Mesoscopic Biochemical Basis of Isogenetic Inheritance and Canalization: Stochasticity, Nonlinearity, and Emergent Landscape

Biochemical reaction systems in mesoscopic volume, under sustained environmental chemical gradient(s), can have multiple stochastic attractors. Two distinct mechanisms are known for their origins: ($a$) Stochastic single-molecule events, such as gene expression, with slow gene on-off dynamics; and ($b$) nonlinear networks with feedbacks. These two mechanisms yield different volume dependence for the sojourn time of an attractor. As in the classic Arrhenius theory for temperature dependent transition rates, a landscape perspective provides a natural framework for the system's behavior. However, due to the nonequilibrium nature of the open chemical systems, the landscape, and the attractors it represents, are all themselves {\em emergent properties} of complex, mesoscopic dynamics. In terms of the landscape, we show a generalization of Kramers' approach is possible to provide a rate theory. The emergence of attractors is a form of self-organization in the mesoscopic system; stochastic attractors in biochemical systems such as gene regulation and cellular signaling are naturally inheritable via cell division. Delbr\"{u}ck-Gillespie's mesoscopic reaction system theory, therefore, provides a biochemical basis for spontaneous isogenetic switching and canalization.Comment: 24 pages, 6 figure

Mesoscopic Kinetic Basis of Macroscopic Chemical Thermodynamics: A Mathematical Theory

From a mathematical model that describes a complex chemical kinetic system of $N$ species and $M$ elementrary reactions in a rapidly stirred vessel of size $V$ as a Markov process, we show that a macroscopic chemical thermodynamics emerges as $V\rightarrow\infty$. The theory is applicable to linear and nonlinear reactions, closed systems reaching chemical equilibrium, or open, driven systems approaching to nonequilibrium steady states. A generalized mesoscopic free energy gives rise to a macroscopic chemical energy function \varphi^{ss}(\vx) where \vx=(x_1,\cdots,x_N) are the concentrations of the $N$ chemical species. The macroscopic chemical dynamics \vx(t) satisfies two emergent laws: (1) (\rd/\rd t)\varphi^{ss}[\vx(t)]\le 0, and (2)(\rd/\rd t)\varphi^{ss}[\vx(t)]=\text{cmf}(\vx)-\sigma(\vx) where entropy production rate $\sigma\ge 0$ represents the sink for the chemical energy, and chemical motive force $\text{cmf}\ge 0$ is non-zero if the system is driven under a sustained nonequilibrium chemostat. For systems with detailed balance $\text{cmf}=0$, and if one assumes the law of mass action,\varphi^{ss}(\vx) is precisely the Gibbs' function $\sum_{i=1}^N x_i\big[\mu_i^o+\ln x_i\big]$ for ideal solutions. For a class of kinetic systems called complex balanced, which include many nonlinear systems as well as many simple open, driven chemical systems, the \varphi^{ss}(\vx), with global minimum at \vx^*, has the generic form $\sum_{i=1}^N x_i\big[\ln(x_i/x_i^*)-x_i+x_i^*\big]$,which has been known in chemical kinetic literature.Macroscopic emergent "laws" are independent of the details of the underlying kinetics. This theory provides a concrete example from chemistry showing how a dynamic macroscopic law can emerge from the kinetics at a level below.Comment: 8 page

Landscapes of Non-gradient Dynamics Without Detailed Balance: Stable Limit Cycles and Multiple Attractors

Landscape is one of the key notions in literature on biological processes and physics of complex systems with both deterministic and stochastic dynamics. The large deviation theory (LDT) provides a possible mathematical basis for the scientists' intuition. In terms of Freidlin-Wentzell's LDT, we discuss explicitly two issues in singularly perturbed stationary diffusion processes arisen from nonlinear differential equations: (1) For a process whose corresponding ordinary differential equation has a stable limit cycle, the stationary solution exhibits a clear separation of time scales: an exponential terms and an algebraic prefactor. The large deviation rate function attains its minimum zero on the entire stable limit cycle, while the leading term of the prefactor is inversely proportional to the velocity of the non-uniform periodic oscillation on the cycle. (2) For dynamics with multiple stable fixed points and saddles, there is in general a breakdown of detailed balance among the corresponding attractors. Two landscapes, a local and a global, arise in LDT, and a Markov jumping process with cycle flux emerges in the low-noise limit. A local landscape is pertinent to the transition rates between neighboring stable fixed points; and the global landscape defines a nonequilibrium steady state. There would be nondifferentiable points in the latter for a stationary dynamics with cycle flux. LDT serving as the mathematical foundation for emergent landscapes deserves further investigations.Comment: 4 figur

Compositional coding capsule network with k-means routing for text classification

Text classification is a challenging problem which aims to identify the category of texts. Recently, Capsule Networks (CapsNets) are proposed for image classification. It has been shown that CapsNets have several advantages over Convolutional Neural Networks (CNNs), while, their validity in the domain of text has less been explored. An effective method named deep compositional code learning has been proposed lately. This method can save many parameters about word embeddings without any significant sacrifices in performance. In this paper, we introduce the Compositional Coding (CC) mechanism between capsules, and we propose a new routing algorithm, which is based on k-means clustering theory. Experiments conducted on eight challenging text classification datasets show the proposed method achieves competitive accuracy compared to the state-of-the-art approach with significantly fewer parameters

Analytical Mechanics in Stochastic Dynamics: Most Probable Path, Large-Deviation Rate Function and Hamilton-Jacobi Equation

Analytical (rational) mechanics is the mathematical structure of Newtonian deterministic dynamics developed by D'Alembert, Langrange, Hamilton, Jacobi, and many other luminaries of applied mathematics. Diffusion as a stochastic process of an overdamped individual particle immersed in a fluid, initiated by Einstein, Smoluchowski, Langevin and Wiener, has no momentum since its path is nowhere differentiable. In this exposition, we illustrate how analytical mechanics arises in stochastic dynamics from a randomly perturbed ordinary differential equation $dX_t=b(X_t)dt+\epsilon dW_t$ where $W_t$ is a Brownian motion. In the limit of vanishingly small $\epsilon$, the solution to the stochastic differential equation other than $\dot{x}=b(x)$ are all rare events. However, conditioned on an occurence of such an event, the most probable trajectory of the stochastic motion is the solution to Lagrangian mechanics with $\mathcal{L}=\|\dot{q}-b(q)\|^2/4$ and Hamiltonian equations with $H(p,q)=\|p\|^2+b(q)\cdot p$. Hamiltonian conservation law implies that the most probable trajectory for a "rare" event has a uniform "excess kinetic energy" along its path. Rare events can also be characterized by the principle of large deviations which expresses the probability density function for $X_t$ as $f(x,t)=e^{-u(x,t)/\epsilon}$, where $u(x,t)$ is called a large-deviation rate function which satisfies the corresponding Hamilton-Jacobi equation. An irreversible diffusion process with $\nabla\times b\neq 0$ corresponds to a Newtonian system with a Lorentz force $\ddot{q}=(\nabla\times b)\times \dot{q}+1/2\nabla\|b\|^2$. The connection between stochastic motion and analytical mechanics can be explored in terms of various techniques of applied mathematics, for example, singular perturbations, viscosity solutions, and integrable systems.Comment: 23 pages, 2 figure

Infinite matrix product states, boundary conformal field theory, and the open Haldane-Shastry model

We show that infinite Matrix Product States (MPS) constructed from conformal field theories can describe ground states of one-dimensional critical systems with open boundary conditions. To illustrate this, we consider a simple infinite MPS for a spin-1/2 chain and derive an inhomogeneous open Haldane-Shastry model. For the spin-1/2 open Haldane-Shastry model, we derive an exact expression for the two-point spin correlation function. We also provide an SU($n$) generalization of the open Haldane-Shastry model and determine its twisted Yangian generators responsible for the highly degenerate multiplets in the energy spectrum.Comment: 5+7 pages, 4 figures, published version, a typo in the twisted Yangian generators corrected (thanks to the authors of arXiv:1511.08613 for pointing out this typo