Ming-Chang Huang

Fluctuations in gene regulatory networks as Gaussian colored noise

The study of fluctuations in gene regulatory networks is extended to the case of Gaussian colored noise. First, the solution of the corresponding Langevin equation with colored noise is expressed in terms of an Ito integral. Then, two important lemmas concerning the variance of an Ito integral and the covariance of two Ito integrals are shown. Based on the lemmas, we give the general formulas for the variances and covariance of molecular concentrations for a regulatory network near a stable equilibrium explicitly. Two examples, the gene autoregulatory network and the toggle switch, are presented in details. In general, it is found that the finite correlation time of noise reduces the fluctuations and enhances the correlation between the fluctuations of the molecular components.

A generalized integral fluctuation theorem for general jump processes

Using the Feynman-Kac and Cameron-Martin-Girsanov formulae, we obtain a generalized integral fluctuation theorem (GIFT) for discrete jump processes by constructing a time-invariable inner product. The existing discrete IFTs can be derived as its specific cases. A connection between our approach and the conventional time-reversal method is also established. Unlike the latter approach that has been extensively employed in the existing literature, our approach can naturally bring out the definition of a time reversal of a Markovian stochastic system. Additionally, we find that the robust GIFT usually does not result in a detailed fluctuation theorem.

Distribution and density of the partition function zeros for the diamond-decorated Ising model.

Exact renormalization map of temperature between two successive decorated lattices is given, and the distribution of the partition function zeros in the complex temperature plane is obtained for any decoration level. The rule governing the variation of the distribution pattern as the decoration level changes is given. The densities of the zeros for the first two decoration levels are calculated explicitly, and the qualitative features about the densities of higher decoration levels are given by conjecture. The Julia set associated with the renormalization map is contained in the distribution of the zeros in the limit of infinite decoration level, and the formation of the Julia set in the course of increasing the decoration level is given in terms of the variations of the zero density.

Conformation-networks of two-dimensional lattice homopolymers

The effect of different Monte Carlo move sets on the folding kinetics of lattice polymer chains is studied from the geometry of the conformation-network. The networks have the characteristics of small-world: the local connections are more clustered than that of the corresponding random lattices, and the characteristic path lengths increase logarithmically with the number of nodes. One of the elementary moves, rigid rotation, has drastic effect on the geometric properties of the network. The move increases greatly the connections and reduces significantly the shortest path lengths between conformations. Including rigid rotation to the move set results in the increase of the dimensionality of the conformation space to the value about 4.

Ferromagnetic phase transitions of inhomogeneous systems modelled by square Ising models with diamond-type bond-decorations

The two-dimensional Ising model defined on square lattices with diamond-type bond-decorations is employed to study the nature of the ferromagnetic phase transitions of inhomogeneous systems. The model is studied analytically under the bond-renormalization scheme. For an n-level decorated lattice, the long-range ordering occurs at the critical temperature given by the fitting function (kBTc/J)n=1.6410+(0.6281)exp[−(0.5857)n], and the local ordering inside n-level decorated bonds occurs at the temperature given by the fitting function (kBTm/J)n=1.6410−(0.8063)exp[−(0.7144)n]. The critical amplitude Asing(n) of the logarithmic singularity in specific heat characterizes the width of the critical region, and it varies with the decoration-level n as Asing(n)=(0.2473)exp[−(0.3018)n], obtained by fitting the numerical results. The cross over from a finite-decorated system to an infinite-decorated system is not a smooth continuation. For the case of infinite decorations, the critical specific heat becomes a cusp with the height c(n)=0.639852. The results are compared with those obtained in the cell-decorated Ising model.

Coupling-anisotropy and finite-size effects in interfacial tension of the two-dimensional Ising model

Exact expressions of the Bloch wall free energy are obtained for the Ising model on a rectangular lattice and infinitely long cylinder. The interfacial tension amplitudes are obtained for different coupling and aspect ratios. Finite-size scaling theory is used to analyse the effects of coupling anisotropy and finite size in the interfacial tension.

Master equation approach to folding kinetics of lattice polymers based on conformation networks.

Based on the master equation with the inherent structure of conformation network, the authors investigate some important issues in the folding kinetics of lattice polymers. First, the topologies of conformation networks are discussed. Moreover, a new scheme of implementing Metropolis algorithm, which fulfills the condition of detailed balance, is proposed. Then, upon incorporating this new scheme into the geometric structure of conformation network the authors provide a theorem which can be used to place an upper bound on relaxation time. To effectively identify the kinetic traps of folding, the authors also introduce a new quantity, which is employed from the continuous time Monte Carlo method, called rigidity factor. Throughout the discussions, the authors analyze the results for different move sets to demonstrate the methods and to study the features of the kinetics of folding.

Steady-state work fluctuations for an overdamped Brownian particle driven by external stochastic force

The steady-state work fluctuations for an overdamped Brownian particle driven by an external stochastic force are analyzed by using the Langevin approach. The first two moments of the work distribu...

Critical probability and scaling functions of bond percolation on two-dimensional random lattices

We locate the critical probability of bond percolation on two-dimensional random lattices as . Because of the symmetry with respect to permutation of the two axes for random lattices, we expect that for an aspect ratio of unity and sufficiently large lattices, the probability of horizontal spanning equals the probability of vertical spanning. This is confirmed by our Monte Carlo simulations. We show that the ideas of universal scaling functions and non-universal metric factors can be extended to random lattices by studying the existence probability and the percolation probability P on finite square, planar triangular, and random lattices with periodic boundary conditions using a histogram Monte Carlo method. Our results also indicate that the metric factors may be the same between random lattices and planar triangular lattices provided that the aspect ratios are 1 and .