Stochastic Fluctuations In Gene Expression

Gene expression is an inherently stochastic process since genes are activated andinactivated by random rare enzyme and many mRNA and protein are present in lownumber per cell. Stochastic noise in gene expression arises as a result of species insmall copy number undergoing transitions between discrete chemical states. Howto measure stochastic fluctuation in gene expression have drawn a great attentionin mathematical biology field. Many genes are transcribed and translated by manyother regulated factors, such as di?erent kinds of enzyme in chemical reaction en-vironment. The issue that how gene expression network regulate and keep all thegene expressing at some range under changeable environment has become the mostimportant in this field. In recent years, more and more mathematical models withdifferent methods and simulation techniques are set up to describe the regulatedmechanism of gene regulatory network and study the underneath work principle ofit. Here the noise in gene network is investigated using theΩ-expansion techniques,Monte Carlo simulation method and matrix theory.In chapter two, we show that the linear noise approximation implies an in-variant relationship between the normalized variances and normalized covariance insteady-state statistics of one single gene expression system with feedback regula-tion. This invariant relationship provides an exactly statistical interpretation forwhy the stochastic noise in gene expression should be measured by the normalizedvarianceσ2/ n 2 (i.e. the variance is normalized by the squared average) ratherthan Fano factor. The nature of the normalized variance reveals the basic relation-ship between the stochasticity and system size in gene expression. The linear noiseapproximation implies also that for both mRNA and protein, the total noise can bedecomposed into two basic components, one concerns the contribution of averagenumber of molecules (intrinsic noise), and other the contribution of interactions be- tween mRNA and protein (extrinsic noise). For the situation with linear feedback,our results clearly show that for two genes with the same average number of proteinmolecules, the gene with negative feedback will have a small protein noise, i.e., thenegative feedback will reduce the protein noise. For the effect of the burst size onthe protein noise, we show also that the protein intrinsic noise will decrease with theincrease of the burst size, but the protein extrinsic noise is independent of the burstsize. After theoretical analysis, we also use Monte Carlo method do correspond-ing simulations since all the biochemical process can be seen as a birth-and-deathprocess. The algorithm of Monte Carlo simulations is from Gillespie for stochasticcoupled chemical reactions in 1977. The results of the corresponding simulationsupport the above conclusions.In chapter three, stochastic fluctuations in a protein synthetic cascade is inves-tigated using standardΩ-expansion technique and matrix theory. A protein cascadeis a ordered sequence of proteins which the transcriptional level of mRNA of ithprotein is only regulated by the (i - 1)th protein and the transcriptional level ofthe first protein is assumed to be a constant. Here,we focus on two aspects, one isthe sensitivity of the steady-state, the other the noise (measured by the normalizedvariance in chapter 2) propagation in protein cascade. For the steady-state sensi-tivity, we focus mainly our attention on the properties of the sequence of proteinmolecule numbers. We show the conditions that result in the ultrasensitive"all-or-none"behavior, it depends on the threshold behavior of input protein concentrationwhich provide a standard switch structure, and the proportion between the productof burst size and regulation strength and decay rates. And for the noise propagation,the noises of proteins are also measured by the normalized variance, we show clearlythat (i) for any one given protein species in this cascade, the contributions of ?uc-tuations in upstream proteins to its noise should be additive, this reveals how thenoises are propagated in a protein cascade; (ii) the output noise levels can vary asa function of the input concentrations and cascade length,and we get the material function expression. We use the normalized covariances between the first and theith protein to measure how the statistical correlations between them. We show thatfor large cascade stage, the variations in input and output proteins may be indepen-dent of each other under some simple conditions. This property exactly matches thenature of the protein molecule number sequence. The results of the correspondingMonte Carlo simulation also support the above conclusions. Our results provide apossible theoretical explanation for the previous experimental studies.