Revisit of Macroscopic Dynamics for Some Non-equilibrium Chemical Reactions from a Hamiltonian Viewpoint
Revisit of Macroscopic Dynamics for Some Non-equilibrium Chemical Reactions from a Hamiltonian Viewpoint
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DOI：
10.1007/s10955-022-02985-5
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发表时间：
2021-08
影响因子：
1.6
通讯作者：
Yuan Gao;Jian‐Guo Liu
中科院分区：
文献类型：
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作者：
Yuan Gao;Jian‐Guo Liu
活细胞中的大多数生物化学反应是通过恒化器与环境相互作用以交换能量和物质的开放系统。在介观尺度下，这些生化反应中的每种物种的数量可以用随机时变泊松过程来模拟。为了表征大数极限下的宏观行为，路径空间中的大数定律确定了描述物种浓度动力学的平均场极限非线性反应速率方程，而化学主方程的WKB展开产生了Hamilton-Jacobi方程，相应的Hamilton-Jacobi方程的Legendre变换给出了大偏差原理中良好的速率函数（作用泛函）。本文利用Hamilton-Jacobi方程的定态解，将一般宏观反应速率方程分解为守恒部分和耗散部分。该稳态解用于确定一般化学反应的能量景观和热力学，特别是在非平衡稳态下保持正熵产生率。证明了在介观和宏观两个层次上的能量耗散规律，并给出了从介观到宏观的通道。在大数极限下，凸介观相对熵泛函呈现出非凸的能量景观，体现了非平衡态的特征。该定常解的存在性由定常Hamilton-Jacobi方程的弱KAM解在不定时间范围内的最优控制表示来保证。此外，我们使用对称Hamilton量来研究一类非平衡酶反应，由于通量分组简并，导致非凸能量景观，并将保守-耗散分解降低为Onsager型强梯度流。这种对称哈密顿量意味着多个稳态（生化反应中的罕见事件）之间的过渡路径是一个修改的时间反转的最小作用路径与相关的路径亲和力和能量障碍。我们说明了这一想法，通过一个催化反应，并计算通过其能量景观连接两个稳态的过渡路径的能量势垒。
Most biochemical reactions in living cells are open systems interacting with environment through chemostats to exchange both energy and materials. At a mesoscopic scale, the number of each species in those biochemical reactions can be modeled by a random time-changed Poisson processes. To characterize macroscopic behaviors in the large number limit, the law of large numbers in the path space determines a mean-field limit nonlinear reaction rate equation describing the dynamics of the concentration of species, while the WKB expansion for the chemical master equation yields a Hamilton–Jacobi equation and the Legendre transform of the corresponding Hamiltonian gives the good rate function (action functional) in the large deviation principle. In this paper, we decompose a general macroscopic reaction rate equation into a conservative part and a dissipative part in terms of the stationary solution to the Hamilton–Jacobi equation. This stationary solution is used to determine the energy landscape and thermodynamics for general chemical reactions, which particularly maintains a positive entropy production rate at a non-equilibrium steady state. The associated energy dissipation law at both the mesoscopic and macroscopic levels is proved together with a passage from the mesoscopic to macroscopic one. A non-convex energy landscape emerges from the convex mesoscopic relative entropy functional in the large number limit, which picks up the non-equilibrium features. The existence of this stationary solution is ensured by the optimal control representation at an undetermined time horizon for the weak KAM solution to the stationary Hamilton–Jacobi equation. Furthermore, we use a symmetric Hamiltonian to study a class of non-equilibrium enzyme reactions, which leads to nonconvex energy landscape due to flux grouping degeneracy and reduces the conservative–dissipative decomposition to an Onsager-type strong gradient flow. This symmetric Hamiltonian implies that the transition paths between multiple steady states (rare events in biochemical reactions) is a modified time reversed least action path with associated path affinities and energy barriers. We illustrate this idea through a bistable catalysis reaction and compute the energy barrier for the transition path connecting two steady states via its energy landscape.