14,767 research outputs found
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: () Stochastic
single-molecule events, such as gene expression, with slow gene on-off
dynamics; and () 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
species and elementrary reactions in a rapidly stirred vessel of size
as a Markov process, we show that a macroscopic chemical thermodynamics
emerges as . 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
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 represents the sink for the chemical energy, and chemical
motive force is non-zero if the system is driven under a
sustained nonequilibrium chemostat. For systems with detailed balance
, and if one assumes the law of mass action,\varphi^{ss}(\vx)
is precisely the Gibbs' function
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 ,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 where is a Brownian
motion. In the limit of vanishingly small , the solution to the
stochastic differential equation other than 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 and Hamiltonian equations with
. 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
as , where is called a large-deviation
rate function which satisfies the corresponding Hamilton-Jacobi equation. An
irreversible diffusion process with corresponds to a
Newtonian system with a Lorentz force . 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() 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
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