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称为神经嵴细胞的细胞家族实现的。神经嵴细胞在胚胎生长早期的许多生物发育中起着至关重要的作用，例如骨骼、软骨的形成以及头发和皮肤的色素沉着。表皮色素沉着是黑色素形成的产物，是通过黑色素细胞实现的。黑色素母细胞是黑色素细胞的早期前体，是负责产生黑色素的色素生成细胞。黑素母细胞起源于神经嵴的主干区域，从那里分层并沿其迁移路径背侧迁移，在发育中的表皮上定植。然后它们分化成黑色素细胞并开始产生黑色素。黑素母细胞的存活依赖于受体Kit及其配体Kitl之间的信号传导。Kit基因的突变可以改变信号机制，导致黑色素细胞行为错误，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时间，使表皮部分失去这些产生色素的细胞。由此产生的神经病变被称为花心症。斑纹老鼠的腹部有一个白点。其他神经嵴细胞的错误行为导致更严重的疾病，如神经纤维瘤病和先天性巨结肠病。较大的哺乳动物也有斑点病，例如牛身上有一块块没有色素的皮肤。我的项目将涉及用数学模型来模拟成黑素细胞的行为。到目前为止，使用基于实验参数化随机主体的模型，我已经能够在生长区域复制小鼠的白色腹部斑点。我们还表明，改变模型的参数值可以产生大型哺乳动物的斑块状图案，例如在某些奶牛物种中看到的图案。我们现在想沿着以下关键路线扩展这项工作：大型哺乳动物的模式形成：尽管基于点阵代理的模型似乎可以成功地在奶牛身上复制斑块，但我们的工作缺乏与这些结果相伴随的数学分析。我们将开发一个数学上更严格的连续体模型，以配合将量化奶牛斑块形成的模拟算法。据观察，与老鼠腹部的斑点相比，奶牛的斑点有更清晰和明确的边界。我们将开发混合确定性-连续体模型，利用图灵的扩散驱动模式形成理论来研究这种现象。间充质真皮细胞向毛囊的定位：间充质真皮细胞在毛囊形态发生中起关键作用。响应某些信号通路，间充质细胞聚集并形成真皮凝聚物的周期性模式。这些冷凝物的位置标志着未来毛囊的位置。在这个项目中，我们对驱动这种周期性模式的机制很感兴趣。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：本博士期间的大部分随机建模将采用gillespis随机模拟算法。Gillespie算法假设细胞周期时间呈指数分布，并表现出无记忆特性。然而，众所周知，在现实中，细胞周期时间并不是由这种分布得出的。先前已经证明，使用Erlang分布可以更准确地模拟小鼠中某些细胞的细胞周期时间。我们将开发模型来真实地模拟成皮细胞的细胞周期时间数据。现有工作的证据表明，子细胞的细胞周期时间与它们的远亲细胞之间存在相关性。我们希望在现有模型的基础上进行扩展，以纳入代际相关性的影响

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