatic phase, spreads over the network in the oscillating phase, and has a single peak in the adiabatic phase. Coherence is shown for $X^\mathrm{ad}=g_1/2k=20$ and varying $X^\mathrm{eq}=f/h$. Panels (a) and (c) use $X^\mathrm{ad}=X^\mathrm{eq}=10$.
• Figure 3: In the non-adiabatic phase, local winding of bound and unbound chains individually predict two peaks. (a) Local winding number is tracked along the bound (blue) and unbound (orange) chains. It is computed under periodic boundary conditions, as illustrated in two cartoons. (b) Steady-state peaks occur at an edge (bound) and a domain wall (unbound), as predicted by local winding numbers. (c) Observed peak locations in the unbound states $n_p$ (orange dots) match predicted domain wall locations $n^*_1$ (gray line) for a range of parameter values. (d) Relative probability between peaks $\Delta p=\sum_n p_{n1}-p_{n0}$ for a range of $X^\mathrm{ad}$ and $X^\mathrm{eq}$; peaks are equal when $X^\mathrm{eq}=2X^\mathrm{ad}$. Insets: right peak dominates when $X^\mathrm{eq}>2X^\mathrm{ad}$, and is smaller when $X^\mathrm{eq}<2X^\mathrm{ad}$. Plots use $X^\mathrm{eq}=X^\mathrm{ad}=20$ and $\omega=0.01$.
• Figure 4: In the adiabatic phase, combined local winding determines domain wall and peak location. (a) Local winding number is tracked along two averaged chains (black). Cartoons illustrate networks with periodic boundary conditions, where either the unbound or bound chain dominates the net winding direction. (b) Steady state peak in the total probability (black) is determined by the domain wall in the local winding number. (c) Observed peak locations $n_p$ (black dots) match predicted domain wall locations $n^*_\mathrm{tot}$ (gray lines) for a range of parameter values. (d) Relative probability between unbound and bound chains $\Delta p=\sum_n p_{n1}-p_{n0}$ for a range of $X^\mathrm{ad}$ and $X^\mathrm{eq}$. probabilities are equal when $X^\mathrm{eq}=X^\mathrm{ad}$. Insets: unbound probability dominates when $X^\mathrm{eq}>X^\mathrm{ad}$, and is smaller when $X^\mathrm{eq}<X^\mathrm{ad}$. Plots use $X^\mathrm{eq}=X^\mathrm{ad}=20$ and $\omega=10$.
• Figure 5: Spectrum of the transition matrix $\mathcal{W}(\chi)$ with manually imposed periodic boundary conditions for a range of $\chi\in[-\pi, \pi]$ and three values of adiabaticity $\omega$. Other system parameters are set to $X^\mathrm{ad}=X^\mathrm{eq}=10$.