The potential and flux landscape theory of evolution

We established the potential and flux landscape theory for evolution. We found explicitly the conventional Wright's gradient adaptive landscape based on the mean fitness is inadequate to describe the general evolutionary dynamics. We show the intrinsic potential as being Lyapunov function(monotonically decreasing in time) does exist and can define the adaptive landscape for general evolution dynamics for studying global stability. The driving force determining the dynamics can be decomposed into gradient of potential landscape and curl probability flux. Non-zero flux causes detailed balance breaking and measures how far the evolution from equilibrium state. The gradient of intrinsic potential and curl flux are perpendicular to each other in zero fluctuation limit resembling electric and magnetic forces on electrons. We quantified intrinsic energy, entropy and free energy of evolution and constructed non-equilibrium thermodynamics. The intrinsic non-equilibrium free energy is a Lyapunov function. Both intrinsic potential and free energy can be used to quantify the global stability and robustness of evolution. We investigated an example of three allele evolutionary dynamics with frequency dependent selection (detailed balance broken). We uncovered the underlying single, triple, and limit cycle attractor landscapes. We found quantitative criterions for stability through landscape topography. We also quantified evolution pathways and found paths do not follow potential gradient and are irreversible due to non-zero flux. We generalized the original Fisher's fundamental theorem to the general (i.e., frequency dependent selection) regime of evolution by linking the adaptive rate with not only genetic variance related to the potential but also the flux. We show there is an optimum potential where curl flux resulting from biotic interactions of individuals within a species or between species can sustain an endless evolution even if the physical environment is unchanged. We offer a theoretical basis for explaining the corresponding Red Queen hypothesis proposed by Van Valen. Our work provides a theoretical foundation for evolutionary dynamics.     

The energy pump and the origin of the non-equilibrium flux of the dynamical systems and the networks

The global stability of dynamical systems and networks is still challenging to study. We developed a landscape and flux framework to explore the global stability. The potential landscape is directly linked to the steady state probability distribution of the non-equilibrium dynamical systems which can be used to study the global stability. The steady state probability flux together with the landscape gradient determines the dynamics of the system. The non-zero probability flux implies the breaking down of the detailed balance which is a quantitative signature of the systems being in non-equilibrium states. We investigated the dynamics of several systems from monostability to limit cycle and explored the microscopic origin of the probability flux. We discovered that the origin of the probability flux is due to the non-equilibrium conditions on the concentrations resulting energy input acting like non-equilibrium pump or battery to the system. Another interesting behavior we uncovered is that the probabilistic flux is closely related to the steady state deterministic chemical flux. For the monostable model of the kinetic cycle, the analytical expression of the probabilistic flux is directly related to the deterministic flux, and the later is directly generated by the chemical potential difference from the adenosine triphosphate (ATP) hydrolysis. For the limit cycle of the reversible Schnakenberg model, we also show that the probabilistic flux is correlated to the chemical driving force, as well as the deterministic effective flux. Furthermore, we study the phase coherence of the stochastic oscillation against the energy pump, and argue that larger non-equilibrium pump results faster flux and higher coherence. This leads to higher robustness of the biological oscillations. We also uncovered how fluctuations influence the coherence of the oscillations in two steps: (1) The mild fluctuations influence the coherence of the system mainly through the probability flux while maintaining the regular landscape topography. (2) The larger fluctuations lead to flat landscape and the complete loss of the stability of the whole system.     

Fluctuation theorem for nonequilibrium reactions

A fluctuation theorem is derived for stochastic nonequilibrium reactions ruled by the chemical master equation. The theorem is expressed in terms of the generating and large-deviation functions characterizing the fluctuations of a quantity which measures the loss of detailed balance out of thermodynamic equilibrium. The relationship to entropy production is established and discussed. The fluctuation theorem is verified in the Schl?gl model of far-from-equilibrium bistability.     

Bistability in the chemical master equation for dual phosphorylation cycles

Dual phospho/dephosphorylation cycles, as well as covalent enzymatic-catalyzed modifications of substrates are widely diffused within cellular systems and are crucial for the control of complex responses such as learning, memory, and cellular fate determination. Despite the large body of deterministic studies and the increasing work aimed at elucidating the effect of noise in such systems, some aspects remain unclear. Here we study the stationary distribution provided by the two-dimensional chemical master equation for a well-known model of a two step phospho/dephosphorylation cycle using the quasi-steady state approximation of enzymatic kinetics. Our aim is to analyze the role of fluctuations and the molecules distribution properties in the transition to a bistable regime. When detailed balance conditions are satisfied it is possible to compute equilibrium distributions in a closed and explicit form. When detailed balance is not satisfied, the stationary non-equilibrium state is strongly influenced by the chemical fluxes. In the last case, we show how the external field derived from the generation and recombination transition rates, can be decomposed by the Helmholtz theorem, into a conservative and a rotational (irreversible) part. Moreover, this decomposition allows to compute the stationary distribution via a perturbative approach. For a finite number of molecules there exists diffusion dynamics in a macroscopic region of the state space where a relevant transition rate between the two critical points is observed. Further, the stationary distribution function can be approximated by the solution of a Fokker-Planck equation. We illustrate the theoretical results using several numerical simulations.     

Extending the fluctuation theorem to describe reaction coordinates

The fluctuation theorem describes the distribution of work done on small systems which have been pushed out of equilibrium in response to an external field. The theorem has recently been a subject of much interest for describing single-molecule experiments and simulations. In this communication, it is shown how the fluctuation theorem can be extended to describe fluctuations not only in the work done on a system, but also in a reaction coordinate. The extension explored in this work allows for a generalized derivation of Hummer and Szabo's expression (G. Hummer and A. Szabo, Proc. Natl. Acad. Sci. 98, 3658 (2001)) for reconstructing the potential of mean force from nonequilibrium trajectories. The derivation demonstrates how implementation of this expression can be more easily facilitated. Atomistic simulations of a biomolecular system are presented which support these results.     

On the flux-force partitioning and non-equilibrium thermodynamics near a steady state

The flux-force linear relationship is a basic building block in the development of irreversible thermodynamics near equilibrium. Here, we explore the fate of this relationship near a steady state in a finer detail by partitioning the fluxes and the forces into time-independent and time-dependent components. To this end, we use a master equation approach without assuming the condition of detailed balance. The connection of the flux-force components with various state functions and path functions provides a detailed picture of the variations of such quantities in terms of the deviation of probabilities from the steady state. Pilot calculations on an exactly-solvable case furnish insights into the energy-balance mechanism for non-equilibrium systems, revealing additionally how an out-of-equilibrium scenario can be favorable in realizing a minimized free energy state.     

Measuring flux distributions for diffusion in the small-numbers limit

For the classical diffusion of independent particles, Fick's law gives a well-known relationship between the average flux and the average concentration gradient. What has not yet been explored experimentally, however, is the dynamical distribution of diffusion rates in the limit of small particle numbers. Here, we measure the distribution of diffusional fluxes using a microfluidics device filled with a colloidal suspension of a small number of microspheres. Our experiments show that (1) the flux distribution is accurately described by a Gaussian function; (2) Fick's law, that the average flux is proportional to the particle gradient, holds even for particle gradients down to a single particle difference; (3) the variance in the flux is proportional to the sum of the particle numbers; and (4) there are backward flows, where particles flow up a concentration gradient, rather than down it. In addition, in recent years, two key theorems about nonequilibrium systems have been introduced: Evans' fluctuation theorem for the distribution of entropies and Jarzynski's work theorem. Here, we introduce a new fluctuation theorem, for the fluxes, and we find that it is confirmed quantitatively by our experiments.     

A model for nonexponential relaxation and aging in dissipative systems

The nonexponential relaxation and aging inherent to complex dynamics manifested in a wide variety of dissipative systems are analyzed through a model of diffusion in phase space in the presence of a nonconservative force. The action of this force establishes a heat flow which maintains the system away from equilibrium. The inability of the system to find its equilibrium state becomes apparent through the presence of an effective temperature field. This is the temperature of the stationary nonequilibrium state reached by the system satisfying a generalized version of the fluctuation-dissipation theorem. The presence of a nonequilibrium temperature leads to a hierarchy of relaxation times responsible for the aging phenomena and to a relation similar to the Vogel-Fulcher-Tammann law [H. Vogel, Phys. Z. 22, 645 (1921); G. S. Fulcher, J. Am. Ceram. Soc. 8, 339 (1925); 8, 789 (1925); G. Tammann and W. Hesse, Z. Anorg. Allg. Chem. 156, 245 (1926)].     

Equilibrium theory for a particle pulled by a moving optical trap

The viscous drag on a colloidal particle pulled through solution by an optical trap is large enough that on experimentally relevant time scales the mechanical force exerted by the trap is equal and opposite the viscous drag force. The rapid mechanical equilibration allows the system to be modeled using equilibrium theory where the effects of the energy dissipation (thermodynamic disequilibrium) show up only in the coordinate transformations that map the system from the laboratory frame of reference, relative to which the particle is moving, to a frame of reference in which the particle is, on average, stationary and on which the stochastic dynamics is governed by a canonical equilibrium distribution function. The simple equations in the stationary frame can be analyzed using the Onsager-Machlup theory for stochastic systems and provide generalizations of equilibrium and near equilibrium concepts such as detailed balance and fluctuation-dissipation relations applicable to a wide range of systems including molecular motors, pumps, and other nanoscale machines.     

A molecular dynamics computer simulation study of room-temperature ionic liquids. II. Equilibrium and nonequilibrium solvation dynamics

The molecular dynamics (MD) simulation study of solvation structure and free energetics in 1-ethyl-3-methylimidazolium chloride and 1-ethyl-3-methylimidazolium hexafluorophosphate using a probe solute in the preceding article [Y. Shim, M. Y. Choi and H. J. Kim, J. Chem. Phys. 122, 044510 (2005)] is extended to investigate dynamic properties of these liquids. Solvent fluctuation dynamics near equilibrium are studied via MD and associated time-dependent friction is analyzed via the generalized Langevin equation. Nonequilibrium solvent relaxation following an instantaneous change in the solute charge distribution and accompanying solvent structure reorganization are also investigated. Both equilibrium and nonequilibrium solvation dynamics are characterized by at least two vastly different time scales--a subpicosecond inertial regime followed by a slow diffusive regime. Solvent regions contributing to the subpicosecond nonequilibrium relaxation are found to vary significantly with initial solvation configurations, especially near the solute. If the solvent density near the solute is sufficiently high at the outset of the relaxation, subpicosecond dynamics are mainly governed by the motions of a few ions close to the solute. By contrast, in the case of a low local density, solvent ions located not only close to but also relatively far from the solute participate in the subpicosecond relaxation. Despite this difference, linear response holds reasonably well in both ionic liquids.     

Analytical theory for permeation of condensable gases in a nanopore

The permeation of a condensable gas mixture in a pressure gradient is examined within a dynamic density functional theory (DDFT). The non-equilibrium density and flux profiles of gas molecules trapped within a nanopore are calculated for each species as a function of feed gas density. Because of important fluid–fluid interaction close to condensation the flux and density gradients are not related by constant transport diffusivities with the thermodynamic correction of uniform density. For long narrow pores the relation of the selectivity to the equilibrium adsorption isotherms is validated. Improved separation is achieved by combining preferential wall interaction and advantageous attraction between gas molecules of different species and examples are discussed. Results from experiments and simulations of permeation in binary mixtures near condensation are still rare and the theory provides a simple basis to study qualitative trends using known parameters.     

Entropy production as a universal functional of reaction rate: chemical networks close to steady states

Entropy production rate (EPR) is the fundamental theoretical quantity in non-equilibrium thermodynamics whereas reaction rate is the primary experimental quantity for a chemical system out-of-equilibrium. In this work, we explore a connection between the above two quantities for general reaction networks. Both cyclic and linear networks of arbitrary dimension are studied, along with a mixed variety. The systems can attain a non-equilibrium steady state (NESS) under chemiostatic condition, which becomes the state of true thermodynamic equilibrium when detailed balance holds. We show that there exists a universal functional relationship of the EPR with reaction rate close to steady states for all the networks considered. Near a NESS, the former varies linearly with the reaction rate. On the other hand, around a true equilibrium, it varies quadratically with the latter. Numerical experiments justify our analytical findings quite transparently.     

Non-equilibrium spin-boson model: counting statistics and the heat exchange fluctuation theorem

We focus on the non-equilibrium two-bath spin-boson model, a toy model for examining quantum thermal transport in many-body open systems. Describing the dynamics within the noninteracting-blip approximation equations, applicable, e.g., in the strong system-bath coupling limit and/or at high temperatures, we derive expressions for the cumulant generating function in both the Markovian and non-Markovian limits by energy-resolving the quantum master equation of the subsystem. For a Markovian bath, we readily demonstrate the validity of a steady-state heat exchange fluctuation theorem. In the non-Markovian limit a "weaker" symmetry relation generally holds, a general outcome of microreversibility. We discuss the reduction of this symmetry relation to the universal steady-state fluctuation theorem. Using the cumulant generating function, an analytic expression for the heat current is obtained. Our results establish the validity of the steady-state heat exchange fluctuation theorem in quantum systems with strong system-bath interactions. From the practical point of view, this study provides tools for exploring transport characteristics of the two-bath spin-boson model, a prototype for a nonlinear thermal conductor.     

非平衡定态化学反应体系对细致平衡偏离的不可逆性之随机热力学量度

在随机描述的层次上,成功地构筑了非平衡定态化学反应体系对细致平衡偏离的不可逆性之随机热力学判据.基于Fokker-Planck方程建立了连续Markov过程系统的随机熵平衡方程,发现在随机态空间中随机熵的时间变率亦可分为源项和流项两部分贡献.对于化学反应体系,作为态空间源项的随机熵产生可作为偏离细致平衡不可逆性的合适度量,此泛函量的零值标志着细致平衡.进一步借助按系统广度量的Ω展开法,通过对随机力及共轭的随机流的分析揭示态空间中的随机熵产生仅源于状态对细致平衡的偏离,并主要来自非Poisson涨落的贡献,因而可作为随机热力学量去判别和量度化学反应体系对细致平衡的偏离.     

SAMPL6 host–guest blind predictions using a non equilibrium alchemical approach

In this paper, we compute, by means of a non equilibrium alchemical technique, called fast switching double annihilation methods (FSDAM), the absolute standard dissociation free energies of the the octa acids host–guest systems in the SAMPL6 challenge initiative. FSDAM is based on the production of canonical configurations of the bound and unbound states via enhanced sampling and on the subsequent generation of hundreds of fast non-equilibrium ligand annihilation trajectories. The annihilation free energies of the ligand when bound to the receptor and in bulk solvent are obtained from the collection of work values using an estimate based on the Crooks theorem for driven non equilibrium processes. The FSDAM blind prediction, relying on the normality assumption for the annihilation work distributions, ranked fairly well among the submitted blind predictions that were not adjusted with a linear corrections obtained from retrospective data on similar host guest systems. Improved results for FSDAM can be obtained by post-processing the work data assuming mixtures of normal components.