Modelling Directional Guidance and Motility Regulation in Cell Migration

Transcription

1 Bulletin of Mathematical Biology (2006) 68: DOI /s x ORIGINAL PAPER Modelling Directional Guidance and Motility Regulation in Cell Migration Anna Q. Cai, Kerry A. Landman, Barry D. Hughes Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia Received: 1 July 2005 / Accepted: 2 September 2005 / Published online: 24 March 2006 C Society for Mathematical Biology 2006 Abstract Although cell migration is an essential process in development, how cells reach their final destination is not well understood. Secreted molecules are known to have a migratory effect, but it remains unclear whether such molecules act as directional guidance cues or as motility regulators. There is potential to use signalling molecules in new medical therapies, so it is important to identify the exact role these molecules play. This paper focuses on distinguishing between inhibitory and repulsive effects produced by signalling molecules, based on recent experiments examining the effect of Slit, a secreted protein, on the migration of neurons from the brain. The primary role of Slit, whether it is an inhibitor or repellent of neurons, is in dispute. We present population-level continuum models and recast these in terms of transition probabilities governing individual cells. Various cellsensing strategies are considered within this framework. The models are applied to the neuronal migration experiments. To resolve the particular role of Slit, simulations of the models characterising different cell-sensing strategies are compared at the population and individual cell level, providing two complementary perspectives on the system. Difficulties and limitations in deducing cell migration rules from time-lapse imaging are discussed. Keywords Signalling molecule Cell migration Motility Directional guidance Diffusion Repulsion Inhibition 1. Introduction Cell migration plays a key role throughout life from embryonic development to death. During embryonic development, cell migration is central to a range of morphogenic processes, from gastrulation (cellular movements occurring at the end of cleavage) to the development of the central nervous system. Birth defects can Corresponding author. address: k.landman@ms.unimelb.edu.au (Kerry A. Landman).

2 26 Bulletin of Mathematical Biology (2006) 68: result from the failure of cells to migrate or migration to the wrong location. After birth, cell migration is essential to wound healing, tissue repair and immune responses, as well as contributing to tumour invasion and metastasis (Franz et al., 2002). Factors that influence cell behaviour during migration include interactions between cells and the environment such as the extracellular-matrix, interactions between cells and chemicals present in the environment, and interactions between cells themselves (Abercrombie, 1980; Lauffenburger and Linderman, 1993; Lauffenburger and Horwitz, 1996) Signalling molecules Extracellular molecules, referred to as signalling molecules, are known to affect the extent of cell migration (Wolpert, 2002). Investigations into the role that signalling molecules play in cell migration are important for understanding normal development processes, as well as for developing potential therapeutic strategies. Signalling molecules that promote cell migration will be important in the treatment of neurological birth defects due to deficiencies of cell migration, the promotion of wound healing, and to building viable tissue cultures on engineered scaffolds. In addition, application of signalling molecules that restrain tumour cell migration would be beneficial to cancer treatment therapy. Cell motility is the ability of a cell to move. Cell motility is affected by motility regulators termed inducersor inhibitors (Masonet al., 2001).Inducersincreasecell motility while inhibitors reduce cell motility. Attractants and repellents affect the directionality of cell motion. Attractants cause cells to migrate in a biased way towards the source of the attractant. Repellents have the opposite affect. Attractants and repellents are only effective through a concentration gradient. For example, a cell that can sense a gradient of a repellent will be biased to move towards a lower repellent concentration. If there is no gradient, cell motion is not biased since the cell has no way to identify a preferred direction. On the other hand, inducers and inhibitors do not strictly require a concentration gradient to be effective. Cells sensing the presence of an inhibitor will have reduced motility even if there is a uniform concentration of inhibitor. The applicability of a signalling molecule for therapeutic treatment depends on its effect on migrating cells. For example, inhibitors, being motility regulators, would be more effective in restraining the spread of cells, as opposed to repellents, being directional guides, which would redirect the cell migration to another region (Ward et al., 2003). Both Mason et al. (2001)andWard et al. (2003) attempt to provide an experimental framework to distinguish between the motility regulatory and directional guidance roles of a particular signalling molecule in neuronal migration. Of course, a signalling molecule may also have a dual role, acting as both an inhibitor and repellent Experimental background A particular signalling molecule, the secreted protein Slit, has beenfound to affect neuronal migration in the brain. But the exact role of Slit is unresolved. Neurons

3 Bulletin of Mathematical Biology (2006) 68: migrate tangentially from the subventricular zone (SVZ) to the olfactory bulb throughout life, for a function related to the sense of smell (Alvarez-Buylla, 1997; Menezes et al., 2002). The effect of Slit has been well documented by Wu et al. (1999), Mason et al. (2001), Menezes et al. (2002)andWard et al. (2003). In experiments by Wu et al. (1999), a circular piece of tissue containing a large number of neurons, called an explant, was removed from the SVZ of postnatal rat brains. An aggregate of Slit-secreting cells was placed on one side of this circular explant. The distribution of the neurons migrating out of the explant was observed to be clearly skewed away from the Slit source and more cells were found on the side away (distal) from the source of Slit compared to the side close (proximal) to that source. This asymmetry was not observed in control experiments where the explant was not exposed to Slit. From these observations, Wu et al. (1999) claimed that Slit was a repellent of neurons. Mason et al. (2001) found the number of neurons migrating out of a similar explant decreased with increasing levels of Slit. These findings suggested that Slit was an inhibitor of neurons. Based on these two experiments, Ward et al. (2003) designed a new co-culture experiment to distinguish between repulsive and inhibititory effects, using a time-delayed application of a Slit-producing aggregate. The experimental set up of Ward et al. (2003) was similar to that of Wu et al. (1999). Circular explants were dissected from SVZ of postnatal rat brains. Neurons were observed to migrate from the explant in a symmetric fashion. Figure 1(a) shows the right half of the neuron distribution at 24 h. At this time, the edge of a control aggregate of non-slit secreting cells was placed along the solid black line. As observed in Fig. 1(b), the cell distribution remained symmetric. In a second experiment with a circular explant, the edge of an aggregate of Slit-secreting cells was placed along the solid line after 24 h. Of course, the neuron distribution is again symmetric before the application of the Slit-producing aggregate, as shown in Fig. 1(c). After a further 24 h, the distribution is now clearly asymmetric, given by Fig. 1(d). Comparison between the control case Fig. 1(b) and (d) shows that the presence of Slit causes a striking difference in the distribution of neurons. The claim by Ward et al. (2003) that Slit is primarily a repellent is supported by the observed reversal of direction of some individual cells when exposed to Slit and a dramatic decrease of cell numbers on the proximal side of the Slit source. Ward et al. (2003) argued that if Slit were an inhibitor, then the cells on the side proximal to the Slit source would slow down rather than reverse direction and move away from Slit. Also if cells were slowed by Slit, then more cells could be expected to accumulate on the proximal side; hence, an increase of cell number should have been found rather than a decrease. Finally, the speed of tracked cells that turned in response to Slit application showed that some cells had increased speed after exposure to Slit and some cells had decreased speed. No significant reduction in the mean speeds was observed (Ward et al., 2003, Fig. 7b). Hence, Ward et al. (2003) argued that there was a lack of evidence for an inhibitory effect and concluded that Slit is primarily a repellent of neurons. Other experiments by Ward et al. (2003) involved a uniform application of Slit to the neuron culture medium. They observed a small but statistically significant reduction in tracked cell speed. The authors believed that the observed reduction of cell speed could be attributed to Slit being a repellent, although their arguments

4 28 Bulletin of Mathematical Biology (2006) 68: Fig. 1 Neuronal cell distribution from a circular explant from Ward et al. (2003). Images of the explant before and after placement of an aggregate; the edge of the aggregate indicated by a black line. (a) Control experiment at 24 h, just before non-slit secreting aggregate placement. (b) Control experiment at 48 h. (c) Slit experiment at 24 h, just before Slit-secreting aggregate placement. (d) Slit experiment at 48 h. The photographs are reproduced with the permission of the Society of Neuroscience. Copyright 2003 Society of Neuroscience. are difficult to follow. From this, we see that the distinction between repellents and inhibitors is far from clear. In this paper, modelling and simulation techniques are applied to unravelling directional guidance and motility regulation. Using the Ward et al. (2003) experiments as a guide, we simulate cell migration from an explant in the presence and absence of a signalling molecule. In Section 2, we consider population-level continuum models based on mass conservation equations. These equations are recast into transition probabilities governing individual cell motility. We consider various strategies whereby a cell senses a signalling molecule and discuss the motility regulation and/or directional guidance effects. In Section 3 we use our models to simulate the Ward et al. (2003) experiments. Different cell-sensing models give rise to differences in population-level distributions and individual cell motility and turning. We compare the simulation results and determine the motility and repulsive effects. Finally, we discuss our findings in relation to the results of Ward et al. (2003) and assess the difficulties and limitations in deducing cell migration rules from time-lapse imaging and/or simulation realisations.

5 Bulletin of Mathematical Biology (2006) 68: Mathematical models We use continuum models based on the mass conservation equation to model the evolution of a cell population. The microscopic description of cell behaviour is derived from the population models via discretising the conservation equation. This gives rise to transition probabilities of nearest-neighbour random walkers, as described by Anderson and Chaplain (1998). These transition probabilities are dependent on the form of the discretisation chosen. The discretisations used here are the simplest possible. Different discretisations give stochastic trajectories that are different over small time intervals only. This modelling approach provides two perspectives on the system and complements the population-level and individual cell-level information provided by time-lapse imaging. A model of migration of a cell population in the absence of any signalling molecule corresponds to the Ward et al. (2003) control experiments, when a Slitsecreting aggregate is not added. We provide an outline of this type of model in Section 2.1 and apply it to the actual control experimental set up in Section 3.1.No comment is made regarding cell proliferation in Ward et al. (2003); here we assume that the explant contains a large number of cells and any proliferation is ignored on the time scale of the experiments (H. Young, 2004, private communication). We consider a variety of ways in which cells can sense Slit, following the work of Othmerand Stevens (1997) and PainterandSherratt (2003). In Section 3, the cellsensing models are applied to the experiments of Ward et al. (2003) to determine the repulsive and inhibitory effects of Slit Cell diffusion For clarity in our initial discussion, we first consider the case of randomly motile cells moving in one dimension. The evolution of the cell density u(x, t) at position x and time t can be modelled by the diffusion equation u t = ( D u ), x x where for the present the diffusivity D is assumed to be constant. Associated with u(x, t) we define a probability density function P(x, t), such that the probability of finding an individual between positions a and b is given by b a P(x, t)dx. Thetwo functions are related as u(x, t) P(x, t) =. (2) u(x, t)dx A microscopic description of individual cell behaviour can be determined by discretising the population model Eq. (1) using x = i x, t = j t to obtain a nearestneighbour random walk expression. We introduce P j i = xu(i x, j t), (3) u(x, t)dx (1)

6 30 Bulletin of Mathematical Biology (2006) 68: and interpret P j i as the probability that a cell is situated at lattice site i after j time steps. It will later be seen that P j i = 1forall j, and therefore the discretised system conserves probability. To obtain the probability P j+1 i from the previous time step P j i, the diffusion equation for u (or equivalently, the diffusion equation for P) is discretised in space using a standard central difference scheme. The technique gives rise to an equation in terms of transition probabilities. A transition probability T(m l) is the probability that a cell moves from site l to site m. This procedure gives, for constant diffusivity, where P j+1 i = T(i i + 1)P j j i+1 + T(i i 1)Pi 1 + T(i i)p j i, (4) T(i ± 1 i) = t D. (5) ( x) 2 With equal left and right transition probabilities, individual cells are unbiased in their choice of direction. The probability of a cell making a transition out of its current site is just the sum of the left and right transition probabilities, that is, Pr(leaving the current site i) = T(i + 1 i) + T(i 1 i). (6) This probability can be interpreted as the motility ofa cell. Furthermore, the probability that the cell remains at the same site is T(i i) = 1 T(i + 1 i) T(i 1 i) = 1 2 t D. (7) ( x) 2 We can see from the transition probabilities that cell motility is a constant and proportional to the diffusivity D. Such a model is applicable when cell cell interactions are negligible, for example when the cells are spaced far enough apart. When cell cell interactions become important, we expect the cell motility to depend on the local cell density and therefore the diffusivity would take the form D(u). With a non-constant D in Eq. (1), the discretisation is carried out with a centred differencing scheme at the mid-points between the lattice points (outlined for a general case in Appendix). With D(u) i = D(u(i x, j t)) the discretised P j+1 i again satisfies (4), with the transition probabilities now generalised to T(i ± 1 i) = t ( ) D(u)i + D(u) i±1, (8) ( x) 2 2 T(i i) = 1 T(i + 1 i) T(i 1 i). (9) For brevity, we suppress the index j in the expressions of the transitions probabilities. The transition probabilities are all evaluated at the current time step j; hence the time stepping algorithm (4) is fully explicit.

7 Bulletin of Mathematical Biology (2006) 68: Here the transition probabilities depend on information at the current and the target sites and hence show a bias of cell motility when cell density is non-uniform in the domain. The direction of the bias depends on the difference between the left and right transition probabilities, that is, directional bias = T(i + 1 i) T(i 1 i). (10) Here the right-hand side of (10) is the mean displacement per step measured in units of x. The actual direction of the bias will depend on the functional form of D(u). This will be discussed in more detail in Section 3.1. Although other discretisations of a non-linear diffusion equation are possible, one should only use discretisations that give non-negative transition probabilities and are properly normalised, that is, T(i + 1 i) + T(i 1 i) + T(i i) = 1. (11) This is the case for all discretisations used in this paper. For all our simulations, the lattice is sufficiently large so that the cells do not walk off the lattice. Therefore, boundary conditions are unnecessary Cell movement in the presence of a signalling molecule To describe the motion of cells one may initially take a population-level view, based on a conservation law u t + j = 0 (12) together with a relation linking the flux j to the cell concentration u and the signalling molecule concentration s. Alternatively, one may work with a stochastic model that carries information concerning random motions of individual cells. These cell motions will be influenced by the signalling molecule distribution and the presence of neighbouring cells. Both approaches have their appeal. We have chosen to start from the population-level conservation law approach, so that we can include the effect of interacting cells in a phenomenological but consistent way based on cell diffusivities that depend on cell density. From the population-level continuum equations we extract a probability model that is discrete both in space and in time that enables us to follow the motions of individual cells, in the manner of Anderson and Chaplain (1998). An alternative approach is to write down a stochastic model for individual cell motion at the outset. If cells are assumed not to interact, this model can be simulated directly, or used to derive continuum, population-level equations by an appropriate limiting procedure. In modelling cell cell interaction, the determination of appropriate rules to describe interactions is not straightforward. Furthermore, when interaction rules are non-trivial, the extraction of useful population-level equations is seldom possible (Liggett, 1985).

8 32 Bulletin of Mathematical Biology (2006) 68: Othmer and Stevens (1997), and Painter and Sherratt (2003) start from stochastic models based on spatially discrete master equations, that is, linear rate equations linking the time derivative of the probability of occupying a site to the probability of currently occupying that site or one of its immediate neighbours. In a master equation formalism, individual cell motions are not immediately apparent, but it is known (Hughes, 1995, chapter 5) that for a wide class of master equations there is a one-to-one correspondence between the solution of the master equation and the solution of a continuous-time random walk process (Montroll and Weiss, 1965) in which a single random walker makes discrete moves at instants separated by random time intervals. For this correspondence to hold, the times between successive moves must be independent, identically distributed, exponential random variables. At long times t, the asymptotic behaviour of the model becomes identical to a discrete-time random walk, when one identifies the number of steps taken as the integer part of the ratio of time t to the mean time between steps. Although in the present work we could have used our continuum population equations to obtain spatially discrete master equations, we have chosen to derive discrete-time equations, since these entail no real loss of generality and are somewhat more convenient in simulations. Cells can sense signalling molecules in a number of ways. The strategies considered here broadly correspond to the cases discussed by Othmer and Stevens (1997), and Painter and Sherratt (2003). We modify the strategies to apply to the signalling molecule alone, rather than to both the signalling molecule and cell cell interactions. We require non-local cell cell interaction effects, described by a nonlinear diffusivity, to apply whether the signalling molecule is present or not. In the absence of the signalling molecule we will require the transition probabilities to collapse to Eq. (8). We consider four cell-sensing strategies: 1. The strictly local model. Here cells sense the concentration of the signalling molecule at their current local position only. 2. The local average model. Cells can sense the concentration of the signalling molecule at their current position and the neighbouring positions. 3. The neighbour-based model. Cells sense the concentration of the signalling molecule at the neighbouring positions only. 4. The gradient-based model. Cells can sense the local difference in the concentration of the signalling molecule at their current position and the neighbouring positions. In each of these models, we determine the transition probabilities that allow to be determined from P j i at the previous time, given by Eq. (4). P j+1 i The strictly local model Consider the population model of the form u t = ( D(u) ) x x [I(s)u], (13)

9 Bulletin of Mathematical Biology (2006) 68: where I(s) is some function of the signalling molecule concentration s(x, t). From the Appendix, the corresponding transition probabilities are T(i ± 1 i) = t ( x) 2 ( ) D(u)i + D(u) i±1 I(s) i. (14) 2 Both transition probabilities depend on the local concentration of the signalling molecule only. It is clear that the presence of the signalling molecule s modulates the motility of cells, through I(s). If the modulation is positive, that is I(s) increases with s, then the molecule is motility inducing, otherwise the molecule is motility inhibiting. In addition, the presence of s doesnot create any bias in the direction ofmovementof individual cells;it modulates,through I(s), any directional bias produced by a non-linear diffusivity The local average model Consider a conservation equation of the form u t = ( D(u, s) u ). (15) x x The function D(u, s) represents cell diffusivity, taking into account its dependence on the local signalling molecule concentration as well as the local cell density. Discretising according to the Appendix, the corresponding transition probabilities are T(i ± 1 i) = t ( x) 2 ( ) D(u, s)i + D(u, s) i±1 2 where D(u, s) i = D(u(i x, j t), s(i x, j t)). When the concentration of s is not uniform, the left and right transition probabilities will reflect different levels of s, producing a bias to the transition probabilities (in addition to any bias caused by cell cell interactions). If, for example, s inhibits motility, that is D(u, s) decreases with s, gradients of s will cause individuals to be biased to move towards a lower concentration of s, giving an effective repulsion. A similar observation can be made if s induces motility and D(u, s) increases with s, except that now the individuals will be attracted to s. Hence, sensing the local averaged concentration of the signalling molecule has the potential to produce both motility regulation and direction guidance effects The neighbour-based model For completeness we include in our study a model of Othmer and Stevens (1997), and Painter and Sherratt (2003) based on the partial differential equation u t (16) = ( D(u)I(s) u ) I(s) D(u)u. (17) x x x

10 34 Bulletin of Mathematical Biology (2006) 68: Although this equation is correctly posed in conservation law form, the associated flux is less naturally interpreted in physical terms than the fluxes in the other three models that we consider. This model gives rise to the transition probabilities T(i ± 1 i) = t ( x) 2 ( D(u)i + D(u) i±1 2 ) I(s) i±1. (18) This model differs from the local average model as the concentration of s at the current occupied site does not contribute to the transition probability out of the current site. The presenceofs regulates cell motility positively when I(s) increases with s, and negatively when I(s) decreases with s. Directional guidance is also possible here if there is a non-uniform concentration of s. Repulsive effects will be produced when I(s) inhibits motility, and attractive effects when I(s) induces motility Gradient-based model The conservation equation with diffusion as well as chemotaxis (Keller and Segel, 1971; Anderson and Chaplain, 1998; Murray, 2002)is u t = ( D(u) u x x + uχ(s) s ), (19) x where χ(s) is referred to as the chemotactic sensitivity. The presence of a chemical gradient causes a directional bias in the cell migration. If χ(s) < 0, then cells are biased to move up a concentration gradient of the signalling molecule, and if χ(s) > 0, cells move down a gradient. These two cases model chemotactic attractants and repellents respectively. Discretising according to the Appendix, the corresponding transition probabilities are T(i ± 1 i) = t ( ) D(u)i + D(u) i±1 s i±1 s i χ(s) ( x) 2 i. (20) 2 2 There is a clear directional bias, proportional to χ(s) i (s i+1 s i 1 ). Hence, if there is a gradient in s, then an attractive effect (χ(s) < 0) or a repulsive effect (χ(s) > 0) will be evident. In addition, the probability of making a transition away from the current site is T(i + 1 i) + T(i 1 i) = t ( x) 2 ( ) D(u)i+1 + 2D(u) i + D(u) i 1 t ( x) 2 χ(s) i 2 ( si+1 2s i + s i 1 2 (21) ). (22) Therefore, the guidance molecule will also have some effect on cell motility, through a contribution related to the second derivative of its concentration s.

11 Bulletin of Mathematical Biology (2006) 68: Unlike the other three strategies, if the distribution of s is uniform there is no effect at all. Here, care must be taken to ensure that the transition probabilities are nonnegative. In an advection diffusion transport context, this is equivalent to ensuring that the Peclet number, v x/d, is less than 2, where the magnitude of the advective velocity v is effectively χ s/ x Simulation process The simulation process in one dimension involves evaluating the transition probabilities for each cell at lattice position i at each time step j. To do this, we generate a random number between 0 and 1. If the number falls between 0 and T(i + 1 i)the cell is moved to the right; if it falls between T(i + 1 i) andt(i + 1 i) + T(i 1 i), the cell is moved to the left; if it falls between T(i + 1 i) + T(i 1 i) and1isremains at site i. The larger the transition probability, the larger the probability that the corresponding movement will be selected. All the cells are moved simultaneously when updating from time step j to time step j + 1. In this section, we have described several models of cell migration in one dimension. All the models generalise naturally to higher dimensions. In two dimensions we use a square lattice, where the movements can be either to the left, right, up or down. Four movement transition probabilities are evaluated, as well as the rest probability. At each time step, five probability ranges are computed. The range into which a random number between 0 and 1 falls will determine a cell s movement. The simulations were programmed in the computer language C. 3. An application: Distinguishing between repulsion and inhibition We now apply the models outlined in the previous section to the experimental set up of Ward et al. (2003). First we shall model the control experiment where neurons migrate from a circular explant in the absence of the signalling molecule Slit. Before tackling the two-dimensional problem, we investigate in one dimension the appropriate way to model the interactions between the cells. After deciding on a diffusivity function, we shall model the application of a Slit source and the resulting movement of cells in response to Slit. In performing simulations, we have attempted to match the qualitative features of the results of Ward et al. (2003). A direct quantitative comparison is not possible, since many aspects of the experiments, such as the actual Slit concentration gradient, are unknown Modelling the control experiments in one dimension Cells were observed to migrate away from the explant in a directed manner before the addition of Slit (Ward et al., 2003). This observation is not surprising, since an explant contains a large number of closely-packed neurons. Directed cell migration suggests that cell cell interactions are an important factor in the experimental

12 36 Bulletin of Mathematical Biology (2006) 68: set up. We noted in Section 2 that a non-linear diffusion equation is appropriate for modelling an interacting cell population. We consider two contrasting nonlinear diffusion models (diffusivity either increasing with density or decreasing with density) and investigate which functional form produces persistent cell movement away from the explant. Interacting population models, where the diffusivity is an increasing function of density, have been used extensively for insect and animal dispersal and invasion (Okubo, 1980; Murray, 2002). Such models are thought to reflect the effect of population pressure. One simple example is u t ( ) u u = D 0, (23) x U 0 x where D 0 and U 0 are positive constants. This model implies increased individual activity with increased local density. Applied to cells in a tissue explant, we predict that cells interior to the explant will have a higher motility compared to the cells at the edge of the explant. From the expression for the transition probabilities in Eq. (8), we deduce that transitions to sites with higher population densities will be preferred. This conclusion holds generally, whatever the details of D(u) so long as D(u) is a strictly increasing function of u. A greater probability of transitions to sites with higher population densities contradicts the experimental observations that cells are biased to move away from the explant, and hence biased towards lower density regions. These arguments lead us to propose an alternative non-linear model, one where the diffusivity is a decreasing function of density. Such a description is based on the contact inhibition of cell locomotion, first proposed by Abercrombie (1980), who observed that the collision of cells usually leads to cell reorientation and a new choice of direction by the cells. It is natural to consider the cell diffusivity decreasing with density, since the presence of other cells will lead to more collisions and therefore hinders and inhibits cell motion. A simple example is provided by u t ( A = D 0 x A+ u u x ), (24) where D 0 and A are positive constants. We predict that individual cells simulated with a contact inhibition model would display persistent migration away from the explant. Painter and Sherratt (2003) also used a diffusion coefficient decreasing with density in their model of two interacting cell populations. Simulations are carried out in one dimension on a linear lattice, with lattice spacing x = 1. A one-dimensional finite-length explant is modelled by 2000 cells placed at each integer site from 5 to5. We now compare results from the three models, as shown in Fig. 2. The density profiles on the left show the cell population spreading away from the center of the explant over time in all three models. As expected, the population pressure model exhibits compact support type solutions (from similarity solutions in the continuum limit, Murray, 2002), while in the contact inhibition model the migration is slowed, since the mobility of cells is small within the explant.

13 Bulletin of Mathematical Biology (2006) 68: Fig. 2 Comparison between the three diffusion models. Left side: Cell number versus position. Right side: The central curve is the mean trajectory (solid line) and the right and left curves represent the mean trajectory plus or minus one standard deviation (dashed lines) for three groups of cells, starting at x = 5, 0, 5. (a) Constant diffusivity model (1) withd = (b) Population pressure model (23) withd 0 = 0.25, U 0 = 4. (c) Contact inhibition model (24) withd 0 = 0.25 A = 10. The cell number profiles are given for t = 100, 200, Here x = t = 1. In order to probe more deeply into differences between the three cases, the trajectories of a group of cells that started from the same site are investigated. They are (i) the 2000 cells that started from the left edge at x = 5, (ii) the 2000 cells that started from the right edge at x = 5, and (iii) the 2000 cells that started at the centre. The graphs on the right in Fig. 2 display three curves for each group of cells. The central curve is the mean trajectory and the right and left curves represent the mean trajectory plus or minus one standard deviation. In the constant diffusion case, the individuals are non-interacting and make random unbiased transitions between sites. For this model, the mean trajectory should be the same for all

14 38 Bulletin of Mathematical Biology (2006) 68: groups. Furthermore, the unbiased nature of the individual movement should give a vertical mean trajectory when displayed on a space time diagram, as in Fig. 2(a). The population pressure model biases individuals to move from low-density regions to high-density regions, demonstrated by the inward mean trajectories of the group of cells that started on the two edges of the simulated explant, as in Fig. 2(b). The bias in the contact inhibition model is from high-density regions to low-density regions and the mean trajectories of cells are found to move outward, as expected and shown in Fig. 2(c). The standard deviation of the paths illustrates the amount of spread of individual cells in the tracked group. Both the population pressure and the constant diffusion model have a large spread compared to the contact inhibition model. Furthermore, this spread results in the cell groups mixing together as time increases. Since the contact inhibition model provides the directed migration observed in the two-dimensional control experiments of Ward et al. (2003), we will limit the nature of the cell cell interaction to this model from here on. In particular, the functional form D(u) = D 0 A/(A+ u) in Eq. (24) will be used. It is appropriate to test the sensitivity of the contact inhibition model to the parameter A. We expect the cells to be more persistent with decreasing values of A, because the non-linear effects are stronger. Simulations are again carried out with an initial configuration of 2000 cells placed at each integer site from x = 5 to x = 5, using various values of A. For very large A values, as in Fig. 3(a), the interaction between cells is reduced so that cells move in an almost independent fashion, similar to the constant diffusion case illustrated in Fig. 2(a). As A is decreased in Fig. 3(b)-(c), the spread of cells decreases significantly and the amount of mixing between cells initially at the centre and cells initially at the edge of the explant decreases significantly, demonstrating increased persistence of cell movement. To check the accuracy of our simulations, the discrete simulation density profiles were compared with the density profiles from the numerical solution of the partial differential equation (24). The results were found to be consistent Modelling the time-delayed application of Slit experiments in two dimensions We have established the contact inhibition model for simulating cell cell interaction in the control experiments. The experiments involving Slit are investigated using the cell-sensing models proposed in Section 2.2, together with the contact inhibition model. For simplicity, the sensitivity to s is chosen in a similar fashion for all four cell-sensing strategies. For the local average model, we choose D(u, s) = D 0 A A+ u e βs, (25) with β>0. This simple exponential functional form decreases in the presence of s giving an inhibitory effect. Furthermore, this form collapses to the control model when Slit is absent (with s = 0) as required. For both the neighbour-based and strictly local models, we choose the function I(s) = e βs. (26)

15 Bulletin of Mathematical Biology (2006) 68: Fig. 3 The dependence on A for the contact inhibition model with D(u) = D 0 A/(A+ u). Left panel: Cell number versus position. Right panel: The central curve is the mean trajectory (solid line) and the right and left curves represent the mean trajectory plus or minus one standard deviation (dashed lines) for three groups of cells, starting at x = 5, 0, 5. (a) A = 10, 000. (b) A = 100. (c) A = 1. The cell number profiles are given for t = 100, 200, Here D 0 = 0.25, x = t = 1. For simplicity, in the gradient-based model we choose the chemotactic sensitivity to be constant, that is, χ(s) = χ, (27) where χ>0 for a repellent. We now turn to modelling the distribution of Slit. Following the experiments of Ward et al. (2003), we choose the source of Slit to be a fixed horizontal distance from the centre of the explant, and assume that it is independent of vertical distance. We assume that the diffusion time scale of Slit is significantly faster than the

16 40 Bulletin of Mathematical Biology (2006) 68: Table 1 Parameter values used in the simulations Explant diameter 200 µm Distance from explant centre to slit source, L 600 µm Time of Slit application, t a 24 h A 0.02 cells/µm 2 Cell diffusivity, D µm 2 /h cm 2 /s Chemotactic sensitivity, χ 10 4 µm 2 /h per unit of s concentration β 20 per unit of s concentration λ 0.01 (µm) 1 Q 2D s λ Lattice sizes, x, y 5 µm Simulation time interval, t h time scale of movement of cells, so that the Slit concentration is at a steady state. If t a is the time of activation of the Slit source, then the equation determining s,for times t > t a, is given by 2 s D s ks + Qδ(x L) = 0, (28) x2 where D s and k are the constant diffusivity and degradation rate of s respectively, L is the horizontal distance from the centre of the explant to the Slit source and Q is a measure of the strength of the Slit source. The resulting Slit concentration is s(x, y, t) = Q 2D s λ e λ x L H(t t a ), (29) where λ = (k/d s ) 1/2 and H is the Heaviside step function. The parameter values used in the simulations are summarised in Table 1. The parameters are estimated from the experiments of Ward et al. (2003) where possible; otherwise, they are chosen to produce qualitatively meaningful results that can be used as a comparison between the models and the experiments. For example, given the signalling molecule concentration gradient and the cell diffusivity, the lattice size must be chosen sufficiently small in the gradient-based model (that is the Peclet number must be less than 2) to ensure that transition probabilities (20) are non-negative Simulation results in two dimensions A circular explant is approximated by a set of 1457 lattice squares. Ward et al. (2003) provided no information regarding the number of neurons in each explant. We place 14 cells at each lattice site, using a total of 20,398 cells. To display the results of the simulations clearly, four lattice squares making up a larger square are binned. For the first 24 h, Slit is absent from all the simulations. Figure 4(a) shows the distribution of cells initially and after 24 h. At t = 0all the cells are in the explant (Fig. 4(a)). At 24 h, a radially symmetric distribution of the cells has formed resulting from the transition probabilities derived from the

17 Bulletin of Mathematical Biology (2006) 68: Fig. 4 Cell distributions. (a) Cells confined to the explant at time t = 0. (b) Cell distribution after migration from the explant at t = 24 h. The horizontal and vertical scales are in µm. The colour scale indicates cell numbers. contact inhibition model for cell cell interaction (Fig. 4(b)). For the control case, this simulation is allowed to run another 24 h with no signalling molecule present. For the case when Slit-secreting cells are placed along the right boundary (x = 600 µm) at 24 h, the transition probabilities are modified to account for a selected type of cell-sensing strategy resulting from the prescribed distribution of s given by Eq. (29). The simulations are run for another 24 h. The resulting cell migration patterns are now discussed Cell distribution Figure 5 illustrates the cell distributions resulting from all the models 48 h after placement of the explant. The control case is given in Fig. 5(a). In each of Fig. 5(b) (e), the Slit source is at the right-hand vertical edge and has been present for 24 h. Comparing the control case with the other four cases, we observe that the control case remains largely symmetric, whereas all the other cases give rise to an asymmetric cell distribution. The asymmetric distributions are skewed away from the source of Slit at x = 600 µm. It is apparent that examination of the distributions of the cell population alone is not sufficient to distinguish between the different models of cell-sensing from Fig. 5(b) (e). In particular, it is impossible to distinguish between inhibitory and repulsive effects from these plots alone. The only notable observation is that the region immediately next to the Slit source appears to have fewer cells in the gradient-based model (Fig. 5(e)) than the strictly local, local average and neighbourhood-based models (Fig. 5(b) (d)). We will first examine another population-based feature before outlining cellbased techniques we have developed to help distinguish between the cell-sensing strategies Mean trajectories We calculated the mean trajectories of cells placed initially at radial positions between 90 and 100 µm from the center of the explant and grouped in 10 uniform

18 42 Bulletin of Mathematical Biology (2006) 68: Fig. 5 Cell distributions 48 h after placement of the explant. (a) Control case. In (b) (e), the source of Slit is placed along the right edge at x = 600 µm at 24 h, indicated by the thick red line. (b) Strictly local model. (c) Local average model. (d) Neighbour-based model. (e) Gradient-based model. The horizontal and vertical scales are in µm. The colour scale indicates cell numbers.

19 Bulletin of Mathematical Biology (2006) 68: Fig. 6 The mean trajectories of groups of cells initially in a thin annular region of the explant from t = 0 48 h. Solid black circle ( ) marks the mean position of the group of cells at t = 24 h. (a) Control case. In (b) (e), the source of Slit is placed along the right edge at x = 600 µmat24h. (b) Strictly local model. (c) Local average model. (d) Neighbour-based model. (e) Gradient-based model. The horizontal (x) and vertical (y) scales are in µm.

20 44 Bulletin of Mathematical Biology (2006) 68: sectors, with approximately 200 cells in each. The position of the centre of each group of cells is represented by a point in Fig. 6. This figure shows the mean trajectories starting from t = 0 over 48h for 10 such groups of cells. The black circle on each trajectory marks the mean position at the time that the Slit source is placed at x = 600 µm in Fig. 6(b) (e), that is, at 24 h. For the first 24 h, the mean trajectories are directed outwards from the center of the explant due to contact inhibition of cell locomotion. After the time-delayed application of Slit, sharp turns in the mean trajectories can be seen in all sensing models in Fig. 6(b) (e). These are consistent with the asymmetries seen in the cell distribution plots of Fig. 5(b) (e), showing cells near the Slit source as most affected by the presence of s. For the control case, the mean trajectories are shorter than in the first 24 h, but essentially remain in the same direction. This is consistent with the contact inhibition model, since the cell cell interaction is less pronounced at later times, hence reducing the directional bias. However, it is worth remarking that the plots do not necessarily signify individual cells turning away from the Slit source. This is clearly demonstrated by considering the strictly local sensing model. Consider the right-hand sectors (x > 0). Cells migrate from this region in the first 24 h before application of Slit. After the placement of the Slit-producing aggregate, cell motility depends on its proximity to this aggregate. Those cells close to the Slit source have less motility than those that are further away. Averaging the movements over the whole group produces a mean trajectory, which points away from the Slit source. It should be noted that the net movement of cells on the distal side of the Slit source (x < 0) is small in all cases. This is expected, since the density is significantly reduced, minimising the bias in the transition probabilities, and therefore producing mean trajectories with near zero net displacement. The mean trajectories show a skewing away from the Slit source and again have only limited usefulness, like the cell distribution plots in Fig. 5. We now outline three techniques to distinguish between the cell-sensing strategies at the individual cell level Where did the cell start from and where did it go? We look for evidence of cells turning away from the Slit source, a main experimental observation by Ward et al. (2003). To do so, we examine a group of cells along a thin vertical strip with fixed horizontal position at 24 h (immediately before the application of Slit), as these cells will experience the same concentration of Slit. Cells positioned too close to the explant are affected more significantly by contact inhibition effects than the Slit effects. Therefore, we chose those cells that are between the distance x = 400 and 450µm at 24h to be the representative group. Figure 7 shows our selected group of cells at 24 h and asks the following question. Where did these cells start from initially and what are their final positions at 48 h? For ease of visualisation, the aspect ratio in these figures is not unity. In Fig. 7, our selected cell group is indicated by solid green circles, corresponding to their position at 24 h just before the s distribution is applied. Their initial positions at t = 0 are denoted by solid blue circles, while their final positions at 48 h are indicated by solid red circles.

21 Bulletin of Mathematical Biology (2006) 68: Fig. 7 A selected group of cells, indicated with solid green circles ( ), at 24 h just before the timedelayed placement of the non-slit producing or Slit-producing aggregate. The initial positions at t = 0 are indicated with solid blue circles ( ), while their final positions at 48 h are indicated with solid red circles ( ). (a) Control case. In (b) (e), the source of Slit is placed along the right edge at x = 600 µm at 24 h. (b) Strictly local model. (c) Local average model. (d) Neighbour-based model. (e) Gradient-based model. The horizontal and vertical scales are in µm. The aspect ratio is not unity here. In the control case in Fig. 7(a) the contact inhibition effect is observed since slightly more cells end up further away from the explant. The gradient-based model in Fig. 7(e) displays significant turning of cells; however, a small number of cells have remained close to their 24 h position, possibly due to the additional small motility effect of this model. Both the local average and the neighbour-based models lead to some cells turning away from the Slit source. This is expected due to the biased nature of the transition probabilities in the presence of Slit. However, the level of turning is less than that produced by the gradient-based model. The strictly local model is not expected to produce significant cell turning. In Fig. 7(b), we see that some cells have turned away from the Slit source back towards the explant due to the stochastic nature of the simulation; however, this number is insignificant compared to the number of cells that have accumulated and positioned

22 46 Bulletin of Mathematical Biology (2006) 68: Fig. 8 Number of cells in sectors. (a) The six equal sectors and their position in relation to the Slit aggregate. (b) Total cell number (including those inside the explant) in each sector for the control and four sensing strategies. themselves slightly closer to the Slit source. In this model, the cells are inhibited by the presence of Slit. Moreover, since they do not sense the Slit gradient, no movement bias is created by the presence of Slit, unlike local average and neighbourbased sensing Number of cells in sectors Ward et al. (2003) observed that the number of cells in the region on the side close to the Slit source is significantly less than the number of cells in the region far away from the Slit source. In order to obtain comparable data, we counted the number of cells at 48 h in six evenly divided sectors with respect to the center of the explant, as illustrated in Fig. 8. A decrease in the number of cells in the region close to Slit (region VI in Fig. 8) is due to turning of cells away from the explant. The gradient-based, neighbourbased and local average models all show the reduced number effects observed by Ward et al. (2003). This drop in cell number correlates with the repulsive effect of Slit Cell speed Ward et al. (2003) found that the mean speed of various tracked cells does not show a statistically significant reduction after the placement of Slit. They observed

23 Bulletin of Mathematical Biology (2006) 68: that some cells increased their speed, some decreased speed, while others showed no real change of speed. However, such measurements are not sufficient to argue that Slit does not have an inhibitory effect on the cells. In considering this, the local cell density around the tracked cells must also be taken into account. From modelling the control experiments, we know that contact inhibition between cells is an important factor in these experiments which involved large numbers of cells migrating out of a closely packed tissue explant. It is perhaps not surprising that randomly tracked cells would have different speeds, as this is dependent on local density. To investigate the effects of Slit on cell motility, we examine the displacements of cells over a short time interval and investigate their dependence on the Slit concentration. Figure 9 shows the displacements of 1000 simulated cells in the first 4 h after the application of Slit. In these figures, the solid circle indicates the position of the cell at 24 h and the line indicates the size and direction of the net displacement over the subsequent 4 h. The erratic actual trajectories during the 4 h time interval are not shown. Hence, these pins are effectively a displacement vector. For ease of visualisation, the magnitude of the displacement is colour coded. For the control case in Fig. 9(a), the cells close to the explant have smaller displacements compared to those far away, due to contact inhibition effects. In the strictly local, local average and neighbour-based models, cell motility is inhibited by the presence of Slit, as seen from the small displacements of cells positioned close to the Slit source (the red short pins) in Fig. 9(b) (d). This contrasts the gradient-based cases Fig. 9(e) where there is no clear decrease in the motility of cells positioned close to Slit. In the gradient-based model, while the motility does not appear to be reduced, the direction of motion is changed, resulting in cell turning due to the presence of Slit. The direction change can be observed on the right-hand side of the distributions, where most pins are pointing inward in Fig. 9(e) compared to outwardpointing pins in the control case Fig. 9(a). This behaviour demonstrates the dominant repulsive effect of this model. The information shown in Fig. 9 can be readily summarised by averaging the results in a particular way. For a given value of the horizontal coordinate x, all of the cells located within a long thin vertical band (40 µm wide with x as the coordinate at the centre of the band) at the time of application of Slit are selected as a group. The displacements of each of the cells in this group over 4 h after Slit application are averaged to give a mean displacement as a function of horizontal position x, as illustrated in Fig. 10. Contact inhibition effects are demonstrated by the smaller mean displacement at the origin compared to all positions to the left of explant. Furthermore, over the whole x < 0 region, the control case and all cell-sensing strategies give similar mean displacements, since the effects of Slit are minimal there. To the right of the explant (on the same side as the Slit aggregate x > 0), we see a clear distinction between the models. The models that produce a significant inhibitory effect are the strictly local, local average and the neighbour-based models; moreover, the magnitude of the inhibition is similar for these three models over the 4-h period. In contrast, the displacement by cells close to Slit source in the gradient-based model is comparable to the control case, indicating no significant

24 48 Bulletin of Mathematical Biology (2006) 68: Fig. 9 Representation of net displacement of individual cells over t = h. The solid circle indicates the position of the cell at 24 h and the line indicates the net displacement over the subsequent 4 h, (after the application of Slit for cases (b) (e)). (a) Control case. In (b) (e), the source of Slit is placed along the right edge at x = 600 µm at 24 h. (b) Strictly local model. (c) Local average model. (d) Neighbour-based model. (e) Gradient-based model. The horizontal and vertical scales are in µm. The colour scale indicates the magnitude of the cell displacement.