Numerical Methods for Models of Biological Systems

在实验和理论之后，计算是科学方法的第三个支柱。特别是在生物学中，计算机模拟被用来理解现象，提出具有启发性和成本效益的实验建议，并做出预测。越来越多地，我们对生物系统的理解被封装在模拟中。大量可用的实验数据和对非常详细的模型的渴望需要快速、准确和可并行的数值方法。数学在创建和分析数值方法中起着至关重要的作用。我正在为描述细胞群体中的电活动和遗传生化反应网络的大型常微分方程系统开发数值方法。结合遗传学、生物化学和电生理学的生物系统的例子是昼夜节律（大约24小时）时钟、视网膜和几乎所有的发育过程。耦合振子群在生物系统中很常见，一直是数学分析和模拟的焦点。振荡器群包括心脏的起搏器、分泌胰岛素的胰岛、萤火虫群和大脑的许多区域。我创造的一种数值方法，人口密度粒子法，能够快速准确地模拟噪声耦合振荡器的人口。我正在努力使这种方法更广泛地适用，提高其性能，并建立良好的理论基础。经过一些修改，该方法还可以用于参数拟合，这是建模人员的常见任务。我也在研究一种基于随机游走的偏微分方程数值方法，用于描述流体力学、电扩散和空间扩展的生化反应。当这些方程为生物系统建模时，复杂的边界和复杂的边界条件往往会阻碍模拟的快速和准确。我希望我的方法能够克服与边界相关的挑战。我将研究数值刚性常微分方程的高阶方法，可并行化方法，预计算技术，使用图形处理单元加速，低秩近似，机器学习技术，无网格方法，粒子方法和随机行走方法来解决模拟生物模型时通常面临的数值挑战。这将导致更快、更准确的模拟，最终使人们能够对许多生物学问题有新的见解。我将公开发布我的数值方法的高质量、可用的实现，这样它们就可以被其他人用作构建模块，创建一个科学发现的社区。

After experimentation and theory, computation is the third pillar of the scientific method. Especially in biology, computer simulations are used to understand phenomena, suggest illuminating and cost-effective experiments, and make predictions. Increasingly, our understanding of biological systems is being encapsulated in simulations. The incredible amount of available experimental data and a desire to have very detailed models necessitates having fast, accurate, and parallelizable numerical methods. Mathematics plays an essential role in creating and analysing numerical methods. I am developing numerical methods for large systems of ordinary differential equations that describe the electrical activity of and genetic-biochemical reaction networks in populations of cells. Examples of biological systems that couple genetics, biochemistry, and electrophysiology are the circadian (approximately 24) clock, the retina, and nearly all developmental processes. Populations of coupled oscillators are common in biological systems and have historically been the focus of mathematical analysis and simulation. Oscillator populations include the pacemaker of the heart, pancreatic islets that secrete insulin, populations of fireflies, and many regions of the brain. A numerical method of my creation, the population density particle method, enables fast and accurate simulation of populations of noisy coupled oscillators. I am working to make this method more broadly applicable, improve its performance, and put it on a sound theoretical foundation. With some modification, this method can also be used for parameter fitting, a common task for modellers. I am also studying a random walk based numerical method for partial differential equations that describe hydrodynamics, electrodiffusion, and spatially extended biochemical reactions. When these equations model biological systems, it is often the case that intricate boundaries and complicated boundary conditions prevent simulations from being fast or accurate. I expect my approach to overcome the challenges associated with boundaries. I will investigate high order methods for numerically stiff ordinary differential equations, parallelizable methods, pre-computation techniques, using graphical processing units for acceleration, low-rank approximation, machine learning techniques, grid-free methods, particle methods, and random walk methods to address numerical challenges commonly faced when simulating biological models. This will result in faster and more accurate simulations that will ultimately enable new insights into many biological problems. I will publicly distribute quality, usable implementations of my numerical methods so that they may be used as building blocks by others, creating a community of scientific discovery.