Stochastic Analysis of Gene Expression

Abstract

In this chapter, stochasticity in gene expression is investigated using $\Omega$-expansion technique. Two theoretical models are considered here, one concern the stochastic fluctuations in a single-gene network with negative feedback regulation, and the other the additivity of noise propagation in a protein cascade. All of these theoretical analyses may provide a basic framework for understanding stochastic gene expression.

Appendix

Derivation of Eq. 43 . To derive Eq. 43, we first introduce two theorems about the matrix equation in the following.

Theorem 1. The matrix equation ${\mathbf{AX + XB = C}}$ with ${\mathbf{A,B,C}} \in {{\mathbf{C}}^{n \times n}}$ has a solution ${\mathbf{X}},$ if and only if $\left[ {\begin{array}{*{20}{c}} {\mathbf{A}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{B}} \\ \end{array} } \right]$ and $\left[ {\begin{array}{*{20}{c}} {\mathbf{A}} & {\mathbf{C}} \\ {\mathbf{0}} & {\mathbf{B}} \\ \end{array} } \right]$ are similar (30).

Theorem 2. If all the eigenvalues of ${\mathbf{A}}$ and ${\mathbf{B}}$ have nonnegative real parts, then the matrix equation ${\mathbf{AX + XB = C}}$ has a unique solution $X = - \int_0^\infty {{{\hbox{e}}^{{\mathbf{A}}t}}{\mathbf{C}}{{\hbox{e}}^{{\mathbf{B}}t}}} {\hbox{d}}t$ (30).

Notice that in Eqs. 40 and 41, the Jacobian matrix ${\mathbf{A}}$ satisfies Theorem 1. Thus, the matrix equation ${\mathbf{A\Xi }} + {\mathbf{\Xi }}{{\mathbf{A}}^T}{\mathbf{+ B = 0}}$ must have a solution. Notice also that all the possible eigenvalues of ${\mathbf{A}}$ are negative. Thus, from Theorem 2, we have Eq. 43.