Nonequilibrium Enhances Adaptation Efficiency of Stochastic Biochemical Systems

<div><p>Adaptation is a crucial biological function possessed by many sensory systems. Early work has shown that some influential equilibrium models can achieve accurate adaptation. However, recent studies indicate that there are close relationships between adaptation and nonequilibrium. In this paper, we provide an explanation of these two seemingly contradictory results based on Markov models with relatively simple networks. We show that as the nonequilibrium driving becomes stronger, the system under consideration will undergo a phase transition along a fixed direction: from non-adaptation to simple adaptation then to oscillatory adaptation, while the transition in the opposite direction is forbidden. This indicates that although adaptation may be observed in equilibrium systems, it tends to occur in systems far away from equilibrium. In addition, we find that nonequilibrium will improve the performance of adaptation by enhancing the adaptation efficiency. All these results provide a deeper insight into the connection between adaptation and nonequilibrium. Finally, we use a more complicated network model of bacterial chemotaxis to validate the main results of this paper.</p></div

The relationship between adaptation and nonequilibrium in the sensory network model.

<p>(a) The response of the MCP receptor to a step increase of the ligand concentration under different choices of <i>α</i>. (b) The dependence of the adaptation efficiency on <i>α</i> under <i>τ</i>_<i>a</i> = 0.1 s (blue line) and <i>τ</i>_<i>a</i> = 200 s (red line). (c) The dependence of the adaptation sensitivity on <i>α</i> under <i>τ</i>_<i>a</i> = 0.1 s (blue line) and <i>τ</i>_<i>a</i> = 200 s (red line). (d) The dependence of the adaptation error on <i>α</i> under <i>τ</i>_<i>a</i> = 0.1 s (blue line) and <i>τ</i>_<i>a</i> = 200 s (red line).</p

The sufficient and necessary conditions such that the equality in Eq (6) holds.

<p>The system is required to have a cyclic topological structure. Moreover, the transition rates along the forward cycle (blue arrows) need to be the same and the transition rates along the backward cycle need to be all zero.</p

Nonequilibrium enhances the adaptation efficiency.

<p>The red lines illustrate the case where the state transitions of the system do not have a separate time scale. In this case, the model parameters are taken as <i>π</i> = (1, 0, 0), <i>μ</i> = (0.1, 0.5, 0.4), <i>g</i> = (0, 2, 1), <i>a</i> = 1.2, <i>b</i> = 0.6, and <i>c</i> = 0.2. The blue lines illustrate the case where state 1 has relatively fast transitions compared to the other two states. In this case, all the model parameters remain the same except that we take <i>μ</i> = (0.0001, 0.5, 0.4999). (a) The peak output versus the net flux. (b) The adaptation efficiency versus the net flux.</p

The number of complex eigenvalues of the generator matrix under different ranges of <i>α</i>.

<p>The number of complex eigenvalues of the generator matrix under different ranges of <i>α</i>.</p

The <i>E. coli</i> chemotactic sensory system.

<p>(a) A schematic diagram of the MCP receptor. (b) The response of the MCP receptor to a step increase of the ligand concentration. (c) The sensory network model of the MCP receptor.</p

Adaptation and overshoot in biochemical systems.

<p>(a) Adaptation. (b) Overshoot and non-overshoot.</p

Simple overshoot and oscillatory overshoot.

<p>(a) Simple overshoot. (b) Oscillatory overshoot. When the number of states is small, the oscillation of the output can hardly be observed. When the number of states is large, the oscillation of the output may become apparent.</p

Three types of phase transitions as the net flux varies.

<p>(a)-(c) The dependence of <i>E</i> (red line) and −<i>λ</i>₂ <i>F</i> (blue line) on the net flux <i>J</i>. (a) The red and blue lines intersect at <i>J</i> = <i>α</i>. (b) The red line is below the blue line. (c) The red line is above the blue line. (d) The change of the output as the net flux increases. (e) The possible directions (blue arrows) and impossible directions (red arrows) of the phase transition as the nonequilibrium driving becomes stronger.</p

Hetero-regulatory modules inference results.

<p>(A) The predicted hetero-regulatory modules recover known MAPK pathways in <i>S.cerevisiae</i>, including filamentous growth (FG) pathway, mating pheromone (MP) pathway, cell wall integrity (CWI) pathway, and high osmolarity glycerol (HOG) pathway. (B) Distribution of function of HeR module's target genes. The size of the pie, which represents the functional distribution of the corresponding target gene set, is proportional to the number of target genes in the module. (C) Analysis of the target genes of HOG kinase (Ssk2, Pbs2, Hog1) related HeR modules reveals cross-talk between HOG pathway and other MAPK pathways, and indicates potential role of Sok2 in HOG pathway. (D) HeR modules related to transcription factors Tec1 and Ste12 inferred a feedback loop in mating pathway. Shown is the logic of the inference.</p