Math & Computational Core

主题D：数学与计算 生物学核心 主题：Nie 其他项目教师：Lowengrub，Xin，Mjolsness 概述 描述主题A-C中涉及的生物过程和相互作用的方程和模型 主要是非线性偏微分方程（PDE），包含随机效应或/和 在某些情况下移动边界。了解每个生物过程在复杂的 生物系统需要我们对这些方程进行详细的研究和分析。因为 我们寻求回答的问题通常涉及定量系统级特征，我们面临 在使用当前的计算和分析工具分析这些方程方面存在重大挑战。为 例如，为了研究输入参数变化的解决方案的灵敏度 在计算上，可能需要执行数百万次计算来收集和分析相关性 溶液模式和生物过程之间的联系。我们进行了这样的计算，以确定新的 策略，以实现稳定状态下的形态发生梯度的鲁棒性， 空间维度和一个扩散配体（参考我们的鲁棒性论文）。但对于更复杂的 涉及时间动态或/和更高（两个和三个）空间维度的新的和有效的系统 需要数值方法。这个主题的前两个项目集中在计算工具上 处理这些类型的系统。 在以往的主题中提出的大多数生物化学反应都具有刚性的反应速率常数 这因此限制了用于时间更新的时间步长的大小。这是一个瓶颈， 大多数刚性反应扩散方程的数值模拟。在第一个项目（D.1）中，我们建议 在二维和三维空间中发展刚性反应扩散方程的有效时间算法 通过减少或甚至去除与刚度相关的时间步长约束来减小尺寸。的 研究计划是基于我们最近成功地开发半隐式算法一维 系统和我们的初步研究纳入这种类型的新方法与两个和 三维系统。项目A.1是一项关于抑制空间噪声的研究， 许多扩散配体的偏微分方程的时间和稳态计算的数量， 和项目A.2，一项关于斑马鱼发育模式的研究，需要快速的时间模拟， 三维几何中的反应扩散方程。开发新的时间方法是 除了项目B.1、B.3、C.1和C.2之外，对两个项目都至关重要，这些项目都涉及类似类型的 方程

Theme D: Mathematical and Computational Biology Core Theme Leader: Nie Other Project Faculty: Lowengrub, Xin, Mjolsness Overview The equations and models describing the biological processes and interactions dealt with in Themes A-C are mainly nonlinear Partial Differential Equations (PDEs), with inclusion of stochastic effects or/and moving boundaries for some cases. Understanding the role of each biological process in a complex biological system requires us to study and analyze these equations in great detail. Because the questions we seek to answer typically involve quantitative system-level features, we face substantial challenges in analyzing those equations using current computational and analytical tools. For example, in order to investigate sensitivities of solutions to variations in input parameters computationally, one may need to perform millions of calculations to collect and analyze correlations between solution patterns and biological processes. We have performed such calculations to identify new strategies to achieve robustness of morphogen gradients at steady-state for a system involving one spatial dimension and one diffusing ligand (refs to our robustness paper). But for more complex systems involving temporal dynamics or/and higher (two and three) spatial dimensions, new and efficient numerical methods are needed. The first two projects in this theme focus on computational tools dealing with these types of systems. Most of the bio-chemical reactions proposed in previous themes have stiff reaction rate constants that consequently restrict the size of time-step for temporal updating. This is a bottle-neck for numerical simulations of most stiff reaction-diffusion equations. In the first project (D.1), we propose to develop efficient temporal algorithms for stiff reaction-diffusion equations in two- and three- spatial dimensions by reducing or even removing the timestep constraint associated with stiffness. The research plan is based on our recent success on developing semi-implicit algorithms for onedimensional systems and our preliminary studies on incorporation of this type of new method with twoand three- dimensional systems. Project A.1, a study on suppression of spatial noise, demands a large number of both temporal and steady-state calculations for PDEs with many diffusing ligands, and Project A.2, a study on developmental patterning of Zebrafish, requires speedy temporal simulations of reaction-diffusion equations in a three-dimensional geometry. Development of new temporal methods is critical to both projects in addition to Projects B.1, B.3, C.1 and C.2 that all involve similar types of equations.