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We develop a framework to discuss stability of epigenetic states as first exit problems in dynamical systems with noise. We consider in particular the stability of the lysogenic state of the lambda prophage, which is known to exhibit exceptionally large stability. The formalism defines a quantative measure of robustness of inherited states. In contrast to Kramers well-known problem of escape from a potential well, the stability of inherited states in our formulation is not a numerically trivial problem. The most likely exit path does not go along a steepest decent of a potential -- there is no potential. Instead, such a path can be described as a zero-energy trajectory between two equilibria in an auxiliary classical mechanical system. Finding it is similar to e.g. computing heteroclinic orbits in celestial mechanics. The overall lesson of this study is that an examination of equilibria and their bifurcations with changing parameter values allow us to quantify both the stability and the robustness of particular states of a genetic control system.
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This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of exit operator under Skorokhod topology, which reveals the intrinsic connection between overfitting Dirichlet boundary and fine topology. As an application, we establish the sub and supersolutions for a class of non-stationary HJB (Hamilton-Jacobi-Bellman) equations with fractional Laplacian operator via Feynman-Kac functionals associated to $alpha$-stable processes, which help verify the solvability of the original HJB equation.
In the scenario of the narrow escape problem (NEP) a particle diffuses in a finite container and eventually leaves it through a small escape window in the otherwise impermeable boundary, once it arrives to this window and over-passes an entropic barrier at the entrance to it. This generic problem is mathematically identical to that of a diffusion-mediated reaction with a partially-reactive site on the containers boundary. Considerable knowledge is available on the dependence of the mean first-reaction time (FRT) on the pertinent parameters. We here go a distinct step further and derive the full FRT distribution for the NEP. We demonstrate that typical FRTs may be orders of magnitude shorter than the mean one, thus resulting in a strong defocusing of characteristic temporal scales. We unveil the geometry-control of the typical times, emphasising the role of the initial distance to the target as a decisive parameter. A crucial finding is the further FRT defocusing due to the barrier, necessitating repeated escape or reaction attempts interspersed with bulk excursions. These results add new perspectives and offer a broad comprehension of various features of the by-now classical NEP that are relevant for numerous biological and technological systems.
We study a principal-agent problem with one principal and multiple agents. The principal provides an exit contract which is identical to all agents, then each agent chooses her/his optimal exit time with the given contract. The principal looks for an optimal contract in order to maximize her/his reward value which depends on the agents choices. Under a technical monotone condition, and by using Bank-El Karouis representation of stochastic process, we are able to decouple the two optimization problems, and to reformulate the principals problem into an optimal control problem. The latter is also equivalent to an optimal multiple stopping problem and the existence of the optimal contract is obtained. We then show that the continuous time problem can be approximated by a sequence of discrete time ones, which would induce a natural numerical approximation method. We finally discuss the principal-agent problem if one restricts to the class of all Markovian and/or continuous contracts.
We determine the dimensional dependence of the percolative exponents of the jamming transition via numerical simulations in four and five spatial dimensions. These novel results complement literature ones, and establish jamming as a mixed first-order percolation transition, with critical exponents $beta =0$, $gamma = 2$, $alpha = 0$ and the finite size scaling exponent $ u^* = 2/d$ for values of the spatial dimension $d geq 2$. We argue that the upper critical dimension is $d_u=2$ and the connectedness length exponent is $ u =1$.
We study a stochastic process $X_t$ related to the Bessel and the Rayleigh processes, with various applications in physics, chemistry, biology, economics, finance and other fields. The stochastic differential equation is $dX_t = (nD/X_t) dt + sqrt{2D} dW_t$, where $W_t$ is the Wiener process. Due to the singularity of the drift term for $X_t = 0$, different natures of boundary at the origin arise depending on the real parameter $n$: entrance, exit, and regular. For each of them we calculate analytically and numerically the probability density functions of first-passage times or first-exit times. Nontrivial behaviour is observed in the case of a regular boundary.
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