Mathematical Model

In the following, we describe in mathemathical terms the actual model problem that we want to solve.

Continuous Problem​

In principle, we want to describe processes which are continuously observable in time and space (e.g. continuum mechanics). Additionally, we are interested in systems of reaction-diffusion equations that are compartmentalized. That is, where each compartment has a different reaction-diffusion process and the interaction between them happens through the intersection or overlap between the two compartments (e.g. a living cell).

Compartments​

We say that each compartment, denoted by $\Omega^k$, is an open, bounded, connected subset of Euclidean space $\mathbb{R}^d$, with Lipschitz boundary $\partial\Omega^k$ and space dimension $d\in\{1,2,3\}$. Moreover, we require that the compartments are the non-overlapping decomposition of an open, bounded, and connected domain $\Omega\subset\mathbb{R}^d$, such that,

where $K$ is the total number of compartments.

Species​

We will distinguish species within the same compartment with a subscript, usually $i$, and species on different compartments with a superscript, usually $k$. For example, $u_i^k$ is the $i$-th species on the $k$-th compartment. Furthermore, we will denote bold letters to refer to all $N^k$ species in the $k$-th compartment, i.e, $\bm{u}^k:=[u^k_1,\dots,u_{N^k}^k]^T$.

Transport​

We then characterize transport with a diffusive flux operator, $\mathcal{D}_i^k$, for the $i$-th species by

where $\mathsf{D}^k_{i\star}$ represents the $i$-th row of the self and cross-diffusion tensor $\mathsf{D}^k$, in other words, $\mathsf{D}^k_{i\star}\nabla\bm{u}^k:=\sum_{i=1}^{N^k}\mathsf{D}^k_{ij}\nabla u^k_j$.

Reaction Network​

A (bio-)chemical reaction is the transformation of one species to another, i.e, $u_i^k\to u_j^k$. There are many natural processes that present a rich reaction network. Particularly, we will represent the deterministic rate of change of $u^k_i$ caused by chemical reactions with the reaction operator $\mathcal{R}_i^k\left(\bm{u}^k\right)$. Such an operator is required to be a Lipschitz function and be mass conservative.

Membrane​

We call membranes all the boundaries of the compartments. In particular, we designate its geometry to be a $(d-1)$-manifold defined with respect to the boundary of the compartments, i.e.,

where $\partial\Omega$ represents the boundaries of the domain $\Omega$.

Transmission Conditions​

The transmission conditions refer to the flux rate at which compartment species are transformed and moved across the membrane. It depends on the concentration of species and the diffusion coefficients at which species can move on the surroundings of the membrane. In our model, we say that this equals to the outer flux $\mathcal{D}_i^k$ of the species $i$ leaving the compartment $k$. It is defined by a general transmission condition, $\mathcal{T}^{kl}_i$, that may take the form of typical Robin boundary conditions as well as complex chemical reaction networks for the different species touching the membrane, i.e.,

where $\mathbf{n}^k$ is the outer normal vector on $\Omega^k$. As is natural, the transmission conditions must be mass conservative.

Strong Formulation​

Joining all definitions from above, we obtain a boundary value problem (BVP) which it reads as follow:

Given an initial condition ${u_{(0)}}_i^k$ and a final time $T$, find $u_i^k$ such that

for every $k,l=1,\ldots,K$, and $i=1,\ldots,N^k$.