MA565 Course Syllabus

Mathematical Models in the Life Sciences

T/TH 11AM-12:15PM, MCS B33

Instructor Information:

Samuel Isaacson

Associate Professor

Department of Mathematics and Statistics

111 Cummington Mall, Room 231 (MCS)

Course Webpage (all homeworks will be posted here):

http://math.bu.edu/people/isaacson/ma565-spring-2018/ma565.html

Office Hours:

Tuesday from 12:15-1:15pm and 4:30-5:30pm in MCS 231.

Textbooks: (We will use material and assign problems from each of the following):

[EK1]: LA Segel and L Edelstein-Keshet, A Primer on Mathematical Models in Biology. (Main textbook)

- Ebook available through the BU library: here.

[EK2]: L Edelstein-Keshet, Mathematical Models in Biology

- Ebook available through the BU library: here.

[SO]: E. Sontag, Lecture Notes on Mathematical Systems Biology.

- PDF available online: here.

Description and Goals:

This course introduces methods for analytically investigating mathematical models of biological systems. An emphasis will be placed on ODE, PDE and stochastic models, along with related mathematical tools for analyzing the models. Mathematical topics will include dimensional analysis, qualitative analysis of ODE models, asymptotic methods, quasi-steady state approximations, multi-stability, oscillatory systems, excitable systems, stochastic differential equations and drift-diffusion models. Biological topics will include chemotherapy, viral and disease dynamics, cellular signaling and receptor dynamics, gene regulation, neural excitation, polymer dynamics and transport within cells.

Prerequisites:

MA 226 or 231 or the equivalent. It is assumed that students are comfortable with solution methods for linear first and second order ODEs, phase-plane methods for systems of ODEs in 2D, and fully recall multivariable calculus (GreenÕs theorem, divergence theorem, partial derivatives, surface integrals, and area/volume integrals). This material will be used throughout the course without review.

10% - Homework assignments (graded only on completeness)

30% - Midterm I (2/22 in class)

30% - Midterm II (4/5 in class)

30% - Final (Tuesday May 8th, 12:30-2:30pm)

Exam Policy:

You are expected to be able to attend class, and as such, makeup exams will not be given for scheduling conflicts. Similarly, it is your responsibility to make sure you both know the official registrarÕs final exam date and can attend the exam. (For example, booking a plane flight before the exam date is not an excusable reason to be unable to take the exam.) If you feel you have an extraordinary reason you will not be able to attend an exam, the instructor must be notified the first week of class (otherwise no consideration will be given to providing a makeup exam). If you miss an exam, a makeup will be provided only if you have missed the exam for medical reasons and notify the instructor within 24 hours of the reason. Makeup exams will generally be oral exams with you answering questions on the blackboard.

Homework Policy:

Homework will be due in class, and will not be accepted late. Homework must be neatly written up, stapled, and have all problems and related work solved in order as listed on the assignment. Homework that does not satisfy this policy will be returned with a grade of zero.

If you feel an exam problem has been incorrectly graded, the professor must be notified the day the assignment/exam is returned to you. After 24 hours all grades are final.

Tentative Course Outline (Subject to change):

1.     Continuous Growth Models

a.     Logistic models in population dynamics.

b.     Chemostats, effect of chemotherapeutic agents on cells.

c.     Non-dimensionalization.

d.     Epidemiology, SIRS models (viral and disease dynamics).

2.     Chemical Kinetics and Cellular Signaling

a.     Types of chemical reactions, simple models.

b.     Enzymatic reactions and quasi-steady state approximations.

c.     Michaelis-Menton approximation and asymptotic analysis.

d.     Allosteric inhibition, cooperativity and multistability.

e.     Hill models and Goldbeter-Koshland approximations.

f.      Applications to signaling networks, receptor systems and gene regulation.

g.     Oscillations and excitable systems. Genetic oscillators, Hodgkin-Huxley, and Fitzhugh-Nagumo models.

3.     Stochastic and PDE Models

a.     From NewtonÕs equations to stochastic differential equations.

b.     Brownian motion and diffusion equation.

c.     Exit time problems and cellular signaling.

d.     Drift-diffusion processes and the Fokker-Planck equation.

e.     Polymer dynamics and microtubule transport.