Power-law distribution of gene expression fluctuations

Large-scale genomic technologies has opened new possibilities to infer gene regulatory networks from time series data. Here, we investigate the relationship between the dynamic information of gene expression in time series and the underlying network structure. First, our results show that the distribution of gene expression fluctuations (i.e., standard deviation) follows a power-law. This finding indicates that while most genes exhibit a relatively low variation in expression level, a few genes are revealed as highly variable genes. Second, we propose a stochastic model that explains the emergence of this power-law behavior. The model derives a relationship that connects the standard deviation (variance) of each node to its degree. In particular, it allows us to identify a global property of the underlying genetic regulatory network, such as the degree exponent, by only computing dynamic information. This result not only offers an interesting link to explore the topology of real systems without knowing the real structure but also supports earlier findings showing that gene networks may follow a scale-free distribution.     

Emergence of the self-similar property in gene expression dynamics

Many theoretical models have recently been proposed to understand the structure of cellular systems composed of various types of elements (e.g., proteins, metabolites and genes) and their interactions. However, the cell is a highly dynamic system with thousands of functional elements fluctuating across temporal states. Therefore, structural analysis alone is not sufficient to reproduce the cell's observed behavior.In this article, we analyze the gene expression dynamics (i.e., how the amount of mRNA molecules in cell fluctuate in time) by using a new constructive approach, which reveals a symmetry embedded in gene expression fluctuations and characterizes the dynamical equation of gene expression (i.e., a specific stochastic differential equation). First, by using experimental data of human and yeast gene expression time series, we found a symmetry in short-time transition probability from time t to time t+1. We call it self-similarity symmetry (i.e., the gene expression short-time fluctuations contain a repeating pattern of smaller and smaller parts that are like the whole, but different in size). Secondly, we reconstruct the global behavior of the observed distribution of gene expression (i.e., scaling-law) and the local behavior of the power-law tail of this distribution. This approach may represent a step forward toward an integrated image of the basic elements of the whole cell.     

Transition state theory: A generalization to nonequilibrium systems with power-law distributions

Transition state theory (TST) is generalized to nonequilibrium systems with power-law distributions. The stochastic dynamics that gives rise to the power-law distributions for the reaction coordinate and momentum is modeled by Langevin equations and corresponding Fokker-Planck equations. It is considered that a system far away from equilibrium does not have to relax to a thermal equilibrium state with Boltzmann-Gibbs distribution, but asymptotically approaches a nonequilibrium stationary state with a power-law distribution. Thus, we obtain a possible generalization of TST rates to nonequilibrium systems with power-law distributions. Furthermore, we derive the generalized TST rate constants for one-dimensional and n-dimensional Hamiltonian systems away from equilibrium, and obtain a generalized Arrhenius rate for systems with power-law distributions.     

The emergence of scale-free networks with a seceding mechanism

In order to explore further the underlying mechanism of scale-free networks, we study stochastic secession as a mechanism for the creation of complex networks. In this evolution the network growth incorporates the addition of new nodes, the addition of new links between existing nodes, the deleting and rewiring of some existing links, and the stochastic secession of nodes. To random growing networks with preferential attachment, the model yields scale-free behavior for the degree distribution. Furthermore, we obtain an analytical expression of the power-law degree distribution with scaling exponent γ ranging from 1.1 to 9. The analytical expressions are in good agreement with the numerical simulation results.     

Maximum Renyi entropy principle for systems with power-law Hamiltonians

The Renyi distribution ensuring the maximum of Renyi entropy is investigated for a particular case of a power-law Hamiltonian. Both Lagrange parameters alpha and beta can be eliminated. It is found that beta does not depend on a Renyi parameter q and can be expressed in terms of an exponent kappa of the power-law Hamiltonian and an average energy U. The Renyi entropy for the resulting Renyi distribution reaches its maximal value at q=1/(1+kappa) that can be considered as the most probable value of q when we have no additional information on the behavior of the stochastic process. The Renyi distribution for such q becomes a power-law distribution with the exponent -(kappa+1). When q=1/(1+kappa)+epsilon (0相似文献   

Fractionally integrated process for transition economics

We analyze the European transition economics and show that many time series of major indices exhibit (i) power-law correlations in their values, (ii) power-law correlations in their magnitudes and (iii) an asymmetric probability distribution. Applying the phase randomization procedure to these time series, we show that magnitude correlations completely vanish. We propose a stochastic model that can generate time series with features (i), (ii) and (iii), and we show by means of numerical simulations that this model is capable of reproducing these three features found in the empirical data.     

How the Number of Alleles Influences Gene Expression

The higher organisms, eukaryotes, are diploid and most of their genes have two homological copies (alleles). However, the number of alleles in a cell is not constant. In the S phase of the cell cycle all the genome is duplicated and then in the G2 phase and mitosis, which together last for several hours, most of the genes have four copies instead of two. Cancer development is, in many cases, associated with a change in allele number. Several genetic diseases are caused by haploinsufficiency: Lack of one of the alleles or its improper functioning. In the paper we consider the stochastic expression of a gene having a variable number of copies. We applied our previously developed method in which the reaction channels are split into slow (connected with change of gene state) and fast (connected with mRNA/protein synthesis/decay), the later being approximated by deterministic reaction rate equations. As a result we represent gene expression as a piecewise deterministic time-continuous Markov process, which is further related with a system of partial differential hyperbolic equations for probability density functions (pdfs) of protein distribution. The stationary pdfs are calculated analytically for haploidal gene or numerically for diploidal and tetraploidal ones. We distinguished nine classes of simultaneous activation of haploid, diploid and tetraploid genes. This allows for analysis of potential consequences of gene duplication or allele loss. We show that when gene activity is autoregulated by a positive feedback, the change in number of gene alleles may have dramatic consequences for its regulation and may not be compensated by the change of efficiency of mRNA synthesis per allele.     

Production, depreciation and the size distribution of firms

Many empirical researches indicate that firm size distributions in different industries or countries exhibit some similar characters. Among them the fact that many firm size distributions obey power-law especially for the upper end has been mostly discussed. Here we present an agent-based model to describe the evolution of manufacturing firms. Some basic economic behaviors are taken into account, which are production with decreasing marginal returns, preferential allocation of investments, and stochastic depreciation. The model gives a steady size distribution of firms which obey power-law. The effect of parameters on the power exponent is analyzed. The theoretical results are given based on both the Fokker-Planck equation and the Kesten process. They are well consistent with the numerical results.     

A contribution to the systematics of stochastic volatility models

We systematically compare several classes of stochastic volatility models of stock market fluctuations. We show that the long-time return distribution is either Gaussian or develops a power-law tail, while the short-time return distribution has generically a stretched-exponential form, but can also assume an algebraic decay, in the family of models which we call “GARCH” type. The intermediate regime is found in the exponential Ornstein-Uhlenbeck process. We also calculate the decay of the autocorrelation function of volatility.     

Zipf's law in gene expression

Using data from gene expression databases on various organisms and tissues, including yeast, nematodes, human normal and cancer tissues, and embryonic stem cells, we found that the abundances of expressed genes exhibit a power-law distribution with an exponent close to -1; i.e., they obey Zipf's law. Furthermore, by simulations of a simple model with an intracellular reaction network, we found that Zipf's law of chemical abundance is a universal feature of cells where such a network optimizes the efficiency and faithfulness of self-reproduction. These findings provide novel insights into the nature of the organization of reaction dynamics in living cells.     

Quantifying cross-correlations using local and global detrending approaches

In order to quantify the long-range cross-correlations between two time series qualitatively, we introduce a new cross-correlations test Q_CC(m), where m is the number of degrees of freedom. If there are no cross-correlations between two time series, the cross-correlation test agrees well with the χ²(m) distribution. If the cross-correlations test exceeds the critical value of the χ²(m) distribution, then we say that the cross-correlations are significant. We show that if a Fourier phase-randomization procedure is carried out on a power-law cross-correlated time series, the cross-correlations test is substantially reduced compared to the case before Fourier phase randomization. We also study the effect of periodic trends on systems with power-law cross-correlations. We find that periodic trends can severely affect the quantitative analysis of long-range correlations, leading to crossovers and other spurious deviations from power laws, implying both local and global detrending approaches should be applied to properly uncover long-range power-law auto-correlations and cross-correlations in the random part of the underlying stochastic process.     

A constructive approach to gene expression dynamics

Recently, experiments on mRNA abundance (gene expression) have revealed that gene expression shows a stationary organization described by a scale-free distribution. Here we propose a constructive approach to gene expression dynamics which restores the scale-free exponent and describes the intermediate state dynamics. This approach requires only one assumption: Markov property.     

Extreme self-organization in networks constructed from gene expression data

We study networks constructed from gene expression data obtained from many types of cancers. The networks are constructed by connecting vertices that belong to each others' list of K nearest neighbors, with K being an a priori selected non-negative integer. We introduce an order parameter for characterizing the homogeneity of the networks. On minimizing the order parameter with respect to K, degree distribution of the networks shows power-law behavior in the tails with an exponent of unity. Analysis of the eigenvalue spectrum of the networks confirms the presence of the power-law and small-world behavior. We discuss the significance of these findings in the context of evolutionary biological processes.     

Distribution of Chinese names

We present a statistical model for the distribution of Chinese names. Both family names and given names are studied on the same basis. With naive expectation, the distribution of family names can be very different from that of given names. One is affected mostly by genealogy, while the other can be dominated by cultural effects. However, we find that both distributions can be well described by the same model. Various scaling behaviors can be understood as a result of stochastic processes. The exponents of different power-law distributions are controlled by a single parameter. We also comment on the significance of full-name repetition in Chinese population.     

Quasi-stationary simulation of the contact process

We review a recently devised Monte Carlo simulation method for the direct study of quasi-stationary properties of stochastic processes with an absorbing state. The method is used to determine the static correlation function and the interparticle gap-length distribution in the critical one-dimensional contact process. We also find evidence for power-law decay of the interparticle distance distribution in the two-particle subspace.     

Statistics of random lasing modes in weakly scattering systems

We investigated the statistics of random lasing modes in colloidal solutions with local pumping. The ensemble-averaged spectral correlation function of single-shot emission spectra exhibits regular oscillations. The statistical distribution of laser emission intensity follows a power-law decay, in comparison with an exponential decay of the statistical distribution of amplified spontaneous emission (ASE) intensity. The dramatic difference between the statistics of lasing peaks and that of the stochastic ASE spikes illustrates their distinct mechanisms.     

Analytical results for a non-Markovian process of gene expression with positive and negative feedbacks

Gene expression is a very complex process and involves many small biochemical reaction steps, resulting in a non-Markovian discrete stochastic process due to molecular memory between individual reactions. At present, this process is successfully investigated by generalized chemical master equation models. However, these models do not consider the role of feedback networks in gene expression. How the interaction between feedbacks and molecular memory affects gene expression still remains not well understood. Here, we establish generalized chemical master equation models of gene expression with positive and negative feedbacks. Assuming that the process of producing proteins follows an Erlang probability distribution, we obtain the analytical expression for this model in a steady state, as well as the measure of the noise of protein numbers. We further find that molecular memory competes with the positive feedback in suppressing the noise of the protein number. For our model with a negative feedback, molecular memory can strengthen the intensity of suppressing this noise. These interesting results imply that molecular memory are as important as the feedbacks to affect gene expression.     

Effect of Correlations on the Exponents for the Power-Law Distributions in Self-Organized Criticality

The origin of power-law distributions in self-organized criticality is investigated by treating the variation of the number of active sites in the system as a stochastic process. An avalanche is mapped to a first-return random-walk process in a one-dimensional lattice. In order to understand the reason of variant exponents for the power-law distributions in different self-organized critical systems, we introduce the correlations among evolution steps. Power-law distributions of the lifetime and spatial size are found when the random walk is unbiased with equal probability to move in opposite directions. It is found that the longer the correlation length, the smaller values of the exponents for the power-law distributions.