Lecture: MW 12:30pm – 1:50pm, Skye Hall, Room 284

Instructor: Qixuan Wang

Office hours and location: By appointment

Course Description: In this course, we will talk about several multiscale modeling topics in biology and physics, with a focus on biology. For each topic, the lecturer will introduce the biology background and mathematical tools, then discuss 1-2 papers together with students.

Pre-requisites for Graduate Students: None.

Course Topics:

1. Introduction: Spatial and temporal scales in nature · Examples of multi-scale models in life sciences

2. Reaction Kinetics: Reaction rate equations · Law of mass action · Gene expression (transcription & translation) · Genotype vs. Phenotype · Small numbers of molecules and noise · Enzyme kinetics · Michaelis and Menten model · Cooperativity · Hill equation

3. Ordinary Differential Equations: Phase plane · Linear stability analysis · Numerical methods: Euler Method and Runge-Kutta Method

4. Stochastic Differential Equations: Wiener process · White noise and Brownian motion · Ito Formula · Black-Scholes diffusion equation · Euler-Maruyama Method and Milstein Method · Chemical Master Equation · Stochastic Simulation Algorithm

5. Nature at Different Scales: Navier-Stokes Equation · Nondimensionaliztion of Navier-Stokes equation · Reynolds number · Low Reynolds number Newtonian fluid · Microswimming and Scallop Theorem · Viscoelastic fluid

6. Singular Perturbation Theory: Regular vs. singular perturbation · Method of matched asymptotics · Boundary layers in in compressible flows ·Nondimensionalization of the enzyme kinetics · Slow manifold · Multi-Time Step Method and Heterogeneous Multiscale Method

7. Gene Regulation and Cell Signaling: Gene regulation · Feedback control · Gene regulatory network · Bifurcation Theory · Boolean network · Cell signaling ·Cell lineage model

8. Growth:

a. General Topics: Growth, remodeling and morphogenesis · Forms of growth (tip, accretive, volumetric) ·Isometric vs. allometric growth

b. Discrete Computational Models: Cellular Automata · Cellular Potts Models · Center Dynamics Models · Vertex Dynamics Models · Subcellular Element Models · Example modeling systems (plant development, hair follicle regeneration)

c. Morphoelasticity Theory: Elasticity · Linear elastic material and Hooke’s law · Neo-Hookean material ·Viscoelastic material · Example: a growing rod in one dimension (pure elastic deformation vs. pure growth) ·Growth with elastic response · Example: growth of a rod between two plates · Slow growth vs. fast elastic response · Morphoelasticity Theory · Example: homogeneous deformation of a cuboid · Example: a growing cuboid · Numerical examples (tumor, brain, leaf)

Students are highly encouraged to suggest any topic as well as papers that they are interested in.

Grading: Course grades are determined by the combined total of the attendance (30%), talk reports (40%), presentation (30%), listed as follows.

Attendance (30%): Signing sheet will be distributed in each lecture. If you miss no more than two lectures, you will get the full credits for attendance.

Talk reports (40%): You need to submit two talk reports (20% each) through the quarter. One from a departmental seminar talk and one from an exterior local conference/meeting talk. Both talks have to be broadly related to multiscale modeling. More details regarding this part will be provided in the first lecture of the course.

Presentation (30%): You need to give a 20-30 minutes presentation in class through or at the end of the quarter. If you are doing multiscale modeling related research, you are encouraged to present your research. Otherwise you can present a paper suggested by the lecturer, or anything you have in mind as long as it is related to the course topics. A short paper should be presented by 1 student alone, while long papers can be presented by a team of 2 students. In either way, each student should control your presentation to be within 20-30 minutes.