#### Appendix G.1. Mathematical Definitions of div_{T} and ∇_{T} and Biological Meanings

For the sake of completeness, we recall here the definitions of gradient and divergence operators, and we discuss what they represent in the biological context presented in our work.

Let

$\mathsf{\Omega}\subset \mathcal{S}$ be an open set of the three-dimensional Euclidean space,

$\mathcal{S}$, and let

$f:\mathsf{\Omega}\to \mathbb{R}$ be a scalar field defined over

$\mathsf{\Omega}$. If all the partial derivatives of

f exist everywhere in

$\mathsf{\Omega}$, the gradient (“grad” or ∇) of

f at a given point

$x\in \mathsf{\Omega}$ is defined as [

65,

66].

where

${\left\{{\mathit{e}}_{k}\right\}}_{k=1}^{3}$ is a Cartesian triad of basis vectors, and the triple of real numbers

$({x}_{1},{x}_{2},{x}_{3})$ denotes the coordinates of

$x\in \mathsf{\Omega}$.

Moreover, given a vector field

$\mathit{u}$ over

$\mathsf{\Omega}\subset {\mathbb{R}}^{3}$, and adopting the decomposition

the triple of real-valued functions

$({u}_{1},{u}_{2},{u}_{3})$ represents the components of

$\mathit{u}$ in the Cartesian vector basis

${\left\{{\mathit{e}}_{k}\right\}}_{k=1}^{3}$. If each function

${u}_{k}$, for

$k=1,2,3$, is differentiable, the “gradient of

$\mathit{u}$” at

$x\in \mathsf{\Omega}$ is defined by

where the symbol “⊗” denotes the dyadic product between vectors. We recall that, for any pair of vectors

$\mathit{v}$ and

$\mathit{w}$, the dyadic product

$\mathit{v}\otimes \mathit{w}$ is the second-order tensor with components

${[\mathit{v}\otimes \mathit{w}]}_{mn}={v}_{m}{w}_{n}$. Finally, the divergence of

$\mathit{u}$ is obtained by contracting the two indices of

$\mathrm{grad}\mathit{u}$ [

65], i.e.,

Note that, if $\mathrm{grad}\phantom{\rule{0.166667em}{0ex}}\mathit{u}\left(x\right)$ exists, $\mathrm{div}\phantom{\rule{0.166667em}{0ex}}\mathit{u}\left(x\right)$ can be computed as the trace of the matrix representation of $\mathrm{grad}\phantom{\rule{0.166667em}{0ex}}\mathit{u}\left(x\right)$.

In our work, we make frequent use of the gradient and divergence operators, since both operators feature in the diffusion–reaction equations on which our model is based. In fact, each of such equations represents the mass balance law for a given species of the model, and can be written as

where

c is the concentration of the considered species,

$\mathit{q}$ is the mass flux vector describing the motion of the species, and

r is the corresponding source or sink term.

Equation (A5) is the local form of the mass balance law of mass, and descends from the integral form

In Equation (A6), $P\subseteq \mathsf{\Omega}$ is a generic part of $\mathsf{\Omega}$, $\partial P$ is its boundary of P, $\mathit{n}$ is the field of unit vectors normal to $\partial P$, and $\mathrm{d}v$ and $\mathrm{d}a$ denote volume and surface measures, respectively. The physical (and biological) meaning of Equation (A6) is that the mass of the species contained in P with the concentration c varies is time in response to the flux of the species through $\partial P$, and to its production or depletion in P. The flux of the species through $\partial P$ is given by the component of $\mathit{q}$ along $\mathit{n}$, obtained by the scalar product $\mathit{q}\xb7\mathit{n}$. By prolonging $\mathit{q}$ to the interior of P, and assuming differentiability, Gauss’ Theorem can be invoked to recast Equation (A6) as

Assuming that the part P of $\mathsf{\Omega}$ is fixed, the time derivative of the volume integral of c can be rewritten as the volume integral of the partial time derivative of c, thereby yielding

Furthermore, requiring that Equation (A8) applies for any choice of P allows rewriting Equation (A8) as reported in Equation (A5).

In summary, the use of the divergence of the mass flux vector, i.e., $\mathrm{div}\mathit{q}$, allows transforming the surface integral of $\mathit{q}\xb7\mathit{n}$ featuring in Equation (A6) into the volume integral of $\mathrm{div}\mathit{q}$ featuring in Equations (A7) and (A8).

Equation (A5), or the integral form in Equation (A6), involves two scalar unknowns, i.e., c and r, and the vectorial unknown $\mathit{q}$, which introduces the triple of unknown scalar components $({q}_{1},{q}_{2},{q}_{3})$. Hence, to close the mathematical problem associated with Equation (A5), it is necessary to provide constitutive relationships connecting $\mathit{q}$ and r with c, or prescribing them from the outset. Here, we focus on the relationship that, on physical grounds, has to be supplied in order to relate $\mathit{q}$ to c.

If we assume that the motion of the considered species occurs by diffusion, we need to describe the spontaneous and irreversible process by which the motion of the species is driven by the spatial non-uniformity of its distribution. In other words, if one prepares an experiment in which the concentration of the species, c, varies in space, a motion of the species can be observed that tends to restore a space-independent concentration. For this reason, and since the descriptor of the spatial variability of c is the concentration gradient, $\nabla c$, we identify $\nabla c$ as the trigger of the species’ motion. If the medium in which the species evolves is isotropic, the easiest way to relate $\mathit{q}$ to $\nabla c$ is to assume a linear relationship of the type $\mathit{q}=-D\nabla c$, where D is referred to as coefficient of diffusion. We remark that many other laws of this kind are to be found in physics, biology and engineering: Fick’s law of molecular diffusion, Fourier’s law of heat conduction, Darcy’s law of fluid filtration through porous media, and Ohm’s law of electric conduction are perhaps the most prominent examples. The expression of $\mathit{q}$ used in our work, i.e., $\mathit{q}=-D\nabla c$, is in fact a particular case of Fick’s law of diffusion. By substituting it into Equation (A5), we find

In Equation (A9), the reaction term, r, can be either prescribed from the outset or related to c through a constitutive law of the type $r=\widehat{r}\left(c\right)$.

In our work, to describe the diffusion of a given substance on the (open) surfaces that represent our computational domains, we use the tangential gradient and the tangential divergence operators. For the scalar field

c, the tangential gradient is denoted by

${\nabla}_{T}c$ and is defined as

where

$\mathit{P}=\mathit{I}-\mathit{n}\otimes \mathit{n}$ is said to the

projection operator,

$\mathit{I}$ is the identity tensor, and

$\mathit{n}$ is field of unit vectors normal to the surface

$\mathcal{S}$ over which the diffusion takes place. For Equation (A10) to make sense,

c has to be restricted to

$\mathcal{S}$. Hence,

c has to be understood as

${c}_{|\mathcal{S}}$ in Equation (A10). Accordingly, the tangential mass flux vector associated with

${c}_{|\mathcal{S}}$ is given by

${\mathit{q}}^{\sigma}=-D{\nabla}_{T}c$.

To write the mass balance law on

$\mathcal{S}$, one has to take the tangential divergence of the mass flux vector. In general, for a vector field

$\mathit{\psi}$ defined on

$\mathcal{S}$, but that is not necessarily tangent to

$\mathcal{S}$, the tangential divergence is defined as [

67]

where

$\mathrm{tr}(\u2022)$ is the trace operator,

${\mathit{\psi}}^{\sigma}=\mathit{P}\mathit{\psi}$ is the projection of

$\mathit{\psi}$ onto

$\mathcal{S}$,

$\kappa $ is the mean curvature of

$\mathcal{S}$, and

${\psi}_{n}$ is the component of

$\mathit{\psi}$ normal to

$\mathcal{S}$. We emphasize that, for a tangential vector field, as is the case for

${\mathit{q}}^{\sigma}$, it holds that

$\mathit{P}{\mathit{q}}^{\sigma}={\mathit{q}}^{\sigma}$, and Equation (A11) becomes independent of the mean curvature, thereby reducing to

Accordingly, the diffusion–reaction equation (Equation (A9)) becomes

and is the local form of the integral equation

where

$\mathcal{A}\subseteq \mathcal{S}$ is a subsurface of

$\mathcal{S}$,

$\partial \mathcal{A}$ is a closed and regular line, contained in

$\mathcal{S}$ and defining the boundary of

$\mathcal{A}$,

$\nu $ is a field of unit vectors normal to

$\partial \mathcal{A}$ and tangent to

$\mathcal{A}$, and

$\mathrm{d}s$ is a line measure.

The biological meaning of the use of the tangential space operators is that they allow for restricting movement of components to a 2D surface embedded within 3D, e.g., to restrict the movement of biological agents such as proteins to the ER surface.

#### Appendix G.2. Biological Motivations to Use the Operators div_{T} and ∇_{T}

In summary, the divergence of the mass flux vector, $\mathrm{div}\mathit{q}$, descends from Gauss’ Theorem, and permits expressing the species’ mass balance law in local form. However, as shown in Equation (A6), one can dispense with $\mathrm{div}\mathit{q}$, if the mass balance law is formulated in integral form. This occurs, for instance, if a finite volume method is applied to solve Equation (A6), as is the case in our simulations. Indeed, when the Finite Element Method is employed, the part P of $\mathsf{\Omega}$ is referred to as finite volume, and the quantity $\mathit{q}\xb7\mathit{n}$ is evaluated only at selected points of $\partial P$, which are said to be integration points. Moreover, upon substituting $\mathit{q}$ with its explicit expression, Equation (A6) becomes

The presence of the concentration gradient $\nabla c$ in Equation (A15) is an essential feature of our model, which aims to resolve explicitly the spatial variability of c. Indeed, if the first term on the right-hand-side of Equation (A15) were suppressed from the outset, our model would boil down to a standard description of virus dynamics, entirely based on ordinary differential equations. Models of this kind supply indications about the time evolution of averaged values of the concentration of a given species, but they are not designed to keep track of the spatial distribution of such a concentration. In this respect, the biological motivation for using $\nabla c$ in our model is the need for resolving in space the variability of c. The knowledge extracted by such information is expected to improve our understanding of the processes underlying the virus dynamics in the cells.

Within our model approach, we use div${}_{T}$ and ${\nabla}_{T}$ rather then the normal div and grad operator. This means that we use only that part of div and grad which belongs to the tangential space of the 2D ER manifold as embedded within the full 3D space.

This technique is motivated by the fact that the nonstructural viral proteins anchor to the ER surface directly after their cleavage from the polyprotein. Their movement is restricted to the ER surface. Hence, our surface PDE approach mimics the restriction of NSP movement to the ER surface. For reasons explained in the main part, we restrict also the movement of the other components to the ER surface. However, this restriction is preliminary and we intend to overcome this limitation within forthcoming model simulations.