Fundamental Of Mathematical
Badomero Schoulund
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Melanoblasts, the embryonic precursors of the pigment producing melanocytes of skin and hair, colonize the developing epidermis during development. Mice heterozygous for mutations of the receptor tyrosine kinase Kit present with a depigmented ventral belly spot long thought to represent a failure of either melanoblast proliferation or migration1. Melanoblasts are specified at embryonic day 9 (E9) in the pre-migratory neural crest2 by downregulation of the transcription factors Foxd3 and Sox2 and upregulation of the master transcription factor Mitf3,4. At E10.5 melanoblasts delaminate from the neural crest, upregulate the melanoblast-specific genes Pmel and Dct5,6, and accumulate in a region known as the migration staging area (MSA)1,7,8. Melanoblast survival in the skin is dependent on signalling between Kit (expressed in melanoblasts) and its ligand Kitl (expressed by dermal fibroblasts and keratinocytes), however initial delamination does not require Kit signalling1,9. At E10.5 melanoblasts begin to leave the MSA in a Kitl- and endothelin 3 (Edn3)-dependent manner and embark on a dorsolateral migratory pathway in the dermis between the developing somites and the ectoderm1,7,8,10. At E12.5 melanoblasts move from the dermis to the epidermis, upregulate E-cadherin, lose their dependence on Edn3 and continue to migrate and proliferate11,12,13. A dermal population also persists whose size remains constant (and so proportionally decreases in relation to the epidermal population)14. Epidermal colonization is complete by around E15.5, after which melanoblasts downregulate E-cadherin and begin to localize to the developing hair follicles11,15,16. Luciani et al.14 measured melanoblast doubling times in the dermis and epidermis, and used a mathematical model to estimate the number of melanoblast progenitors specified in the pre-migratory neural crest14.
We hypothesized that undirected melanoblast movement and proliferation, in tandem with tissue growth (Fig. 2a and Supplementary Table 2) are sufficient for melanoblast colonization and that this simple mechanism can explain the patterns observed in chimeras, individually labelled clones and Kit mutants. We used our observations to parameterize a stochastic model of melanoblast colonization of the trunk (Methods). Our modelling framework only considers the growth of the trunk region and its colonization by the migrating melanoblast population. The domain is limited axially to the region between and not including the limb buds and encompasses the complete dorsoventral length (Fig. 2a and Methods). We assume that no new melanoblasts are specified after E10.5 and therefore that the growing melanoblast population is produced solely by the proliferation of this founder population. Melanoblasts migrate first within the developing dermis between E10.5 and E12.5, and subsequently within the epidermis and dermis between E12.5 and E15.5. We collectively refer to the dermal and epidermal layers that can support melanoblast survival as the dorsoventral integument (DVI; Methods, Supplementary Fig. 2) and assume that melanoblast behaviour in these compartments is equivalent. We describe above through analysis of cell orientations in Dct::lacZ embryos and time-lapse experiments that there is no directed migration in either compartment. In our simulations we employed an agent-based, discrete-space random-walk model on a growing two-dimensional lattice employing volume exclusion, whereby at most one agent (melanoblast) occupies each square lattice site, and melanoblasts cannot migrate or proliferate into occupied sites40. The stochastic events are simulated using the Gillespie algorithm (Methods)41.
Using the parameters generated from our experimental observations (Methods and Supplementary Table 3) our model was able to replicate the relationships between cell density, the diffusion coefficient and cell cycle time described above (red lines in Fig. 2e,f). It predicts colonization of the growing domain. That is, the averaged cell density in the model at all stages of embryonic development closely fits our experimental data (Fig. 3a). Examples of domain colonization are provided in Supplementary Movie 6 and Supplementary Fig. 4a.
Using stochastic individual-level modelling we have examined the importance of density-dependent diffusion and proliferation for colonization of the DVI and conclude that colonization is most sensitive to changes in proliferation. This is in agreement with Zhang et al.29 who explored the interaction between neural crest migration and proliferation using an on-lattice model for the colonization of the gut by enteric ganglia progenitors29. One weakness of our model is that it assumes that melanoblast behaviour is equivalent in the relatively sparsely packed three-dimensional dermal environment between E10.5 and E12.5, and in the more tightly packed two-dimensional epidermal environment between E12.5 and E15.5. Experimentally, we demonstrate that this is qualitatively the case but there will certainly be minor differences. The on-lattice approach we use is more appropriate for the latter of these scenarios. However, to represent these two environments separately would require a computationally intensive hybrid model and a number of new and potentially inaccessible parameters, which would complicate the model and hamper the investigation of the patterning questions we chose to address. Our model assumes that all melanoblasts arise by proliferation of the differentiated melanoblasts present at E10.5. This may not be the case as further cells fated to be melanoblasts may differentiate after E10.5. Another source of melanoblasts may be from Schwann cell precursors (SCPs) emanating from the dorsal ramus from E12.5 onwards as has been proposed by Adameyko et al.52. However, as the lineage tracing approach that identified these cells has been questioned53,54 and we have no access to the key parameters of their possible behaviour, incorporating Schwann cell precursor-derived melanoblasts into the present model is not feasible.
Cheeseman et al.26,27 investigated the dominance of sub-lineages in a lattice-based discrete model. They found that in many cases the progeny of two cells (of the 500 they initialized) could contribute in the order of 25% of the cells in the final population. This effect was mediated by a process of sequential isolation of individual lineages deprived of space to proliferate into26,27. This stochastic drift in clone size has been demonstrated experimentally and explored mathematically in the mouse intestine55,56,57. Selection of dominant lineages is relatively weak in our simulations of melanoblast domain colonization owing to the more diffuse initial conditions. More cells are able to establish a significant lineage because they have the required space to proliferate initially and consequently fewer lineages become spatially isolated. This implies that the stripes seen in our model are predominantly formed by the coalescence of multiple like-coloured subclones, and not by the presence of dominant lineages. Furthermore, in our model, the domain grows in both the dorsoventral and axial directions, whereas in Cheeseman et al.26,27 domain growth is only in the dorsoventral direction or is absent. The two-dimensional growth in our model further reduces the role of dominant lineages since cells, which may previously have been isolated, can gain space into which they may proliferate through domain growth events. Our modelling shows that the generation of rare clones22 and chimeric patterns19 can proceed through a common mechanism employing tissue expansion and density-dependent movement and proliferation. Further experimental clonal analyses, using stochastic labelling methods such as brainbow/confetti57,58, are required to explore whether our predictions of the behaviour of melanoblast subclones are accurate.
Embryonic skin culture was performed as described in Mort et al.31. Briefly, up to six cultures were imaged in parallel per experiment. Skin was sampled from the flank of E13.5, 14.5 and 15.5 mouse embryos. The dorsoventral position varied but was never taken at the ventral extreme. The skin samples were mounted on a clip filled with 1% w/v agarose (in PBS) and secured with suture thread. The clip was then inserted into a custom designed six-well chamber so that the skin was sandwiched against a lummox gas-permeable membrane (Greiner). The wells were filled with DMEM (no phenol red) supplemented with 1 Glutamax (Gibco), 1% v/v penicillin/streptomycin and 10% v/v fetal calf serum. Whole E11.5 embryos were embedded in 1% w/v agarose (in PBS) in a large custom-made imaging clip so that the dorsal region of the flank was just protruding above the surface of the agarose. The clip was then inserted into a custom designed six-well chamber so that the protruding region of the embryo was pressed against a lummox gas-permeable membrane (Greiner). The wells were filled with DMEM (no phenol red) supplemented with 1 Glutamax (Gibco), 1% v/v penicillin/streptomycin and 10% v/v fetal calf serum.
Initial number and position of cells. We defined the number of progenitor melanoblasts by counting the melanoblasts in the trunk of Dct::lacZ embryos at E10.5; a mean (95% CI) of 20.325.95 melanoblasts (Fig. 2c). In our Dct::lacZ embryos we noted an under-representation of melanoblasts in the centre of the trunk region, although not always clear at E10.5 this was most striking at E11.5 (Supplementary Fig. 1c). We therefore weighted our initial distribution in a similar manner, initializing 21 agents such that on average one-third are between sites 12 and 32 of the axial axis, and the remaining two-thirds are evenly distributed between sites 1 and 11, and 33 and 43 corresponding to a slight under-representation in the middle of the axial axis. These agents are distributed so that 95% are between sites 8 and 17 of the dorsoventral axis. All agents are distributed between sites 8 and 19 of the dorsoventral axis.
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