Structured models of cell migration incorporating molecular binding processes

Abstract

The dynamic interplay between collective cell movement and the various molecules involved in the accompanying cell signalling mechanisms plays a crucial role in many biological processes including normal tissue development and pathological scenarios such as wound healing and cancer. Information about the various structures embedded within these processes allows a detailed exploration of the binding of molecular species to cell-surface receptors within the evolving cell population. In this paper we establish a general spatio-temporal-structural framework that enables the description of molecular binding to cell membranes coupled with the cell population dynamics. We first provide a general theoretical description for this approach and then illustrate it with three examples arising from cancer invasion.

Appendices

Appendix 1: A measure theoretic setting

A measure theoretical justification of the binding and unbinding rates introduced to define the structural flux given in (14) is as follows. Let ${\mathfrak {B}}({\mathcal {P}})$ denote the Borel $\sigma$-algebra of the $i$-state space ${\mathcal {P}}$. In our model, given a density of molecular species ${\mathbf {m}}(t,x)$, the structural measure of their binding rate to the total cell density C(t, x) is denoted by $\eta _{{\mathbf {b}}}(\cdot ;{\mathbf {m}}):{\mathfrak {B}}({\mathcal {P}})\rightarrow {\mathbb {R}}^p$ and is assumed to be absolutely continuous with respect to the Lebesgue measure on ${\mathcal {P}}$. Then the induced Lebesgue-Radon-Nikodym density

$$\begin{aligned} \mathbf {b}(\cdot ;{\mathbf {m}}) = \begin{pmatrix}b_1(\cdot ;{\mathbf {m}})\\ \vdots \\ b_p(\cdot ;{\mathbf {m}})\end{pmatrix} : {\mathcal {P}}\rightarrow {\mathbb {R}}^p. \end{aligned}$$

(43)

is uniquely defined by

$$\begin{aligned} \eta _{{\mathbf {b}}} (W;{\mathbf {m}})&= \int \limits _W {\mathbf {b}}(\gamma ;{\mathbf {m}}) {\mathrm {d}}\gamma , \quad \forall W\in {\mathfrak {B}}({\mathcal {P}}), \end{aligned}$$

(44)

(Halmos 1978), and represents the binding rate of the molecular species ${\mathbf {m}}$ to the cell population density c.

Similarly, the structural measure of their unbinding rate of the bound molecular species ${\mathbf {n}}(t,x)$ is denoted by $\eta _{{\mathbf {d}}}$ and is again assumed to be absolutely continuous with respect to the Lebesgue measure on ${\mathcal {P}}$. Thus, this leads to an unbinding rate depending only on the $i$-state given by the Lebesgue-Radon-Nikodym density

$$\begin{aligned} {\mathbf {d}}(\cdot ) = \begin{pmatrix}d_1(\cdot )\\ \vdots \\ d_p(\cdot )\end{pmatrix} : {\mathcal {P}}\rightarrow {\mathbb {R}}^p \end{aligned}$$

(45)

is uniquely defined by

$$\begin{aligned} \eta _{{\mathbf {d}}} (W)&= \int \limits _W {\mathbf {d}}(\gamma ) {\mathrm {d}}\gamma , \quad \forall W\in {\mathfrak {B}}({\mathcal {P}}). \end{aligned}$$

(46)

Appendix 2: The source term for arbitrary Borel sets $W\subset {\mathcal {P}}$

Let $W \subset {\mathcal {P}}\subset {\mathbb {R}}^p$ be an arbitrary Borel set and define $zW := \{zw: w\in W\}$ for $z\in {\mathbb {R}}$. If $z \ne 0$, we can also write $zW = \{\tilde{w}: \frac{1}{z} \tilde{w} \in W\}$. Assume that W, 2W, and $\frac{1}{2} W$ are pairwise disjoint as shown in Fig. 1. Then, the source of cells in the structural region W that was obtained in (10) reads as

$$\begin{aligned} \int \limits _W S(t,x,y) {\mathrm {d}}y&= 2 \int \limits _{2W} \varPhi (\tilde{y},{\mathbf {u}}) c(t,x, \tilde{y}){\mathrm {d}}\tilde{y} - \int \limits _W \varPhi (y,{\mathbf {u}}) c(t,x,y) {\mathrm {d}}y. \end{aligned}$$

(47)

Note that we may have to invoke Convention 1 in the evaluation of the integral over 2W. The purpose of this appendix is to show that Eq. (47) also holds for arbitrary Borel sets $W\subset {\mathcal {P}}$. We start with the following technical lemma.

Lemma 1

Consider a set A such that $A\cap 2A = \emptyset$. Then it holds that $\frac{1}{2} A\cap A=\emptyset$ and, provided that A is also convex, $\frac{1}{2} A\cap 2A=\emptyset$.

Proof

Suppose there exists an $x\in \frac{1}{2} A\cap A$. Then $x\in A$ and it exists a $y\in A$ such that $x=\frac{1}{2} y$. This implies that $y=2x$ is also an element of 2A and thus $y\in A\cap 2A$, a contradiction. Thus $\frac{1}{2} A\cap A=\emptyset$ must hold.

Now suppose there exists an $x\in \frac{1}{2} A\cap 2A$. Then there exist $y,z\in A$ such that $x=\frac{1}{2} z$ and $x = 2y$. Now observe that $x=\alpha y + (1-\alpha )z$ for $\alpha =\frac{2}{3}\in (0,1)$ and thus x can be written as a convex combination of y and z. Since A is convex, we also have $x\in A$. But then, $z=2x\in A\cap 2A$, a contradiction. Thus $\frac{1}{2} A\cap 2A=\emptyset$ must hold. $\square$

This enables us now to prove the following theorem.

Theorem 1

Let W be an arbitrary convex and compact subset of ${\mathcal {P}}$. Then Eq. (47) holds.

Proof

Since $W \subset {\mathcal {P}}$ is an arbitrary convex and compact set, the sets W, 2W, and $\frac{1}{2} W$ might not be pairwise disjoint. Since W is compact, we have that the Lebesgue measure $\lambda (W)<\infty$. Furthermore, it holds that $\lambda (zW) = z^{p} \lambda (W)$ for all $z\in {\mathbb {R}}$. We define the sequence of sets

$$\begin{aligned} W_k := \bigcap \limits _{i=0}^k 2^{-i} W, \quad k=0,1,2,\ldots . \end{aligned}$$

These sets are, as intersection of convex sets, convex and have the following properties

$$\begin{aligned} W_0 = W,\quad W_j \subseteq W_i\text { for all } j\ge i,\quad \text {and}\quad \lambda (W_k)\le 2^{-pk} \lambda (W). \end{aligned}$$

Note that if $0\not \in W$ then there exists a finite K such that $W_k=\emptyset$ for all $k\ge K$, otherwise, if $0\in W$ then $0\in W_k$ for all k and $\lim _{k\rightarrow \infty }W_k=\{0\}$. Therefore, combining both cases, define $W_\infty :=\{0\}\cap W$; clearly $\lambda (W_\infty )=0$. Thus we can write

where $A_k:=W_k {\setminus } W_{k+1}$ for $k=0,1,\ldots$. From the definition of the sets $W_k$ we can also deduce the following relation

$$\begin{aligned} 2W_k = 2W\cap W_{k-1}\quad \text { for } k=1,2,\ldots . \end{aligned}$$

Now, for all $k=0,1,2,\ldots$ we obtain

$$\begin{aligned} A_k \cap 2 A_k&= (W_k {\setminus } W_{k+1}) \cap (2 W_k) {\setminus } (2 W_{k+1})\\&= (W_k {\setminus } W_{k+1}) \cap (2 W_k) {\setminus } (2W\cap W_{k})\\&= (W_k {\setminus } W_{k+1}) \cap \Bigl [\underbrace{((2 W_k){\setminus } (2W))}_{=\emptyset } \cup \, ((2W_k){\setminus } W_k) \Bigr ]\\&= (W_k {\setminus } W_{k+1}) \cap ((2W_k){\setminus } W_k) = \emptyset . \end{aligned}$$

Following the first part of Lemma 1 it now also follows that $\frac{1}{2} A_k \cap A_k =\emptyset$ for $k=0,1,2,\ldots$. The second part of Lemma 1 is not applicable here since $A_k$ is in general not convex. However, note that the derivation above also shows that

$$\begin{aligned} 2 A_k = (2W_k){\setminus } W_k \text { for } k=0,1,2,\ldots . \end{aligned}$$

Using that relation, once directly and once multiplied by $\frac{1}{4}$, we now obtain, for all $k=0,1,2,\ldots$,

$$\begin{aligned} \frac{1}{2}A_k \cap 2 A_k&=\Biggl (\Bigl (\frac{1}{2}W_k\Bigr ){\setminus } \Bigl (\frac{1}{4}W_k\Bigr )\Biggr ) \cap \bigl ((2W_k){\setminus } W_k\bigr ). \end{aligned}$$

Now assume that there exists an $x\in \frac{1}{2}A_k \cap 2 A_k$. Then necessarily, $x\in \frac{1}{2}W_k$ and $x\in 2 W_k$. As in the proof of the second part of Lemma 1 it follows, thanks to the convexity of $W_k$, that also $x\in W_k$. However, then $x\not \in (2W_k){{\setminus }} W_k$ and thus $x\not \in \frac{1}{2}A_k \cap 2 A_k$, a contradiction. Thus it also holds that $\frac{1}{2}A_k \cap 2 A_k=\emptyset$.

In summary, it holds that, for each $k=0,1,2,\ldots$, the sets $A_k$, $2 A_k$, and $\frac{1}{2} A_k$ are pairwise disjoint and hence Eq. (47) holds with W replaced by $A_k$.

Now we can conclude for our arbitrary convex and compact set $W \subset {\mathcal {P}}$, that

$\square$

Since we have shown that Eq. (47) holds for arbitrary convex and compact subsets of ${\mathcal {P}}$, it in particular also holds for all rectangles, which are a family of generators of the Borelian $\sigma$-algebra on ${\mathcal {P}}$ (Halmos 1978). Hence it holds for all Borel subsets of ${\mathcal {P}}$.

Appendix 3: Non-dimensionalisation and parameter tables

Based on a typical cancer cell volume of $1.5 \cdot 10^{-8}\hbox {cm}^{3}$, see Anderson (2005) and references cited there, we set

$$\begin{aligned} \vartheta _c = 1.5 \cdot 10^{-8}\hbox {cm}^{3}/\hbox {cell} \end{aligned}$$

and define the scaling parameter $c_*= 1/\vartheta _c = 6.7 \cdot 10^{7}\hbox {cells}/\hbox {cm}^{3}$ as the inverse of $\vartheta _c$, i.e. taken as the maximum cell density such that no overcrowding occurs. Assuming that a cell is approximately a sphere, we obtain a surface area of $\varepsilon = 2.94 \cdot 10^{-5}\hbox {cm}^{2}/\hbox {cell}$. In Lodish et al. (2007), the amount of surface receptors is given by a range from 1000 to 50,000 molecules per cell. We take the upper limit which is translated to $50{,}000\,\hbox {molecules}/\hbox {cell}$ = $8.3 \cdot 10^{-14}\upmu \hbox {mol}/\hbox {cell}$ and gives a reference surface density of

$$\begin{aligned} y_*= \frac{8.3 \cdot 10^{-14}\,\upmu \hbox {mol}/\hbox {cell}}{2.94 \cdot 10^{-5}\,\hbox {cm}^{2}/\hbox {cell}} = 2.82 \cdot 10^{-9}\,\upmu \hbox {mol}/\hbox {cm}^{2}. \end{aligned}$$

In Abreu et al. (2010) it is stated that the collagen density in engineered provisional scaffolds should be between 2 and $4\,\hbox {mg}/\hbox {cm}^{3}$ for in vivo delivery. We take the upper limit as scaling parameter $v_*$ for the ECM density. Assuming that ECM at this density fills up all available physical space, we obtain $1=\rho (0,v_*)=\vartheta _v v_*$ and thus

$$\begin{aligned} \vartheta _v:=\frac{1}{v_*}. \end{aligned}$$

Note that with these these scalings we have already for the volume fraction of occupied space, cf. (5),

$$\begin{aligned} \rho (C,v)=\vartheta _c C + \vartheta _v v = \vartheta _c c_*\tilde{C} + \vartheta _v v_*\tilde{v} = \tilde{\vartheta }_c\tilde{C}+\tilde{\vartheta }_v\tilde{v} = \tilde{C}+\tilde{v} = \tilde{\rho }(\tilde{C},\tilde{v}). \end{aligned}$$

The scaling parameters $\tau =10^{4}\hbox {s}$ and $L=0.1\hbox {cm}$ are chosen as in Gerisch and Chaplain (2008) and Domschke et al. (2014) and, as in loc. cit., the value of the scaling parameter $m_*$ remains unspecified. Table 1 shows the model parameters with units and their non-dimensionalised counterparts, and intermediate quantities of these can be found in Table 2.

Table 1 Parameters p of the general model (18) with their unit and their non-dimensionalised counterparts $\tilde{p}$

Table 2 Intermediate model quantities p of the general model (18) with their unit and their non-dimensionalised counterparts $\tilde{p}$