# Introduction to Computational Models in Biology

AMATH 422/522

WINTER 2015

MW 4:30-5:50
Room B027; of ICL lab, Communications Building

Here, you will learn about models that arise in the life sciences 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. No prior programming experience is assumed.

Images: Terrence Sanger, Medical College of Georgia, Steve Coombes, Yulia Timofeeva.

Instructor |
---|

Eric Shea-Brown |

325 Lewis Hall |

Office hours: M 9:00-10:00 |

**Required text:** "Dynamic Models in Biology,".
by Stephen Ellner and John Guckenhiemer (called **EG** below). .

**Optional text:** "Matlab: A Practical Introduction to Programming and Problem Solving," by Stormy Attaway is a very helpful programming guide and reference, for use as needed.

**Codes and data for lectures and lab exercises**

** Required tutorials, lab "manuals," and handouts will be given in class **

## Syllabus and readings

**(1) Course overview, introduction to programming, and introduction to mathematical models in the life sciences**

**.**

- Modeling objectives: prediction and theory development
- Introduction to programming: vectors, matrices, loops, logic, plotting
- Exponential and chaotic population growth in a simple system

READINGS:

- EG Chapter 1

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

- Introduction to population biology
- Euler-Lotka formula and root-finding for age-class models: "Leslie Matrices"
- Matrix multiplication and eigenvalues in MATLAB
- The Perron-Frobenius Theorem, dominant eigenvalues and population growth
- Eigenvalue sensitivity formulas and applications in ecology

READINGS:

EXTRA TUTORIALS:

- EG Chapter 2
- Literature: crouse.pdf

- For those wanting a Linear Algebra tutorial / referesher: geometric introduction to linear algebra by Eero Simoncelli (SVD section optional)

**(3) Stochastic models**(2.5 weeks)

- Introduction to electrical membranes and neurons
- Random variables and probability
- Channel statistics: the binomial distribution
- Transition probabilities and Markov chains
- Equilibrium states -- dominant eigenvalues and Perron-Frobenius return!
- Central limit theorem and deterministic limits
- Applications to ion channels and reverse-engineering molecular configurations

READINGS:

- EG Chapter 3.1-3.3, EG Lab manual (section 11).
- Anderson and Stevens (1973)

**(4) Continuous time models and biological networks**(3 weeks)

- Ordinary differential equations and vector "arrow" fields -- visualizing flows in MATLAB
- Nullclines
- Equilibria: Newton's method in MATLAB and R
- Stability and oscillations
- Numerical solution methods, MATLAB and R implementation
- Applications in gene networks, oscillators, and genetic toggle switches
- Stochastic differential equations and biological memory models

READINGS:

- EG Chapter 4, 5.1-5.4, 5.7, 6.1-6.3.
- Shen-Orr et al, network motifs 2002
- Alon, Nature Reviews genetics 2007

**(5) Fitting and testing models**

- Least squares fits, maximum likelihood
- Cross validation and model selection
- Model optimization in MATLAB and R
- Building models: model vs. parameter error

READINGS:

- EG Chapter 9

## Course structure and grading

Course grades are composed of: Problem Sets 35%, Class participation 10%, Case study presentation 10%, Project presentation 10%, Project paper 35%.Problem sets will be due on select Wednesdays (roughly every other week) at the start of class. Formatting: in your writeup please present all material for a given problem together -- e.g. under "Problem II" you'd have code, plots, any analysis and results, then we'd go to the next problem. Late policy: 50% credit if turned in late but within 2 days of deadline; not accepted otherwise.

Class participation credit given for participating in in-class computer exercises that will be interspersed with lectures.

Case study: Each student or group of 2 students (your choice) will give a brief in-class presentation of a paper that applies the modeling and computational techniques we have learned in the course. Paper suggestions, guidelines and grading criteria will be distributed. **Presentation dates: Feb.
23 and 25.**

Project: These case studies will be developed into course projects, consisting of a brief in-class presentation and paper writeup. **Presentation dates: Mar. 9 and 11.**
Papers due to Shea-Brown mailbox, Lewis Hall: **WEDS. MAR 18, 5 PM**.

## Computing

In this course, we will make extensive use of the Matlab ( The MathWorks, Inc) programming language. As an OPTIONAL bonus, the textbook's website also provides many codes in the language R, which has closely related syntax and is used extensively in some computational biology communities.There is access to MATLAB at the ICL labs on campus.

Additionally, you can access MATLAB remotely by following the links to "terminal server" -- or follow this link to instructions for how to log in. Two tips: on mac, you might need to select "millions of colors." AND, PLEASE BE CAREFUL: on any platform, make sure you know where you are saving your files before you logout -- if they are on a remote machine, you might not be able to access them easily or at all again. You can also email or "dropbox" or "google drive" the files to yourself before logout. Or, depending on platform/setup, you might need to check a box such as "disk drives" to gain direct access to local harddrive or flash drive.

Another option is to purchase the student version of MATLAB for your personal computer -- this is available for a very heavily discounted price.