Numerical Methods for Models of Biological Systems

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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.