A General Model of Structured Cell Kinetics
Michael Grinfeld
Nigel Mottram
Jozsef Farkas
GMF1202.tex
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\title{A General Model of Structured Cell Kinetics} 

\author{Michael Grinfeld}
\address{Department of Mathematics and
    Statistics, University of Strathclyde, 26 Richmond Street, Glasgow
    G1 1XH, UK}
 \email{m.grinfeld@strath.ac.uk}

 \author{Nigel Mottram}
 \address{School of Mathematics and Statistics,
   University of Glasgow, University Place, Glasgow G12 8QQ, UK}
\email{Nigel.Mottram@glasgow.ac.uk}

  \author{Jozsef Farkas}
 \address{Division of Computing Science and
   Mathematics, University of Stirling, Stirling FK9 4LA, UK}
 \email{jozsef.farkas@stir.ac.uk}
 

\begin{document}

\begin{abstract}
  \noindent We present a modelling framework for the dynamics of cells
  structured by the concentration of a micromolecule they
  contain. We derive general equations for the evolution of the cell
  population and of the extra-cellular concentration of the molecule
  and apply this approach to models of silicosis and quorum sensing in
  Gram-negative bacteria.\\

  \noindent {\bf Keywords:} structured populations, transport
  equations, integro-differential equations, silicosis
\end {abstract}

\maketitle

\section{Introduction}

Many physical systems involve the interaction of micro-scale objects
and macro-scale objects within a region. For instance, in biology, the
micro-scale objects could be molecules of a particular chemical with
the macro-scale-objects being cells and the region a Petri dish or an
organ. This region, a domain of volume $W$, may be of fixed size or
change with time but we assume that the micro-scale objects, of a
species $X$ (which from now we call molecules for brevity), have no
volume while the macro-scale objects (which from now on we call cells)
each have volume $V_0$. The $X$ molecules may be present inside or
outside the cells, with the concentration of $X$ varying between the
cells. We also assume that the molecules of $X$ can participate in any
subset of the following processes: they can be injected into or be
removed from the domain, they can enter and exit cells and they can be
produced, processed and destroyed by cells. Suppose also that the fate
of a cell is dependent on the amount of $X$ that it contains. The goal
of the present paper is to introduce, using ideas of Metz and Diekmann
\cite{MD} and of Brown \cite{Brown}, a modelling framework for such
situations.

Below we describe in detail two specific examples of systems where our
modelling approach is appropriate, the dynamics of silicosis, the
biological background for which can be found in \cite{Tran}, and
quorum sensing in Gram-positive bacteria following Brown
\cite{Brown}. However, our framework is suitable for many other
situations, some of which are briefly considered in
Section~\ref{Conc}.  We are confident there are many other examples,
in both biological and non-biological systems, where the proposed
philosophy may be useful.

The structure of the paper is as follows. In Section~\ref{Sil} we
introduce the biological background of silicosis and argue that the
mathematical model of \cite{Tran}, which is couched in terms of
coagulation-fragmentation equations, does not give a correct
description of the dynamics; this is the original motivation for
developing the present approach. In Section~\ref{Mod} we show how to
derive the required equations in the general case. In
Section~\ref{Appl} we complete the specification of the silicosis
model and briefly consider quorum sensing in Gram-positive bacteria,
for which our framework is in fact an ``overkill'', but which may
prove useful in the process of testing numerical methods. Finally, in
Section~\ref{Conc} we suggest other areas of application of our
framework, discuss mathematical issues and draw conclusions.

Note that the present paper is purely methodological and that all
results on existence, uniqueness, or asymptotic behaviour of solutions
of the type of equations we derive here, are left to future work.

\section{Silicosis} \label{Sil}

Let us summarise the 1995 silicosis model of Tran {\em et
  al.}~\cite{Tran}, which should be consulted for references. The
biological background is as follows: quartz particles are ingested and
arrive in the lung. There they may be picked up by macrophages, with
the intent of being removed together with their quartz load via the
muco-ciliary escalator. However, if a macrophage accumulates too large
a quartz load, it becomes immobile and eventually dies by apoptosis in
the lung, releasing its quartz load.

The variables in the model of Tran {\em et al.}~are: free quartz dust in
the lungs in concentration $x(t)$ and concentrations of macrophages
$M_k(t)$ containing $k$ {\em particles} of quartz. They write down an
equation for the evolution of free quartz particle concentration and
for $M_k$, equations of the form
\begin{equation}
  \frac{dM_k}{dt} = \alpha_{k-1}x M_{k-1}- \alpha_k x M_k + \cdots ,\label{tran}
\end{equation}
where $\alpha_{k-1}$ and $\alpha_{k}$ are the kinetic constants for
the process of macrophages with $k-1$ and $k$ particles of quartz,
respectively, ingesting one additional quartz particle.

In eq.~(\ref{tran}), the $\cdots$ stand for the two different
``death'' processes: the disappearance of cells together with their
quartz load via the the muco-ciliary elevator; or cell apoptosis
accompanied by the release of the quartz load into the lungs. To make
this model fully specified, it must be complemented by an equation for
the production of na\"{i}ve macrophages, $M_0(t)$; this rate in
general depends on the quartz load in the lungs. See
Section~\ref{silm} below for a reasonable form of such an equation.

This model was later considered in \cite{CDG}; mathematically it is
interesting and falls in the framework of coagulation-fragmentation
equations (see \cite{BLL} for an up-to-date exposition of this area of
infinite-dimensional dynamical systems). The global existence of its
solutions has been proven in later work by da Costa and coauthors
\cite{CPS}, and many mathematical questions connected with the model
of Tran {\em et al.}~are still open.

To understand our objections to the modelling of silicosis as a
coagulation--fragmentation system, consider a typical
coagulation--fragmentation reaction scheme,
\[
  c_k+ c_1 \rightleftharpoons c_{k+1}.
\]
Here $c_k$, $c_1$, $c_{k+1}$ are {\em concentrations} of $k$-mers,
monomers, and $(k+1)$-mers of some chemical species, respectively. The
equation for the evolution of $c_{k+1}$ corresponding to this reaction
scheme is
\[
  \frac{dc_{k+1}}{dt} = \alpha_{k}c_k c_1 - \beta_{k+1} c_{k+1},
\]
where $\alpha_{k}$ is the kinetic constant for the coagulation
reaction between $k$-mers and monomers and $\beta_{k+1}$ is the
kinetic constant of the fragmentation of a molecule of $(k+1)$-mer
into a monomer and a $k$-mer. This equation is simply mass-action
kinetics, and $\alpha_k$ and $\beta_{k+1}$ are assumed to be functions
of $k$ only.

If we now compare this coagulation--fragmentation process with
eq.~(\ref{tran}), we would suppose that the underlying reaction scheme
is
\[
    M_k + x \rightleftharpoons M_{k+1}.
\]  
This situation is subtly different in that the reaction here
is between the molecules of quartz outside the cells and the {\em
  content} of the cells. Though $M_k$, $x$, $M_{k+1}$ have units of
concentration, the reaction encoded in this scheme in general involves
the {\em concentration} of quartz, and not the number of quartz
particles, inside the cells. As an example, consider the case of
exchange of quartz between the outside and the inside of a cell driven
by passive (Fickian) diffusion. Then the rate of exchange of molecules
is proportional to $(x-q_I)$, where $q_I$ is the internal
concentration of quartz.

However, the $k$ in $M_k$ denotes the {\em number} of quartz molecules
in a cell, not their concentration. To define internal concentration
of quartz, we cannot assume that a cell is a point object, and have to
endow it with a volume, say $V_0$. But if cells now have finite
volumes, it becomes clear that the concentration of free quartz is
also a function of the number of cells which is not the case in the
coagulation--fragmentation setup. This is precisely the kind of
confusion of units that is avoided in the type of model proposed
below.

\section{Derivation of Equations} \label{Mod}

\begin{figure}[htp]
%\begin{center}  
%\begin{picture}(100,170)
\centerline{\includegraphics[width=0.5\textwidth]{setup1.eps}}
%\end{picture}
\caption{The situation being modelled: the domain $\Omega$ contains
  species $X$ (red dots) and cells (black circles). The possible
  processes involved are: (a) na\"{i}ve cells enter the domain; (b)
  species $X$ enters the domain; species $X$ (c) enters or (d) exits
  cells; (e) $X$-laden cells exit the domain and (f) cells release
  molecules of $X$ in $\Omega$; also present, but not indicated in
  this figure, are synthesis and degradation of $X$.}
\label{fig:1}
%\end{center}
\end{figure} 

We are modelling the situation sketched in Figure~\ref{fig:1} and
described as follows. The system consists of a domain $\Omega$ of
volume $W$ that contains molecules of $X$ (denoted by red dots in
Fig.~\ref{fig:1}) with extracellular concentration $x$, and a
population of cells (the black circles in Fig.~\ref{fig:1}). We assume
that the volume of each cell is $V_0$ and treat the molecules as
having no volume.

The cells differ in their $X$ content, and we define the
time-dependent density of the cells with internal concentration of $X$
being $y$ to be $M(y,t)$. That is, the number of cells with internal
concentrations of $X$ between $y_1$ and $y_2$, with $y_1 < y_2$, is
  \[
    {\displaystyle \int_{y_1}^{y_2} M (y, t) \, {\rm d}y}.
   \] 
Then 
\[
  V(t)= V_0 \int_0^\infty M (y, t) \,{\rm d}y
\]
is the total volume occupied by the cells and we assume that for all
time $t$, $V (t) < W$. By this we mean that if $V(0) < W$, the
dynamics of the system ensures that $V(t) < W$.

Furthermore, inside each cell we have a process that involves the
substance $X$, which can be described by an ordinary differential
equation such as
 \begin{equation}\label{ieq}
   \frac{dy}{dt} = f (x, y) = J(x,y) + g(y),
 \end{equation}
 where we have separated the transport term $J(x,y)$, that may depend
 on both the internal and external concentration of $X$, and the
 intracellular synthesis and degradation term $g(y)$. This type of
 equation is called an $i$-equation (for individual) by Metz and
 Diekmann \cite{MD}.

 Metz and Diekmann \cite{MD} also show (in many different ways) how to
 derive the equation governing the evolution of $M(y,t)$, what they
 call the $p$-equation (for population), 
 \begin{equation}\label{peq}
   M_t (y, t) + (f (x, y)M (y, t))_y = P (M (y, t), y) + Q(M(y,t),y),
 \end{equation} 
 where in the right-hand side the terms $P$ and $Q$ encode all the
 population level processes (such as birth and death). Specifically,
 we denote by $Q(M (y, t), y)$, the population level processes that
 feed back into the $x$ dynamics (e.g.~when cells die in $\Omega$
 releasing their contents). The model is then closed by specifying the
 $x$ dynamics, adding suitable initial
 conditions $x(0)$ and $M(y,0)$, and a boundary condition for
 na\"{i}ve cells entering $\Omega$, usually of the form
\[
  f (x(t),0) M (0, t) = s(\cdot),
\]
where the function $s$ can depend on a variety of variables. Thus the
unknown dependent variables are $x(t)$ and $M(y,t)$.

Deriving the equation governing the evolution of $x(t)$ is
algorithmic, by keeping account of the total number of molecules of
$X$ in the extracellular space. In general, the result is an
integro-differential equation. The derivation of the equation for
$x(t)$ crucially uses the thinking of Brown \cite{Brown}, which we now
discuss.

Brown \cite{Brown} considers the system described above but with the
simplification that every cell contains the same number of molecules,
so that the concentration of $X$ in all cells is $y$, a constant.
If $K(t)$ is the number of cells at time $t$, then the total volume
occupied by cells is $V(t)=K(t)V_0$, the cell-free volume is
$W-V(t)$ and the total number of molecules outside of cells is
$N_E=(W-V(t))x(t)$.

We will work in two stages. First we write equations for the transport
of $X$ molecules between the interior and exterior of the cells,
assuming that the flux is given by $J(x,y)$ and that it is positive
when molecules enter the cells. We have
\begin{equation}\label{eqne}
      \frac{dN_E}{dt} = -VJ(x,y).
\end{equation}
An obvious example of a flux would be $J(x,y)=D(x-y)$, with $D$
a diffusion constant, but in examples of interest, mechanisms
involving facilitated transport or phagocytosis should be  considered.  

The proportionality to $V$ in eq.~(\ref{eqne}) comes from the
consideration that the rate of change should be proportional to the
available surface area of the cells, which, given we assume a fixed
individual cell volume and surface area, is proportional to the number
of cells and thus proportional to the total volume of cells.

Now let us rewrite these equations in terms of concentrations
only. Since $N_E=x(W-V)$, we have 
\[
    \frac{dx}{dt} = -\frac{V}{W-V} J(x,y) + \frac{x}{W-V}
    \frac{dV}{dt}.
\]
Now we add to this molecular transport equation terms involving
synthesis and degradation terms which we collect in one term,
$H(x,y)$; we have 
\begin{equation}\label{eqbr}
         \frac{dx}{dt} = -\frac{V}{W-V} J(x,y) + \frac{x}{W-V}
         \frac{dV}{dt} + H(x,y).
\end{equation}
Here $H(x,y)$ incorporates all extracellular production and
degradation of $X$ and all the population level processes that feed
back into the extracellular concentration $x(t)$.

We now consider the more general case in which the cells may have
different internal concentrations of $X$. This necessitates a number
of changes. First of all, the total number of molecules being released
per unit time by cellular processes, i.e~$Q$ in eq.~(\ref{peq}), is
${\displaystyle V_0 \int_0^\infty y Q(M(y,t),y) \, dy}$ and hence we
can write 
\[
  H(x,y) = h(x) + \frac{V_0}{W-V} \int_0^\infty y Q(M(y,t),y) \, dy,
\]
where $h(x)$ is the rate of extracellular production and
degradation of $X$, which depends on the particular modelling context. 

Secondly, the term $V(t)J(x,y)=V_0 K(t) J(x,y)$ is replaced by the
integral ${\displaystyle V_0 \int_0^\infty M(y,t) J(x,y) \, dy}$. With
these changes, the equation for the evolution of $x(t)$ becomes   
 \begin{equation}\label{newx}
    \frac{dx}{dt} = h(x) + \frac{1}{W-V} \left[
      -V_0\int_0^\infty J(x,y)M(y,t) \, dy\right. + \left. V_0\int_0^\infty
     y\,Q(M(y,t),y)\, dy + x \frac{dV}{dt} \right].
 \end{equation} 

 Therefore our modelling framework consists of the equation governing
 the cell population, the $p$-equation, eq.~(\ref{peq}); the equation
 governing the concentration of $X$ external to the cells, the
 $x$-equation, eq.~(\ref{newx}); and the data for any particular model
 set through the biologically determined terms $J(x,y)$, $g(x)$,
 $P(M(y),t), y)$, $Q(M(y),t), y)$, $h(x)$ and the boundary condition
 function $s(\cdot)$.

\section{Applications}\label{Appl}
In this section we formulate in detail a new model of silicosis 
and discuss a quorum-sensing model in Gram-positive bacteria. In
Section~\ref{Conc} we will mention some other situations that are
covered by our framework.   

\subsection{Silicosis}\label{silm} 
As in Tran {\em et al.}~\cite{Tran}, we assume that quartz is being
ingested at a constant rate. We set $M(y,t)$ to be the density of
macrophages having internal quartz concentration $y$ at time $t$.  As
in \cite{Tran}, we assume that new macrophages are produced at rate
$s$ determined by the quartz load,
${\displaystyle L(t) := \int_0^\infty yM(y,t)\, dy}$, such that
\begin{equation}\label{eqs}
  s(L(t)) = s_0 + u(L(t)),
\end{equation}
where $s_0$ is a background level of recruitment of na\"{i}ve cells
into the domain when quartz is not present in any cells and $u(\cdot)$
is a bounded function, with $u(0)=0$.

Cells with internal concentration of quartz $y$ are removed by the
muco-ciliary escalator at a rate $p(y)$, where $p(y)$ is a decreasing
function of $y$ since, as their quartz content increases, cells are
increasingly immobile. Cells are also more liable to die by apoptosis
as their quartz content increases, and so the rate of them releasing
their contents inside the lungs, $q(y)$, is an increasing function of
$y$.


As there is no intracellular processing of quartz, so that $g(y)=0$ in
eq.~(\ref{ieq}), we only need to specify the transport
mechanism. Phagocytosis of quartz particles cannot be described by
simple diffusion, so we set
\begin{equation}\label{i2}
  \frac{dy}{dt} = J(x,y), 
\end{equation}
where the function $J$ is non-negative, bounded, increasing in
$x$ and decreasing in $y$, as is also assumed in \cite{Tran}. An
example would be
\[
  J(x,y) = \frac{\gamma x}{x+y+x_{1/2}},
\]
where $\gamma$ is the flux when $x\rightarrow\infty$ and $x_{1/2}$ at
which the value of $x$ when the flux for na\"{i}ve cells (i.e., $y=0$)
is half the maximum value, both positive constants.


So far we have all the information needed to specify the $p$-equation
for $M$, which is therefore
\begin{equation}\label{p2}
M_t(y,t) + (J(x,y) M(y,t))_y = - (p(y)+q(y))M(y,t). 
\end{equation}

Now we need to formulate the equation for $x(t)$. From (\ref{newx}) it
follows that all we need to do is to specify the function $h(x)$, the
rate of change of concentration of quartz particles in the
extracellular region due to introduction from outside the domain. If
we assume that $A$ particles of quartz are ingested per unit time, we
have
\[
h(x) = \frac{A}{W-V},
\]
and hence 
\begin{equation}\label{eqx}
  \frac{dx}{dt} = \frac{1}{W-V} \left[ A - V_0\int_0^\infty
    J(x,y)M(y,t)
    \,dy + 
    V_0\int_0^\infty yq(y)M(y,t)\, dy + x\frac{dV}{dt} \right].
\end{equation}

We can derive an expression for $dV/dt$ in terms of the variables
$M(y,t)$ to substitute in  (\ref{eqx}). Since $dV/dt$ is given by
\[
  \frac{dV}{dt} = V_0\int_0^\infty M_t(y,t) \, dy,
\]  
integrating the $p$-equation by parts and assuming that 
that for all times $t>0$ we have that
\[
  \lim_{y \to \infty} J(x,y) M(y,t)=0,
\]
we obtain
\begin{equation}\label{dvdt}
  \frac{dV}{dt} = V_0\left(s(L(t)) -\int_0^\infty (p(y)+r(y)) M(y,t)\,
    dy\right).
\end{equation}

In addition, we must specify suitable initial conditions $x(0)$ and
$M(y,0)$ as well as the boundary condition at $y=0$, which, using
eq.~(\ref{eqs}), is
\begin{equation}\label{bc}
  J(x,0)M(0,t)= s_0 + u(L(t)).
\end{equation}

We note that the resulting system is a linear transport equation (for
the population variable $M$) coupled to an nonlinear
integro-differential differential equation for the extracellular
quartz concentration $x$.


Having assumed that the load
\[
    L(t)= \int_0^\infty y M(y,t) \, dy
\]
is finite for all time, the correct setting for the theory is a space
of positive Radon measures with a finite first moment. Particular
choices of the functions $p(y)$ and $r(y)$ and the input function
$s(\cdot)$ must ensure that if $V(0)<W$, then $V(t)<W$. For example,
this can be shown to be the case if we assume that $p(y)+r(y)$ is
bounded below and that $s(\cdot)$ is bounded above by constants.



\subsection{Quorum sensing in Gram-negative bacteria}

For background on quorum sensing in bacteria and a number of models
that assume that the internal concentration of the signal molecule
$X$, i.e., $y$, is the same in all cells, see \cite{Brown}.  The case
of Gram-negative bacteria is an interesting test case of our approach,
as it involves a significant simplification: we can
assume that the number of cells is constant, so $V \equiv \hbox{const}$ is
now a parameter and $dV/dt=0$.  Fickian diffusion of $X$ between the
cells and the extracellular region is assumed, so that the
$i$-equation is
\[
  \frac{dy}{dt}=D(x-y)+g_s(y)+g_d(y),
\]
where $D$ is a diffusion constant, and we included terms modelling
intracellular synthesis $g_s(y)$ (which in \cite{Brown} is taken to be
a constant term plus a Hill-type term) and intracellular degradation
$g_d(y)$ (which in \cite{Brown} is taken to be linear, so that
$g_d(y)=-m_Iy$, parameterised by the intracellular degradation rate
$m_I$). The $p$-equation is
\[
  M_t(y,t) + ([D(x-y)+g_s(y)+g_d(y)]M(y,t))_y =0.
\]
The boundary condition is $M(0,t)=0$ and the $x$ equation becomes 
\[
  \frac{dx}{dt} = \frac{V_0 D}{W-V}\int_0^\infty (y-x)M(y,t)\, dy - 
m_E x, 
\]
where $m_E$ is the extracellular degradation rate. 

\section{Remarks} \label{Conc}

We start by briefly mentioning other possible applications. The list
is clearly incomplete.

\begin{itemize}
\item $X$ can be a drug that interacts with specific cells, so this is
  a suitable framework for chemotherapy modelling;
\item $X$ may be bacteria that are ingested by neutrophiles, and can
  multiply inside the cell. This framework is therefore a possible
  model of bacterial infection and resistance \cite{MSR};
\item $X$ can be a mitogen, giving a model of stem cell number
  maintenance in which cells that ingest enough mitogen will multiply,
  and divide their mitogen label among the daughter cells, while cells
  that do not have enough mitogen, die~\cite{Kitadata}.
\item As mentioned in the Introduction, this framework can also be
  used to model non-biological systems, one example of which is the
  decontamination of a chemostat using zeolites. There are many
  examples of this type of system, showing that it is not necessary to
  interpret ``macro-scale objects'' as biological cells.
\end{itemize}

Clearly the $i$-equation for $y$ could have been an stochastic
differential equation; then the $x$-equation would have been an
integro-stochastic differential equation, and the $M$ equation a
stochastic partial differential equation. Furthermore, extensions to
multidimensional $x$ and $y$ are straightforward although
incorporating spatial structure seems to us much more challenging (as
it is for coagulation--fragmentation equations).


To summarise, we have presented a modelling framework that seems to
cover a vast number of possible modelling contexts beyond the reach of
coagulation--fragmentation equations. Such a framework necessitates
the analysis of complicated mathematical objects and so work on
existence, structure of equilibria, convergence to equilibria and
their regularity, etc., is required when specific examples are
considered. The search will be for measure-valued solutions, and
relevant work in this direction has been undertaken by Carrillo,
Gwiazda and co-workers; see, for example, \cite{CCGU, GM}. In terms of
possible numerical solutions to the governing equations, there are no
off-the-shelf numerical methods that we know of, although it seems
that escalator box train (EBT) methods could be adapted to the problem
(see related work of Carrillo, Gwiazda and Ulikowska \cite{CGU}).

Finally, we note that in many of the biological settings for which
this framework could be used, deeper understanding of the active
transport of molecules in and out of cells may be needed. Good models
of transport across membranes (facilitated transport, phagocytosis,
pumps etc.) are relatively sparse in the literature (though see, e.g.,
\cite{Naftalin,Rea}) and further work on such models would be of
significant benefit.






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\end{document}



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