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.
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Acknowledgements
PD was supported by the Northern Research Partnership PECRE scheme and the Deutsche Forschungsgemeinschaft under the grant DO 1825/1-1. DT and AG would like to acknowledge Northern Research Partnership PECRE scheme. DT and MAJC gratefully acknowledge the support of the ERC Advanced Investigator Grant 227619, “M5CGS—From Mutations to Metastases: Multiscale Mathematical Modelling of Cancer Growth and Spread”. The authors PD, DT, AG, and MAJC would like to thank the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme “Coupling Geometric PDEs with Physics for Cell Morphology, Motility and Pattern Formation” supported by EPSRC Grant Number EP/K032208/1.
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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
is uniquely defined by
(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
is uniquely defined by
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
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
These sets are, as intersection of convex sets, convex and have the following properties
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
Now, for all \(k=0,1,2,\ldots \) we obtain
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
Using that relation, once directly and once multiplied by \(\frac{1}{4}\), we now obtain, for all \(k=0,1,2,\ldots \),
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
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
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
Note that with these these scalings we have already for the volume fraction of occupied space, cf. (5),
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.
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Domschke, P., Trucu, D., Gerisch, A. et al. Structured models of cell migration incorporating molecular binding processes. J. Math. Biol. 75, 1517–1561 (2017). https://doi.org/10.1007/s00285-017-1120-y
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DOI: https://doi.org/10.1007/s00285-017-1120-y