# A framework for modeling cells spreading, motility and the relocation of
proteins on advecting lipid membranes

M. Serpelloni<EMAIL_ADDRESS>M. Arricca<EMAIL_ADDRESS>C.
Bonanno<EMAIL_ADDRESS>A. Salvadori<EMAIL_ADDRESS>Università degli Studi di Brescia, Department of Mechanical and Industrial
Engineering, Brescia, 25123, Italy  Università degli Studi di Brescia,
Department of Civil, Environmental, Architectural Engineering and Mathematics,
Brescia, 25123, Italy  The Mechanobiology research center at Università degli
Studi di Brescia , Brescia, 25123, Italy

###### Abstract

The response of cells during spreading and motility is dictated by several
multi-physics events, which are triggered by extracellular cues and occur at
different time-scales. For this sake, it is not completely appropriate to
provide a cell with classical notions of the mechanics of materials, as for
“rheology” or “mechanical response”. Rather, a cell is an alive system with
constituents that show a reproducible response, as for the contractility for
single stress fibers or for the mechanical response of a biopolymer actin
network, but that reorganize in response to external cues in a non-exactly-
predictable and reproducible way. Aware of such complexity, in this note we
aim at formulating a multi-physics framework for modeling cells spreading and
motility, accounting for the relocation of proteins on advecting lipid
membranes.

##  1 Introduction

Receptors dynamic along cell membrane is a key factor in several biological
phenomena, as for angiogenesis, tumor metastasis, endocytosis and exocytosis.
Angiogenesis is a multistep process in which endothelial cells are affected by
several extracellular stimuli, including growth factors, extracellular matrix,
and parenchymal and stromal cells. In this process, growth factor receptors as
well as adhesion receptors convey the extracellular signaling in a coordinate
intracellular pathway promoting cell proliferation, migration, and their
reorganization in active vessels [1]. Integrins are a family of cell adhesion
receptors that support and modulate several cellular functions required for
tumor metastasis. They can directly contribute to the control and progress of
metastatic dissemination. During tumor development, changes in this family of
receptors impact upon the ability of tumor cells to interact with their
environment and enable metastatic cells to convert to a migratory and invasive
phenotype. Integrins regulate each step of the metastasis and affect tumor
cell survival and interaction with changing environments in transit from the
primary tumor to distant target organs [2]. Receptor-mediated endocytosis is a
process by which cells absorb metabolites, hormones, proteins – and, in some
cases, viruses – by the inward budding of the plasma membrane (invagination).
This process forms vesicles containing the absorbed substances and is strictly
mediated by receptors on the surface of the cell [3].

Whereas uncountable papers have been published on the biology of cells
spreading, motility and the relocation of proteins on advecting lipid
membranes, the mathematical modeling definitely lags behind experiments and
overall received much less attention. Although nowadays a widespread
literature in mechanobiology exists, the relocation of proteins and their
interaction with the reorganizing cytoskeleton in the biological phenomena
mentioned above is still an ongoing research topic, let alone the formulation
of efficient algorithms and computational solvers for three-dimensional
simulations.

In this note, we attempt at defining a multi-physics scheme for the modeling
of cells spreading, motility and the relocation of proteins on advecting lipid
membranes, framing the mathematical setting within the mechanics and
thermodynamics of continua [4], stemming from seminal works [5, 6, 7] and
accounting for recent literature, either connected to the endocytosis of virus
in human and animal cells [8, 9, 10] or ligand-receptor mediated raft
formation [11], chemotaxis [12], surface-associated caveolae
mechanotransduction [13].

The paper is designed as follows. After a nomenclature of the main symbols and
the definition of operators in a Lagrangian setting, the paper focuses in
section 4 upon the relocation and reaction of receptors on a lipid membrane
that advects. The topic is purposely presented in a broad sense, in order to
be applicable to several possible receptors-ligands interactions: specific
applications - carried out in [14], [15] and in the companion paper [16] \-
deals with the relocation of vascular endothelial growth factor receptors and
integrins during endothelial cell adhesion and spreading. In spite of the
generality, section 4 is self-contained and includes the description of
Reynold’s theorem on a surface that advects, of the equations that rule
proteins transport on an advecting lipid membrane, and eventually of the
receptors-ligand interactions, in form of chemical reactions, that take place
concurrently with relocation. A rather similar approach has been taken in
section 5, which concerns the relocation and reaction of actin to form
biopolymers within the cytosol. The mechanical evolution of the cell is
discussed afterwards in section 6: besides stating the classical balance laws
(of linear and angular momentum), the section is accompanied by an extensive
discussion on boundary conditions, aimed at showing that Neumann type of
conditions, due to electrostatic interactions, are most likely not responsible
for cell spreading and motion in view of the modest amount of energy involved
in those interactions compared to the bulk energy of a cell. We concluded
therefore that spreading is a result of extensional and contractile forces
exerted by pseudopodia and the cytoskeleton machinery [17]. Those forces have
been investigated further in section 7, where the thermodynamics of receptors
motion on the membrane was studied at first up to the constitutive theory and
the receptors-ligand interactions kinetics. The analysis of the thermo-chemo-
mechanics of cells is the last section of this work: in it, we highlight the
role of strain and stress decompositions in order to model cell adhesion,
protrusion, and contractility. A bibliographic review is presented in a rather
extensive paragraph, showing various approaches pursued in the literature to
cover the multiscale scenario of cell viscoelasticity and identifying missing
pieces within the theoretical framework that we set in the present note.

##  2 Nomenclature

###  2.1 Notation

Vectors $\vec{a}$ will be denoted by an over-right-arrow, second order tensors
${\bm{A}},{\bm{a}}$ by bold face. This notation does not apply to operators.

###  2.2 Operators

\- the symbol ${\rm div}\left[\,{-}\,\right]$ denotes the divergence operator
in the current configuration, i.e. ${\rm
div}\left[\,{\vec{f}}\,\right]={\partial f_{i}}/{\partial x_{i}}$
\- the symbol ${\rm Div}\left[\,{-}\,\right]$ denotes the referential
divergence operator, i.e. ${\rm Div}\left[\,{\vec{f}}\,\right]={\partial
f_{i}}/{\partial X_{i}}$
\- the symbol ${\rm\nabla}\left[\,{-}\,\right]$ denotes the gradient operator
in the current configuration
\- the symbol ${\rm Grad}\left[\,{-}\,\right]$ denotes the referential
gradient operator
\- the symbol $\cdot$ denotes the single contraction of two vectors
\- the symbol $:$ denotes the double contraction of two tensors
\- the symbols $||\vec{x}||^{2}$, $||{\bm{x}}||^{2}$ denote the squared norm
of vector $\vec{x}$ or tensor ${\bm{x}}$
\- the symbol T denotes transposition of a tensor
\- the symbol -1 denotes the inverse of a tensor

###  2.3 Variables and fields

\- the symbol $t$ denotes time
\- the symbol $\Omega(t)\in\mathbb{R}^{3}$ denotes a volume that advects
\- the symbol $\partial\Omega(t)$ denotes the surface of $\Omega(t)$
\- the symbol ${\cal P}(t)\subset\partial\Omega(t)$ denotes a part of
$\partial\Omega(t)$
\- the symbol $\vec{v}_{adv}({\vec{x}},t)$ denotes the velocity of advection
at place ${\vec{x}}$ and time $t$
\- the symbol $\vec{n}({\vec{x}},t)$ denotes the outward normal at place
${\vec{x}}$ and time $t$
\- the symbol ${\bm{l}}({\vec{x}},t)$ denotes the velocity gradient at place
${\vec{x}}$ and time $t$
\- the symbol ${\bm{d}}({\vec{x}},t)$ denotes the stretching at place
${\vec{x}}$ and time $t$
\- the symbol ${\bm{F}}({\vec{X}},t)$ denotes the deformation gradient at
point ${\vec{X}}$ and time $t$
\- the symbol ${\bm{C}}({\vec{X}},t)$ denotes the right Cauchy-Green tensor at
point ${\vec{X}}$ and time $t$
\- the symbol ${\bm{P}}({\vec{X}},t)$ denotes the first Piola stress tensor at
point ${\vec{X}}$ and time $t$
\- the symbol $J({\vec{X}},t)$ denotes the determinant ${\rm det}[{\bm{F}}]$
at point ${\vec{X}}$ and time $t$
\- the symbol $\vec{n}_{R}({\vec{X}},t)$ denotes the outward normal at point
${\vec{X}}$ and time $t$
\- the symbol $m_{a}$ denotes the molar mass of species $s$
\- the symbol $c_{a}$ denotes the molarity of species $s$
\- the symbol $\rho_{a}$ denotes the density of species $s$
\- the symbol ${\overline{s}}_{a}$ denotes the mass supply of species $s$
\- the symbol ${s}_{a}$ denotes the molar supply of species $s$
\- the symbol $\vec{\hbar}_{a}$ denotes the density flux of species $s$
\- the symbol $\vec{h}_{a}$ denotes the molar flux of species $s$

(a) The reference body $\Omega_{R}$ and the deformed body $\Omega(t)$. Note
that $\vec{x}\in{{\cal P}_{(}t)}$ implies $\vec{X}\in{{\cal P}_{R}}$.

(b) Frenet frame at point $\vec{y}\in\partial{\cal P}(t)$ and the normal
vector $\vec{n}$ at point $\vec{x}\in{\cal P}(t)$.

Figure 1: Notation

##  3 Definitions

Denote with $\Omega(t)$ a volume that advects, and with $\partial\Omega(t)$
its surface. A point $\vec{x}\in\Omega(t)$ is defined as the image of a point
$\vec{X}$ in a reference configuration $\Omega_{R}$ through a smooth function
${\chi}(\vec{X},t)$ termed motion. Following [4], we will name deformation the
snapshot of a motion at a fixed time $t$:

${{\chi}}_{t}(\vec{X})={\chi}(\vec{X},t)\;.$

The deformation is assumed to be a one-to-one map. In addition, denoting the
deformation gradient with

${\bm{F}}={{\rm Grad}\left[\,{\chi_{t}}\,\right]}\;,$

the requirement $J={\det}\left[\,{{\bm{F}}}\,\right]>0$ holds. Define on the
surface a part ${\cal P}(t)\subset\partial\Omega(t)$ as in Fig. 1, and
consider a scalar function $f({\vec{x}},t)$ with ${\vec{x}}\in{\cal P}(t)$.
Denote with

$\vec{v}_{adv}({\vec{x}},t)={\rm d}\vec{x}/{\rm d}t$

the velocity of advection at location ${\vec{x}}$ and time $t$; such a
velocity has an arbitrary direction, i.e. it is not necessarily tangent to
$\partial\Omega(t)$.

The Frenet-Serret reference frame at a generic point $\vec{y}\in\partial{\cal
P}(t)$ is defined as in Fig. 1, in terms of the two unit vectors
$\vec{t}_{\|}(\vec{y},t)$ (tangent) and $\vec{t}_{\bot}(\vec{y},t)$ (normal).
The vector $\vec{n}(\vec{y},t)$ (binormal) is here taken of non-unit length,
being the imagine in $\Omega(t)$ of a unit vector $\vec{n}_{R}$ in the
reference configuration $\Omega_{R}$, by means of the contravariant
transformation

$\vec{n}={\bm{F}}^{-T}\,\vec{n}_{R}\;.$

On the other hand, the following covariant transformations hold:

$\vec{t}_{\|_{R}}={\bm{F}}^{-1}\,\vec{t}_{\|}\;,\qquad\vec{t}_{\bot_{R}}={\bm{F}}^{-1}\,\vec{t}_{\bot}\;,$

with the obvious implication that $\vec{t}_{\|_{R}}$ and $\vec{t}_{\bot_{R}}$
are not unit vectors. The Frenet formulae holds, namely:

$\kappa\;\vec{t}_{\bot}=-\partial\vec{t}_{\|}/\partial
s\;,\qquad\tau\;\vec{t}_{\bot}=\partial\frac{\vec{n}}{|\vec{n}|}/\partial
s\;,\qquad\kappa\;\vec{t}_{\|}-\tau\;\frac{\vec{n}}{|\vec{n}|}=\partial\vec{t}_{\bot}/\partial
s\;,$

where $\kappa$ denotes the curvature and $\tau$ the torsion.

The projected gradient operator of a scalar field $f$ on a surface $\cal P$ is
defined as follows

$\displaystyle{\rm\nabla}_{{\cal
P}}\left[\,{f}\,\right]={\rm\nabla}\left[\,{f}\,\right]-\;\frac{\vec{n}\cdot{\rm\nabla}\left[\,{f}\,\right]}{|\vec{n}|^{2}}\,\vec{n}\;,$
(1a) in the current configuration, whereas in the reference configuration it
reads $\displaystyle{\rm Grad}_{{\cal P}}\left[\,{f}\,\right]={\rm
Grad}\left[\,{f}\,\right]-\;\vec{n}_{R}\cdot{\rm
Grad}\left[\,{f}\,\right]\,\vec{n}_{R}\;,$ (1b)

The projected divergence operator of a vector field $\vec{v}$, which has an
arbitrary direction, on a surface $\cal P$ is defined as follows

$\displaystyle{\rm div}_{{\cal P}}\left[\,{\vec{v}}\,\right]={\rm
div}\left[\,{\vec{v}}\,\right]-\;\frac{\vec{n}\cdot{\bm{l}}\vec{n}}{|\vec{n}|^{2}}\;,$
(2a) $\displaystyle{\rm Div}_{{\cal P}_{R}}\left[\,{\vec{v}_{R}}\,\right]={\rm
Div}\left[\,{\vec{v}}\,\right]-\;{\vec{n}_{R}\cdot{{\rm
Grad}\left[\,{\vec{v}_{R}}\,\right]}\vec{n}_{R}}\;,$ (2b)

in the current and reference configurations, respectively. Tensor ${\bm{l}}$
is the gradient of $\vec{v}$,
${\bm{l}}={\rm\nabla}\left[\,{\vec{v}}\,\right]$. Note that ${\bm{l}}$ in eq.
(2a) can be replaced by its symmetric part ${\bm{d}}={\rm
sym}\left[\,{{\bm{l}}}\,\right]$, since for any skew-symmetric tensor
${\bm{w}}$ it holds $\vec{n}\cdot{\bm{w}}\vec{n}=0$ . Alternative forms for
the projected divergence operators are

$\displaystyle{\rm div}_{{\cal
P}}\left[\,{\vec{v}}\,\right]={\rm{curl}}\left[\,{\frac{\vec{n}}{|\vec{n}|}\times\vec{v}}\,\right]\cdot\frac{\vec{n}}{|\vec{n}|}\;,\qquad{\rm
Div}_{{\cal
P}_{R}}\left[\,{\vec{v}}\,\right]={\rm{Curl}}\left[\,{\frac{\vec{n}_{R}}{|\vec{n}_{R}|}\times\vec{v}_{R}}\,\right]\cdot\frac{\vec{n}_{R}}{|\vec{n}_{R}|}\;.$
(3a)

Provided sufficient smoothness, the divergence theorem holds also for
advecting membranes, in the form:

$\int_{{\cal P}(t)}\,{\rm div}_{{\cal P}}\left[\,{\vec{g}}\,\right]\;{\rm
d}a=\int_{\partial{\cal P}(t)}\,\vec{g}\cdot\vec{t}_{\bot}\,{\rm d}\ell\;.$
(4)

The proof of this theorem, as well as for all other theorems not explicitly
stated in this paper, can be found in [18].

##  4 Relocation and reaction of receptors on a lipid membrane that advects

###  4.1 Reynold’s theorem on a surface that advects

Reynold’s theorem on ${\cal P}(t)$ reads as follows [18]:

$\frac{{\rm d}}{{\rm d}t}\int_{{\cal P}(t)}\,f\,{\rm d}a=\int_{{\cal
P}(t)}\,\frac{\partial f}{\partial t}\,+\;{\rm div}_{\cal
P}\left[\,{f\,\vec{v}_{adv}}\,\right]\;{\rm d}a\;,$ (5)

where $\vec{v}_{adv}({\vec{x}},t)$ is the velocity of advection at location
${\vec{x}}$ and time $t$. By taking $f=1$, eq. (5) depicts the area evolution
of ${{\cal P}(t)}$as

$\frac{{\rm d}}{{\rm d}t}\int_{{\cal P}(t)}\,{\rm d}a=\int_{{\cal
P}(t)}\,\;{\rm div}_{\cal P}\left[\,{\vec{v}_{adv}}\,\right]\;{\rm d}a\;.$

It is intuitive that advection with velocity in the tangent plane has the
potential of modifying the surface area, however even
$\vec{v}_{adv}({\vec{x}},t)\propto\vec{n}({\vec{x}},t)$ can do so, as for the
homothetic expansion of a rubber balloon. Reynold’s theorem (5) can be also
restated as

$\displaystyle\frac{{\rm d}}{{\rm d}t}\int_{{\cal P}(t)}\,f({\vec{x}},t)\,{\rm
d}a=\int_{{\cal P}(t)}\,\frac{{\rm d}\,f({\vec{x}},t)}{{\rm
d}t}\,+\,f({\vec{x}},t)\;{\rm div}_{{\cal
P}}\left[\,{\vec{v}_{adv}}\,\right]\;{\rm d}a\;.$ (6)

and is a restriction on surfaces of the classical Reynold’s transport relation
on volumes ( see [4], section 16 among others ).

###  4.2 Mass transport on a surface that advects

####  4.2.1 Mass balance in the current configuration for a convecting species

Consider a generic species $a$ at a point $\vec{x}$ on the surface
$\partial\Omega(t)$. Species $a$ convects with velocity
$\vec{v}_{a}(\vec{x},t)$. The latter entails the dragging, or advection,
velocity $\vec{v}_{adv}(\vec{x},t)$ and another velocity that is due to many
possible physics, as for diffusion or migration. If internalization of species
from the membrane is not allowed, the net velocity $\vec{v}_{a}-\vec{v}_{adv}$
lays in the tangent plane of the membrane and

$(\vec{v}_{a}-\vec{v}_{adv})\cdot\vec{n}=0\;.$ (7)

Since species are modeled on a membrane, which is a two-dimensional manifold,
the surface density $\rho_{a}$ of species $a$ measures the mass of the species
per unit surface. The density flux vector of species $a$, denoted with
$\vec{\hbar}_{a}$, is the product of the surface density times the net
velocity of species $a$, i.e.

$\vec{\hbar}_{a}=\rho_{a}\;(\vec{v}_{a}-\vec{v}_{adv})\;.$ (8)

Define on the surface a part ${\cal P}(t)\subset\partial\Omega(t)$ as in Fig.
1. The flux of species $a$ across the boundary $\partial{\cal P}(t)$ is

$\int_{\partial{\cal P}(t)}\,\vec{\hbar}_{a}\cdot\vec{t}_{\bot}\;{\rm d}\ell$

and the mass balance of species $a$ in the advecting configuration ${\cal
P}(t)$ reads

$\frac{{\rm d}}{{\rm d}t}\int_{{\cal P}(t)}\,\rho_{a}({\vec{x}},t)\,{\rm
d}a+\int_{\partial{\cal P}(t)}\,\vec{\hbar}_{a}\cdot\vec{t}_{\bot}\;{\rm
d}\ell=\int_{{\cal P}(t)}\,{\overline{s}}_{a}({\vec{x}},t)\,{\rm d}a\;,$ (9)

where ${\overline{s}}_{a}({\vec{x}},t)$ is the surface mass supply111As an
example, in biology cells may produce proteins that move to the lipid
membranes from the cytosol.  of species $a$. By means of the divergence
theorem (4) and of Reynold’s transport theorem in the form (6), balance law
(9) becomes

$\int_{{\cal P}(t)}\,\frac{{\rm d}\rho_{a}}{{\rm d}t}\,+\rho_{a}\;{\rm
div}_{{\cal P}}\left[\,{\vec{v}_{adv}}\,\right]+{\rm div}_{{\cal
P}}\left[\,{\vec{\hbar}_{a}}\,\right]\;{\rm d}a=\int_{{\cal
P}(t)}\,{\overline{s}}_{a}({\vec{x}},t)\,{\rm d}a\;.$

Since it holds for all ${\cal P}(t)$, it eventually localizes as

$\,\frac{{\rm d}\rho_{a}}{{\rm d}t}\,+\rho_{a}\;{\rm div}_{{\cal
P}}\left[\,{\vec{v}_{adv}}\,\right]+{\rm div}_{{\cal
P}}\left[\,{\vec{\hbar}_{a}}\,\right]\;=\,{\overline{s}}_{a}({\vec{x}},t)\;.$
(10)

This formulation of the mass conservation law has been considered also in
[19]. The mass balance can be finally written in terms of surface molarity
$c_{a}$ (in moles or molecules per unit surface), by division by the molar or
molecular mass ($m_{a}$) of species $a$. By denoting with
$c_{a}=\rho_{a}/m_{a}$, $s_{a}={\overline{s}}_{a}/m_{a}$, and
$\vec{h}_{a}=\vec{\hbar}_{a}/m_{a}$ the local balance (10) becomes

$\,\frac{{\rm d}c_{a}}{{\rm d}t}\,+c_{a}\;{\rm div}_{{\cal
P}}\left[\,{\vec{v}_{adv}}\,\right]+{\rm div}_{{\cal
P}}\left[\,{\vec{h}_{a}}\,\right]\;=\,{s}_{a}({\vec{x}},t)\;.$ (11)

####  4.2.2 Mass balance in the reference configuration for a convecting
species

The mass balance (11) can be rephrased in the reference configuration at point
$\vec{X}$ and time $t$. To this aim, define the reference molarity of species
$a$ as

${c_{a_{R}}}({\vec{X}},t)=\,c_{a}(\,{\vec{x}}({\vec{X}},t\,),t\,)\;j({\vec{X}},t)\;,$
(12)

the reference flux vector ${\vec{h}}_{a_{R}}({\vec{X}},t)$ and the reference
mass supply $s_{a_{R}}({\vec{X}},t)$ as

${\vec{h}}_{a_{R}}=\,j\,{\bm{F}}^{-1}\;{\vec{h}_{a}}(\,{\vec{x}}({\vec{X}},t\,),t\,)\;,\qquad
s_{a_{R}}=\,j\,s_{a}(\,{\vec{x}}({\vec{X}},t\,),t\,)\;,$ (13)

respectively, where [4, 20]:

$j=J\;|{\bm{F}}^{-T}\vec{n}_{R}|=J\;\sqrt{\vec{n}_{R}\cdot{\bm{C}}^{-1}\vec{n}_{R}}\;.$
(14)

The referential form of the mass balance (11) can be derived from the mass
balance in the form (9), and reads

$\frac{\partial{c_{a_{R}}}}{\partial t}+\,{\rm Div}_{{\cal
P}_{R}}\left[\,{{\vec{h}}_{a_{R}}}\,\right]=s_{a_{R}}\;.$ (15)

For the sake of brevity, the proof has been here omitted, interested readers
may find it in [18].

###  4.3 Relocation and reaction

The association and formation of a protein complex follow a two-steps
mechanism; the formation of an encounter complex, in which previously free
proteins show few specific interactions and assume many orientations, and the
evolution of the encounter complex in the final complex. The encounter
complex, which therefore represents the ensemble of orientations of proteins,
is mostly dominated by electrostatic interactions. Under certain conditions it
evolves in the final complex, when protein perfectly match each other,
otherwise it dissociates and proteins return to be free [21, 22].

The two steps mechanism which describes the formation of a protein complex
reads:

${\rm R}+{\rm L}\operatornamewithlimits{\rightleftarrows}_{k_{-1}}^{k_{1}}{\rm
C}^{*}\operatornamewithlimits{\rightleftarrows}_{k_{-2}}^{k_{2}}{\rm C}$ (16)

where $\rm{R}$ and $\rm{L}$ are the receptors $({\rm R})$ and ligands $({\rm
L})$ free proteins, $\rm{C}^{*}$ represents the encounter complex and $\rm{C}$
is the final complex. In Equation (16), $k_{1}$ and $k_{-1}$ are the rate of
formation and dissolution of the encounter complex, $\rm{C}^{*}$, whereas
$k_{2}$ and $k_{-2}$ are the forward and reverse rate constants for formation
of the final complex, $\rm{C}$, from $\rm{C}^{*}$.

Assuming that the formation of the encounter complex occurs whenever $\rm{R}$
and $\rm{L}$ are separated by an encounter distance smaller than $r$, then
$k_{1}=2\pi[D({\rm{R}})+D(\rm{L})]$,
$k_{-1}=2[D({\rm{R}})+D({\rm{L}})]r^{-2}$. Here $D(\rm{R})$ and $D(\rm{L})$
are the translational diffusion constants for protein motion in the membrane
and the equilibrium constant for the encounter step, $K_{d}=\pi r^{2}$,
represents the area of a disk of radius $r$ [23]. If the concentration of
${\rm C}^{*}$ is smaller than the concentration of free proteins or final
complexes, it is a good approximation to set ${\rm d}{C}^{*}/{\rm d}t=0$,
leading to the binding-unbinding interaction

${\rm R}+{\rm L}\operatornamewithlimits{\rightleftarrows}_{k_{b}}^{k_{f}}{\rm
C}$ (17)

most commonly used [23]. A similar approach has been taken in [14, 24] for the
relocation of VEGFR-2 receptors and in [15] for integrins. Coefficients
$k_{f}$ and $k_{b}$ are the kinetic constants of the forward and backward
reactions, respectively. The rate of reaction (17), denoted with
$w^{\eqref{eq:chem_react}}$and measured in $[\frac{mol}{m^{2}s}]$, quantifies
the net formation of (C) on the advecting membrane as the difference between
the forward and backward reactions. Equation (15) shall be extended to account
for the reaction (17) and tailored to species $a=R,L,C$.

Receptors (either free or bound into the complex) are distributed along the
membrane together with other lipid species and proteins. They are assumed to
freely move laterally, effects due to steric hindrance are not accounted for.
The amount of proteins per unit area that can be placed at a membrane location
$\vec{x}$ is thus limited by the actual size of the protein itself. This
evidence ushers the definition of a saturation limit for the species,
${c_{a}^{max}}({\vec{x}},t)$.

During their life, cells and their membranes undergo major macroscopic
mechanical deformations. Studies on the red blood cell [25] suggest that the
membrane deformation occur at constant area, but this evidence does not appear
to be supported by experiments in endothelial cells during spreading [17].
Individual protein and phospholipid can easily move laterally within the
membrane, which results in a very low shear stiffness. The fluid mosaic model
[26] captures this evidence, adding a questionable high resistance to areal
expansion. Indeed the mechanisms that are in charge of areal expansion during
cell spreading are complex and involve the micro-structural
topology222Multiscale investigations, however, fall out of the scope of the
present work. of the membrane (as for flattening of invaginated membrane
domains [27], i.e. the role of the caveolae as membrane surface repository
readily made available for fast geometrical evolution as during filopodia
extension). The structure of the lipid membranes, however, induce to suppose
that the saturation concentration ${c_{a}^{max}}({\vec{x}},t)$, i.e. the
maximum number of moles or molecules per unit area for any species $a$,
remains unchanged in time in the current configuration. This choice in turn
entails that the number of moles or molecules per unit area in the reference
configuration is not constant and evolves in time following eq. (12), i.e.

${c_{a_{R}}^{max}}({\vec{X}},t)=\,c_{a}^{max}({\vec{x}}({\vec{X}},t),t)\,j({\vec{X}},t)\;.$
(18)

Accordingly, the value of the non-dimensional ratio between the concentration
of species $a$ and its amount ${c_{a}^{max}}$ at saturation,

$\vartheta_{a}={c_{a}}/{c_{a}^{max}}$ (19)

in the current configuration remains unchanged in the reference configuration

$\vartheta_{a_{R}}({\vec{X}},t)=\vartheta_{a}({\vec{x}},t)\;.$ (20)

The kinetics of reaction $\eqref{eq:chem_react}$ is modeled as for ideal
systems via the law of mass action [28]

$w^{(\ref{eq:chem_react})}=k_{f}\,\frac{\vartheta_{L}}{(1-\vartheta_{L})}\,\frac{\vartheta_{R}}{(1-\vartheta_{R})}-\,k_{b}\,\frac{\vartheta_{C}}{(1-\vartheta_{C})}\;.$
(21)

At chemical equilibrium, as $w^{(\ref{eq:chem_react})}=0$, the concentrations
obey the relation

$\frac{k_{f}}{k_{b}}=\frac{\vartheta_{C}^{\rm eq}}{(1-\vartheta_{C}^{\rm
eq})}\,\frac{(1-\vartheta^{\rm eq}_{R})}{\vartheta^{\rm
eq}_{R}}\,\frac{(1-\vartheta^{\rm eq}_{L})}{\vartheta^{\rm eq}_{L}}=K_{\rm
eq}^{(\ref{eq:chem_react})}$ (22)

which defines the constant of equilibrium $K_{\rm eq}^{(\ref{eq:chem_react})}$
of reaction (17).

Far from the saturation limit, $(1-\vartheta_{a})\sim 1$ for all $a$.
Accordingly, the mass action law (21) simplifies as

$w^{(\ref{eq:chem_react})}=\tilde{k}_{f}\,c_{L}\,c_{R}-\,\tilde{k}_{b}\,c_{C}$
(23)

once the new constants

$\tilde{k}_{f}=k_{f}({c_{L}^{max}}{c_{R}^{max}})^{-1}\;,\qquad\tilde{k}_{b}=k_{b}({c_{C}^{max}})^{-1}$

are defined.

The diffusion of receptors and the viscous evolution of the cell during
adhesion and migration appear to be much slower than the interaction kinetics,
i.e. the time required to reach chemical equilibrium is orders of magnitude
smaller than the time-scale of other processes. For this reason, thermodynamic
equilibrium may be invoked in place of a transient evolution, thus inferring
the constraint $w^{(\ref{eq:chem_react})}=0$ to the concentrations of species
at all times. Far from saturation, equating (23) to zero implies that

${c_{C}}=\frac{c_{R}\,c_{L}}{\alpha}\;,$ (24)

having denoted with $\alpha$ the following constant:

$\alpha=\frac{\tilde{k}_{b}}{\tilde{k}_{f}}=\frac{c^{max}_{R}\,c^{max}_{L}}{c^{max}_{C}}\;\frac{1}{K_{\rm
eq}^{(\ref{eq:chem_react})}}\;.$ (25)

In view of identity (24), the two concentrations $c_{R}$ and $c_{L}$ describe
the problem in full, and the concentration of the complex can be deduced a
posteriori.

In vivo experiments show that the complex molecules usually have a much
smaller mobility than receptors, perhaps induced by their size. For in vitro
experiments [14, 24, 15], ligands are prevented to flow onto the substrate:
given that complex molecules result from the interaction with immobile
ligands, they are macroscopically steady as well. Since receptors move along
the membrane, reaction (17) traps mobile receptors and vice-versa [29]. In
this work, analogously to [30], ligands and complex are assumed to be
motionless, i.e.

$\vec{h}_{L}=\vec{h}_{C}=\vec{0}\;.$ (26)

The reaction rate $w^{\eqref{eq:chem_react}}({\vec{x}},t)$, being a mass
supply, shall transform as ${s}_{a}({\vec{x}},t)$ according to eq. (13). The
invariance of $\vartheta_{a}$ with the configuration and the analysis of the
mass action law (21) imply that the forward and backward “constants”, which
encompass the dimensionality of $w^{\eqref{eq:chem_react}}({\vec{x}},t)$, are
not actually constants in the reference configuration. They rather change with
time and with point $\vec{X}$ according to

$k_{f_{R}}({\vec{X}},t)=\,j({\vec{X}},t)\,k_{f}\;,\qquad
k_{b_{R}}({\vec{X}},t)=\,j({\vec{X}},t)\,k_{b}\;\,$ (27)

with $j({\vec{X}},t)$ as in (14). The equilibrium constant in the reference
configuration, being the ratio of $k_{f_{R}}$ and $k_{b_{R}}$ remains
independent upon the configuration. Eventually, the mass action law (21) in
the reference configuration writes

$w^{(\ref{eq:chem_react})}_{R}=k_{f_{R}}\,\frac{\vartheta_{L}}{(1-\vartheta_{L})}\,\frac{\vartheta_{R}}{(1-\vartheta_{R})}-\,k_{b_{R}}\,\frac{\vartheta_{C}}{(1-\vartheta_{C})}\;.$
(28)

In view of all considerations made so far, the local form (15) of the mass
balance specify as follows ( omitting the dependency upon ${\vec{X}}$ and $t$
):

$\displaystyle\frac{\partial{c_{R_{R}}}}{\partial t}+\,{\rm Div}_{{\cal
P}_{R}}\left[\,{{\vec{h}}_{R_{R}}}\,\right]+\,w^{(\ref{eq:chem_react})}_{R}\;=s_{R_{R}}\;,$
(29a) $\displaystyle\frac{\partial{c_{L_{R}}}}{\partial
t}+\,w^{(\ref{eq:chem_react})}_{R}\;=0\;,$ (29b)
$\displaystyle\frac{\partial{c_{C_{R}}}}{\partial
t}-\,w^{(\ref{eq:chem_react})}_{R}\;=0\;.$ (29c)

Equation (29a) is defined on the membrane surface $\partial\Omega_{R}$, where
the receptors flow. The supply $s_{R_{R}}$ accounts for internalization or
generation of proteins: it is the amount of receptors that are generated
within the cell and reach the membrane or that internalize. It can be related
to the change in the membrane area through a parameter $\kappa_{R_{R}}$ as

$\displaystyle s_{R_{R}}({\vec{X}},t)$
$\displaystyle=\kappa_{R_{R}}\frac{\partial j}{\partial t}$
$\displaystyle=\kappa_{R_{R}}\left[\;|{\bm{F}}^{-T}\,\vec{n}_{R}|\,J\,{\rm
tr}\left[\,{{\bm{l}}}\,\right]\,-\frac{J}{2}\,\frac{1}{|{\bm{F}}^{-T}\,\vec{n}_{R}|}\;\vec{n}_{R}\cdot{\bm{C}}^{-1}\,\frac{\partial{\bm{C}}}{\partial
t}\,{\bm{C}}^{-1}\,\vec{n}_{R}\right]\;.$ (30)

At all points at which ligands and receptors do not interact, the reaction
rate $w^{(\ref{eq:chem_react})}_{R}$ vanishes. Equation (29b) is rather
defined in the location where ligands stand. In vitro, a given amount of
ligands (which can be thought of as the initial condition of eq. (29b) are
spread upon a microscope slide. Finally, eq. (29c) is defined in the contact
zone between the cell and the slide where reaction (17) takes place.

It is convenient to rephrase eq. (29b) in terms of the “ligands made available
for the reaction” in place of the “ligands spread on the slide”. The former
ligands are the ones “felt” at a point on the membrane as the distance from
such a point and the substrate, where ligands are spread out, becomes
sufficiently small.

Such a distance can be understood as a cutoff, within which the formation of
an encounter complex, $\rm{C}^{*}$, becomes possible as a consequence of
diffusion, as made clear in [21, 22, 23, 31]. Despite the size of the cutoff
distance remains inaccurately estimated, it was established to be on the order
of tens nanometers [23, 5]. It arises form the interplay of attractive and
repulsive forces between either two cells or a cell and a substrate. Indeed,
negative electrical charge carried by cells generates repulsive electrostatic
forces - repulsive barrier \- which is further enriched by an additional
resistance provided by the compression of the glycocalyx proteins. Rather,
electrodynamic van der Waals forces are expected to be attractive [23]. Both
van der Waals and compressive forces are characterized as non-specific long
ranged forces, whereas cell adhesion is generally mediated by the specific
short ranged receptor-ligand interactions, which can cause cell adhesion much
more tightly than the non-specific electrical forces [23, 30]. Cells separated
by a distance less than, or equal to, the cutoff distance should form a zone
of adhesion with the substrate by means of local fluctuations in receptors
density, so that small regions of increased density can penetrate through the
resisting potential to react with the source of ligands on substrate [5].

This point of view, which corresponds to the picture of tight receptor-ligand
bond as a set of weak non covalent physical interactions [32], is made
explicit by a supply function $s_{L_{R}}$, that vanishes at long ranges and
rapidly reaches the initial concentration of ligands available for the
reaction at short distances

$\displaystyle\frac{\partial{c_{L_{R}}}}{\partial
t}+\,w^{(\ref{eq:chem_react})}_{R}\;=s_{L_{R}}\;.$ (31)

The ligand supply $s_{L_{R}}({\vec{X}},t)$ becomes available for the reaction
during the spreading of the cell. It seems to be logically related to: i) a
gap function between the substrate rich in ligands and the cell membrane in
the current configuration; ii) a lag in time, namely a point-wise function of
an internal variable that activates when the gap function is below some
threshold and is related to the chemical kinetics of the binding-unbinding
reaction (17). In this form, all three equations (29a), (29c), (31) can be
written on the membrane $\vec{X}\in\partial\Omega_{R}$.

Assuming that the time scale of the chemical reaction is much faster than
other processes, the concentrations of species may be governed by
thermodynamic equilibrium at all times. The concentration of complex
$c_{C_{R}}$ relates then to the others by the equation
$w^{(\ref{eq:chem_react})}=0$, which leads to eq. (24) in the current
configuration. Making use of mapping (12), eq. (24) relates the concentration
of complex in the reference configuration $c_{C_{R}}$ to the concentration of
ligands and receptors in the same configuration $c_{L_{R}}$, $c_{R_{R}}$ as
follows

${c_{C_{R}}}=\frac{c_{R_{R}}\,c_{L_{R}}}{\alpha_{R}({\vec{X}},t)}\;,\qquad\alpha_{R}({\vec{X}},t)=\alpha\,j({\vec{X}},t)\;,$
(32a)

with constant $\alpha$ defined in eq. (25). Transformation (32a) is consistent
with the assumption (18) made on how saturations transform.

In conclusion, exploiting identity (32a), the two concentrations $c_{R_{R}}$
and $c_{L_{R}}$ fully describe the problem in the assumption of infinitely
fast kinetics, whereas the concentration of the complex can be deduced a
posteriori. The two governing equations descend from eqs.(29) and read:

$\displaystyle\frac{\partial{c_{R_{R}}}}{\partial
t}+\,\frac{\partial{c_{C_{R}}}}{\partial t}+\,{\rm Div}_{{\cal
P}_{R}}\left[\,{{\vec{h}}_{R_{R}}}\,\right]\;=s_{R_{R}}\;,\qquad\vec{X}\in\partial\Omega_{R}\;,$
(32b) $\displaystyle\frac{\partial{c_{L_{R}}}}{\partial
t}+\,\frac{\partial{c_{C_{R}}}}{\partial
t}\;=s_{L_{R}}\;,\qquad\vec{X}\in\partial\Omega_{R}\;.$ (32c)

Equations (4.3), with associated initial conditions

$\displaystyle c_{R_{R}}(\vec{X},0)=c^{0}_{R_{R}}(\vec{X})\;,\qquad
c_{L_{R}}(\vec{X},0)=0\;,\qquad c_{C_{R}}(\vec{X},0)=0\;$

and Dirichlet-Neumann boundary conditions define the relocation of receptors
that undergo binding-unbinding reactions on the reference configuration of a
membrane that advects. These are balance equations and as such hold for any
constitutive behavior for the mass flux. These equations are coupled to the
mechanical evolution of the cell (i.e. adhesion, spreading, migration) through
the function $s_{L_{R}}(\vec{X},t)$, which “transfers” ligands on the membrane
according to the geometry of the cell.

##  5 Relocation and reaction of actin to form biopolymers

The extensive mathematical description made in section 4 will guide the
modeling of the relocation and reaction of actin to form biopolymers in the
cytosol, which will be summarized here in a shorter shape.

Biopolymers are composed of actin, a protein termed globular or G-actin in its
monomeric form and F-actin when it forms filamentous polymers. In turn, actin
filaments can bundle to form stress fibers, or cross-link to form polymer
networks that allow the movement of the cell. Polymerization is usually
triggered by extracellular signals. In the case of cell locomotion, for
instance, the cell extends finger-like protrusions by which the cell “feels”
the surrounding surface. As done in [33], the precise details of the signaling
pathways are here ignored. Rather, the level of signaling is assumed given in
the reference configuration by a function

${\cal
C}(\vec{X},t)=\gamma_{i}\,\exp\left[{-|\vec{x}(\vec{X},t)-\vec{y}_{i}|}\right]\,\exp\left[{-\frac{t-\tau_{i}}{\theta}}\right]$
(33)

that accounts for the location of discrete signaling points $\vec{y}_{i}$ in
the surroundings emitting signals of intensity $\gamma_{i}$ at time
$\tau_{i}$; $\theta$ is the decay constant of the signal. This approach in
modeling the external stimulus is similar to the membrane activator in [34].

The transduction of the signal results in the polymerization of the actin
filaments and their cross-linking or bundling. The formation of single actin
filaments can be modeled as a bimolecular reaction similar to (16), as in
[35]; in this note, the biopolymer turn-over will be described at a larger
scale, involving the interplay between fundamental units and stress-fibers or
pseudopodia, in the form

${\rm G}\operatornamewithlimits{\rightleftarrows}_{k_{b}}^{k_{f}}{\rm F}$ (34)

with ${\rm F}$ denoting either one of the two biopolymers. The network or
fiber formation rate of reaction (34), denoted with
$w^{(\ref{eq:actin_polymerization})}$, is influenced by mechanical stresses:
stress fibers stability is favored by tension, for instance. For this reason,
the stress tensor enters the chemical potential and the dissociation reaction
of biopolymers. The kinetics of reaction $\eqref{eq:actin_polymerization}$ is
modeled via the law of mass action, properly extended to account for
signaling:

$w^{(\ref{eq:actin_polymerization})}(\vec{X},t)={\cal
C}(\vec{X},t)\;k_{f}\,\frac{\vartheta_{G}}{(1-\vartheta_{G})}-\,{\cal{D}}(\vec{X},t)\,k_{b}\,\frac{\vartheta_{F}}{(1-\vartheta_{F})}\;,$
(35)

having already discussed the meaning of the ratio $\vartheta$ in eq. (19).
Function $\cal{D}$ accounts for the role of the stress in the dissociation of
biopolymers, see for instance [33].

###  5.1 Mass transport in the cytosol

Consider a generic species $a$ at a point $\vec{x}$ in the cytosol
$\Omega(t)$. The mass balance of species $a$ in the advecting configuration
${\cal Q}(t)$ localizes as

$\,\frac{{\rm d}\rho_{a}}{{\rm d}t}\,+\rho_{a}\;{\rm
div}\left[\,{\vec{v}_{adv}}\,\right]+{\rm
div}\left[\,{\vec{\hbar}_{a}}\,\right]\;=\,{\overline{s}}_{a}({\vec{x}},t)\;,$
(36)

with $\vec{\hbar}_{a}$ and $\vec{v}_{adv}$ defined earlier in section 4.2.1,
$\rho_{a}$ is the density of species $a$. The mass balance can be restated in
terms of molarity $c_{a}$ (in moles or molecules per unit volume), by division
by the molar or molecular mass ($m_{a}$) of species $a$. By denoting with
$c_{a}=\rho_{a}/m_{a}$, $s_{a}={\overline{s}}_{a}/m_{a}$, and
$\vec{h}_{a}=\vec{\hbar}_{a}/m_{a}$ the local balance (36) becomes

$\,\frac{{\rm d}c_{a}}{{\rm d}t}\,+c_{a}\;{\rm
div}\left[\,{\vec{v}_{adv}}\,\right]+{\rm
div}\left[\,{\vec{h}_{a}}\,\right]\;=\,{s}_{a}({\vec{x}},t)\;.$ (37)

The latter can be rephrased in the reference configuration at point $\vec{X}$
and time $t$. To this aim, define the reference molarity of species $a$ as

${c_{a_{R}}}({\vec{X}},t)=\,c_{a}(\,{\vec{x}}({\vec{X}},t\,),t\,)\;J({\vec{X}},t)\;,$
(38)

the reference flux vector ${\vec{h}}_{a_{R}}({\vec{X}},t)$ and the reference
mass supply $s_{a_{R}}({\vec{X}},t)$ as [4]

${\vec{h}}_{a_{R}}=\,J\,{\bm{F}}^{-1}\;{\vec{h}_{a}}(\,{\vec{x}}({\vec{X}},t\,),t\,)\;,\qquad
s_{a_{R}}=\,J\,s_{a}(\,{\vec{x}}({\vec{X}},t\,),t\,)\;,$ (39)

respectively. The reaction rate
$w^{\eqref{eq:actin_polymerization}}({\vec{x}},t)$, being a mass supply, shall
transform according to eq. (39)b. The invariance of $\vartheta_{a}$ with the
configuration and the analysis of the mass action law (35) imply that the
forward and backward “constants”, which encompass the dimensionality of
$w^{\eqref{eq:actin_polymerization}}({\vec{x}},t)$, are not actually constants
in the reference configuration. They rather change with time and with point
$\vec{X}$ according to

$k_{f_{R}}({\vec{X}},t)=\,J({\vec{X}},t)\,k_{f}\;,\qquad
k_{b_{R}}({\vec{X}},t)=\,J({\vec{X}},t)\,k_{b}\;\,$ (40)

The ratio $k_{f_{R}}/k_{b_{R}}$ remains independent upon the configuration.
The referential form of the mass balance equations eventually reads

$\displaystyle\frac{\partial{c_{G_{R}}}}{\partial t}+\,{\rm
Div}\left[\,{{\vec{h}}_{G_{R}}}\,\right]+\,w^{(\ref{eq:actin_polymerization})}_{R}\;=s_{G_{R}}\;,$
(41a) $\displaystyle\frac{\partial{c_{F_{R}}}}{\partial t}+\,{\rm
Div}\left[\,{{\vec{h}}_{F_{R}}}\,\right]-\,w^{(\ref{eq:actin_polymerization})}_{R}\;=s_{F_{R}}\;.$
(41b)

As for the complex molecules, filaments usually have a much smaller mobility
than monomers and might be assumed to be motionless, i.e.

$\vec{h}_{F}={\vec{h}}_{F_{R}}=\vec{0}\;.$ (42)

The diffusion of monomers appears to be much slower than the interaction
kinetics and the concentrations of species may be governed by thermodynamic
equilibrium at all times [36]. The concentration of filaments $c_{F_{R}}$
relates then to the monomers by the equation
$w^{(\ref{eq:actin_polymerization})}=0$, mediated by the local amount of
signaling and stress. Equations (41), with associated initial conditions

$\displaystyle c_{G_{R}}(\vec{X},0)=c^{0}_{G_{R}}(\vec{X})\;,\qquad
c_{F_{R}}(\vec{X},0)=c^{0}_{F_{R}}(\vec{X})\;$

and Dirichlet-Neumann boundary conditions define the relocation of monomers
that undergo polymerization reactions in the reference configuration.

##  6 Mechanical evolution of the cell

Based upon the selection of the mechanisms that are supposed to govern the
structural response of the cell, the balance laws of linear and angular
momentum come out. Literature provides two basic approaches, whether the
structural functions are demanded entirely to the cell membrane [37, 38, 39,
40, 41] or to the development of a cytoskeletal structure within the bulk of
the cell [33, 7, 42, 43, 44, 45, 46, 47, 48]. The influence of curvature on
the elastic stiffness of the membrane appears to be related to the size of the
cell [49] and seems to be negligible for endothelial cells of diameter $\sim
10\mu{\rm m}$. These two evidences lead to consider the reorganization of the
cytoskeleton through a network of actin and intermediate filaments and
microtubules the main responsible for the mechanical response of endothelial
cells, coupled to a passive behavior dictated by the viscosity of the cytosol
as in [33, 7, 48]. Accordingly, balance of linear and angular momentum will be
formulated for the bulk of the cell rather than the membrane.

Forces in continuum mechanobiology are described spatially by contact forces
between adjacent spatial regions (as for the forces exchanged by the substrate
and the cell during adhesion), surface forces exerted on the boundary of the
cell by the environment (as for the receptor-ligand attractive interaction
[23, 50] and repulsive electrostatic interactions), body forces exerted on the
interior points by the environment (as for the gravity or pseudopodia forces
that preside migration). Contact and surface forces, acting on
$\partial\Omega(t)$ will be denoted henceforth with ${\vec{t}}(\vec{x},t)$
whereas body forces will be denoted with ${\vec{b}}(\vec{x},t)$. Their
referential counterparts will inherit the subscript R.

Throughout the rest of the paper we will neglect inertia forces, although some
authors [51] pinpointed the role of inertia forces during migration.
Accordingly, the balance of linear and angular momentum, which are assumed to
hold at each time for all spatial regions ${\cal Q}(t)\subseteq\Omega(t)$,
read:

$\displaystyle\int_{\partial{\cal Q}(t)}{\vec{t}}(\vec{x},t)\;{\rm
d}a+\int_{{\cal Q}(t)}{\vec{b}}(\vec{x},t)\;{\rm d}v=\vec{0}\;,$ (43a)
$\displaystyle\int_{\partial{\cal
Q}(t)}{\vec{r}}\times{\vec{t}}(\vec{x},t)\;{\rm d}a+\int_{{\cal
Q}(t)}{\vec{r}}\times{\vec{b}}(\vec{x},t)\;{\rm d}v=\vec{0}$ (43b)

with $\vec{r}$ denoting the position vector with respect to an arbitrary pole.
Classical arguments of continuum mechanics lead to localize eqs. (43) in the
reference configuration, in terms of the (first) Piola stress tensor
${\bm{P}}$ and of the body forces measured per unit volume in the reference
body

${\vec{b}}_{R}(\vec{X},t)=J(\vec{X},t)\;{\vec{b}}(\vec{x}(\vec{X},t),t)\;.$

The referential local form of the balance of linear momentum reads

${\rm
Div}\left[\,{{\bm{P}}}\,\right]+\vec{b}_{R}=\vec{0}\;,\qquad\vec{X}\in\Omega_{R}\;.$
(44a) The first Piola stress tensor ${\bm{P}}$ must satisfy the local angular
momentum balance ${\bm{P}}{\bm{F}}^{T}={\bm{F}}{\bm{P}}^{T}\;.$ (44b)

###  6.1 Boundary conditions

Contact and surface forces are boundary conditions for problem (44a). They
emanate from electrostatic long or short range interactions, from receptor-
ligand adhesion forces, as well as from contact tractions after adhesion. A
vast literature [52, 53, 35] has been devoted to quantify the forces involved
in these interaction mechanisms. It emerges that uncertainties remain in the
establishment of realistic values for attraction forces, not surprisingly due
to the complexity of the required experimental tasks.

Studies on the influence of non-specific traction forces in cell adhesion were
performed at different time scales, from minutes - as for the spreading of a
mouse embryonic fibroblasts on a matrix-coated surface [54] \- to several
hours - as for a bovine aortic endothelial cells on polyacrylamide gels [17]
\- for different cell sizes. Analyses refer mostly to the early stage of
adhesion: as pointed out in [30], traction models are helpful under specific
conditions and particularly in predicting isotropic early stage of cell
adhesion, which is essentially independent on cytoskeleton remodeling.
Isotropic spreading is made possible by higher ligands densities; at lower
densities of ligands, cells tend to spread anisotropically, by extending
pseudopodia randomly along the cell membrane [17]. This has been made clear
also in modeling micropipette-manipulated red blood cell attachment-detachment
from a substrate [55], which was performed in $\approx 50\ ms$ showing that
after approximately a third of the adhesion-spreading time, the adhesion-
traction forces level off and to further increase spreading area, receptor
diffusion from remote area of the cell to the spreading front is required.

Roughly the same concept has been explored in [56], dealing with charged
flexible particles that adhere to an oppositely charged rigid substrate due to
electrostatic attraction forces. Surface forces drive the adhesion of small
particles. The cell radius in the reference, unstressed configuration was
considered in the micron/sub-micron range $1\ \mu m$ in [56] or even smaller
$12.5\ nm$ in [49].

According to [6], adhesion and spreading also require transport of receptors
from the apical to the basal part of the cell in order to generate attractive
forces.

In this paper we do not account explicitly for integrins, as done in [46]
among others, yet we will use the approaches in [46, 49] to discuss the
magnitude of traction forces in cell spreading. According to [49], Neumann
tractions emanate from short-range, noncovalent interactions between one
receptor and one ligand due to polarization of a non-polar ligand molecule in
the electrostatic field of a charged receptor. The binding force on the
membrane per unit area in the current configuration was given as

$\vec{t}(\vec{x})=-C(Kg_{N}+1)\,((Kg_{N}+1)^{2}+1)\,g_{N}^{-5}\,\exp(-2Kg_{N})\,\rho_{rl}(\vec{x}){\vec{e}}_{2}$
(45)

where: $g_{N}$ is the gap between receptors and ligands, $\rho_{rl}(\vec{x})$
is the minimum concentration of receptors and ligands at location $\vec{x}$,
$C$ is the number of weak noncovalent sub-bonds which form the interaction
between one receptor and one ligand, $K$ is the inverse of the Debye length.
It is of course particularly complex to provide parameters with high accuracy:
assuming that the values provided in [49] apply also to endothelial cells, one
would set $C=1.17\times 10^{-7}{\rm fN}\mu{\rm m}^{-5}$, $K=1$.

The minimum concentration $\rho_{rl}(\vec{x})$ selected in [49] was quite high
($10^{5}$ receptors per $\mu{\rm m}^{2}$) compared to the concentrations of
species that have been measured in [14]. Note also that the term
$\rho_{rl}(\vec{x})$ should not be considered as constant, unless it refers to
all receptors on the membrane, which seems illogical. Therefore, although the
maximum number of moles or molecules per unit area for any species remains
unchanged in time in the current configuration as stated in (18), the
transport processes affect the amount $\rho_{rl}(\vec{x})$ and induce a strong
coupling between mechanical processes in the bulk and chemo-transport
processes on the membrane. Giving these numbers for granted, the resulting
behavior of the Neumann electrostatic attractive tractions is plotted in Fig.
2.

(a) According to [49] (continuum) compared to [46] (dashed)

(b) According to [46] (continuum) compared to [49] (dashed)

Figure 2: Comparison between attractive forces.

According to equation (45), attractive forces are inversely proportional to
the distance between receptors and ligands, and those forces are infinitely
high at contact. To get rid of this paradoxical statement, a strictly positive
lower bound $h_{0}$ shall be defined together with a gap $g_{N}=h-h_{0}$ with
$h_{0}$ being the gap between the cell and the substrate at contact; authors
in [49] suggest $h_{0}=9.0\times 10^{-3}\mu{\rm m}$. Repulsive forces are
expected for distances below such a bound, as in Lennard-Jones potentials, yet
this is not the case of equation (45).

On account of the values provided in [49], accepting also the questionably
high concentration $\rho_{rl}(\vec{x})$ that has been selected therein,
attractive forces turn out to be remarkably high at $h_{0}$, as they concern
integrins binding forces. Nonetheless, attractive forces decay rapidly and at
a distance of ${\rm 0.5}\mu{\rm m}$ they amount to a few $\rm fN/\mu m^{2}$.

Numerical simulations, to appear in a companion publication, show also that
those attractive forces, their range being so short, are not able to cause the
cell spreading unless the characteristic size of the latter becomes very
small. Indeed, authors in [49] considered a cell with radius ($\rm 12.5nm$)
three orders of magnitude smaller than the measured radius of an endothelial
cell in suspension (about $\rm 10{\mu}m$). Size effects in mechanobiology are
well known, and we argue that the role of attractive forces in virus receptors
mediated endocytosis that has been pointed out [9] and in nano-scale cells
studied in [49] does not apply to the spreading of a micron-size endothelial
cell. This remark is somewhat confirmed by analyzing the attractive forces
used in [46], namely

$\vec{t}(\vec{x})=-Q\frac{g_{N}}{\delta_{p}}\,\exp(-\frac{g_{N}}{\delta_{p}})\,{\vec{e}}_{2}$
(46)

with $Q,\delta_{p}$ calibrated as $\rm 50kPa=5*10^{7}fN/{\mu m}^{2}$ and $\rm
0.13\mu m$, respectively. They turn out to be 4 orders of magnitude higher
than (45), in order to allow cell spreading. We could not find justification
in the literature for such a huge value of the ligand-receptor binding force
acting on such a long range extent, hence we argue again that interaction
forces of electrostatic nature are directly responsible of long range
attraction and of spreading. Rather, these interactions are followed by the
extension of pseudopodia from the cell body. As the cell begins to flatten
against the substrate, it forms additional bonds, rearranges its cytoskeleton
to form actin filaments and bundles, creating new focal adhesions. Spreading
thus is a result of extensional and contractile forces exerted by pseudopodia
and the cytoskeleton machinery [17].

##  7 Thermodynamics

The quest of the right thermodynamic principles in mechanobiology is, on one
hand, far from being understood and, from a wider perspective, it paves the
way to boundless questions of philosophical and ethical nature, as for the
establishment of a thermodynamics of life [57], which fall completely out of
the scope of present paper. Major accomplishments have been recently achieved
[58] in formulating fresh concepts that deviate from classical results of
thermodynamics of non equilibrium. In this scientific area, which is nowadays
flourishing, new fundamentals assertions are expected in the years to come.

Being aware of these deficiencies, we admit that our formulation of non
equilibrium thermodynamics [28, 29] may not be able to capture some principles
of mechanobiology that rule the dynamic of receptors - as for the homeostatic
constraint - and we are prone to deepen our formulation in future studies.

###  7.1 Thermodynamics of receptors motion on the membrane

####  7.1.1 Energy Balance

As in section 3, denote with $\Omega(t)$ the advecting cell, and with
$\partial\Omega(t)$ its lipid membrane. Consider an arbitrary region ${\cal
P}(t)\subset\,\partial\Omega(t)$. The first law of thermodynamics represents
the balance of the interplay among the internal energy of ${\cal P}(t)$, the
heat transferred in ${\cal P}(t)$ and the power due to mass exchanged by
receptor dynamics on ${\cal P}(t)$. The energy balance for the problem at hand
reads:

$\frac{{\rm d}\,{\cal U}}{{\rm d}t}({{\cal P}})={\cal Q}_{u}({{\cal P}})+{\cal
T}_{u}({{\cal P}})\;,$ (47)

where ${\cal Q}_{u}$ is the power due to heat transfer and ${\cal T}_{u}$ is
the power due to mass transfer. Denoting with $\partial{\cal P}(t)$ the
bounding closed curve of ${\cal P}(t)$ (see Fig. 1), they read:

$\displaystyle{\cal Q}_{u}$ $\displaystyle=\int_{\cal P}s_{q}\,{\rm
d}a-\oint_{\partial{\cal P}}\vec{q}\cdot\vec{t}_{\perp}\,{\rm d}\ell$ (48a)
$\displaystyle{\cal T}_{u}$ $\displaystyle=\int_{\cal
P}{\mu_{L}^{u}}\,s_{L}+{\mu_{R}^{u}}\,s_{R}\,{\rm d}a-\oint_{\partial{\cal
P}}{\mu_{R}^{u}}\,\vec{h}_{R}\cdot\vec{t}_{\perp}\,{\rm d}\ell$

The time variation of net internal energy ${\cal U}$ thus corresponds to the
power expenditure of two external agents: a heat contribution ${\cal Q}_{u}$
where $s_{q}$ is the heat supplied by external agents and $\vec{q}$ is the
heat flux vector; a mass contribution ${\cal T}_{u}$ in which the scalar
${\mu^{u}_{\beta}}$ denotes the change in specific energy provided by a unit
supply of moles of species $\beta=L,R$. Mass supply $s_{L}$ is the push-
forward of the ligand supply $s_{L_{R}}({\vec{X}},t)$ defined in eq. (31) and
$\vec{h}_{R}$ is the flux of receptors along the membrane in the current
configuration.

The net internal energy can be denoted in terms of specific internal energy
$u$ per unit surface, namely:

${\cal U}({\cal P})=\int_{\cal P}u\,{\rm d}a\,.$ (49)

Applying the surface divergence theorem (4) and mass balances leads from (48)
to

$\displaystyle{\cal Q}_{u}=\int_{\cal P}s_{q}\,{\rm d}a-\int_{{\cal P}}{\rm
div}_{\cal P}\left[\,{\vec{q}}\,\right]\,{\rm d}a\;\,\qquad{\cal
T}_{u}=\int_{\cal P}{\mu_{L}^{u}}\,s_{L}+{\mu_{R}^{u}}\,s_{R}\,{\rm
d}a-\int_{{\cal P}}{\rm div}_{\cal
P}\left[\,{{\mu_{R}^{u}}\,\vec{h}_{R}}\,\right]\,{\rm d}a\;,$ (50a)

whence the first law of thermodynamics arises333Since it must hold for any
region ${\cal P}(t)$, the current configuration local form of the first
principle can be derived exploiting Reynold’s theorem (5) on ${\cal P}(t)$
$\frac{\partial u}{\partial t}\,+\;{\rm div}_{\cal
P}\left[\,{u\,\vec{v}_{adv}}\,\right]=s_{q}-{\rm div}_{\cal
P}\left[\,{\vec{q}}\,\right]+{\mu_{R}^{u}}\frac{\partial{c_{R}}}{\partial
t}+{\mu_{L}^{u}}\frac{\partial{c_{L}}}{\partial
t}+{\mu_{C}^{u}}\frac{\partial{c_{C}}}{\partial
t}-\vec{h}_{R}\cdot{\rm\nabla}_{\cal
P}\left[\,{\mu_{R}^{u}}\,\right]+\left({\mu_{R}^{u}}+{\mu_{L}^{u}}-{\mu_{C}^{u}}\right)w^{(\ref{eq:chem_react})}\;.$
(51)

$\frac{{\rm d}}{{\rm d}t}\int_{\cal P}u\,{\rm d}a=\int_{\cal P}s_{q}\,-{\rm
div}_{\cal P}\left[\,{\vec{q}}\,\right]\,-{\rm div}_{\cal
P}\left[\,{{\mu_{R}^{u}}\,\vec{h}_{R}}\,\right]\,+{\mu_{L}^{u}}\,s_{L}+{\mu_{R}^{u}}\,s_{R}\,{\rm
d}a\;.$

It can be pulled back to the reference configuration in view of definitions of
reference molarity of species ${c_{a_{R}}}({\vec{X}},t)$ in eq. (12), of the
reference flux vector ${\vec{h}}_{a_{R}}({\vec{X}},t)$ and of the reference
mass supply $s_{a_{R}}({\vec{X}},t)$ in eq. (13), which readily extends to
heat fluxes and supplies

$\frac{{\rm d}}{{\rm d}t}\int_{{\cal P}_{R}}u_{R}\,{\rm d}A=\int_{{\cal
P}_{R}}\,s_{q_{R}}\,-{\rm Div}_{{\cal
P}_{R}}\left[\,{\vec{q_{R}}}\,\right]\,-{\rm Div}_{{\cal
P}_{R}}\left[\,{{\mu_{R_{R}}^{u}}\,\vec{h}_{R_{R}}}\,\right]\,+{\mu_{L_{R}}^{u}}\,s_{L_{R}}+{\mu_{R_{R}}^{u}}\,s_{R_{R}}\,{\rm
d}A\;.$ (52)

Since it must hold for any region ${\cal P}_{R}$, the local form of the first
principle can be derived exploiting the mass balance equations (29a), (29c),
(31) in the reference configuration

$\displaystyle\frac{{\rm d}u_{R}}{{\rm d}t}\,$ $\displaystyle=s_{q_{R}}-{\rm
Div}_{{\cal P}_{R}}\left[\,{\vec{q_{R}}}\,\right]-\vec{h}_{R_{R}}\cdot{\rm
Grad}_{{\cal P}_{R}}\left[\,{\mu_{R_{R}}^{u}}\,\right]$
$\displaystyle+{\mu_{R_{R}}^{u}}\,\frac{\partial{c_{R_{R}}}}{\partial
t}+{\mu_{L_{R}}^{u}}\,\frac{\partial{c_{L_{R}}}}{\partial
t}+{\mu_{C_{R}}^{u}}\,\frac{\partial{c_{C_{R}}}}{\partial
t}+\left({\mu_{R_{R}}^{u}}+{\mu_{L_{R}}^{u}}-{\mu_{C_{R}}^{u}}\right)w_{R}^{(\ref{eq:chem_react})}\;.$
(53)

####  7.1.2 Entropy balance equations

The second law of thermodynamics represents the balance of the interplay among
the internal entropy of ${\cal P}$ and the entropy transferred in ${\cal P}$
due to mass exchange and heat transferred on ${\cal P}$. The entropy balance
for the problem at hand reads:

$\frac{{\rm d}S}{{\rm{d}}t}({\cal
P})\,-\,\frac{{\rm{d}}S_{irr}}{{\rm{d}}t}({\cal P})=\,{\cal Q}_{\eta}({{\cal
P}})+{\cal T}_{\eta}({{\cal P}})\;,$ (54)

where $S$ is the net internal entropy of ${\cal P}$, $S_{irr}$ is the entropy
produced inside ${\cal P}$, $Q_{\eta}$ the entropy per unit time due to heat
transfer, $T_{\eta}$ the entropy per unit time due to mass transfer. The
individual contributions read:

$\displaystyle Q_{\eta}$ $\displaystyle=\int_{{\cal P}}\frac{s_{q}}{T}\,{\rm
d}A\,-\,\oint_{\partial{\cal P}}\frac{\vec{q}}{T}\cdot\vec{t}_{\perp}\,{\rm
d}\ell\;,$ (55a) $\displaystyle T_{\eta}$ $\displaystyle=\int_{\cal
P}{\mu_{L}^{\eta}}\,s_{L}+{\mu_{R}^{\eta}}\,s_{R}\,{\rm
d}A-\oint_{\partial{\cal
P}}\mu_{R}^{\eta}\,\vec{h}_{R}\cdot\vec{t}_{\perp}\,{\rm d}\ell\;.$ (55b)

The scalar $\mu_{\beta}^{\eta}$ denotes the change in specific entropy
provided by a unit supply of moles of species $\beta$. Equation
$\eqref{eq:globalentropybalance}$ stems from the non-trivial assumption that
mechanics does not contribute directly to the total entropy flow in the
entropy balance equation [29]. The second law of thermodynamics states that:

$\frac{{\rm{d}}S_{irr}}{{\rm{d}}t}\geq 0.$ (56)

Analogously to the energy counterpart, we define the specific internal entropy
$\eta$ per unit volume and write the entropy imbalance in the reference
configuration as

$\frac{{\rm d}}{{\rm d}t}\int_{{\cal P}_{R}}\,\eta_{R}\,{\rm d}A+\int_{{\cal
P}_{R}}-\frac{s_{q_{R}}}{T}+{\rm Div}_{{\cal
P}_{R}}\left[\,{\frac{\vec{{q_{R}}}}{T}}\,\right]\,-\,{\mu_{L_{R}}^{\eta}}\,s_{L_{R}}-{\mu_{R_{R}}^{\eta}}\,s_{R_{R}}+{\rm
Div}_{{\cal
P}_{R}}\left[\,{\mu_{R_{R}}^{\eta}\,\vec{h}_{R_{R}}}\,\right]\,\,{\rm
d}A\,\geq 0\;.$

After multiplication by $T\geq 0$, replacing $-s_{q_{R}}+\,{\rm Div}_{{\cal
P}_{R}}\left[\,{\vec{q_{R}}}\,\right]$ by means of the energy balance (7.1.1),
and some simple algebra, the local form of the entropy imbalance becomes

$\displaystyle T\,\frac{{\rm d}\eta_{R}}{{\rm d}t}$ $\displaystyle-\frac{{\rm
d}u_{R}}{{\rm d}t}+\,\frac{\partial{c_{R_{R}}}}{\partial
t}\,\left[{\mu_{R_{R}}^{u}}-T\,{\mu_{R_{R}}^{\eta}}\right]+\,\frac{\partial{c_{L_{R}}}}{\partial
t}\,\left[{\mu_{L_{R}}^{u}}-T{\mu_{L_{R}}^{\eta}}\right]+\,\frac{\partial{c_{C_{R}}}}{\partial
t}\left[{\mu_{C_{R}}^{u}}-T{\mu_{C_{R}}^{\eta}}\right]+$ (57)
$\displaystyle-\frac{1}{T}\,\vec{q_{R}}\cdot{\rm Grad}_{{\cal
P}_{R}}\left[\,{T}\,\right]+T\,\vec{h}_{R_{R}}\cdot{\rm Grad}_{{\cal
P}_{R}}\left[\,{\mu_{R_{R}}^{\eta}}\,\right]-\vec{h}_{R_{R}}\cdot{\rm
Grad}_{{\cal P}_{R}}\left[\,{\mu_{R_{R}}^{u}}\,\right]$
$\displaystyle+\left({\mu_{R_{R}}^{u}}-T\,{\mu_{R_{R}}^{\eta}}+{\mu_{L_{R}}^{u}}-\,T{\mu_{L_{R}}^{\eta}}-\mu_{C_{R}}^{u}+\,T{\mu_{C_{R}}^{\eta}}\right)w_{R}^{(\ref{eq:chem_react})}\geq
0$

Denote with $\beta=R,L,C$ and with the symbols $\mu_{\beta_{R}}$,
$A_{R}^{\eqref{eq:chem_react}}$ the quantities

$\mu_{\beta_{R}}=\mu_{\beta_{R}}^{u}-T\,\mu_{\beta_{R}}^{\eta}$ (58)
$A_{R}^{\eqref{eq:chem_react}}=-\mu_{R_{R}}-\mu_{L_{R}}+\mu_{C_{R}}\;.$ (59)

By noting that:

$T\,\vec{h}_{R_{R}}\cdot{\rm Grad}_{{\cal
P}_{R}}\left[\,{\mu_{R_{R}}^{\eta}}\,\right]=\vec{h}_{R_{R}}\cdot{\rm
Grad}_{{\cal
P}_{R}}\left[\,{T\,{\mu_{R_{R}}^{\eta}}}\,\right]-\vec{h}_{R_{R}}\cdot{\rm
Grad}_{{\cal P}_{R}}\left[\,{T}\,\right]\,{\mu_{R_{R}}^{\eta}}$

one finally writes the entropy imbalance as:

$\displaystyle T\,\frac{{\rm d}\eta_{R}}{{\rm d}t}$ $\displaystyle-\frac{{\rm
d}u_{R}}{{\rm d}t}+\,\frac{\partial{c_{R_{R}}}}{\partial
t}\,{\mu_{R_{R}}}+\,\frac{\partial{c_{L_{R}}}}{\partial
t}\,{\mu_{L_{R}}}+\,\frac{\partial{c_{C_{R}}}}{\partial t}{\mu_{C_{R}}}+$ (60)
$\displaystyle-\left(\frac{1}{T}\,\vec{q_{R}}+\,{\mu_{R_{R}}^{\eta}}\vec{h}_{R_{R}}\right)\cdot{\rm
Grad}_{{\cal P}_{R}}\left[\,{T}\,\right]-\vec{h}_{R_{R}}\cdot{\rm Grad}_{{\cal
P}_{R}}\left[\,{\mu_{R_{R}}}\,\right]$ $\displaystyle-
A_{R}^{\eqref{eq:chem_react}}\,w_{R}^{(\ref{eq:chem_react})}\geq 0\;.$

####  7.1.3 Helmholtz Free Energy and thermodynamic restrictions

The referential specific Helmholtz free energy per unit volume is defined as:

$\psi_{R}=u_{R}-T\,\eta_{R}$ (61)

and is taken as a function of temperature and concentrations,
$\psi_{R}\left(T,c_{R_{R}},c_{L_{R}},c_{C_{R}}\right)$. It thus hold:

$\,T\,\frac{{\rm d}\eta_{R}}{{\rm d}t}-\,\frac{{\rm d}u_{R}}{{\rm
d}t}=-\,\frac{{\rm d}\psi_{R}}{{\rm d}t}\,-\eta_{R}\,\frac{\partial
T}{\partial t}=-\frac{\partial\psi_{R}}{\partial c_{L_{R}}}\frac{\partial
c_{L_{R}}}{\partial t}-\frac{\partial\psi_{R}}{\partial
c_{R_{R}}}\frac{\partial c_{R_{R}}}{\partial
t}-\frac{\partial\psi_{R}}{\partial c_{C_{R}}}\frac{\partial
c_{C_{R}}}{\partial t}-\left(\eta_{R}+\frac{\partial\psi_{R}}{\partial
T}\right)\frac{\partial T}{\partial t}$

which can be plugged in $\eqref{eq:ref_entropybalance}$ to derive the entropy
imbalance in the Clausius-Duhem form:

$\displaystyle\left(-\frac{\partial\psi_{R}}{\partial
c_{R_{R}}}+\mu_{R_{R}}\right)\frac{\partial{c_{R_{R}}}}{\partial
t}+\left(-\frac{\partial\psi_{R}}{\partial
c_{L_{R}}}+\mu_{L_{R}}\right)\frac{\partial{c_{L_{R}}}}{\partial
t}+\left(-\frac{\partial\psi_{R}}{\partial
c_{C_{R}}}+\mu_{C_{R}}\right)\frac{\partial{c_{C_{R}}}}{\partial
t}-\left(\eta_{R}+\frac{\partial\psi_{R}}{\partial T}\right)\frac{\partial
T}{\partial t}+$ (62)
$\displaystyle\qquad-\frac{1}{T}\vec{\underline{q_{R}}}\cdot{\rm Grad}_{{\cal
P}_{R}}\left[\,{T}\,\right]-A_{R}^{(\ref{eq:chem_react})}w_{R}^{(\ref{eq:chem_react})}-\vec{h}_{R_{R}}\cdot{\rm
Grad}_{{\cal P}_{R}}\left[\,{\mu_{R_{R}}}\,\right]\,\geq 0$

with
$\vec{\underline{q_{R}}}=\vec{q_{R}}+\,T\,{\mu_{R_{R}}^{\eta}}\vec{h}_{R_{R}}$.
This inequality must hold for any value of the time derivative of the
temperature and of the referential concentrations $c_{R_{R}}$, $c_{L_{R}}$,
and $c_{C_{R}}$. Since they appear linearly in the inequality, the factors
multiplying them must be zero, as otherwise it would be possible to find a
value for the time derivatives that violate the inequality. Therefore, the
following restrictions apply

$\mu_{R_{R}}=\frac{\partial\psi_{R}}{\partial
c_{R_{R}}},\qquad\mu_{L_{R}}=\frac{\partial\psi_{R}}{\partial
c_{L_{R}}},\qquad\mu_{C_{R}}=\frac{\partial\psi_{R}}{\partial
c_{C_{R}}},\qquad\eta_{R}=-\frac{\partial\psi_{R}}{\partial T}\;.$ (63)

In view of formula (63), the amount $\mu_{\beta}$ declared in eq. (58)
acquires the meaning of chemical potential and hence the term
$A^{(\ref{eq:chem_react})}$ in eq. (59) turns out to be the affinity of the
reaction (17). Further remarks on this thermodynamic approach can be found in
[29].

Equation (63) yields to the so called Clausius-Plank inequality:

$\qquad-\frac{1}{T}\vec{\underline{q_{R}}}\cdot{\rm Grad}_{{\cal
P}_{R}}\left[\,{T}\,\right]-A_{R}^{(\ref{eq:chem_react})}w_{R}^{(\ref{eq:chem_react})}-\vec{h}_{R_{R}}\cdot{\rm
Grad}_{{\cal P}_{R}}\left[\,{\mu_{R_{R}}}\,\right]\,\geq 0$ (64)

that splits under the assumptions of Curie’s principle in the following set of
inequalities:

$\displaystyle\frac{1}{T}\vec{\underline{q_{R}}}\cdot{\rm Grad}_{{\cal
P}_{R}}\left[\,{T}\,\right]+\vec{h}_{R_{R}}\cdot{\rm Grad}_{{\cal
P}_{R}}\left[\,{\mu_{R_{R}}}\,\right]\leq 0\;,$ (65a) $\displaystyle
A_{R}^{(\ref{eq:chem_react})}\,w_{R}^{(\ref{eq:chem_react})}\leq 0\;.$ (65b)

####  7.1.4 Constitutive theory

We will assume henceforth that the lipid membrane is in thermal equilibrium,
i.e. ${\rm Grad}_{{\cal P}_{R}}\left[\,{T}\,\right]=\vec{0}$, and that the
Helmholtz free energy density is additively decomposed into three separate
parts:

$\psi_{R}\left(c_{R_{R}},c_{L_{R}},c_{C_{R}}\right)=\psi_{R}^{R}(c_{R_{R}})+\psi_{R}^{L}(c_{L_{R}})+\psi_{R}^{C}(c_{C_{R}})$
(66)

meaning that the contributions of species are uncoupled, neglecting molecular
friction that would lead to a Maxwell-Stefan description of transport. The
free energy density of mobile guest atoms interacting with a host medium is
described by an ideal solution model, which stems from a statistical mechanics
description of the entropy for isolated systems in terms of the density of
states, i.e. the number of possible molecular configurations [59] in the case
of two-state systems. Making recourse to Stirling’s approximation, one finds
that the formula for combinations provides the following free energy density
for the continuum approximation of mixing [59] of the generic species
$\beta=R,L,C$

$\psi_{R}^{\beta}(c_{\beta_{R}})=\mu_{\beta_{R}}^{0}\,c_{\beta_{R}}+R\,Tc_{\beta_{R}}^{max}\left[\vartheta_{\beta_{R}}\ln\vartheta_{\beta_{R}}+(1-\vartheta_{\beta_{R}})\ln(1-\vartheta_{\beta_{R}})\right]\;,$
(67)

with $\vartheta_{\beta_{R}}$ defined in (20) as the ratio between the
concentration and the saturation limit for each species in the reference
configuration. The chemical potential descends from eq. (63)

$\mu_{\beta_{R}}=\frac{\partial\psi_{R}^{\beta}}{\partial
c_{\beta_{R}}}=\mu_{\beta_{R}}^{0}+R\,T\left(\ln\vartheta_{\beta_{R}}-\ln\left(1-\vartheta_{\beta_{R}}\right)\right)\;.$
(68)

A strategy to meet the thermodynamic restriction (65a) is to model the flux of
receptors by Fickian-diffusion, that linearly correlates $\vec{h}_{R_{R}}$ to
the gradient of its chemical potential ${\mu}_{R_{R}}$:

$\vec{h}_{R_{R}}=-{\bm{M}}_{R}(c_{R})\;{\rm Grad}_{{\cal
P}_{R}}\left[\,{{\mu}_{R_{R}}}\,\right]$ (69)

by means of a positive definite mobility tensor ${\bm{M}}_{R}$. The following
isotropic non linear specialization for the mobility tensor ${\bm{M}}_{R}$ is
chosen [60]

${\bm{M}}_{R}(c_{R_{R}})=\mbox{${\rm
u}\mskip-8.0mu|\,$}_{R}\,c_{R_{R}}^{max}\;\vartheta_{R_{R}}\,\left(1-\vartheta_{R_{R}}\right)\;\mathds{1}\;,$
(70)

where $c_{R_{R}}^{max}$ is the saturation limit for receptors, and
$\mbox{${\rm u}\mskip-8.0mu|\,$}_{R}>0$ is the mobility of receptors.
Definition (70) represents the physical requirement that both the pure
($c_{R_{R}}=0$) and the saturated ($c_{R_{R}}=c_{R_{R}}^{max}$) phases have
vanishing mobilities. Neither the mobility $\mbox{${\rm
u}\mskip-8.0mu|\,$}_{R}$ nor the saturation concentration $c_{R_{R}}^{max}$
are assumed to change in time. Whereby experimental data indicate an influence
of temperature, stresses, or concentrations, such a limitation can be removed
without altering the conceptual picture. Noting that

${\rm Grad}_{{\cal
P}_{R}}\left[\,{{\mu}_{R_{R}}}\,\right]=\frac{R\,T}{c_{R_{R}}^{max}}\,\frac{1}{\vartheta_{R_{R}}(1-\vartheta_{R_{R}})}\;{\rm
Grad}_{{\cal P}_{R}}\left[\,{c_{R_{R}}}\,\right]\;,$

Fick’s Law (69) specializes as

$\vec{h}_{R_{R}}=-\mbox{${\rm D}\mskip-8.0mu|\,$}_{R}\,{\rm Grad}_{{\cal
P}_{R}}\left[\,{c_{R_{R}}}\,\right]\;,$ (71)

where $\mbox{${\rm D}\mskip-8.0mu|\,$}_{R}=\mbox{${\rm
u}\mskip-8.0mu|\,$}_{R}\,R\,T$ is the receptor diffusivity.

####  7.1.5 Chemical kinetics

The chemical kinetics of reaction $\eqref{eq:chem_react}$ is modeled via the
law of mass action (28). Experimental evidences [14] show that: (i) the
equilibrium constant (22) is high, thus favoring the formation of ligand-
receptor complex and the depletions of receptors and ligands; (ii) the
diffusion of receptors on the cell membrane is much slower than interaction
kinetics. Accordingly, it can be assumed that the reaction kinetics is
infinitely fast, in the sense that the time required to reach chemical
equilibrium is orders of magnitude smaller than the time-scale of other
processes. For these reasons we assume that the concentrations of species are
ruled by thermodynamic equilibrium at all times, and the concentration of
complex ${c_{C_{R}}}$ is related to the others by the equation (32a). This
very same equation could be derived imposing

$A^{(\ref{eq:chem_react})}=0\;.$

Simple algebra allows deriving eq. (32a), provided that to the equilibrium
constant $K_{\rm eq}^{(\ref{eq:chem_react})}$ the alternative definition

$K_{\rm eq}^{(\ref{eq:chem_react})}=\,\exp\left(-\frac{\Delta
G^{0}}{R\,T}\right)$ (72)

is given, where $\Delta G^{0}=\mu_{C}^{0}-\mu_{L}^{0}-\mu_{R}^{0}$ is the
standard Gibbs free energy.

###  7.2 Thermo-chemo-mechanics of cells

Endothelial cells show two main paradigmatic mechanical attitudes: active and
passive. Active response is related to the ability of the cell to change, as a
result of external cues, its own cytoskeletal conformation, i.e. to reorganize
the morphology of the biopolymers net that provides the structural resistance
during adhesion (to the ECM or to other cells), migration (e.g. chemotaxis,
mechanotaxis, and durotaxis) and division (eg. mitosis). Passive, instead,
refers to the mechanical response that each component of the cell has inasmuch
material bodies, in accordance with their own internal structure and as a
result of external actions.

####  7.2.1 Energy balance

Define in the bulk an arbitrary region ${Q}(t)\subset\Omega(t)$. The energy
balance for the problem at hand, using the notation introduced in [29], reads:

$\frac{{\rm d}\,{\cal U}}{{\rm d}t}({Q})={\cal W}_{u}({Q})+{\cal
Q}_{u}({Q})+{\cal T}_{u}({Q})\;,$ (73)

with $\cal U$ the net internal energy of ${Q}$, ${\cal W}_{u}$ the mechanical
external power, ${\cal Q}_{u}$ the power due to heat transfer, ${\cal T}_{u}$
the power due to mass exchanged by actin dynamics on ${Q}(t)$. It is assumed
that each of these processes is energetically separable in the balance. The
individual contributions read:

$\displaystyle{\cal W}_{u}({Q})$ $\displaystyle=$
$\displaystyle\int_{Q}\vec{b}\cdot\vec{v}\;{\rm d}\Omega+\int_{\partial
Q}\,\vec{t}\cdot\vec{v}\;{\rm d}\Gamma\;,$ (74a) $\displaystyle{\cal
Q}_{u}({Q})$ $\displaystyle=$ $\displaystyle\int_{Q}\,s_{q}\,{\rm
d}\Omega-\int_{\partial Q}\,\vec{q}\cdot\vec{n}\;{\rm d}\Gamma\;,$ (74b)
$\displaystyle{\cal T}_{u}({Q})$ $\displaystyle=$
$\displaystyle\int_{Q}\,\mu^{u}_{G}\,s_{G}\,+\,\mu^{u}_{F}\,s_{F}\;{\rm
d}\Omega-\int_{\partial Q}\,\mu^{u}_{G}\,\vec{h}_{G}\cdot\vec{n}\,{\rm
d}\Gamma\;.$ (74c)

Assumption (42) has been accounted for in the mass transfer contribution
${\cal T}_{u}({Q})$.

The time variation of net internal energy $\cal U$ corresponds to the power
expenditure of external agencies: a mechanical contribution ${\cal W}_{u}$ due
to body forces $\vec{b}$ and surface tractions $\vec{t}$ that do work on
velocities $\vec{v}$; a heat contribution ${\cal Q}_{u}$ where $s_{q}$ is the
heat supplied by external agencies and $\vec{q}$ is the heat flux vector; a
mass contribution ${\cal T}_{u}$ in which the scalar $\mu^{u}_{\beta}$ denotes
the change in specific energy provided by a unit supply of moles of
$\beta=G,F$ actin. Mass supply $s_{\beta}$ is the push-forward of the supply
$s_{\beta_{R}}({\vec{X}},t)$ defined in eq. (39) and $\vec{h}_{G}$ is the flux
of G-actin in the current configuration.

Standard application of the divergence theorem and of balance equations leads
from (74a) to

$\displaystyle{\cal W}_{u}({Q})$ $\displaystyle=$
$\displaystyle\int_{Q}\,{\bm{\sigma}}:{{\bm{l}}}\,{\rm d}\Omega\;.$ (75)

where ${{\bm{l}}}$ is the gradient of velocity tensor, i.e.
${{\bm{l}}}={{\rm\nabla}\left[\,{\vec{v}}\,\right]}$ and ${\bm{\sigma}}$ is
the Cauchy stress tensor. Since it is well known that

${\bm{\sigma}}:{{\bm{l}}}\,{\rm d}\Omega={\bm{P}}:{\dot{{\bm{F}}}}\,{\rm
d}\Omega_{R}={\bm{S}}:{\dot{{\bm{E}}}}\,{\rm d}\Omega_{R}\;,$

the mechanical power expenditure can be written in terms of the first Piola-
Kirchoff stress ${\bm{P}}=J\,{\bm{\sigma}}\,{\bm{F}}^{-T}$ or of the second
Piola-Kirchoff stress ${\bm{S}}={\bm{F}}^{-1}{\bm{P}}$ in the referential
configuration. Analogously, by defining the referential heat flux
$\vec{q}_{R}=J\,{\bm{F}}^{-1}\,\vec{q}$ and making use of Nanson’s formula, it
holds

$\vec{q}\cdot\vec{n}\;{\rm d}\Gamma=\vec{q}_{R}\cdot\vec{n}_{R}\;{\rm
d}\Gamma_{R}\;.$ (76)

As usual in the thermodynamics of continua, see e.g. [4], one can make use of
the specific internal energy $u_{R}$ per unit volume in the reference
configuration to write the referential local form of the first principle as

$\displaystyle\frac{{\rm d}u_{R}}{{\rm d}t}$
$\displaystyle={\bm{S}}:{\dot{{\bm{E}}}}\ +s_{q_{R}}-{\rm
Div}\left[\,{\vec{q}_{R}}\,\right]-\vec{h}_{G_{R}}\cdot{\rm
Grad}\left[\,{\mu_{G_{R}}^{u}}\,\right]$ (77)
$\displaystyle+{\mu_{G_{R}}^{u}}\,\frac{\partial{c_{G_{R}}}}{\partial
t}+{\mu_{F_{R}}^{u}}\,\frac{\partial{c_{F_{R}}}}{\partial
t}+\left({\mu_{G_{R}}^{u}}-{\mu_{F_{R}}^{u}}\right)w_{R}^{(\ref{eq:actin_polymerization})}\;.$
(78)

####  7.2.2 Entropy imbalance

The second law of thermodynamics represents the balance of the interplay among
the internal entropy of $Q$ and the entropy transferred in it due to mass
exchange and heat transferred. We make the non-trivial assumption that
mechanics does not contribute directly to the total entropy flow in the
entropy balance equation, as profoundly elaborated in [28, 61]. The entropy
balance for the problem at hand reads:

$\frac{{\rm d}\,{\cal S}}{{\rm d}t}({Q})-\frac{{\rm d}\,{\cal S}_{i}}{{\rm
d}t}({Q})={\cal Q}_{\eta}({Q})+{\cal T}_{\eta}({Q})\;,$ (79)

where $\cal S$ is the net internal entropy of ${Q}$, ${\cal S}_{i}$ is the
entropy produced inside ${Q}$, ${\cal Q}_{\eta}$ the entropy per unit time due
to heat transfer, ${\cal T}_{\eta}$ the entropy per unit time due to mass
transfer. The individual contributions read:

$\displaystyle{\cal Q}_{\eta}({Q})$ $\displaystyle=$
$\displaystyle\int_{Q}\,\frac{s_{q}}{T}\,{\rm d}\Omega-\int_{\partial
Q}\,\frac{\vec{q}}{T}\cdot\vec{n}\;{\rm d}\Gamma\;,$ (80) $\displaystyle{\cal
T}_{\eta}(Q)$ $\displaystyle=$
$\displaystyle\int_{Q}\,\mu^{\eta}_{G}\,s_{G}\,+\,\mu^{\eta}_{F}\,s_{F}\;{\rm
d}\Omega-\int_{\partial Q}\,\mu^{\eta}_{G}\,\vec{h}_{G}\cdot\vec{n}\,{\rm
d}\Gamma\;.$ (81)

The second law of thermodynamics states that

$\frac{{\rm d}\,{\cal S}_{i}}{{\rm d}t}({Q})\geq 0\;.$

As for the energy, one can make use of the specific internal entropy
$\eta_{R}$ per unit referential volume to localize and rephrase the entropy
imbalance in terms of internal energy taking advantage of identity (77) and of
the sign definiteness of temperature :

$\displaystyle T\;\frac{\rm d}{{\rm d}t}\;\eta_{R}\,-\frac{{\rm d}u_{R}}{{\rm
d}t}+{\bm{S}}:{\dot{{\bm{E}}}}\ +\frac{\partial c_{G_{R}}}{\partial
t}\,\mu_{G_{R}}+\frac{\partial c_{F_{R}}}{\partial t}\,\mu_{F_{R}}+$ (82)
$\displaystyle\qquad-\left(\frac{1}{T}\,\vec{q_{R}}+\,{\mu_{G_{R}}^{\eta}}\vec{h}_{G_{R}}\right)\cdot{\rm
Grad}\left[\,{T}\,\right]-\,\vec{h}_{G_{R}}\cdot{\rm
Grad}\left[\,{\mu_{G_{R}}}\,\right]-A_{R}^{\eqref{eq:actin_polymerization}}\,w_{R}^{(\ref{eq:actin_polymerization})}\geq
0\;,$

having denoted with $\beta=G,F$ and with the symbols $\mu_{\beta_{R}}$,
$A_{R}^{\eqref{eq:actin_polymerization}}$ the quantities

$\mu_{\beta_{R}}=\mu_{\beta_{R}}^{u}-T\,\mu_{\beta_{R}}^{\eta}$ (83)
$A_{R}^{\eqref{eq:actin_polymerization}}=-\mu_{G_{R}}+\mu_{F_{R}}\;.$ (84)

####  7.2.3 Helmholtz free energy and thermodynamic restrictions

The referential specific Helmholtz free energy per unit volume
$\psi_{R}\left(T,c_{G_{R}},c_{F_{R}},{\bm{C}},{\bm{\xi}}\right)$, defined as
in (61), is taken as a function of temperature, strains (either ${\bm{C}}$ or
${\bm{E}}$), concentrations $c_{G_{R}},c_{F_{R}}$, and of some kinematic
internal variables ${\bm{\xi}}$ that compare with the usual meaning in
inelastic constitutive laws [4, 20, 61, 62, 63, 64]. It follows that

$\,T\,\frac{{\rm d}\eta_{R}}{{\rm d}t}-\,\frac{{\rm d}u_{R}}{{\rm
d}t}=-\,\frac{{\rm d}\psi_{R}}{{\rm d}t}\,-\eta_{R}\,\frac{\partial
T}{\partial t}\;,$ (85)

which can be inserted in (88) to derive the entropy imbalance in final form:

$\displaystyle-\,\frac{{\rm d}\psi_{R}}{{\rm d}t}\,-\eta_{R}\,\frac{\partial
T}{\partial t}+{\bm{S}}:{\dot{{\bm{E}}}}\ +\frac{\partial c_{G_{R}}}{\partial
t}\,\mu_{G_{R}}+\frac{\partial c_{F_{R}}}{\partial t}\,\mu_{F_{R}}+$ (86)
$\displaystyle\qquad-\left(\frac{1}{T}\,\vec{q_{R}}+\,{\mu_{G_{R}}^{\eta}}\vec{h}_{G_{R}}\right)\cdot{\rm
Grad}\left[\,{T}\,\right]-\,\vec{h}_{G_{R}}\cdot{\rm
Grad}\left[\,{\mu_{G_{R}}}\,\right]-A_{R}^{\eqref{eq:actin_polymerization}}\,w_{R}^{(\ref{eq:actin_polymerization})}\geq
0\;.$

In view of the stated functional dependency of the free energy, its total
derivative with respect to time reads:

$\frac{{\rm d}}{{\rm
d}t}{\psi_{R}(T,c_{G_{R}},c_{F_{R}},{{\bm{C}}},{\bm{\xi}})}=\frac{\partial\psi_{R}}{\partial
T}\,\frac{\partial T}{\partial t}+\frac{\partial\psi_{R}}{\partial
c_{G_{R}}}\,\frac{\partial c_{G_{R}}}{\partial
t}+\frac{\partial\psi_{R}}{\partial c_{F_{R}}}\,\frac{\partial
c_{F_{R}}}{\partial
t}+\frac{\partial\psi_{R}}{\partial{\bm{C}}}\,:\,\dot{{\bm{C}}}+\frac{\partial\psi_{R}}{\partial{\bm{\xi}}}\,:\,\frac{\partial{\bm{\xi}}}{\partial
t}$ (87)

The Clausius-Duhem inequality yields:

$\displaystyle\left(-\frac{\partial\psi_{R}}{\partial
c_{G_{R}}}+\mu_{G_{R}}\right)\frac{\partial{c_{G_{R}}}}{\partial
t}+\left(-\frac{\partial\psi_{R}}{\partial
c_{F_{R}}}+\mu_{F_{R}}\right)\frac{\partial{c_{F_{R}}}}{\partial
t}+\,\frac{\partial T}{\partial
t}\,\left(-\eta_{R}-\frac{\partial\psi_{R}}{\partial
T}\right)+\;\dot{{\bm{C}}}:\left(\frac{1}{2}\,{\bm{S}}-\frac{\partial\psi_{R}}{\partial{\bm{C}}}\right)+$
$\displaystyle\qquad-\frac{\partial\psi_{R}}{\partial{\bm{\xi}}}\,:\,\frac{\partial{\bm{\xi}}}{\partial
t}-\left(\frac{1}{T}\,\vec{q_{R}}+\,{\mu_{G_{R}}^{\eta}}\vec{h}_{G_{R}}\right)\cdot{\rm
Grad}\left[\,{T}\,\right]-\,\vec{h}_{G_{R}}\cdot{\rm
Grad}\left[\,{\mu_{G_{R}}}\,\right]-A_{R}^{\eqref{eq:actin_polymerization}}\,w_{R}^{(\ref{eq:actin_polymerization})}\geq
0\;.$ (88)

This inequality must hold for any value of the time derivative of the
temperature, the referential concentrations, the strain tensor. Since they
appear linearly in the inequality, the factors multiplying them must be zero,
as otherwise it would be possible to find a value for the time derivatives
that violate the inequality. Therefore, the following prescriptions apply

${\bm{S}}=2\,\frac{\partial\psi_{R}}{\partial{\bm{C}}}\;,\qquad\eta_{R}=-\frac{\partial\psi_{R}}{\partial
T}\;,\qquad\mu_{G_{R}}=\frac{\partial\psi_{R}}{\partial
c_{G_{R}}}\;,\qquad\mu_{F_{R}}=\frac{\partial\psi_{R}}{\partial c_{F_{R}}}\;.$
(89a) The internal force, conjugate to ${\bm{\xi}}$, will be denoted with the
symbol ${\bm{\chi}}$, i.e.
${\bm{\chi}}_{R}=-\frac{\partial\psi_{R}}{\partial{\bm{\xi}}}\;.$ (89b)

Equation (89a) yields to the Clausius-Plank inequality, which under the
assumptions of Curie symmetry principle [28], can be written as

$\displaystyle{\bm{\chi}}_{R}\,:\,\dot{{\bm{\xi}}}\geq 0$ (90a)
$\displaystyle\left(\frac{1}{T}\,\vec{q_{R}}+\,{\mu_{G_{R}}^{\eta}}\vec{h}_{G_{R}}\right)\cdot{\rm
Grad}\left[\,{T}\,\right]+\,\vec{h}_{G_{R}}\cdot{\rm
Grad}\left[\,{\mu_{G_{R}}}\,\right]\leq 0$ (90b) $\displaystyle
A_{R}^{\eqref{eq:actin_polymerization}}\,w_{R}^{(\ref{eq:actin_polymerization})}\leq
0$ (90c)

####  7.2.4  Decompositions.

The stress filed ${\bm{S}}$ will be additively decomposed in the sum of the
active and passive contributions, analogously to generalized Maxwell models

${\bm{S}}={\bm{S}}_{active}+{\bm{S}}_{passive}\;.$ (91)

Active response is related to cytoskeletal reorganization in stress fibers and
pseudopodia, whereas the passive response reflects the mechanical behavior
that each component of the cell has inasmuch material bodies.

We base the theory for pseudopodia on a multiplicative decomposition of the
deformation gradient

${\bm{F}}=\,{\bm{F}}^{e}\,{\bm{F}}^{c}\;.$ (92)

Tensor ${\bm{F}}^{c}$, named swelling distortion is the local distortion of
the material neighborhood of a point due to a volumetric swelling (de-
swelling) due to the phase change of actin, from monomeric to a network of
filaments and vice-versa. Its representation will be taken as
${\bm{F}}^{c}=\lambda^{c}\,\mathds{1}$, assuming therefore that a dense
network of actin filaments form in pseudopodia. This approach conforms well
for lamellipodia filament networks, although it might result inappropriate for
slender and highly oriented microstructures seen in filopodia, which might be
better captured by the protrusion-contraction uniaxial tensors presented in
[51] or [65]. The following identities can be easily assessed:

${\det}\left[\,{{\bm{F}}^{c}}\,\right]=J^{c}={\lambda^{c}}^{3}\;,\qquad{\dot{J}}^{c}/J^{c}=3\,{\dot{\lambda}}^{c}/{\lambda^{c}}\;,\qquad{\bm{l}}^{c}={\dot{{\bm{F}}}}^{c}\,{{\bm{F}}^{c}}^{-1}={\dot{J}}^{c}/(3J^{c})\mathds{1}\;.$
(93)

We assume that changes in $J^{c}$ occur because of changes in filaments
$J^{c}=J^{c}(c_{F_{R}})$ and define the partial molar volume of the
pseudopodia as

$\Omega_{C}(c_{F_{R}})=\frac{{\rm d}J^{c}}{{\rm d}c_{F_{R}}}$ (94)

and it holds

${\dot{J}}^{c}=\Omega_{C}(c_{F_{R}})\,\frac{{\partial}c_{F_{R}}}{{\partial}t}\;.$
(95)

The decomposition (92) leads to a multiplicative decomposition for the left
Cauchy-Green tensor, too:

${\bm{C}}=\,{\bm{C}}^{e}\,{\bm{C}}^{c}\;$ (96)

with the swelling factor ${\bm{C}}^{c}={J^{c}}^{2/3}\;\mathds{1}$ and the
elastic factor ${\bm{C}}^{e}={J^{c}}^{-2/3}\;{\bm{C}}\;.$ A classical [60]
specification of $J^{c}(c_{F_{R}})$ is the affine map

$J^{c}(c_{F_{R}})=1+(c_{F_{R}}-c_{F_{R}}^{0})\,\Omega_{C}$ (97)

with a constant partial molar volume $\Omega_{C}>0$.

In the realm of viscoelasticity, it is also common to perform a multiplicative
decomposition of the deformation gradient ${\bm{F}}^{e}$ into volumetric
${\bm{F}}^{e^{v}}$ and isochoric ${\bm{F}}^{e^{i}}$ factors

${\bm{F}}^{e}=\,{\bm{F}}^{e^{v}}\,{\bm{F}}^{e^{i}}\;.$ (98)

The volumetric factor ${\bm{F}}^{e^{v}}={J^{e}}^{1/3}\;\mathds{1}$ turns out
to be completely identified by the determinant of ${\bm{F}}^{e}$, whereas the
isochoric factor ${\bm{F}}^{e^{i}}={J^{e}}^{-1/3}\;{\bm{F}}^{e}$ obeys to the
constraint ${\det}\left[\,{{\bm{F}}^{e^{i}}}\,\right]=1$. The decomposition
(98) leads to a multiplicative decomposition for the left Cauchy-Green tensor,
too:

${\bm{C}}^{e}=\,{\bm{C}}^{e^{v}}\,{\bm{C}}^{e^{i}}\;,$ (99)

with volumetric factor ${\bm{C}}^{e^{v}}={J^{e}}^{2/3}\;\mathds{1}$ and the
isochoric factor ${\bm{C}}^{e^{i}}={J^{e}}^{-2/3}\;{\bm{C}}^{e}\;.$

####  7.2.5 Constitutive theory

Two among the several ways to satisfy the thermodynamic restriction (90b) have
been discussed in [29] in the framework of trapping. Here, we proceed as for
the membrane imposing that the cytosol stands in thermal equilibrium, whereby
${\rm Grad}\left[\,{T}\,\right]=\vec{0}$. The flow of actin monomers is
linearly related to the gradient of their chemical potential by Fick’s
assumption, consistently with the thermodynamic restriction (90b):

$\displaystyle\vec{h}_{G_{R}}=-{\bm{M}}_{G_{R}}(c_{G_{R}})\;{\rm
Grad}\left[\,{{\mu}_{G_{R}}}\,\right]\;.$ (100a)

The following isotropic non linear specialization for the mobility tensor
${\bm{M}}_{G_{R}}$ is chosen [60]

${\bm{M}}_{G_{R}}(c_{G_{R}})=\mbox{${\rm
u}\mskip-8.0mu|\,$}_{G_{R}}\,c_{G_{R}}^{max}\;\vartheta_{G_{R}}\,\left(1-\vartheta_{G_{R}}\right)\;\mathds{1}\;,$
(101)

where $c_{G_{R}}^{max}$ is the saturation limit for receptors, and
$\mbox{${\rm u}\mskip-8.0mu|\,$}_{G_{R}}>0$ is the mobility of actin monomers.
Assuming that the trapped species $F$ has vanishing mobility is an alternative
view of modeling the absence of their flux.

The Helmholtz free energy density $\psi_{R}$ is modeled by decomposing it into
separate parts: a thermal contribution $\psi_{R}^{th}$, a diffusive
contribution $\psi_{R}^{diff}$, an elastic contribution $\psi_{R}^{el}$, and
an inelastic (also called configurational ) counterpart $\psi_{R}^{in}$

$\psi_{R}(T,c_{G_{R}},c_{F_{R}},{{\bm{C}}},{\bm{\xi}})=\psi_{R}^{th}(T)+\psi_{R}^{diff}(c_{G_{R}},c_{F_{R}})+\psi_{R}^{el}(c_{F_{R}},{{\bm{C}}})+\psi_{R}^{in}(c_{F_{R}},{{\bm{E}}},{\bm{\xi}})\;.$
(102)

This splitting is here taken for granted without motivation. We will not
indulge in the description of $\psi_{R}^{th}$ (see [29] in case of interest)
and we’ll rather focus on the remaining parts.

Statistical mechanics depicts the entropy for isolated systems in terms of the
density of states, the number of possible molecular configurations [59].
Making recourse to Stirling’s approximation and since the entropy transforms
with the volume by means of $J$, one finds that the following well-known
expression of the entropy of mixing in the reference configuration arises:

$\eta_{\beta_{R}}^{diff}=-R\,J\,c_{\beta}^{max}\,\left(\vartheta_{\beta}\,\ln[\vartheta_{\beta}]+(1-\vartheta_{\beta})\,\ln[1-\vartheta_{\beta}]\right)\;,$
(103)

the universal gas constant $R$ being the product of Boltzmann constant $k_{B}$
and Avogadro’s number and having denoted with $\beta=G,F$ and with
$\vartheta_{\beta_{R}}$ the ratio

$\vartheta_{\beta_{R}}({\vec{X}},t)={c_{\beta_{R}}}/{c_{\beta_{R}}^{max}}\;.$
(104)

We argued in eq. (18) that, in view of the structure of the lipid membranes,
the maximum number of moles or molecules per unit area for any species remains
unchanged in time in the current configuration. The same argument does not
seem to apply for the bulk, hence we take henceforth that

${c_{\beta_{R}}^{max}}({\vec{X}},t)=\,c_{\beta}^{max}({\vec{x}}({\vec{X}},t),t)\,J({\vec{X}},t)$
(105)

is constant and write the free energy density for the continuum approximation
of mixing [59] as

$\displaystyle\psi_{R}^{diff}(c_{G_{R}},c_{F_{R}})$
$\displaystyle=\;\mu_{G_{R}}^{0}\,c_{G_{R}}+R\,Tc_{G_{R}}^{max}\left[\vartheta_{G_{R}}\ln\vartheta_{G_{R}}+(1-\vartheta_{G_{R}})\ln(1-\vartheta_{G_{R}})\right]$
(106)
$\displaystyle\;+\mu_{F_{R}}^{0}\,c_{F_{R}}+R\,Tc_{F_{R}}^{max}\left[\vartheta_{F_{R}}\ln\vartheta_{F_{R}}+(1-\vartheta_{F_{R}})\ln(1-\vartheta_{F_{R}})\right]\;.$

Note that if the saturation is constant in the current configuration, an
explicit coupling of the free energy of mixing with the deformation arises by
means of $J$. A new stress would come out, in view of the thermodynamic
prescription (89a).

Following [61], we will define visco-elastic materials based on the
multiplicative decomposition (99). Specifically, the free energy for visco-
elastic materials will be defined as follows

$\psi_{R}^{el}(c_{F_{R}},{{\bm{C}}})+\psi_{R}^{in}(c_{F_{R}},{{\bm{E}}},{\bm{\xi}})=\psi_{R}^{el,vol}(c_{F_{R}},{\bm{C}}^{e^{v}})+\psi_{R}^{el,iso}(c_{F_{R}},{\bm{C}}^{e^{i}})+\psi_{R}^{in}(c_{F_{R}},{\bm{E}}^{e}-{\bm{\xi}})\;.$
(107)

with $\psi_{R}^{in}$ depending upon ${{\bm{E}}}^{e}$ by means of
${{\bm{C}}^{e^{i}}}$. The volumetric part of the elastic free energy is
defined through $J^{e}$, highlighting the role of the swelling tensor and of
the concentration of pseudopodia, since

${\bm{C}}^{e^{v}}={J^{e}}^{2/3}\;\mathds{1}={J}^{2/3}\;{J^{c}}^{-2/3}\;\mathds{1}=\left(\frac{J}{1+(c_{F_{R}}-c_{F_{R}}^{0})\,\Omega_{C}}\right)^{2/3}\;\mathds{1}$
(108)

in view of eq. (97). On the other end, it holds

${\bm{C}}^{e^{i}}={\bm{C}}^{e}\,{J^{e}}^{-2/3}={\bm{C}}\,{J^{c}}^{-2/3}\,{J^{e}}^{-2/3}={\bm{C}}\,{J}^{-2/3}$
(109)

hence ${\bm{C}}^{e^{i}}$ depends merely upon the state of deformation and not
upon the concentration of species. This outcome reverberates upon the
energetic contributions $\psi_{R}^{el,iso}$ and $\psi_{R}^{in}$. The latter is
such that

$\frac{\partial\psi_{R}^{in}}{\partial{\bm{E}}}=-\frac{\partial\psi_{R}^{in}}{\partial{\bm{\xi}}}\;,$
(110)

a property physically grounded in the rheological model of Maxwell, for which
we refer to [61] or [66].

Provided that the above holds, the selection for $\psi_{R}^{el}$ and
$\psi_{R}^{in}$ is arbitrary. Their selection shall be different in modeling
the passive behavior or the active response of pseudopodia and stress fibers.
The elastic, reversible behavior that occurs once the viscous effects vanish
(ideally at $t\rightarrow\infty$ ) is captured by $\psi_{R}^{el}$. The
inelastic free energy accounts for the non-equilibrium response due to
viscosity - the so called dissipation potential. By thermodynamic restrictions
(89b) and identity (110)

$\displaystyle{\bm{\chi}}_{R}$
$\displaystyle=-\frac{\partial\psi_{R}^{in}}{\partial{\bm{\xi}}}=\frac{\partial\psi_{R}^{in}}{\partial{\bm{E}}}$
(111a) $\displaystyle{\bm{S}}$
$\displaystyle=2\,\frac{\partial\psi_{R}^{el}}{\partial{\bm{C}}}+{\bm{\chi}}_{R}\;.$
(111b)

According to eq. (111b), tensorial internal forces ${\bm{\chi}}_{R}$ can be
interpreted as a non-equilibrium stress tensor of second Piola-Kirchoff kind,
that accounts for the viscous response.

Inelastic internal entropy production (90a) was described by the internal flux
variables ${\bm{\xi}}$ and by their energy-conjugate forces ${\bm{\chi}}_{R}$.
A simple way to satisfy constraint (90a) is choosing a positive definite
operator ${\mathds{L}}$ such that

$\displaystyle{\bm{\chi}}_{R}$
$\displaystyle={\mathds{L}}\,\dot{{\bm{\xi}}}\;.$ (112)

In case of isotropy, the fourth order operator ${\mathds{L}}$ restricts to the
scalar viscosity $\nu$ times the identity operator. Equations (111a), (112)
provide evolution equations for ${\bm{\chi}}_{R}$ that allow the algorithmic
integration of the constitutive law once a selection for the free energy
densities $\psi_{R}^{el}$ and $\psi_{R}^{in}$ is made.

The chemical potential of G-actin monomers and of F-actin networks descends
from thermodynamic prescriptions (89a), in the form

$\displaystyle\mu_{G_{R}}$
$\displaystyle=\frac{\partial\psi_{R}^{diff}(c_{G_{R}},c_{F_{R}})}{\partial
c_{G_{R}}}$ (113a) $\displaystyle\mu_{F_{R}}$
$\displaystyle=\frac{\partial\psi_{R}^{diff}(c_{G_{R}},c_{F_{R}})}{\partial
c_{F_{R}}}+\frac{\partial\psi_{R}^{el,vol}(c_{F_{R}},{\bm{C}}^{e^{v}})}{\partial
c_{F_{R}}}+\frac{\partial\psi_{R}^{el,iso}(c_{F_{R}},{\bm{C}}^{e^{i}})}{\partial
c_{F_{R}}}+\frac{\partial\psi_{R}^{in}(c_{F_{R}},{\bm{E}}^{e}-{\bm{\xi}})}{\partial
c_{F_{R}}}\;.$ (113b)

While the chemical potential of actin monomers has merely an entropic nature,
mechanical contributions enter the definition of the chemical potential of
actin networks. Specifically, mechanics affects $\mu_{F_{R}}$ in the
volumetric contribution $\psi_{R}^{el,vol}$ through the swelling tensor
${\bm{C}}^{e^{v}}$ (108), whereas the isochoric tensor ${\bm{C}}^{e^{i}}$ was
proven to be independent upon the concentration of species in eq. (109).
Nonetheless, the parameters of the viscoelastic loading-unloading law are
expected to depend upon the extent of the polymerization reaction by means of
the network concentration $c_{F_{R}}$ in all terms of the mechanical free
energy.

The mechanical effect on the chemical potential does not propagate into the
mass flux because the mobility of actin network is assumed to be negligible.
Mechanics however enters the affinity of polymerization reaction (34) in view
of definition (84). The stress state is expected to favor polymerization
nearby the lipid membrane and depolymerization towards the nucleus.

####  7.2.6 The multiscale scenario of cell viscoelasticity

Although the mechanical framework of the free energy depicted above is rather
clear, a specialization of the constitutive equations has not been attempted
here and in many cases (as for stress fibers and microtubules) it has not been
attempted in the literature, to the best of our knowledge. The complexity
leads in the multiscale scenario of cell viscoelasticity: while the mechanical
behavior and properties of intermediate filaments, actin filaments, and
microtubules has been nowadays quite clarified, at least in terms of relative
stiffness and strengths, bundles of the filaments, their response,
polymerization, shape and time evolution are not yet captured by comprehensive
models at the “macroscopic” scale through appropriate free energies. As a
consequence, the ability of models to capture the mechanics of fundamental
cellular processes (as chemotaxis, cell sprouting, junction and
differentiation, endocytosis and exocytosis to cite a few) still requires
abundant research before gaining predicting capabilities in simulations.

The cytoskeleton, an interconnected network of regulatory proteins and
filamentous biological polymers, undergoes massive reorganization during cell
deformation, especially after cell rolling and adhesion [35, 67] and in
mediating, sensing and transduction of mechanical cues from the micro-
environment [68]. Homogenized models for the mechanical response of a cell
shall include in effective, macroscopic properties the
polymerisation/depolimerisation of filaments, the process of cross-linking
that determine the architecture of cytoskeletal filaments, and the passive
mechanical properties of the cytosol. In view of the above, the thermodynamics
of statistically-based continuum theories for polymers with transient networks
[69, 70, 71, 72, 73] appear to be good candidates for the selection of free
energies $\psi_{R}^{el}(c_{F_{R}},{{\bm{C}}})$ and
$\psi_{R}^{in}(c_{F_{R}},{{\bm{E}}},{\bm{\xi}})$. The need of statistical
approaches to model the time-dependent response of polymers with reversible
cross-links emerges, since the overall response is influenced by rate of
assembly and disassembly of cross-linking factors that is controlled at
molecular level by actin nucleation, capping, severing factors and by the
activity of molecular motors such myosin-II, which, in combination with cross-
linkers, appears to be responsible of the viscoelastic properties of the
cytoskeleton [74]. At present however, such a comprehensive model has not yet
been proposed for the pseudopodia driven cell motion. Classical models as
hyperelastic Saint-Venant [51] or newtonian viscous fluids [75] eventually
surrounded by a hyperelastic, zero-thickness membrane [76] have been used for
the pseudopodia, whereas a very large amount of literature concerns pseudopod
dynamics ( see for instance [77] and the large literature therein ) or ameboid
motion [78] with no account for their mechanical response. Different
approaches to cell motility, as for active gel theory coupled to the classical
theory of thin elastic shells, are also widely used [79], but are not
discussed in this work. The framework described herein, including myosin
dynamics, phase transformations between G-actin and F-actin, has been depicted
in a set of publications by the group of H. Gomez [34, 80]. The flow of the
F-actin network was treated as a Newtonian fluid and directed by its velocity.
A one dimensional yet comprehensive model has been proposed in [81]. Not
surprisingly, the nucleus and its meshwork of intermediate filaments formed
mostly of proteins (nuclear lamina), contribute to the viscoelasticity of
cells [82]. Depending upon the content of Lamins, the nucleus becomes more or
less stiff, impacting cell migration: nuclear deformation facilitates cell
migration through complex environments, whereas its stiffness may act as a
mechanical barrier for a migratory cell [83]. Cells are capable to modify
their viscoelasticity while migrating across confined spaces [84], a very
intriguing mechanism yet complex to be captured macroscopically in view of its
multiscale nature.

The multiscale scenario is invoked also for cell contractility. There are
evidences [85] that the interaction among filaments, motors, and cross-linkers
is mechanically stimulated. As reported in [68], myosin binding to actin
fibers occurs in a force-dependent manner, as well as the contractile response
of actomyosin to extracellular stiffness. According to [86], force feedback
controls motor activity and increases density and mechanical efficiency of
self-assembling branched actin networks, thus suggesting that those feedbacks
could allow migratory cells adjusting their viscoelastic properties to favor
migration. Mass transport and cell contractility have been accounted for in
several publications with different degree of complexity [72, 48, 65]: to the
best of our knowledge, however, the force transmission has always been modeled
stemming from the similarity between the sarcomeric structure of stress fibers
and the actin-myosin interactions in muscle cells. In [33] a multi-dimensional
network of stress fibers was built on the notion of a representative volume
element, in which stress fibers can form in any direction with equal
probability. An average macroscopic stress is then recovered from the fiber
tension, which in turn is generated by the cross-bridging cycles and described
by a Hill-like relation [87] of viscoelastic nature. Anisotropic stress fibers
distributions have been considered in [72], making use of Von Mises
distribution functions at the “microscale” coupled to a directional averaging
operator. The active contraction is described in terms of the change of fiber
length and its rate of change, with a product formula of viscoelastic origin.
Experimental evidences, however, seem to show that such a resemblance might be
questionable in the dynamics and mechanics of endothelial cell spreading [17]
and hence that the predictive capability of this family of models might be
poor for this family of cells.

Finally, the passive response of the cytosol, provided mainly by the
intermediate filaments attached to the nuclear and plasma membranes, has been
modeled by several authors by means of classical models as linear elasticity
[72], the finite strain generalization of Hooke’s law [33] or a Neo-Hookean
potential energy

$\psi_{R}^{el}({\bm{C}}^{e})=\frac{G_{0}}{2}\,(I_{1}({\bm{C}}^{e})-3)\;,\qquad\psi_{R}^{in}({{\bm{E}}}^{e},{\bm{\xi}})=\frac{G_{0}-G_{\infty}}{G_{0}}\;\psi_{R}^{el}({\bm{E}}^{e}-{\bm{\xi}})\;,$
(114)

where $G_{0}$ is the initial shear modulus and $G_{\infty}$ is the shear
modulus at the end of the viscous processes. This classical choice of
Helmholtz free energy is associated to efficient integration schemes, depicted
in [66].

##  8 Concluding remarks

In this note, a multi-physics framework of protein relocation on the advecting
lipid membrane during cells spreading and motion has been put forward. It sets
the (continuum) thermodynamic background for simulations of receptor
recruitment during migration: simulations carried out in [14] stem from a
simplified form of the framework and described the limiting factors in
vascular endothelial growth factor receptors relocation; similarly, we
discussed in [15] the relocation of integrins on the membrane and their
interactions with growth factor receptors; a companion paper [16] deals with
the relocation of vascular endothelial growth factor receptors on advecting
lipid membrane during endothelial cell adhesion and spreading. Those
simulations may have a significant impact in biology and in the
pharmacological treatment of cancer, either in view of their predictive nature
in virtual experiments, or by clearly identifying the sequence of processes
that limit the relocation of targeted proteins during in vitro experiments.

The present work still has significant limitations, yet by illustrating a
complex and rigorous scenario it might be a cornerstone to account for several
further processes. To cite a major phenomenon that has been insufficiently
discussed here, the proteins transport on the membrane is crucially coupled to
the cytoskeleton reorganization, which is related to the motion of integrins
on the membrane: the formation of focal adhesion sites is preliminary to
stress fibers generation and contractility. Internalization of complexes is
another occurence not included in this work. Further publications, therefore,
will be devoted to extend this framework to these and others challenging
tasks.

We also aimed in this paper at recollecting recent publications from several
schools on cell mechanics, encasing them in a unified framework, being aware
that a comprehensive account of publications is significantly hard in view of
the broadness of the literature in the field. We clarified that for some
processes, as for contractility and protrusion, either a thermodynamically
consistent formulation has not been devised yet or it stems upon simplistic
models that do not account for the microstructural evolution of the
biopolymers. Even in this fascinating field, the last word is far from being
spoken.

Acknowledgements

This work has been supported by grants from the company Ferriera Valsabbia
through a liberal donation to fund studies in the field of Mechanobiology, and
from Fondazione Berlucchi to Mattia Serpelloni. We gratefully acknowledge
pleasant scientific discussions with S. Mitola, E. Grillo, and C. Ravelli from
the DMMT at the University of Brescia.

References

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