Collective Cell Behaviour with Neighbour-Dependent Proliferation, Death and Directional Bias

Abstract

Collective cell migration and proliferation are integral to tissue repair, embryonic development, the immune response and cancer. Central to collective cell migration and proliferation are interactions among neighbouring cells, such as volume exclusion, contact inhibition and adhesion. These individual-level processes can have important effects on population-level outcomes, such as growth rate and equilibrium density. We develop an individual-based model of cell migration and proliferation that includes these interactions. This is an extension of a previous model with neighbour-dependent directional bias to incorporate neighbour-dependent proliferation and death. A deterministic approximation to this individual-based model is derived using a spatial moment dynamics approach, which retains information about the spatial structure of the cell population. We show that the individual-based model and spatial moment model match well across a range of parameter values. The spatial moment model allows insight into the two-way interaction between spatial structure and population dynamics that cannot be captured by traditional mean-field models.

Appendices

Appendix 1: Definition of Spatial Moments

For a stationary spatial point process in a square domain $\Omega =[0,L]\times [0,L]$, the first spatial moment $Z_1(t)$ is simply the expected density of agents (i.e. expected population size divided by domain area $L^2$) at time t. The second spatial moment $Z_2(\varvec{\xi },t)$ is the expected density of pairs of agents separated by displacement $\varvec{\xi }$ at time t. For brevity, the argument t will be omitted in the following. To give a rigorous definition of the second moment, we first define the random variable $N_h(A)$ to be the number of agents in a region $A\subset \Omega$ and let $D_h(x)\subset \Omega$ denote the disc of radius h centred on $\mathbf{x}\in \Omega$. Since we are dealing with a stationary point process, we can assume without loss of generality that one of the agents in a pair (or triplet) is located at $\mathbf{x}=\varvec{0}$. We then define

$$\begin{aligned} Z_2(\varvec{\xi }) = \lim _{h\rightarrow 0} \frac{ E\left[ N(D_h(\varvec{0}))N_h(D_h(\varvec{\xi })) - N( D_h(\varvec{0})\cap D_h(\varvec{\xi }))\right] }{h^2}. \end{aligned}$$

(18)

If the discs $D_h(\varvec{0})$ and $D_h(\varvec{\xi })$ are non-overlapping, the numerator in Eq. (18) reduces to

$$\begin{aligned} E\left[ N(D_h(0))N(D_h(\varvec{\xi }))\right] , \end{aligned}$$

which, in the limit $h\rightarrow 0$, is equivalent to the probability that there is an agent in $D_h(\varvec{0})$ and an agent in $D_h(\varvec{\xi })$. The second term in the expectation in Eq. (18) is necessary to remove self-pairs (Law and Dieckmann 2000; Plank and Law 2015). The third moment (density of triplets) is defined similarly as

$$\begin{aligned} Z_3(\varvec{\xi },\varvec{\xi }')= & {} \lim _{h\rightarrow 0} \frac{1}{h^3} E\left[ N(D_h(\varvec{0}))N(D_h(\varvec{\xi }))N(D_h(\varvec{\xi }'))\right. \nonumber \\&-\, N(D_h(\varvec{0})\cap D_h(\varvec{\xi }))N(D_h(\varvec{\xi }')) \nonumber \\&\left. -\, N(D_h(\varvec{0})\cap D_h(\varvec{\xi }'))N(D_h(\varvec{\xi })) - N(D_h(\varvec{\xi })\cap D_h(\varvec{\xi }'))N(D_h(\varvec{0})) \right. \nonumber \\&\left. +\, 2N(D_h(\varvec{0})\cap D_h(\varvec{\xi }) \cap D_h(\varvec{\xi }')) \right] . \end{aligned}$$

(19)

Again, the extra terms in the expectation are needed to remove non-distinct triplets. Definitions (18) and (19) are equivalent to those of Illian et al. (2008), who referred to them as product densities.

The probability that there is a neighbouring agent located at displacement $\varvec{\xi }$, given that there is a focal agent at $\varvec{0}$ is

$$ \begin{aligned}&\lim _{h\rightarrow 0}\frac{P\left( N(D_h(\varvec{\xi }))=1 | N(D_h(\varvec{0}))=1 \right) }{h} \\&\quad = \lim _{h\rightarrow 0} \frac{P\left( N(D_h(\varvec{\xi }))=1 \& N(D_h(\varvec{0}))=1 \right) }{h P\left( N(D_h(\varvec{0})=1) \right) },\\&\quad = \frac{Z_2(\varvec{\xi })}{Z_1}. \end{aligned}$$

Similarly, probability that there is a neighbouring agent located at displacement $\varvec{\xi }'$, given that there is a focal agent at $\varvec{0}$ and a neighbouring agent at $\varvec{\xi }$ is

$$ \begin{aligned}&\lim _{h\rightarrow 0} \frac{ P\left( N(D_h(\varvec{\xi }'))=1 | N(D_h(\varvec{0}))=1 \& N(D_h(\varvec{\xi }))=1 \right) }{h} \\&\quad = \lim _{h\rightarrow 0} \frac{P\left( N(D_h(\varvec{\xi }'))=1 \& N(D_h(\varvec{\xi }))=1 \& N(D_h(\varvec{0}))=1 \right) }{h P\left( N(D_h(\varvec{\xi }))=1 \& N(D_h(\varvec{0}))=1 \right) }, \\&\quad = \frac{Z_3(\varvec{\xi },\varvec{\xi }')}{Z_2(\varvec{\xi })}. \end{aligned}$$

Appendix 2: Numerical Methods

The IBM is simulated using the Gillespie algorithm as follows. If the population size is N(t), there are 3N(t) possible types of transition, corresponding to either movement, proliferation or death of one agent. The event rates for these 3N(t) transition types are calculated from Eqs. (1) and (2), and the aggregate event rate is

$$\begin{aligned} \lambda (t)=\sum _{n=1}^{N(t)}\left( \hat{M}_n + \hat{P}_n + \hat{D}_n\right) . \end{aligned}$$

(20)

The time increment $\tau$ between consecutive events is exponentially distributed with mean $1/\lambda (t)$. One of the 3N(t) possible types of transitions is then chosen to occur at time $t+\tau$, with a probability that is proportional to the rate of that transition. If the event is a movement, the new location for the motile agent is chosen according to the bias vector for that agent in Eq. (3) and the movement kernel in Eq. (4). If the event is a proliferation, the location for the new daughter agent is chosen from the dispersal kernel $\mu ^{(p)}$ and the population size increases to $N(t+\tau )=N(t)+1$. If the event is a death, the agent is removed and the population size decreases to $N(t+\tau )=N(t)-1$. The population is initially of size $N(0)=N_0$ and is distributed according to a spatial Poisson process, i.e. with no initial spatial structure, in a square domain of size $L\times L$. Periodic conditions are implemented at the boundaries of the spatial domain, meaning that agents’ horizontal and vertical coordinates are taken to be modulo L.

To compare the results of the IBM with the spatial moment model, we need to calculate the first and second moments of the spatial point process arising from a given realisation of the IBM. The first moment is simply the average agent density and is calculated from the IBM as $N(t)/L^2$. To compare the second moment, we calculate the pair correlation function (PCF) C(r) (Illian et al. 2008) by calculating the distance $r=|\mathbf {x}_j-\mathbf {x}_i|$ ($i\ne j$) between every pair of agents, allowing for the periodic boundaries. The PCF is constructed by counting the distances that fall into an interval $[r-\frac{\delta r}{2}, r+\frac{\delta r}{2}]$, i.e. binning distances using a bin width $\delta r$. We normalise each bin count by $N(t)(N(t)-1)(2\pi r \delta r)/L^2$; this normalisation means that, in the absence of spatial structure (i.e. a Poisson spatial pattern), we have $C(r)\equiv 1$ (Binny et al. 2016). When $C(r)>1$ at short distances r, this indicates a clustered pattern. In contrast, $C(r)<1$ at short distances r indicates segregation. The choice of $\delta r$ is important because very small values can yield a PCF dominated by fluctuations, while values that are too large result in an overly smooth function which may mask spatial structure (Binder and Simpson 2015). The PCF C(r) is compared to a radial slice through the two-dimensional function $Z_2(\varvec{\xi })/Z_1^2$, where $r=|\varvec{\xi }|$ (Binny et al. 2016). For each set of parameter values, we perform 20 (unless otherwise stated) repeated realisations of the stochastic IBM and compute an average over the ensemble of realisations for the first moment and the PCF.

Equation (12) for the dynamics of the second moment is solved using the method of lines. This involves discretisation of $\varvec{\xi }=(\xi _1,\xi _2)$ with grid spacing $\Delta$ over the two-dimensional domain $\{-\xi _{{\mathrm {max}}} \le \xi _1, \xi _2 \le \xi _{{\mathrm {max}}} \}$, where $\xi _{{\mathrm {max}}}$ is large enough so that $Z_2(\varvec{\xi }) \approx Z_1^2$ at the boundary. We truncate the tails of the kernels such that $w(\varvec{\xi })=\mu (\varvec{\xi })=0$ for $|\varvec{\xi }|>\xi _{{\mathrm {max}}}/2$ and $h(r)=0$ for $r>\xi _{\mathrm {max}}/2$. The integral terms in Eqs. (5), (6), (8)–(10) and (12) are approximated using the trapezium rule over values of the integration variable for which the relevant kernel is nonzero. These integrals still require values of $Z_2(\varvec{\xi })$ that lie outside the computational domain. For example, in Eq. (8) for $M_2(\varvec{\xi })$ at say $\varvec{\xi }=(0.9\xi _{\mathrm {max}},0)$, the kernel $w^{(m)}(\varvec{\xi }')$ will be nonzero at values of $\varvec{\xi }'$ for which $\varvec{\xi }+\varvec{\xi }'$ lies beyond the right-hand boundary of the domain. Where this happens, we use the value of $Z_2$ on the corner of the computational domain, i.e. we substitute $Z_2(\xi _{{\mathrm {max}}},\xi _{{\mathrm {max}}})$ for the required value of $Z_2(\varvec{\xi })$. The PDF for movement $\mu _2^{(m)}(\varvec{\xi },\varvec{\xi }')$ is normalised numerically using the trapezium rule (Binny et al. 2016). This means that, for each fixed value of $\varvec{\xi }'$, the PDF $\mu _2^{(m)}(\varvec{\xi },\varvec{\xi }')$ is normalised so that $\int _T \mu _2^{(m)}(\varvec{\xi },\varvec{\xi }'){\mathrm {d}} \varvec{\xi }=1$, where $\int _T$ denotes the trapezium rule approximation to the integral. Similarly, the dispersal PDF $\mu _2^{(p)}(\varvec{\xi })$ is normalised so that $\int _T \mu _2^{(p)}(\varvec{\xi }){\mathrm {d}} \varvec{\xi }=1$. We ensure that $\mu _s + 4\sigma _s \le 4\sigma _m, 4\sigma _p,4\sigma _b$, which means that typical steps lengths are much smaller than the spatial scale over which agents interact.

The method of lines converts Eq. (12) into a system of $m\times m$ ordinary differential equations, which are then solved using MATLAB’s inbuilt ode23 solver. The results are insensitive to a reduction in grid spacing $\Delta$. As noted above, the computational domain is large enough such that the conditions at its boundary are approximately mean-field, i.e. $Z_2(\varvec{\xi }) \approx Z_1^2$ on the boundary. Therefore, we do not solve Eq. (7) for the dynamics of the first moment directly, which would lead to an overdetermined system of $(m\times m)+1$ equations in $m\times m$ unknowns. Instead, we use the value of $Z_2$ on the boundary of the computational domain to set the value of the first moment: $Z_1=\sqrt{Z_2(\xi _{\mathrm {max}},\xi _{\mathrm {max}})}$. The initial condition for the dynamics of the second moment is $Z_2(\varvec{\xi })=Z_1^2$ at $t=0$.

Appendix 3: Moment Closure Definitions

Symmetric power-1 closure (Murrell et al. 2004):

$$\begin{aligned} Z_3(\varvec{\xi },\varvec{\xi }') = Z_1\left( Z_2(\varvec{\xi }'-\varvec{\xi }) + Z_2(\varvec{\xi }')+Z_2(\varvec{\xi })\right) - 2Z_1^3. \end{aligned}$$

(21)

Power-2 closure (Murrell et al. 2004; Raghib et al. 2011):

$$\begin{aligned} Z_3(\varvec{\xi },\varvec{\xi }')= & {} \frac{1}{\alpha +\gamma }\left( \alpha \frac{Z_2(\varvec{\xi })Z_2(\varvec{\xi }')}{Z_1} + \beta \frac{Z_2(\varvec{\xi })Z_2(\varvec{\xi }'-\varvec{\xi })}{Z_1}\right. \nonumber \\&\left. + \gamma \frac{Z_2(\varvec{\xi }')Z_2(\varvec{\xi }'-\varvec{\xi })}{Z_1} - \beta Z_1^3\right) . \end{aligned}$$

(22)

The symmetric power-2 closure has $\alpha =\beta =\gamma$, and the asymmetric power-2 closure used in this paper has $(\alpha ,\beta ,\gamma )=(4,1,1)$ (Law et al. 2003).

Power-3 closure, also known as the (Kirkwood 1935) closure:

$$\begin{aligned} Z_3(\varvec{\xi },\varvec{\xi }') = \frac{Z_2(\varvec{\xi })Z_2(\varvec{\xi }')Z_2(\varvec{\xi }'-\varvec{\xi })}{Z_1^3}. \end{aligned}$$

(23)