Yu Qiwei, Tu Yuhai
Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, New Jersey 08544, USA.
IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA.
Phys Rev E. 2022 Apr;105(4-1):044140. doi: 10.1103/PhysRevE.105.044140.
Nonequilibrium reaction networks (NRNs) underlie most biological functions. Despite their diverse dynamic properties, NRNs share the signature characteristics of persistent probability fluxes and continuous energy dissipation even in the steady state. Dynamics of NRNs can be described at different coarse-grained levels. Our previous work showed that the apparent energy dissipation rate at a coarse-grained level follows an inverse power-law dependence on the scale of coarse-graining. The scaling exponent is determined by the network structure and correlation of stationary probability fluxes. However, it remains unclear whether and how the (renormalized) flux correlation varies with coarse-graining. Following Kadanoff's real space renormalization group (RG) approach for critical phenomena, we address this question by developing a state-space renormalization group theory for NRNs, which leads to an iterative RG equation for the flux correlation function. In square and hypercubic lattices, we solve the RG equation exactly and find two types of fixed point solutions. There is a family of nontrivial fixed points where the correlation exhibits power-law decay, characterized by a power exponent that can take any value within a continuous range. There is also a trivial fixed point where the correlation vanishes beyond the nearest neighbors. The power-law fixed point is stable if and only if the power exponent is less than the lattice dimension n. Consequently, the correlation function converges to the power-law fixed point only when the correlation in the fine-grained network decays slower than r^{-n} and to the trivial fixed point otherwise. If the flux correlation in the fine-grained network contains multiple stable solutions with different exponents, the RG iteration dynamics select the fixed point solution with the smallest exponent. The analytical results are supported by numerical simulations. We also discuss a possible connection between the RG flows of flux correlation with those of the Kosterlitz-Thouless transition.
非平衡反应网络（NRNs）是大多数生物学功能的基础。尽管它们具有多样的动态特性，但即使在稳态下，NRNs也具有持续概率流和连续能量耗散的标志性特征。NRNs的动力学可以在不同的粗粒化水平上进行描述。我们之前的工作表明，粗粒化水平下的表观能量耗散率遵循与粗粒化尺度的幂律反比关系。标度指数由网络结构和平稳概率流的相关性决定。然而，目前尚不清楚（重整化后的）流相关性是否以及如何随粗粒化而变化。遵循Kadanoff针对临界现象的实空间重整化群（RG）方法，我们通过为NRNs发展一种状态空间重整化群理论来解决这个问题，这导致了一个关于流相关函数的迭代RG方程。在正方形和超立方晶格中，我们精确求解了RG方程，发现了两种类型的不动点解。存在一族非平凡不动点，其相关性呈现幂律衰减，其特征是幂指数可以在一个连续范围内取任意值。还存在一个平凡不动点，在该点处相关性在最近邻之外消失。当且仅当幂指数小于晶格维度n时，幂律不动点是稳定的。因此，只有当细粒化网络中的相关性衰减比r^{-n}慢时，相关函数才会收敛到幂律不动点，否则收敛到平凡不动点。如果细粒化网络中的流相关性包含多个具有不同指数的稳定解，RG迭代动力学将选择指数最小的不动点解。数值模拟支持了这些分析结果。我们还讨论了流相关性的RG流与Kosterlitz-Thouless转变的RG流之间可能的联系。
