Eddy current and coupled landscapes for nonadiabatic and nonequilibrium complex system dynamics

Abstract

Physical and biological systems are often involved with coupled processes of different time scales. In the system with electronic and atomic motions, for example, the interplay between the atomic motion along the same energy landscape and the electronic hopping between different landscapes is critical: the system behavior largely depends on whether the intralandscape motion is slower (adiabatic) or faster (nonadiabatic) than the interlandscape hopping. For general nonequilibrium dynamics where Hamiltonian or energy function is unknown a priori, the challenge is how to extend the concepts of the intra-and interlandscape dynamics. In this paper we establish a theoretical framework for describing global nonequilibrium and nonadiabatic complex system dynamics by transforming the coupled landscapes into a single landscape but with additional dimensions. On this single landscape, dynamics is driven by gradient of the potential landscape, which is closely related to the steadystate probability distribution of the enlarged dimensions, and the probability flux, which has a curl nature. Through an example of a self-regulating gene circuit, we show that the curl flux has dramatic effects on gene regulatory dynamics. The curl flux and landscape framework developed here are easy to visualize and can be used to guide further investigation of physical and biological nonequilibrium systems.nonequilibrium landscape | adiabaticity | nonadiabaticity H eterogeneity among coupled processes is a hallmark of complex behaviors of physical and biological systems. A photoexcited molecule, for example, may relax into different low-energy states depending on the difference among time scales of electronic and atomic motions. Molecular motors are fueled by ATP hydrolysis and move into the different structural states, where the heterogeneous distribution of time scales of reactions and structural changes should characterize the motor performance. Dynamical complexity owing to the interplay of heterogeneous processes with multiple time scales can give rise to rich phenomena.In a system having multiple time scales, some dynamical quantities may take discrete values whereas others are continuous. An example of such heterogeneity is found in the electron-transfer reaction, in which the electronic state is discrete and atomic motions are continuous (1). Then, the system can be represented by multiple electronic energy surfaces and the change in atomic positions is motion along individual surfaces. Complex behaviors of the system are explained by the combined process of intrasurface motion along the same and intersurface hopping between different electronic energy surface(s). If the intrasurface motion is slower (faster) than the intersurface hopping, the process is called adiabatic (nonadiabatic). We here extend this notion, the significance of adiabatic and nonadiabatic effects, beyond the Hamiltonian systems to general nonequilibrium problems. Indeed, important complex systems such as reaction networks, gene switches, and molecular motors are co...

Landscape and flux reveal a new global view and physical quantification of mammalian cell cycle

“…Higher Barrier Center leads to more robust cell cycle oscillation. Additionally, to quantify the curl probabilistic flux, we define Flux IntLoop as the flux integration along the loop trajectory of the cell cycle oscillation divided by the loop length in gene expression state space (Flux IntLoop = H L J · dl= H L dl; see SI Appendix for details) (28,29).…”

Stem cell differentiation as a many-body problem

“…2, the transition probability Pðn f ; τ n i ; 0Þ of finding the protein concentrations n f = ðn 1 f ; ⋯; n M f Þ at a final time t = τ from n i = ðn 1 i ; ⋯; n M i Þ at t = 0 can be written as Pðn f ; τ n i ; 0Þ = hn f expðΩτÞjn i i. Next, following Zhang et al (27), using a resolution of the identity generalized from the coherent state basis set, the transition probability can be further represented in a path integral form involving an action that follows from the master equation…”

The Regulatory Landscape of Lineage Differentiation in a Metazoan Embryo

“…Metaphorically, the process is often depicted as Waddington’s landscape with marbles rolling downhill in canalized trajectories (Enver et al, 2009; Zhou and Huang, 2011). Such a view is supported by theoretical analysis of small-scale gene networks (Foster et al, 2009; Zhang et al, 2013) and gene expression profiling of cells (Chang et al, 2008; Huang et al, 2005). However, it remains an open question whether canalization is a general feature of in vivo development, as systematic mapping of the landscape and regulation of lineage differentiation is still technically challenging.…”