Diffusion on Surfaces and the Boundary Periodic Unfolding Operator with an Application to Carcinogenesis in Human Cells

Abstract

In the context of periodic homogenization based on the periodic unfolding method, we extend the existing convergence results for the boundary periodic unfolding operator to gradients defined on manifolds. These general results are then used to homogenize a system of five coupled reaction-diffusion equations, three of which are defined on a manifold. The system describes the carcinogenesis of a human cell caused by Benzo-[a]-pyrene molecules. These molecules are activated to carcinogens in a series of chemical reactions at the surface of the endoplasmic reticulum. The diffusion on the endoplasmic reticulum, modeled as a Riemannian manifold, is described by the Laplace-Beltrami operator. The binding process to the surface of the endoplasmic reticulum is modeled in a nonlinear way taking into account the number of free receptors.

Homogenization results for a coupled system of reaction–diffusion equations

“…Proof. We can argue in a similar manner as in [25] (see, also [27], [29], and [30]). Indeed, we can use Schaefer's fixed point theorem.…”

Existence, Uniqueness and Homogenization of Nonlinear Parabolic Problems with Dynamical Boundary Conditions in Perforated Media

“…For the sake of clarity, we split the proof in three parts. Firstly, we pass to the limit, as ε → 0, in (49) in order to get the limit equation satisfied by u. Secondly we identify ξ making use of the solutions of the cell-problems (10), and finally we prove that u is uniquely determined.…”

“…The homogenization of problems which involve the Laplace-Beltrami operator has been considered in recently articles, but using techniques different from those used in the present article. In particular, in [10], Graf and Peter extend the convergence results for the boundary periodic unfolding operator to gradients defined on manifolds. These results are then used to homogenize a system of five coupled reaction-diffusion equations, three of which are defined on a manifold, including diffusion on a biological membrane, modeled as a Riemannian manifold, which is described by the Laplace-Beltrami operator.…”

Existence, uniqueness and concentration for a system of PDEs involving the Laplace–Beltrami operator

“…However, with abuse of notation, we prefer not to invoke w 1 , thus following the same notation as in (5.1)-(5.4). Differentiating (5.17) with respect to t, we obtain 18) where, for t ≥ 1, we have λ 2 j e −2(λ j −λ 1 )t ≤ λ 2 j e −2(λ j −λ 1 ) ≤ λ 2 1 + e −2(1−λ 1 ) =: λ , ∀j ≥ 1 .…”