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:

• Lecture 1: AMATH410_W08_Lec1_cell reproduction.pdf , cell_reproduction.m , cell_reproduction_nonlinear.m
• easterling_et_al_arsenic_pharmacokinetics.pdf
• Lectures 2-3: AMATH410_W08_Lec2-3_compartmental_models.pdf

(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

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

• Lecture 4: AMATH410_W08_Lec4_population_models.pdf , eulot.m, eulot_plot.m
• Lecture 5: AMATH410_W08_Lec5_population_models.pdf , Leslie_iterate.m
• crouse_et_al_sea_turtles.pdf
• Lecture 6: AMATH410_W08_Lec6-7_population_models.pdf
• Lectures 7-8: AMATH410_W08_Lec7-8_population_models.pdf , loggerhead_stage_class_model.m

(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

• Lectures 9-11: AMATH410_W08_Lec9_basics_of_probability.pdf
• introduction_to_neurons_1.pdf
• AMATH410_W08_Lec10-11_stochastic_models_in_neuroscience.pdf
• COLLECTED_SLIDES_ON_stoch_ion_channels_and_quantal_synapses.pdf
• anderson_and_stevens.pdf
• AMATH410_W08_Lec12-13_stochastic_models_in_neuroscience.pdf
• AMATH410_W08_Lec14_stochastic_models_in_neuroscience.pdf
• Voltage-Gated_channels_and_Hodgkin_Huxley_Equations.pdf

(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