Potential and flux field landscape theory. I. Global stability and dynamics of spatially dependent non-equilibrium systems

Socioeconomic agents as active matter in nonequilibrium Sakoda-Schelling models

How robust are socioeconomic agent-based models with respect to the details of the agents' decision rule? We tackle this question by considering an occupation model in the spirit of the Sakoda-Schelling model, historically introduced to shed light on segregation dynamics among human groups. For a large class of utility functions and decision rules, we pinpoint the nonequilibrium nature of the agent dynamics, while recovering an equilibrium-like phase separation phenomenology. Within the mean-field approximation we show how the model can be mapped, to some extent, onto an active matter field description. Finally, we consider nonreciprocal interactions between two populations and show how they can lead to nonsteady macroscopic behavior. We believe our approach provides a unifying framework to further study geography-dependent agent-based models, notably paving the way for joint consideration of population and price dynamics within a field theoretic approach.

Quantifying cancer cell plasticity with gene regulatory networks and single-cell dynamics

Phenotypic plasticity of cancer cells can lead to complex cell state dynamics during tumor progression and acquired resistance. Highly plastic stem-like states may be inherently drug-resistant. Moreover, cell state dynamics in response to therapy allow a tumor to evade treatment. In both scenarios, quantifying plasticity is essential for identifying high-plasticity states or elucidating transition paths between states. Currently, methods to quantify plasticity tend to focus on 1) quantification of quasi-potential based on the underlying gene regulatory network dynamics of the system; or 2) inference of cell potency based on trajectory inference or lineage tracing in single-cell dynamics. Here, we explore both of these approaches and associated computational tools. We then discuss implications of each approach to plasticity metrics, and relevance to cancer treatment strategies.

Nonequilibrium Equation of State for Open Hamiltonian Systems Maintained in Nonequilibrium Steady States

We establish the nonequilibrium equation of state in the stochastic thermodynamics framework for open Hamiltonian systems in contact with multiple heat baths governed by the Langevin equation and the Fokker-Planck equation. The derived nonequilibrium equation of state is an integral part of the nonequilibrium steady-state thermodynamics, and it reduces to the equilibrium equation of state when all the heat baths have the same temperature. We find that the nonequilibrium equation of state can be separated into two parts. One part has the form of the equilibrium equation of state, with the equilibrium temperature replaced by the average temperature of the heat baths. The other part depends on the temperature differences of the heat baths and represents nonequilibrium corrections. For systems with harmonic potentials, the nonequilibrium correction part is linear in the temperature differences of the heat baths. For more general open Hamiltonian systems, it may contain higher powers of the temperature differences. Another important feature of the nonequilibrium equation of state is that, in addition to the temperatures of the heat baths, it also depends on the friction coefficients arising from system-bath interactions in general. This suggests that it represents a relational condition between the system and the heat baths instead of the intrinsic properties of the system itself. In addition, when the potential energy is a homogeneous function, we find that the product of the generalized force and the generalized coordinate is proportional to the nonequilibrium internal energy, which is an alternative form of the nonequilibrium equation of state. Our findings may provide insights into the nonequilibrium equation of state for more general nonequilibrium open systems, including active matter and living organisms. Results in this work with suitable extension may find potential applications in nanoelectronics and biophysics including molecular motors and cells.

Perspectives on the landscape and flux theory for describing emergent behaviors of the biological systems

We give a review on the landscape theory of the equilibrium biological systems and landscape-flux theory of the nonequilibrium biological systems as the global driving force. The emergences of the behaviors, the associated thermodynamics in terms of the entropy and free energy and dynamics in terms of the rate and paths have been quantitatively demonstrated. The hierarchical organization structures have been discussed. The biological applications ranging from protein folding, biomolecular recognition, specificity, biomolecular evolution and design for equilibrium systems as well as cell cycle, differentiation and development, cancer, neural networks and brain function, and evolution for nonequilibrium systems, cross-scale studies of genome structural dynamics and experimental quantifications/verifications of the landscape and flux are illustrated. Together, this gives an overall global physical and quantitative picture in terms of the landscape and flux for the behaviors, dynamics and functions of biological systems.

Landscape-Flux Framework for Nonequilibrium Dynamics and Thermodynamics of Open Hamiltonian Systems Coupled to Multiple Heat Baths

We establish a nonequilibrium dynamic and thermodynamic formalism in the landscape-flux framework for open Hamiltonian systems in contact with multiple heat baths governed by stochastic dynamics. To systematically characterize nonequilibrium steady states, the nonequilibrium trinity construct is developed, which consists of detailed balance breaking, nonequilibrium potential landscape, and irreversible probability flux. We demonstrate that the temperature difference of the heat baths is the physical origin of detailed balance breaking, which generates the nonequilibrium potential landscape characterizing the nonequilibrium statistics and creates the irreversible probability flux signifying time irreversibility, with the latter two aspects closely connected. It is shown that the stochastic dynamics of the system can be formulated in the landscape-flux form, where the reversible force drives the conservative Hamiltonian dynamics, the irreversible force consisting of a landscape gradient force and an irreversible flux force drives the dissipative dynamics, and the stochastic force adds random fluctuations to the dynamics. The possible connection of the nonequilibrium trinity construct to nonequilibrium phase transitions is also suggested. A set of nonequilibrium thermodynamic equations, applicable to both nonequilibrium steady states and transient relaxation processes, is constructed. We find that an additional thermodynamic quantity, named the mixing entropy production rate, enters the nonequilibrium thermodynamic equations. It arises from the interplay between detailed balance breaking and transient relaxation, and it also relies on the conservative dynamics. At the nonequilibrium steady state, the heat flow, entropy flow, and entropy production are demonstrated to be thermodynamic manifestations of the nonequilibrium trinity construct. The general nonequilibrium formalism is applied to a class of solvable systems consisting of coupled harmonic oscillators. A more specific example of two harmonic oscillators coupled to two heat baths is worked out in detail. The example may facilitate connection with experiments.

Resilience analysis of the nexus across water supply, power generation and environmental systems from a stochastic perspective

Effects of external disturbances such as the population change on dynamics of water supply, power generation and environmental (WPE) systems have seldom been investigated. Following the WPE nexus profiled in the study of Feng et al. (2016), this study incorporated stochasticity of population, water supply and power generation into the modeling of the dynamical system in the Hehuang region of China, and further quantified resilience measures to understand the system's ability to withstand stochastic disturbances. First, the stochastic differential equations were used to improve the simulation of stochasticity in the WPE nexus. Next, the transient probability distribution functions (pdfs) of system variables, obtained by Monte Carlo simulation, were used to describe the evolutionary process of the system. Finally, the stationary pdfs of variables which reflect stable states of the system were derived to calculate four resilience measures. It is shown that: (1) The system approached a stable state after Year 2400 by calculating the L2 norm of the difference between transient and stationary pdfs. (2) The environmental system was identified as the most vulnerable subsystem because of its long convergence time. (3) The water supply system did not change greatly and it would remain stable at its current low level, i.e., water consumption per capita would be less than 80m³. The method adopted in this study is conducive to avoiding risk and the results provide valuable insights for regional management of a WPE nexus.

Automatic error control during forward flux sampling of rare events in master equation models

Enhanced sampling methods, such as forward flux sampling (FFS), have great capacity for accelerating stochastic simulations of nonequilibrium biochemical systems involving rare events. However, the description of the tradeoffs between simulation efficiency and error in FFS remains incomplete. We present a novel and mathematically rigorous analysis of the errors in FFS that, for the first time, covers the contribution of every phase of the simulation. We derive a closed form expression for the optimally efficient count of samples to take in each FFS phase in terms of a fixed constraint on sampling error. We introduce a new method, forward flux pilot sampling (FFPilot), that is designed to take full advantage of our optimizing equation without prior information or assumptions about the phase weights and costs along the transition path. In simulations of both single and multidimensional gene regulatory networks, FFPilot is able to completely control sampling error. We then discuss how memory effects can introduce additional error when relaxation along the transition path is slow. This extra error can be traced to correlations between the FFS phases and can be controlled by monitoring the covariance between them. Finally, we show that, in sets of simulations with matched error, FFPilot is on the order of tens-to-hundreds of times faster than direct sampling and noticeably more efficient than previous FFS methods.

Effects of molecular noise on bistable protein distributions in rod-shaped bacteria

The distributions of many proteins in rod-shaped bacteria are far from homogeneous. Often they accumulate at the cell poles or in the cell centre. At the same time, the copy number of proteins in a single cell is relatively small making the patterns noisy. To explore limits to protein patterns due to molecular noise, we studied a generic mechanism for spontaneous polar protein assemblies in rod-shaped bacteria, which are based on cooperative binding of proteins to the cytoplasmic membrane. For mono-polar assemblies, we find that the switching time between the two poles increases exponentially with the cell length and with the protein number. This feature could be beneficial to organelle maintenance in ageing bacteria.

Perspective: Reaches of chemical physics in biology

Chemical physics as a discipline contributes many experimental tools, algorithms, and fundamental theoretical models that can be applied to biological problems. This is especially true now as the molecular level and the systems level descriptions begin to connect, and multi-scale approaches are being developed to solve cutting edge problems in biology. In some cases, the concepts and tools got their start in non-biological fields, and migrated over, such as the idea of glassy landscapes, fluorescence spectroscopy, or master equation approaches. In other cases, the tools were specifically developed with biological physics applications in mind, such as modeling of single molecule trajectories or super-resolution laser techniques. In this introduction to the special topic section on chemical physics of biological systems, we consider a wide range of contributions, all the way from the molecular level, to molecular assemblies, chemical physics of the cell, and finally systems-level approaches, based on the contributions to this special issue. Chemical physicists can look forward to an exciting future where computational tools, analytical models, and new instrumentation will push the boundaries of biological inquiry.