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Degenerate Convergence of Semigroups Related to a Model of Stochastic Gene Expression

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(1) Institute of Mathematics of the Polish Academy of Sciences, Katowice branch, Bankowa 14, 40-007 Katowice, Poland on leave from Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, Nadbystrzycka 38A, 20-618, Lublin, Poland

Abstract:

We consider the Cauchy problem related to the system of equations:

$\frac {\partial f(t,x,y)}{\partial t} = -x\frac {\partial f(t,x,y)}{\partial x}+ r (x-y) \frac {\partial f(t,x,y) }{\partial y} - a\alpha(x,y) f(t,x,y)+ a\alpha(x,y) g(t,x,y),\qquad (1)$

$\frac {\partial g (t,x,y)}{\partial t} = (1-x) \frac {\partial g(t,x,y)}{\partial x} + r (x-y)\frac {\partial g(t,x,y)}{\partial y} + b\beta (x,y) f(t,x,y) -b\beta (x,y) g(t,x,y),$

$x,y\in 0,1],$ where positive constants a and b, and continuous, non-negative functions α and β on 0,1]² are given. This system describes the evolution in time of the distribution of messenger RNA (x) and protein (y) levels in the model of stochastic gene expression introduced recently by Lipniacki et al. 25]. To be more exact, (1) is the Kolmogorov backward equation for the involved Markov process. We give a semigroup-theoretic proof of two hypotheses stated in 25]. The first of them says that if a and b tend to infinity in such a way that a/b tends to a constant c, the solutions to (1) tend to these of

$\frac {\partial f(t,x,y)}{\partial t} = \left (\frac {c \alpha}{c\alpha+\beta}-x \right ) \frac{\partial f (t,x,y)}{\partial x} + r (x-y) \frac {\partial f(t,x,y) }{\partial y} \quad x,y\in 0,1].$

The second hypothesis states that if r tends to infinity, the solutions converge to these of

$\frac {\partial f(t,x)}{\partial t} = -x\frac{\partial f (t,x)}{\partial x} - a\alpha _x f(t,x)+ a\alpha_ x g(t,x),$

$\frac {\partial g (t,x)}{\partial t} = (1-x) \frac {\partial g(t,x)}{\partial x} + b\beta_x f(t,x)-b\beta_x g(t,x), \quad x\in 0,1],$

where $\alpha_x (x) = \alpha (x,x)$ and $\beta_x (x) = \beta (x,x).$ The problem of convergence is seen as that of degenerate convergence of semigroups of linear operators, a notion introduced in our paper 3], where a family of equibounded semigroups defined in a Banach space ${\Bbb B}$ converges to a semigroup that acts only on a subspace ${\Bbb B}_0$ of ${\Bbb B}$ .

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