The Mean And Noise Of Protein Numbers In Stochastic Gene Expression And Their Dynamical Behaviors

It is well known that gene expression plays a central role in the activity of life. Nor-mal life activities, such as stem cell differentiation, embryo development, physical growth,and immune response, depend on correct decoding of genes. Moreover, generation andaggravation of many diseases are caused by gene mutation or abnormal alteration of geneexpression. Therefore, the study of gene expression has been a core subject in molecularbiology, as well as an important research branch in the current study of life science.Gene expression transfers genes stored in DNA molecules to proteins. It consists oftwo main processes which are transcription and translation. In the first step, accordingto the principle of complementary base pairing, genes are transcribed to message RNA(mRNA) molecules. In the second step, mRNA molecules are translated into proteins. Itwas widely accepted in biology that genes were expressed in a deterministic and contin-uous way. In recent years，thanks to the development of computer image processing andfluorescent protein technique, biologists have shown that genes are expressed randomlyand discontinuously. In a single cell, the discontinuity of gene expression is reflected byrandom transition between gene off and gene on states. The stochasticity of gene expres-sion is manifested in variations of the durations in gene on or gene off states. In orderto measure quantitatively the stochasticity of gene expression, biologists have introduceda variety of statistical concepts, such as noise and noise strength. Many scientists havestudied the noise and noise strength in gene expression by using the two-state model. Inthese studies, the sojourn times in the gene on and gene off states are often assumed tobe independently and exponentially distributed. To better describe how the environmentalsignals contribute to the stochasticity of gene transcription, the three-state model of genetranscription has been recently proposed and broadly studied, see for example[17,80–82,85,86].In this thesis, we calculate and analyze the mean, the noise, and the noise strengthof stochastic gene expression by employing the three-state model, and discuss how thesevalues are regulated by gene transcription, random birth and death processes of mRNAand protein. The main innovations of this thesis are as follows:1. It is the first time in the literature to obtain the exact formula of the mean of protein numbers by using the three-state model, and show that the mean is of oscillatory behavior.As the two-state model always produces monotone growing curve for the mean expressionlevel, it is also the first time to provide the simulation of oscillatory phenomena observedin eukaryotic cells by the oscillatory dynamics exhibited by the three-state model.2. We show that, given the same average gene off duration, the protein noise in-creases with the difference|κ-λ|under the equilibrium state (where κ and λ denote theinduction strength and the activation strength of the environmental signals respectively).The noise is minimal when κ=λ, and is maximal when either κ=∞orλ=∞,for which the three-state model is reduced to the two-state model. For the same averagegene off duration, the three-state model always produces less noises than the two-statemodel does. We conjecture that, to avoid higher expression noise, which may be deleteri-ous to cells, natural selection may favor two or more rate-limiting steps to complete onetranscription cycle in a given time.3. Our analysis also gives a decomposition of the protein noise under the equilib-rium state, the far-reaching decomposition helps us explain why protein noise has beenrepeatedly found to be greater than or equal to the inverse of the mean protein levels inNewman el al.(2006，Nature), Yu el al.(2006，Science), Raj el al.(2010，Nature) andTaniguchi el al.(2010，Science). Also, it explains the striking findings of Taniguchi elal.(2010) that the numbers of protein and mRNA for any given gene within the individualcells are uncorrelated.4. We show that the interesting decomposition of the protein noise is not alwaysvalid under the nonequilibrium state. Our results indicate that the protein noise of thethree-state model is capable of producing non-monotonic dynamical complexity.This thesis is organized as follows: In Chapter1, we introduce the biological back-ground of stochastic gene expression and the three-state model. In Chapter2, we calculatethe exact formula of the average protein level from the three-state model, and then showthat it oscillates in some cases. This indicates, in principle, that the three-state modelis capable of producing non-monotonic dynamical complexity. In Chapter3, we obtainthe exact formulae of the noise and noise strength from the three-state model. We ex-plain many important experimental phenomena by analyzing our main results. In Chapter4, the dynamical behaviors of the noise and noise strength are discussed. We find that,comparing with the equilibrium state, the dynamical behavior is more complex.