Landscape and flux decomposition for exploring global natures of non-equilibrium dynamical systems under intrinsic statistical fluctuations Chunhe Li a,b , Erkang Wang a,⇑ , Jin Wang a,c,⇑ a State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin 130022, PR China b Graduate School of the Chinese Academy of Sciences, Beijing 100039, PR China c Department of Chemistry and Physics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA article info Article history: Received 18 December 2010 In final form 8 February 2011 Available online 12 February 2011 abstract We develop a theoretical framework for exploring global natures of non-equilibrium dynamical systems under intrinsic statistical fluctuations. We found the underlying driving force can be decomposed into three terms: gradient of potential landscape, curl probability flux, inhomogeneity of diffusion. We studied a limit cycle oscillation model and found that the potential landscape has a Mexican hat close ring valley shape and attracts the system down to the ring while the curl probability flux on the ring drives the coherent oscillation. The barrier heights characterizing the landscape topography provide a quantitative measure for global stability of non-equilibrium dynamical systems. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction The global nature of non-equilibrium dynamical systems is still a challenge to explore. The time evolution of dynamical system can be described in a either deterministic or stochastic way under external fluctuations from highly dynamical and inhomogeneous environments or the intrinsic fluctuations from finite number of the molecules or species within [1–10]. Many studies have been concentrated on the local dynamics. The corresponding stability analysis are often focused on linear regime near the fixed points or attractors. The global nature of the system is hard to see from the local analysis. Instead of following the local trajectories, the global informa- tion can be obtained from the probability distribution. Under external fluctuations, the probability evolution of system can be determined by probabilistic diffusion equation [11,12]. Several important biological systems such as budding yeast cell cycle, cir- cadian clock, cell fate decision, etc. have been studied [11,13–15] for understanding the global stability and function. However, the global natures of the non-equilibrium dynamical systems under intrinsic statistical fluctuations are still challenging to explore. The challenge is how to quantify the global stability and robustness of the non-equilibrium dynamical systems. Intrin- sic fluctuations are important for many biological systems in mesoscopic scales such as gene regulatory and protein networks. In this work, we will uncover the underlying landscape and prob- ability flux for the global natures of the non-equilibrium dynami- cal systems under the intrinsic fluctuations. We found the driving force of the non-equilibrium dynamical systems F can be decomposed into three terms: the gradient of the probabilistic potential landscape, the probability curl flux and the gradient of diffusion coefficient matrix D. The negative gradient of potential landscape drives the system down to the oscillation attractor, the curl flux drives the system spiraling around. The third term related to the gradient of diffusion coefficient matrix D pro- vides a correction coming from the inhomogeneity of the diffusion through molecular or species number dependency of diffusion coefficient matrix D under intrinsic fluctuations. Therefore, in the non-equilibrium dynamical systems, the potential landscape and the associated probabilistic flux together characterize the global properties and the dynamics of the system. This is different from the case in the equilibrium system where the force F can be repre- sented purely as a gradient of a potential, and the flux flow is zero. We illustrated our idea by using a simple two-variable limit cycle oscillation model [16], often called Selkov model. By means of stochastic simulations [17,18], we investigated the long time steady state properties and furthermore obtain the steady state probability landscape and flux. We found that the landscape has a Mexican hat close ring valley shape and attracts the system down to the ring while the curl flux on the ring drives the coherent oscil- lation. We also calculated the barrier heights at different internal fluctuations characterized by the molecule (or population) num- bers, which provide a quantitative measure for the global stability of the non-equilibrium dynamical system. Furthermore, the 0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2011.02.020 ⇑ Corresponding authors at: State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changc- hun, Jilin 130022, PR China. E-mail addresses: lihe@ciac.jl.cn (C. Li), ekwang@ciac.jl.cn (E. Wang), jin. wang.1@stonybrook.edu (J. Wang). Chemical Physics Letters 505 (2011) 75–80 Contents lists available at ScienceDirect Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett