Mathematical modelling of mammalian pigmentation patterns: Stochastic modelling of melanoblast neural crest cells.

胚胎发生定义了胚胎发育的早期阶段。许多这样的发育是通过一种称为神经脊细胞的细胞家族实现的。在胚胎发育的早期阶段，神经脊细胞在许多生物发育中起着至关重要的作用，如骨骼的形成、软骨的形成以及毛发和皮肤的着色。黑素母细胞是黑素细胞的早期前体，是负责产生黑色素的色素产生细胞。黑色素母细胞位于神经脊的主干区域，从那里剥离并沿其迁徙路径向背侧迁移，以定植发育中的表皮。然后，它们分化成黑素细胞，开始产生黑色素。黑素母细胞的存活依赖于受体Kit和其配体Kit之间的信号传递。Kit基因的突变可以改变信号机制，导致黑素母细胞行为错误，最终导致表皮的不完全定植。最近的研究表明，Kit突变小鼠的黑素母细胞表现出更长的细胞周期，使部分表皮失去了这些产生色素的细胞。由此产生的神经衰退症称为花斑症。一只花斑老鼠的肚子上有一个白色斑点。其他神经脊细胞的错误行为会导致更严重的疾病，如神经纤维瘤病和先天性巨结肠。较大的哺乳动物也会出现斑疹，例如牛身上的斑块无色素皮肤。我的项目将涉及使用数学模型来模拟黑色素母细胞的行为。到目前为止，使用基于实验参数的随机代理模型，我已经能够在农业领域复制小鼠的白色肚皮斑点。我们还表明，改变参数值，我们的模型可以产生较大哺乳动物的斑块状图案，例如在一些奶牛物种中看到的那些。我们现在想沿着以下关键路线扩展这项工作：在较大的哺乳动物中形成图案：尽管基于晶格上试剂的模型似乎可以成功地在奶牛身上复制斑块，但我们的工作缺乏数学分析来伴随这些结果。我们将开发一种更严格的数学连续介质模型来伴随模拟算法，该算法将量化奶牛的斑块形成。已经观察到，与老鼠的腹部斑点相比，奶牛斑块的边界更清晰、更清晰。为了研究这一现象，我们将开发混合确定性-连续体模型，该模型利用图灵的扩散驱动图案形成理论。间充质真皮细胞在毛囊中的定位：间充质真皮细胞在毛囊形态发生中起关键作用。作为对某些信号通路的响应，间充质细胞聚集在一起，形成一种周期性的真皮凝集物。这些冷凝物的位置标志着未来毛囊的位置。在这个项目中，我们感兴趣的是驱动这种周期性模式的机制。我们将使用模拟模型来理解这种模式。真实的细胞周期时间分布：在此PHD期间的许多随机建模将使用吉列斯佩奇随机模拟算法。吉莱斯皮算法假定细胞周期时间服从指数分布，表现出无记忆特性。然而，众所周知，细胞周期时间并不是从现实中的这种分布中得出的。先前已经表明，小鼠体内某些细胞的细胞周期时间可以更准确地用Erlang分布来建模。我们将开发模型来真实地模拟成釉细胞的细胞周期数据。现有研究的证据表明，子代细胞和它们的远亲细胞的细胞周期时间之间存在相关性。我们想要扩展现有的模型，以纳入世代之间的相关性的影响

Embryogenesis defines the early stages of embryonic development. Many such developments are attained via afamily of cells called the neural crest cells. Neural crest cells play a vital role in many biological developments in theearly stages of the growing embryo, e.g. formation of bones, cartilage & pigmentation of hair & skin.Epidermal pigmentation is a product of melanogenesis which is achieved via melanocytes. Melanoblasts, the earlyprecursors of melanocytes, are the pigment-producing cells responsible for producing melanin. Melanoblastsoriginate in the trunk region of the neural crest from which they delaminate & migrate dorsoventrally along theirmigratory pathway to colonise the developing epidermis. They then differentiate into melanocytes & start producingmelanin. Survival of melanoblasts is dependent on signalling between the receptor Kit & its ligand Kitl. Mutations inthe Kit gene can alter the signalling mechanism causing melanoblasts to behave erroneously which ultimately leadsto incomplete colonisation of the epidermis. Recent research suggests that the melanoblasts of Kit mutant miceexhibit longer cell-cycle times, leaving parts of the epidermis deprived of these pigment-producing cells. The resultingneurocristopathy is called piebaldism. A piebald mouse shows a white spot on the belly. Erroneous behaviour ofother neural crest cells leads to more serious conditions such as Neurofibromatosis & Hirschsprung's disease.Larger mammals also show piebaldism, e.g. patches of unpigmented skin in cows.My project will concern modelling melanoblast behaviour using mathematical models. So far, using experimentallyparameterised stochastic agent-based models I have been able to replicate the white belly spots in mice on agrowing domain. We also showed that varying the parameter values our model can produce patch-like patterns oflarger mammals such as those seen in some cow species. We would now like to expand on this work along the following key linesPattern formation in larger mammals: Although the on-lattice agent-based model could seemingly successfullyreplicate patches in cows, our work lacks mathematical analysis to accompany these results. We will be developing amore mathematically rigorous continuum model to accompany the simulation algorithm which will quantify the patchformation in cows. It has been observed that cow patches have sharper & well-defined boundaries in contrast tothe belly spots in mice. We will develop hybrid deterministic-continuum models which exploit Turing's theory ofdiffusion-driven pattern formation in order to investigate this phenomenon.Localisation of mesenchyme dermal cells to hair follicles: Mesenchymal dermal cells play a key role in hair folliclemorphogenesis. Responding to certain signalling pathways the mesenchymal cells aggregate & form a periodicpattern of dermal condensates. The locations of these condensates mark the positions of future hair follicles. In thisstrand of the project, we are interested in the mechanisms which drive this periodic pattern. We will use simulationmodels to understand this patterning.Realistic cell-cycle time distribution: Much of the stochastic modelling during this PhD will employ the GillespieStochastic Simulation Algorithm. The Gillespie algorithm assumes that cell-cycle times are exponentially distributed& exhibit the memoryless property. However, it is well-known that cells-cycle times are not drawn from thisdistribution in reality. It has been shown previously that cell-cycle times of certain cells in mice are more accuratelymodelled using an Erlang distribution. We will be developing models to realistically model cell-cycle time data formelanoblasts. Evidence from existing work suggests correlations between cell cycle times of daughter cells & theirmore distant relatives. We would like to expand on the existing models to incorporate the effects of correlationbetween generations of