SLN 19428, MTWF 8:30-9:20

LOCATION: MWF, GUG 416. ***Tues, Room B022 of ICL lab ** .

Description and Objectives

In AMATH 410, you will learn about models that arise in biology in chemistry and how they're analyzed using modern mathematical and computational techniques. We will cover statistical models, discrete- and continuous- time dynamical models, and stochastic models. Applications will sample a wide range of scales, from biomolecules to population dynamics, with an emphasis on common mathematical concepts and computational techniques. Throughout, our themes will include interpretation of existing data and predictions for new experiments.

MATLAB will be used for numerical computation, visualization, and data analysis -- and mathematical tools taught in parallel with their computational implementation.

This course is designed for students in a wide variety of departments and with backgrounds across the sciences. A working knowledge of calculus is assumed, together with a desire to learn more about the underlying science, mathematics, or both.

Textbooks, Notes, and Course Resources

The required text for this course is "Dynamic Models in Biology," by Stephen Ellner and John Guckenhiemer (called EG below). A few chapters (including CHAPTER 1) are available free online, here. The text should be in the bookstore, is available on Amazon, and is on reserve in the library.

The "Lab Manual" for this course is also required reading. Freely available from the webpage here, this manual introduces MATLAB and computational methods -- and how to use them to solve and analyze the models and problems in the main text.

OTHER COURSE RESOURCES -- Mathematical models and analysis

These useful texts are also on course reserve in the library:

A course in Mathematical Biology, by de Vries, Hillen, Lewis, Muller, Schonfisch

Mathematical Models in Biology, by Leah Edelstein-Keshet

OTHER COURSE RESOURCES -- Computational methods and MATLAB

The notes of Prof. Nathan Kutz for AMATH 301 are a valuable course reference. Prof. Kutz has provided them online here: (.pdf)

There are a variety of MATLAB resource books available at the library. An excellent one is "Matlab Guide," by Desmond and Nicholas Higham. It is currently on "Math Reserve" in the library ( QA297 .H5217 2005).

For other MATLAB resources, including online tutorials, see below.

Syllabus

(1) Course overview and introduction to mathematical models in the life sciences (1 week).

Reading: EG Chapter 1, EG lab manual sections 1-6.

• Modeling objectives: prediction and theory development
• Rate equations, inflow-outflow models
• Michaelis-Mentin Kinetics: enzyme-mediated chemical reactions
• Complex models -- pharmacokinetics
• Introduction to MATLAB: vectors, matrices, loops, logic, plotting

Partial lecture notes, codes, and sampling of papers referred to in class:

(2) Matrix models -- discrete time, linear maps (2 weeks)

Reading: EG Chapter 2.

• Introduction to population biology
• Linear algebra concepts: matrix multiplication and eigenvalues in MATLAB
• Dominant eigenvalues and population growth in MATLAB
• Euler-Lotkerra formula and root-finding in MATLAB

(3) Stochastic models (2.5 weeks)

Reading: EG Chapter 3.1-3.3, EG Lab manual (section 11). Also: Anderson and Stevens (1973), below. Additional resources on stochastic ion channels: (1) Foundations of Cellular Neurophysiology, by Johnston and Wu, (2) Introduction to Theoretical Neurobiology, Volume 2, by H. Tuckwell.

• Coin flipping and binomial distribution in MATLAB
• Transition probabilities and Markov chains
• Equilibrium states -- dominant eigenvalues return!
• Central limit theorem and deterministic limits
• Diffusion equation: applications in neuroscience of decision making

(4) Continuous time models (3.5 weeks)

Reading: EG Chapter 4, 5.1-5.4, 5.7, 6.1-6.3. Also: section 5 of Amath301 notes by N. Kutz (see link above).

• Ordinary differential equations and vector "arrow" fields -- visualizing flows in MATLAB
• Nullclines
• Equilibria: Newton's method in MATLAB
• Stability and oscillations
• Numerical solution methods, MATLAB implementation
• Applications in gene networks, oscillators, and genetic toggle switches
• Applications in population biology and epidemiology
• Applications in neuroscience models and action potentials

(5) More on fitting and testing models (1 weeks)

Reading: EG Chapter 9

• Least squares fits, concept of maximum likelihood
• Model optimization in MATLAB
• Building models: model vs. parameter error

Schedule and Homework

Follow links in the table below to obtain a copy of the homework in Adobe Acrobat (.pdf) format. You may also obtain here solutions to some of the homework and exam problems. For additional information regarding viewing and printing the homework and solution sets, click here.

Homework, Exams, and Events	Date	Problem sets, etc.
First day of class	Monday, Jan. 7
Homework#1	Due Wed. Jan 16, 8:30 AM	Homework #1 (.pdf) Homework #1 solutions (.pdf)
M.L. King Day	Monday, January 21	no classes
Homework#2	Due Wed. Jan 30, 8:30 AM
Homework#3 (note due date)	Due Fri. Feb 8, 8:30 AM
Midterm Review	Mon. Feb 11
Midterm	Wed. Feb 13, in class	Midterm
No class	Fri. Feb 15	Start preparing case studies
President's Day	Monday, February 18	no classes
Case study presentations	IN CLASS Mon. Feb 25 and Weds. Feb 27 8:30 AM
Homework#4	Due Wed. March 5, 8:30 AM	Homework#5	Due Fri. March 14, 8:30 AM
Last day of classes	Friday, March 14
Final Presentations	Tuesday, March 18, 9:00-10:20am	Final Presentations Location: GUG 416

Homework submission and course policies

Homework is due at the beginning of class on the days above. Please take careful note of these dates, as they are somewhat irregularly spaced. Please also note that these dates might change somewhat as the course progresses -- webpage and email updates will be provided.

No late homework will be accepted. However, for all students, I will drop the lowest ONE homework grade (i.e., one out of the five) when calculating the homework average.

 

Grading

Your course grade will be calculated via the following weights:

Homework 40%

Midterm 20%

Case study presentation 5%

Course project 35%

The test schedule is in the table above.

Case study and final project presentation -- follow link to more details.

Each student will give a brief in-class presentation of a paper that applies the modeling and computational techniques we have learned in the course (case study). These studies will be developed into course projects. FOLLOW THIS LINK TO PROJECT AND CASE STUDY DETAILS.

Matlab Resources

In this course, we will make extensive use of Matlab, a technical computing environment for numerical computation and visualization produced by The MathWorks, Inc. Computers running with MATLAB, as well as the accompanying documentation, are available in the ICL Lab. If you are working in the Windows environment, be sure to check out the Matlab notebook feature that integrates Matlab with Microsoft Word.

There are many additional Matlab resources available on the net, such as the below and many tutorials.

• Matlab Hypertext Reference, Portland State University
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