Escape through a small opening: receptor trafficking in a synaptic membrane. (English) Zbl 1087.82018

The following problem \[ \begin{aligned} D\Delta u(x)=-1 &\quad\text{for }x\in\Omega, \tag{1}\\ \frac{\partial u (x)}{\partial n}=0 &\quad\text{for }x\in\partial\Omega -\partial\Omega_a, \tag{2}\\ u(x)=0 &\quad\text{for }x\in\partial\Omega_a \tag{3} \end{aligned} \] is related to the biological processes mentioned in the title. Here \(\Omega\) is the confinement domain for the receptor (performing a Brownian motion). The boundary \(\partial \Omega\) is impervious to it, except for the portion \(\partial \Omega_a\) which is instead absorbing. The relevant biological quantity is the confinement type, i.e. the average time spent by a receptor in \(\Omega\). The peculiarity of problem (1)–(3) in the context is that \(\partial\Omega_a\) is ”small”. The problem is analyzed in various situations, including the one in which the receptor trajectory can be terminated at an anchoring site. Interesting conclusions are reached.


82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
92C40 Biochemistry, molecular biology
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