10,635 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
Sensitivity Amplification in the Phosphorylation-Dephosphorylation Cycle: Nonequilibrium steady states, chemical master equation and temporal cooperativity
A new type of cooperativity termed temporal cooperativity [Biophys. Chem. 105
585-593 (2003), Annu. Rev. Phys. Chem. 58 113-142 (2007)], emerges in the
signal transduction module of phosphorylation-dephosphorylation cycle (PdPC).
It utilizes multiple kinetic cycles in time, in contrast to allosteric
cooperativity that utilizes multiple subunits in a protein. In the present
paper, we thoroughly investigate both the deterministic (microscopic) and
stochastic (mesoscopic) models, and focus on the identification of the source
of temporal cooperativity via comparing with allosteric cooperativity.
A thermodynamic analysis confirms again the claim that the chemical
equilibrium state exists if and only if the phosphorylation potential
, in which case the amplification of sensitivity is completely
abolished. Then we provide comprehensive theoretical and numerical analysis
with the first-order and zero-order assumptions in
phosphorylation-dephosphorylation cycle respectively. Furthermore, it is
interestingly found that the underlying mathematics of temporal cooperativity
and allosteric cooperativity are equivalent, and both of them can be expressed
by "dissociation constants", which also characterizes the essential differences
between the simple and ultrasensitive PdPC switches. Nevertheless, the degree
of allosteric cooperativity is restricted by the total number of sites in a
single enzyme molecule which can not be freely regulated, while temporal
cooperativity is only restricted by the total number of molecules of the target
protein which can be regulated in a wide range and gives rise to the
ultrasensitivity phenomenon.Comment: 42 pages, 13 figure
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
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