Dongliang Zhang

Synchronization among a group of active agents is ubiquitous in nature. Although synchronization based on direct interactions between agents described by the Kuramoto model is well understood, the other general mechanism based on indirect interactions among agents sharing limited resources are less known. Here, we propose a minimal thermodynamically consistent model for the altruistic resource-sharing (ARS) mechanism wherein resources are needed for an individual agent to advance but a more advanced agent has a lower competence to obtain resources. We show that while differential competence in ARS mechanism provides a negative feedback leading to synchronization it also breaks detailed balance and thus requires additional energy dissipation besides the cost of driving individual agents. By solving the model analytically, our study reveals a general trade-off relation between the total energy dissipation rate and the two key performance measures of the system: average speed and synchronization accuracy. For a fixed dissipation rate, there is a distinct speed-accuracy Pareto front traversed by the scarcity of resources: scarcer resources lead to slower speed but more accurate synchronization. Increasing energy dissipation eases this trade-off by pushing the speed-accuracy Pareto front outward. The connections of our work to realistic biological systems such as the KaiABC system in cyanobacterial circadian clock and other theoretical results based on thermodynamic uncertainty relation are also discussed.

Accurate and robust spatial orders are ubiquitous in living systems. In 1952, Turing proposed a general mechanism for pattern formation exemplified by a reaction–diffusion model with two chemical species in a large system. However, in small biological systems such as a cell, the existence of multiple Turing patterns and strong noise can lower the spatial order. Recently, a modified reaction–diffusion model with an additional chemical species is shown to stabilize the Turing pattern. Here, we study non-equilibrium thermodynamics of this three-species reaction–diffusion model to understand the relationship between energy cost and the performance of self-positioning. By using computational and analytical approaches, we show that beyond the onset of pattern formation the positioning error decreases as energy dissipation increases. In a finite system, we find that a specific Turing pattern exists only within a finite range of total molecule number. Energy dissipation broadens this range, which enhances the robustness of Turing pattern against molecule number fluctuations in living cells. The generality of these results is verified in a realistic model of the Muk system underlying DNA segregation in Escherichia coli , and testable predictions are made for the dependence of the accuracy and robustness of the spatial pattern on the ATP/ADP ratio.

The circadian clock of cyanobacteria is based on an approximately 24h rhythm in the phosphorylation level of KaiC, a hexameric ATPase. This oscillation can be reconstituted in vitro by incubating three proteins together with ATP. Like all chemical oscillators, this system must include a nonlinear, or switch-like, feedback loop, whose nature has been unclear. Here, by using single particle cryo-EM at near-atomic resolution we identified two major conformational states of KaiC subunits, denoted as the exposed state and the buried state, which may provide a structural basis of how the KaiC hexamer changes its conformation during the (day-night) phosphorylation-dephosphorylation cycle. We classify the abundance and pattern of exposed and buried states within hexamers for more than 160,000 KaiC particles. The statistics of the spatial arrangement of the two states in hexamers can be quantitatively fit by a simple statistical physics model with an interaction energy between neighboring subunits and a local field that depends on phosphorylation state. Our study shows that phosphorylation shifts the probability of each conformation and reveals that there is substantial cooperativity between neighboring subunits, which can allow a KaiC hexamer to respond in an ultrasensitive, switch-like manner to changes in the phosphorylation level.

Living systems are open systems, where the laws of nonequilibrium thermodynamics play the important role. Therefore, studying living systems from a nonequilibrium thermodynamic aspect is interesting and useful. In this review, we briefly introduce the history and current development of nonequilibrium thermodynamics, especially that in biochemical systems. We first introduce historically how people realized the importance to study biological systems in the thermodynamic point of view. We then introduce the development of stochastic thermodynamics, especially three landmarks: Jarzynski equality, Crooks’ fluctuation theorem and thermodynamic uncertainty relation. We also summarize the current theoretical framework for stochastic thermodynamics in biochemical reaction networks, especially the thermodynamic concepts and instruments at nonequilibrium steady state. Finally, we show two applications and research paradigms for thermodynamic study in biological systems.

The energy dissipation rate in a nonequilibrium reaction system can be determined by the reaction rates in the underlying reaction network. By developing a coarse-graining process in state space and a corresponding renormalization procedure for reaction rates, we find that energy dissipation rate has an inverse power-law dependence on the number of microscopic states in a coarse-grained state. The dissipation scaling law requires self-similarity of the underlying network, and the scaling exponent depends on the network structure and the probability flux correlation. Existence of the inverse dissipation scaling law is shown in realistic biochemical systems such as biochemical oscillators and microtubule-kinesin active flow systems.

A model of coupled molecular biochemical oscillators is proposed to study non-equilibrium thermodynamics of synchronization. Under general considerations, we find that chemical interactions within an ensemble of autonomous oscillators break detailed balance and thus cost energy. This extra energy cost, in addition to the energy dissipated for driving each individual oscillator, is necessary to power the coupling interactions such as oscillator–oscillator exchange reactions, which are responsible for correcting the phase error in each individual noisy oscillator with respect to the collective oscillation of the whole ensemble. By solving the steady-state distribution of the many-oscillator system analytically and numerically, we show that the system reaches its synchronized state through a non-equilibrium phase transition as energy dissipation increases. The critical energy dissipation per period depends on both the frequency and strength of the exchange reaction, which reveals an optimal (efficient) design for achieving maximum synchronization with a fixed energy budget. We apply our general theory to the Kai system in the cyanobacterial circadian clock and predict a relationship between the KaiC ATPase activity and synchronization of the KaiC hexamers. The theoretical framework established here can be extended to study thermodynamics of collective behaviours in other non-equilibrium active systems.

After a brief historical review of ergodicity and mixing in dynamics, particularly in quantum dynamics, we introduce definitions of quantum ergodicity and mixing using the structure of the system's energy levels and spacings. Our definitions are consistent with the usual understanding of ergodicity and mixing. Two parameters concerning the degeneracy in energy levels and spacings are introduced. They are computed for right triangular billiards and the results indicate a very close relation between quantum ergodicity (mixing) and quantum chaos. At the end, we argue that, besides ergodicity and mixing, there may exist a third class of quantum dynamics which is characterized by a maximized entropy.