Entropy-production-driven oscillators in simple nonequilibrium networks

The development of tractable nonequilibrium simulation methods represents a bottleneck for efforts to describe the functional dynamics that occur within living cells. We here employ a nonequilibrium approach called the $\lambda$ ensemble to characterize the dissipative dynamics of a simple Markovian network driven by an external potential. In the highly dissipative regime brought about by the $\lambda$ bias, we observe a dynamical structure characteristic of cellular architectures: The entropy production drives a damped oscillator over state populations in the network. We illustrate the properties of such oscillations in weakly and strongly driven regimes, and we discuss how control structures associated with the “dynamical phase transition” in the system can be related to switches and oscillators in cellular dynamics.

Figure 1

Schematic illustration of the nonequilibrium triangle network. Trajectories are driven around the total cycle according to an imbalance between rates $x$ and $y$, which govern the traversal of $N$ miniature, triangular cycles. The symmetry-breaking link (parameterized by a relatively slow rate, $\sigma )$ between the right and left sides gives rise to interesting behavior in the observed entropy production. Rates are translated into transition counts for the transition probability framework; several self-transition counts are added to each node to improve numerical stability.

Figure 2

Mean of the entropy production rate, $\Omega$, as a function of the biasing parameter $\lambda$ in a 50-cycle triangle network. A dynamical phase transition (characterized by the sharp jump in the entropy production rate near $\lambda =1)$ separates two distinct regimes of dissipative behavior.

Figure 3

Finite-size scaling of the mean entropy production rate as a function of $\lambda$ in triangle networks. As the number of cycles in the network, $N$, is increased, the dynamical phase transition grows uniformly sharper but maintains its qualitative characteristics. Since mean entropy production rates vary slightly as $t_{\mathrm{obs}}$ is modulated, $\langle \Omega \rangle$ is calculated at $t_{\mathrm{obs}}=2N$ for purposes of comparison.

Figure 4

Steady-state probability distribution for the unbiased, strongly driven triangle network. The peak in probability at the extreme right (near the symmetry-breaking link) is truncated to preserve scale. States are indexed according to a left-to-right orientation of Fig. 1, where the slow link connects states 1 and 101. No dynamics as a function of trajectory length are observed in this unbiased case.

Figure 5

Finite-size scaling of the steady-state, “bottleneck” probability peak height (defined as the probability of the right-most state) in unbiased triangle networks. In the lower plot, probabilities are scaled by the number of states in each respective network; all values shown correspond to strong driving conditions.

Figure 6

Steady-state probability distributions for the four $\lambda$-biased networks analyzed compared in the text. As always, the driving direction is from left to right in each of the subplots. The curves shown here describe probabilities in the long-time, ergodic limit; the dynamical processes illustrated in Figs. 7–9 converge to these steady states at respective low and high levels of entropy production.

Figure 7

Dynamics within the low-entropy-production phases (at $\lambda =0.7)$ observed for the (a) weakly and (b) strongly driven networks. In both cases, the dynamics describe a “localization transition” onto states near the slow-link boundary. Probabilities are calculated for times $t$ according to $t=t_{\mathrm{obs}}/2$, and the probability spike at the right side of the network is truncated to preserve scale. Probabilities in the central states decay monotonically as time advances.

Figure 8

Dissipative dynamics in the high-entropy-production phase $\left(\lambda =1.4\right)$ for the strongly driven triangle network. Once more, probabilities are calculated for times $t=t_{\mathrm{obs}}/2$. Oscillations persist over the course of the 12 frames.

Figure 9

Dissipative dynamics in the high-entropy-production phase $\left(\lambda =1.4\right)$ for the weakly driven triangle network. Once more, probabilities are calculated for times $t=t_{\mathrm{obs}}/2$. Oscillations are damped more quickly and occur over a longer period than in the strongly driven case.

Figure 10

Finite-size scaling of the first central probability peak height in oscillations observed under the constraint of high entropy production $\left(\lambda =1.4\right)$ and under strong driving conditiions. In the lower plot, probabilities are once more scaled by state cardinalities.

Figure 11

Probability oscillations measured explicitly as a function of time within a single ensemble of long $\left(t_{\mathrm{obs}}=10000\right)$ trajectories at $\lambda =1.4$ and under strong driving conditions. Deviating from the $t=t_{\mathrm{obs}}/2$ convention in the $\lambda$-ensemble yields results biased by future outcomes; in this case, oscillation wave forms are rightward-leaning compared to the curves presented in the Figs. 8 and 9. The period and peak amplitude of this oscillator (approximately 200 time steps and 0.03, respectively), however, are very similar to values extracted from time points at $t_{\mathrm{obs}}/2$.

Figure 12

Dominant eigenvectors corresponding to the auxiliary operator, ${\mathbb{W}}_{\lambda }^{\mathrm{aux}}$, in the high-entropy-production phase. Real eigenvector components are illustrated at left, and imaginary eigenvector components are shown at right; each couple of vectors coincides with a complement of complex eigenvalues, appearing in complex conjugate pairs. The eigenvectors presented here exhibit the four largest real eigenvalue components that contribute to the auxiliary dynamics. These eigenvectors collectively form a set of near-harmonic modes that describe the oscillatory behavior seen at high levels of mean entropy production. In particular, the slowest mode (top) can be seen in the evolution of dynamics in Fig. 8.

Figure 13

Real components of dominant eigenvectors corresponding to the auxiliary operator, ${\mathbb{W}}_{\lambda }^{\mathrm{aux}}$, in the low-entropy-production phase. While some fine structure appears near the right side of the cycle, most oscillatory character observed above the phase transition disappears in the low-entropy-production phase. Below the phase transition, the dominant eigenvectors instead serve to transfer density toward the network's bottleneck in a nearly uniform fashion. Imaginary eigenvector components are also localized to the left side of the bottleneck.

Figure 14

Illustration of probability oscillations over time in the middle and extreme right portions of the network. In both the (a) weakly driven and (b) strongly driven cases, oscillations follow a regular period, with peak probability amplitudes for extreme rightward states (red, higher amplitude) exceeding the peak amplitudes for central state (blue, lower amplitude) probabilities. In both cases, the two probabilities are phase-locked in opposition and evolve over time according to a smooth waveform.