Abstract
Mathematical models are often applied to describe cell migration regulated by diffusible signalling molecules. A typical feature of these models is that the spatial and temporal distribution of the signalling molecule density is reported by solving a reaction–diffusion equation. However, the spatial and temporal distributions of such signalling molecules are not often reported or observed experimentally. This leads to a mismatch between the amount of experimental data available and the complexity of the mathematical model used to simulate the experiment. To address this mismatch, we develop a discrete model of cell migration that can be used to describe a new suite of co–culture cell migration assays involving two interacting subpopulations of cells. In this model, the migration of cells from one subpopulation is regulated by the presence of signalling molecules that are secreted by the other subpopulation of cells. The spatial and temporal distribution of the signalling molecules is governed by a discrete conservation statement that is related to a reaction–diffusion equation. We simplify the model by invoking a steady state assumption for the diffusible molecules, leading to a reduced discrete model allowing us to describe how one subpopulation of cells stimulates the migration of the other subpopulation of cells without explicitly dealing with the diffusible molecules. We provide additional mathematical insight into these two stochastic models by deriving continuum limit partial differential equation descriptions of both models. To understand the conditions under which the reduced model is a good approximation of the full model, we apply both models to mimic a set of novel co–culture assays and we systematically explore how well the reduced model approximates the full model as a function of the model parameters.
1 Introduction
Random motility is widely recognised as the key mechanism driving in vitro cell migration in highly idealised homogeneous environments (Huang et al. 2005; Treloar et al. 2014). However, in more realistic situations, cell migration is often regulated by external signals such as diffusible molecules. Cell migration regulated by signalling molecules plays an important role in embryonic development (Behar et al. 1996; Simpson et al. 2006), cancer metastasis (Kucia et al. 2004; Müller et al. 2001) and wound healing (Flegg et al. 2015; Pettet et al. 1996). In these situations, cell migration is often activated by signalling molecules binding to receptors on the cell surface (Yoon et al. 2016). Signalling molecules can be present in the environment or secreted by other cells (Luster 1998; Wright et al. 2005). In Figure 1(a) we show an example of such a system where a signalling molecule called stromal cell–derived factor 1 (SDF-1) binds to the C-X-C motif chemokine receptor 4 (CXCR4) expressed on the surface of a mesenchymal stem cell (MSC). This process can regulate migration of MSCs (Yoon et al. 2016).
There are two key mechanisms that give rise to cell migration regulated by diffusible signalling molecules: (i) chemokinesis is where undirected cell migration is regulated by the local density of a particular signalling molecule (Liu and Klominek 2004; Cai et al. 2006); and (ii) chemotaxis is where the direction of cell migration is influenced by the spatial gradient of a signalling molecule (Keller and Segel 1971). The primary difference between chemokinesis and chemotaxis is that, at the individual level, chemokinesis influences the rate of undirected random cell movement without explicitly introducing a directional bias, whereas chemotaxis explicitly stimulates directional cell movement (Cai et al. 2006). Various experimental methods, such as transwell assays (Chen et al. 2006), microfluidic devices (Son et al. 2015), chemokinesis and chemotaxis assays (Richards et al. 2004; Rosoff et al. 2004), and co-culture migration assays (Chung et al. 2009; Frimberger et al. 2006) are used to study the role of chemokinesis and chemotaxis. However, these experimental approaches suffer from many important limitations. Two key limitations are: (i) signalling molecules are technically difficult to visualise in real time (Tokoyoda et al. 2004), and (ii) the spatial gradient of the signalling molecules is difficult to quantify (Chung et al. 2009).
Mathematical models have been widely used to mimic experimental observations relating to chemokinesis and chemotaxis (Brumley et al. 2019; Simpson et al. 2006). In the mathematical modelling literature, perhaps the most well–known model describing chemotaxis is the Keller–Segel partial differential equation (PDE) model proposed in 1971 (Keller and Segel, 1971). This fundamental continuum model has since been generalised to describe both chemokinesis and chemotaxis simultaneously (Balding and McElwain 1985; Byrne et al. 1998; Hillen and Painter 2009; Sherratt 1994). Further extensions of these continuum models include: (i) incorporating multiple cell populations (Stinner et al. 2014); (ii) explicitly modelling receptor–molecule binding (Sherratt et al. 1993); (iii) treating aggregates of cells as multiple interacting phases (Byrne and Owen 2004); and (iv) modelling responses with multiple signalling molecules (Painter et al. 2000). Apart from applying continuum PDEs to study cell migration stimulated by signalling molecules, discrete stochastic models have also been employed (Khain and Sander 2014; Pillay et al. 2018). Compared to continuum models, discrete models can be used to describe individual cell–level behaviour, and to specify how individual cells respond to signalling molecules. Using discrete models can be advantageous when comparing model predictions with experimental images that focus on individual cell level behaviour.
A key limitation of standard modelling frameworks is that typical models of chemokinesis and chemotaxis explicitly describe spatial and temporal distributions of the signalling molecules, often using a reaction–diffusion equation (Painter et al. 2000; Stinner et al. 2014). This is an important limitation because information about the spatial and temporal distributions of signalling molecules is rarely available from experiments (Chung et al. 2009; Tokoyoda et al. 2004). For example, we show an immunofluorescence image in Figure 1(b), with the green fluorescence indicating the expression of CXCR4. However, it is impossible to quantify the number of receptors or the spatial variability of the density of signalling molecules in this kind of standard experimental image. Therefore, it is unclear whether it is useful to mimic such an experiment with a mathematical model that explicitly describes the spatial and temporal variations in signalling molecule density. If one was to use a classical modelling approach, such as the Keller–Segel model, we would have no way of testing whether the spatial and temporal distributions of signalling molecules is accurate since these details are not available from standard experiments.
Motivated by new co–culture migration assays that we report in Section 2, the aim of this work is to develop an agent–based modelling framework that can be used to describe the dynamics of two interacting subpopulations of cells in a co–culture assay. In this model the movement of one type of agents is stimulated by the presence of signalling molecules that are produced by the other type of agents. The spatial and temporal distribution of the signalling molecules is governed by discrete conservation statement that is related to a reaction–diffusion PDE. We refer to this new model as the full discrete model since we explicitly describe the spatial and temporal distributions of agents and signalling molecules. To make the full discrete model more compatible with experimental data, we simplify the model by assuming that the dynamics of the signalling molecules is much faster than the time scale of cell migration. This simplification enables us to explore how the spatial distribution of signalling molecules affects cell movement without having to solve the underlying conservation equation for the signalling molecules. We refer to this simplified model as the reduced discrete model. The reduced model is both simpler to apply than the full discrete model since there are less parameters to estimate, as well as being more consistent with experimental observations in which the details of the signalling molecules are not reported. To provide additional mathematical insight into these two different stochastic models we also explore the continuum limit descriptions of both the full and reduced discrete models are derived through mean field analysis. This leads to new PDE models of signalling molecule–stimulated cell migration.
2 Co-culture experimental motivation
To motivate our modelling work we perform and report typical data from a series of in vitro co–culture ring barrier migration assays (Das et al. 2015). The full experimental protocol is documented in the Supplementary Material. Briefly, this type of co-culture assay involves uniformly seeding one type of cells inside a circular ring barrier and uniformly seeding another type of cells uniformly outside of the ring barrier (Das et al. 2015). Interactions between the two cell types can give rise to either a chemokinetic or chemotactic effect, depending upon the particular cell lines used in the experiment. In our expeiments, hepatocytes are seeded inside the ring barrier and MSCs are seeded outside the ring barrier in each experimental well in a 12–well tissue culture plate (Figure 2(a)–(b)). After seeding, the tissue culture plate is placed in an incubator overnight to allow cells to attach to the substrate. After attachment, the ring barrier is removed, leaving a vacant annulus of width approximately 1 mm (Figure 2(c)). Observations of the resulting cell migration are recorded by taking images of a small field–of–view over a 24 h period and recording the coordinates of particular cell trajectories over this period. Results in Figure 2(f)–(g) compare the endpoints of 20 typical MSC trajectories in two different experiments. The trajectories in Figure 2(f) are taken from a control experiment in which hepatocytes are omitted and we see that the MSCs appear to migrate randomly, with no obvious preferred direction. In contrast, the trajectories in Figure 2(g) are taken from an experiment that includes hepatocytes and we see clear evidence that the MSC migration is directed towards the location of the hepatocytes. Typical experimental results, such as those in Figure 2, do not provide any information about the temporal or spatial distribution of signalling molecules.
Many experimental studies indicate a role for chemokinesis or chemotaxis in co-culture experiments but do not show any spatial or temporal information about distributions of signalling molecules (Das et al. 2015). Therefore, we are motivated to model such experiments in a different way. For simplicity we apply our models to a small rectangular subregion as illustrated by the orange rectangle in Figure 2(c). Migration of cells in this subregion is predominantly horizontal, and to be consistent with this we develop our models in a one–dimensional geometry. Typical doubling times of MSC cell are over 50 h (Gruber et al. 2012), and since we only focus on relatively short time experiments we neglect the contribution of cell proliferation in our models.
3 Discrete models
The experimental data in Figure 2 provides strong evidence that MSC migration is biased in the presence of hepatocytes. Since hepatocytes are known to produce signalling molecules, such as SDF-1, we hypothesize that the directed migration of MCSs in Figure 2 is driven by a chemical signal. However, the data in Figure 2 does not indicate whether the directed migration arises from chemokinesis or chemotaxis, since both of these mechanisms can give rise to directed migration in the presence of a gradient of signalling molecule (Cai et al. 2006; Painter and Sherratt 2003). The modelling framework developed in this study can be used to examine either chemokinesis, chemotaxis, or a combination of chemokinesis and chemotaxis. For simplicity, we present the details by focusing on modelling chemokinesis in the main document. Additional results for modelling chemotaxis are presented in the Supplementary Material.
3.1 Full discrete model
We consider an agent–based model on a one–dimensional lattice where each site is indexed i ∈ [1, I] and has position x = (i − 1)Δ, where Δ is the lattice spacing that we take to be a typical cell diameter. The lattice is occupied by two different types of agents that represent the two different types of cells in the co-culture experiment: Subpopulation 1 which secretes signalling molecules, such as the hepatocytes in Figure 2, and Subpopulation 2 which senses and responds to the signalling molecules, such as the MSCs in Figure 2. The model is an exclusion process, meaning that each lattice site can be occupied by, at most, one agent. Therefore, in any single realisation of the model the occupancy of agents from Subpopulation 1 is given by Ai ∈ {0, 1}. If site i is occupied by an agent from Subpopulation 1 we have Ai = 1, and Ai = 0 otherwise. Similarly, in any single realisation of the model the occupancy of agents from Subpopulation 2 is given by Bi ∈ {0, 1}. The total number of agents from Subpopulation 1 and Subpopulation 2 are N1 and N2, respectively.
Since signalling molecules are many orders of magnitude smaller than cells, we allow each lattice site to be occupied by an arbitrary number of molecules, and we describe the density of signalling molecules at site i as Ci ∈ [0, ∞), where Ci is a continuous function of time. We assume that Ci is measured in some appropriate unit, such as µM. Such hybrid models that treat cells as discrete objects and signalling molecules as continuous densities is standard in the mathematical biology literature (Alacón et al. 2003; Mallet and de Pillis 2006). We will now specify how agents on the lattice move in response to the signalling molecules.
3.1.1 Agent movement
Within a particular time step of duration τ, agents from Subpopulation 1 and Subpopulation 2 attempt to undergo a nearest neighbour random walk with probability f1(Ci) ∈ [0, 1] and f2(Ci) ∈ [0, 1], respectively. The functional forms of f1(Ci) and f2(Ci) determine how the agents respond to the density of signalling molecules. For example, if f1(Ci) is increasing, the signalling molecules amplify the migration rate of Subpopulation 1. We will discuss the particular choice of f1(Ci) and f2(Ci) in Section 5.2.
Agent movement is simulated using a random sequential update method. During a time step of duration τ, N1 + N2 agents are randomly selected, one at a time, with replacement (Jin et al. 2016a). If an agent from Subpopulation 1 at site i is selected, that agent attempts to undergo a nearest neighbour random walk with probability f1(Ci). Similarly, if an agent from Subpopulation 2 at site i is selected, that agent attempts to undergo a nearest neighbour random walk with probability f2(Ci). In all cases the target site is chosen at random, and potential motility events are aborted if the target site is occupied. Reflecting boundary conditions are applied.
3.1.2 Signalling molecules
To be consistent with our experimental observations in Figure 1, we assume the signalling molecules are secreted by agents from Subpopulation 1 at a particular rate. The signalling molecules diffuse, undergo decay, and are taken up by agents from Subpopulation 2. We suppose that the spatial and temporal distribution of signalling molecules is governed by discrete conservation statement, where Dc [µm2/h] is the molecular diffusivity, λ [µM/h] is the secretion rate, κ [/h] is the uptake rate and µ [/h] is the intrinsic decay rate. We solve Equation (1) numerically as outlined in the Supplementary Material.
3.2 Reduced discrete model
We now formulate a reduced discrete model that retains key elements of the full discrete model without the need to explicitly solve for the spatial and temporal distribution of the signalling molecules. To distinguish between the two models we write the site occupancy of Subpopulation 1 and Subpopulation 2 as Ui ∈ {0, 1} and Vi ∈ {0, 1}, respectively. The diffusivity of signalling molecules is approximately three orders of magnitude greater than a typical cell diffusivity (Jin et al. 2017; Mac Gabhann and Popel 2004). This motivates us to simplify the model by assuming we have quasi-steady conditions since the diffusive transport evolves on much faster timescale than the source terms on the right of Equation (1). If the magnitude of the source terms in Equation (1) are negligible relative to the diffusive transport term, at steady state we have Ci+1 − 2Ci + Ci−1 = δCi = 0. Setting Ci+1 − 2Ci + Ci−1 = 0 and δCi = 0 in Equation (1) gives Ci = λUi/(µ + κVi), which could be a useful way to indirectly represent the effect of the signalling molecules as a function of the spatial arrangement of the agents on the lattice. This kind of quasi-steady assumption is often used to simplify continuum mathematical models where some kind of diffusible signal (e.g. Cai et al. 2006) or diffusible nutrient (e.g. Breward et al. 2002) is assumed to approach steady state much faster than the dynamics of some population of cells. The consequences of making such assumptions in a stochastic framework are rarely, if ever, examined in detail.
Since we have an exclusion process, each lattice site can be occupied by a single agent. Therefore, simply applying Ci = λUi/(µ + κVi) leads to Ci = 0 at any site with Ui = 0, or Ci = λ/µ for any site with Ui = 1. To make this approximation more realistic, we take the occupancy of lattice site i to be the average of the nearest neighbour lattice sites, Ûi = (Ui−1 + Ui+1)/2 and , giving
Therefore, in the reduced model, we take Gi to approximate density of the signalling molecule density at site i. Using this approximation in our discrete modelling framework allows us to implicitly simulate the role of the signalling molecules without needing to solve Equation (1). This approach has three clear advantages over the full discrete model: (i) the reduced discrete model involves less parameters than the full discrete model; (ii) the reduced discrete model is faster to computer than the full discrete model since there is no need to solve the evolution equation for Ci; and (iii) the reduced discrete model is more consistent with typical experimental observations that do not measure or report spatial and temporal distributions of the signalling molecules.
The reduced discrete model is implemented computationally using a similar random sequential update method. The only difference is that in the reduced discrete model we apply f1(Gi) and f2(Gi) instead of f1(Ci) and f2(Ci), and there is no need to solve the evolution equation for Ci. Of course, the key question that we are interested in now is to establish when the reduced model provides a good approximation to the full model. Intuitively we expect that the reduced discrete model will be a good approximation of the full model when Ci is accurately approximated by Gi. However, to explore this quantitatively we need to compare the performance of the two models over a series of biologically relevant parameter values. Before we consider this comparison, we also provide more mathematical insight into the two models by deriving approximate continuum limit descriptions of the two discrete modelling frame-works.
4 Continuum limit descriptions
We begin the continuum limit derivation by assuming we have access to a large number of identically prepared realisations of the full discrete model, and we denote the average occupancy of Subpopulation 1 and Subpopulation 2 at site i by Āi ∈ [0, 1] and , respectively. Similarly, the average density of signalling molecules at site i is given by ). Invoking a mean-field assumption and accounting for all possible events that alter the occupancy of site i over a time step of duration τ, we obtain where δĀi and are the change in occupancy at site i of Subpopulation 1 and 2, respectively, and is the total average occupancy at site i. To convert these discrete conservation statements into continuous expressions we identify the discrete variables with appropriate continuous variables, Āi(t) = a(x, t), and . Expanding each term in Equations (3)–(4) about site i using a Taylor series and neglecting terms of 𝒪(Δ3), we divide both sides of the resulting expressions by τ and take the limit Δ → 0 and τ → 0 jointly, with the ratio Δ2/τ held constant, to give where Da = Δ2f1(c)/ (2τ) and Db = Δ2f2(c)/ (2τ) are the diffusion coefficients for Subpopulation 1 and Subpopulation 2, respectively and s(x, t) = a(x, t)+ b(x, t). We refer to Equations (5)–(7) as the full continuum model.
The continuum limit description of the reduced discrete model can be obtained using a very similar approach. The approximate conservation statements for the two subpopulations can be written as, where all terms have a similar interpretation to those in Equations (3)–(4). We proceed to the continuum limit in the same way, arriving at where u(x, t) and v(x, t) are the densities of Subpopulation 1 and Subpopulation 2, respectively. Here, Da = Δ2f1(g)/ (2τ) and Db = Δ2f2(g)/ (2τ) are the diffusion coefficients for Subpopulation 1 and Subpopulation 2, respectively. We refer to Equations (10)–(11) as the reduced continuum model.
5 Results and Discussion
In this section we explore solutions of the full and reduced models, both discrete and continuum, for a typical geometry and timescale that reflect the co–culture assay in Figure 2. For the discrete models we set τ = 0.01 h and Δ = 20 µm to reflect a typical cell diameter. To simulate the width of the experimental field-of-view in Figure 2(c) we choose I = 201. We initialise the discrete simulations by placing agents from Subpopulation 1 to the left of the domain and agents from Subpopulation 2 to the right of the domain at t = 0. All sites with i ≤ 76 are randomly populated with probability 0.6 by agents from Subpopulation 1 and all sites with i ≥ 126 are randomly populated with probability 0.6 by agents from Subpopulation 2. This initial condition leaves 1000 µm of vacant space in the middle of the domain which is consistent with the initial width of the annulus of free space in Figure 2. In the full discrete model we assume that Ci = 0 at all sites at t = 0.
The full and reduced continuum models are solved numerically as outlined in the Supplementary Material. The initial condition in the continuum model is consistent with the discrete models by setting a(x, 0) = 0.6 for 0 ≤ x ≤ 1500 µm, b(x, 0) = 0.6 for 2500 ≤ x ≤ 4000 µm, and a(x, 0) = b(x, 0) = 0 elsewhere. In the full continuum model we set c(x, 0) = 0 for 0 ≤ x ≤ 4000 µm and we will comment on this choice of initial conditions later.
5.1 Choice of model parameters
To make our simulations consistent with experimental observations we note that MSCs are known to respond to diffusible molecules secreted by hepatocytes in co–culture assays, whereas the migration of hepatocytes are unaffected by the presence of MSCs in co–culture (Novo et al. 2011; Yoon et al. 2016). Accordingly, in the discrete models we assume that agents from both subpopulations undergo unbiased migration when Ci = 0 and that Ci has no impact upon the migration of Subpopulation 1 so we set f1(Ci) to be a constant. In contrast, we choose f2(Ci) to be a smooth increasing function, given by where α ≥ 0 specifies the strength of the chemokinetic response, and H is a constant relating to the migration rate of Subpopulation 2 in the absence of signalling molecules. In this section we choose H = 9, which gives f2(0) = 1/10. We set f1(Ci) = f2(0) = 1/10 so that in the absence of the chemical signal, agents from both subpopulations undergo unbiased random migration at the same rate. In terms of the continuum limit description, our choices of Δ, τ, f1(Ci) and f2(Ci) correspond to Da = Db(0) = Du = Dv(0) = 2000 µm2/h which is a typical value of cell diffusivity in low density tissue culture (Jin et al. 2016b).
There are five free parameters in the full and reduced models: Dc, λ, κ, µ, and α. We note that the diffusivity of typical diffusible molecules is approximately 105 µm2/h (Mac Gabhann and Popel 2004; Veldkamp et al. 2009). Experimental observations of the half life of diffusible molecules is around 0.5 h (Kirkpatrick et al. 2010), which corresponds to an exponential decay rate of approximately 1 /h. Therefore, we set Dc = 105 µm2/h and µ = 1 /h. We are unaware of any detailed experimental measurements of production and uptake rates of SDF-1 for co–culture experiments with hepatocytes and MSC so we choose λ = 1 µM/h and κ = 1 /h, to be of the same order as the decay rate. Later we will vary these choices of parameter values to gain insight into the sensitivity of the model predictions to these choices of parameter values.
5.2 Comparisons of the full and reduced models
Results in Figure 4(a)–(b) show snapshots of the time evolution of agent positions in the full and reduced discrete models, respectively. In these preliminary simulations we specify a weak chemokinetic effect, α = 1. Comparing the distribution of agents in different rows of the subfigures shows that the two subpopulations migrate into the initially–vacant space over time. We estimate the expected behaviour of the simulations by averaging the occupancy of each lattice site using 500 identically–prepared realisations of the stochastic models and show the averaged density profiles in Figure 4(c) where we see that the averaged density profiles from the reduced discrete model compares very well.
To investigate how the comparison between the full and reduced discrete models depends upon the strength of the chemokinetic effect we present additional results in the in Figure 4 (d)–(e) for α = 100 and in Figure 4 (g)–(h) for α = 1000. Comparing the averaged density profiles at t = 24 h shows that we maintain reasonably good agreement between the reduced and full models for the moderate chemokinetic effect in Figure 4(f) but we see that the reduced discrete model does not approximate the full discrete model very well when the chemokinetic effect is strong, as in Figure 4(i). In addition, in the Supplementary Material we compare the averaged density profiles at t = 48 h to allow more time, beyond the typical experimental timescale, for the two subpopulations to interact. These additional results over a longer time scale are consistent with the results in Figure 4.
All results in Figure 4 correspond to discrete results. We now examine how well the averaged data from the two discrete models compare with the numerical solution of the associated continuum limit descriptions. Results in Figure 5(a)–(c) compare averaged density profiles from the full model with corresponding solutions of Equations (5)–(7) for α = 1, 100 and 1000, respectively. These results show that the new PDE models provide an accurate approximation of the averaged behaviour of the full discrete model when α = 1 and α = 100, but that the solution of the continuum limit PDE does not provide an accurate approximation of the averaged data from the full discrete model when chemokinesis is sufficiently strong, α = 1000. Similarly, results in Figure 5(d)–(f) compare averaged density profiles from the reduced model with corresponding solutions of Equations (10)–(11) for α = 1, 100 and 1000, respectively. Again, we see that the solution of the continuum limit PDE models provides a good approximation of the averaged behaviour of the reduced discrete model when α = 1 and α = 100, but we observe some discrepancy when the chemokinesis is sufficiently strong, α = 1000. Therefore, while the continuum limit PDEs can provide a good description of the average behaviour of the discrete model for certain parameter choices, they do not always provide a good approximation of the discrete models and this discrepancy is associated with the failure of the mean-field approximation (Simpson et al. 2010). Therefore, for the remainder of this study we will focus on using the discrete models and explore the differences in the performance of the full and reduced discrete models.
We now quantitatively explore the difference between the full and reduced discrete models for a range of signalling molecules diffusivity (Dc = 10, 105, 106 µm2/h) and a range of chemokinetic strengths (α = 1, 100, 1000). To quantify the quality-of-match between the full and reduced models, we compute a measure of the least–squares difference between the averaged density profiles, where m is an index indicating the number of identically-prepared realisations and M = 500 is the total number of identically–prepared realisations considered. For each combination of Dc and α that we consider, we compute E1(24; Dc, α, λ, κ, µ) and E2(24; Dc, α, λ, κ, µ) with fixed values of λ = 1 µM/h and κ = µ = 1 /h, and we plot the averaged density profiles at t = 24 h in Figure 6. Results in Figure 6 indicate that E1 is relatively small and insensitive to the parameter values we consider. In contrast, E2 increases with both α and Dc. In particular we see that the reduced discrete model can provide a very good approximation of the full discrete model when α is sufficiently small, but the approximation becomes poor when α increases.
In addition to comparing averaged agent density profiles for the full and discrete models, the insets provided in each subfigure of Figure 6 show the spatial distributions of and at t = 24 h. We see that is accurately approximated by when Dc = 10 and 105 µm2/h, whereas the comparison is poor when Dc = 106 µm2/h. As a result we have relatively good agreement between the full and reduced averaged density profiles in Figure 6(a)–(e) since is reasonably well approximated by . However, results in Figure 6(f) shows that even with a relatively good match between and , the match between the averaged density profiles of the full and reduced discrete models can still be poor when the strength of chemotaxis is sufficiently large, here α = 1000. Results in Figure 6(h)–(i) correspond to cases where and do not match well and in all these cases we see that the average density profiles in the reduced discrete model do not provide a good approximation of the averaged density profiles in the full discrete model.
Overall, comparing the average density profiles in Figure 6 confirms that the reduced discrete model can be used to approximate the full discrete model for certain parameter choices. In general we see that the quality of match between the two models tends to decreases with α and the performance of the reduced model is also sensitive to other parameters such as Dc. To provide further insight into how the performance of the reduced discrete model depends upon the choice of parameters we compute E1 and E2 at t = 24 h over a range of µ, λ, α and Dc. For each choice of α and Dc, we construct two-dimensional heat maps showing E1 and E2 as a function of µ and λ. The heat maps, shown in Figure 7, indicate that the reduced model provides a reasonably good approximation of the full model provided we have a sufficiently small α and Dc. Comparing the magnitude of E1 and E2 as a function of λ and µ indicates that the accuracy of the reduced discrete model is less sensitive to variation in λ and µ than it is to variations in α and Dc. Similar results (not shown) also indicate that E1 and E2 are relatively insensitive to the choice of κ for the choice of initial condition to mimic the co–culture experiments in Figure 2. Therefore, we have chosen to focus our examination of the performance of the reduced discrete model to α, Dc, λ and µ.
6 Conclusion and Outlook
Typical mathematical models of cell migration stimulated by signalling molecules involve some kind of reaction–diffusion equation to explicitly describe the spatial and temporal distribution of the signalling molecules. However, such information is rarely available from experimental observations since signalling molecules are challenging to record and image. Motivated by a suite of new co– culture cell migration assays, we develop new mathematical modelling tools to describe the cell migration regulated by signalling molecules in an attempt to avoid the need for working directly with a description of the spatial and temporal distribution of signalling molecules. We first develop a full discrete model that describes the migration and interactions of two subpopulations of cells, in which the movement of one subpopulation is regulated by the presence of signalling molecules secreted by cells in the other subpopulation. In this model, the spatial and temporal distribution of the signalling molecules is governed by a discrete conservation statement that is related to a reaction–diffusion equation. To make this description consistent with experimental observations, we simplify the full discrete model by invoking a quasi–steady state assumption in the reaction–diffusion equation governing the spatial and temporal distribution of the signalling molecules. With this simplification, we obtain a reduced discrete model which implicitly describes a similar interaction between the two cell populations without needing to solve the underlying conservation statement. To provide additional mathematical insight into these two models we obtain continuum limit descriptions of both models, leading to new PDE models.
In the full discrete model we suppose that the migration rates of agents from Subpopulation 1 and Subpopulation 2 are given by functions f1(Ci) and f2(Ci), respectively, where Ci is the density of signalling molecules at site i. Similarly, in the reduced discrete model the migration rates of agents from Subpopulation 1 and Subpopulation 2 are given by f1(Gi) and f2(Gi), respectively, where Gi is the approximate density of signalling molecules at site i. Choosing particular functional forms for f1 and f2 allows us to specify whether cell migration is stimulated or inhibited by the signalling molecule. We choose forms of f1 and f2 that are relevant to the hepatocype–MSC co–culture experiments in Figure 2, and we compare the performance of the full discrete model and reduced discrete model for a typical experimental geometry, timescale, and parameter choices, and we focus on comparing the full and reduced models for different strengths of the chemokinesis effect. This comparison indicates particular situations where the reduced discrete model could be used in place of the full discrete model. In general we find that the reduced discrete model performs particularly well when the strength of chemokinesis is sufficiently small, whereas for sufficiently strong chemokinesis the comparisons indicate that the reduced model is not always a good approximation. Without making such comparisons, it is not obvious when it would be reasonable to use the reduced model.
There are several features of this study that could warrant further investigation: (i) For simplicity, we focus on developing one–dimensional models to describe cell migration regulated by signalling molecules, and these one– dimensional models can be extended to two–dimensional geometries where necessary; (ii) In all comparisons we assume Ci = 0 at t = 0 in the full model description. This assumption is reasonable given that typical experiments do not provide any information about the spatial and temporal distribution of signalling molecules. If, instead, the initial distribution for Ci was known or measurable, all comparisons in this work could be repeated making use of that information; (iii) In this work we choose particular forms of f1 and f2 that are relevant to the hepatocyte–MSC co–culture experiments in Figure 2. Other choices of f1 and f2 could be made for different co–culture systems as relevant; and (iv) In the current modelling framework we assume that cells sense signalling molecules locally, at the same site i. However, in the cell biology literature there have been different hypotheses put forward about non-local sensing over different spatial ranges (Hopkins and Camley 2019). Such non-local sensing could be introduced into our modelling framework by making appropriate adjustments to the discrete models and then examining how these changes manifest in the continuum limit description.
Acknowledgments
This work is supported by the Australian Research Council (DP170100474) and the National Health and Medical Research Council (APP1126091, APP1141121). WJ is supported by a QUT Vice-Chancellor’s Research Fellowship.