#
Introduction to Computational Models in Biology

AMATH 422/522

Fall 2017

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

COURSE DISCUSSION BOARD: Piazza (link should arrive in email, contact me if not)

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, but it will be useful.

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

Instructor |
---|

Kameron Decker Harris |

325 Lewis Hall |

Office hours: Monday, 1-2:30 pm in Lewis 320 |

**Required text:**

Dynamic Models in Biology
by Stephen Ellner and John Guckenhiemer (**EG** below)

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

## 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:TUTORIALS: For those who are new to MATLAB

- EG Chapter 1 -- overview of biological models

**(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
- The Perron-Frobenius Theorem, dominant eigenvalues and population growth
- Eigenvalue sensitivity formulas and applications in ecology

READINGS:

- EG Chapter 2 Summary here
- Literature: crouse.pdf
EXTRA TUTORIALS:

- 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)
EXTRA TUTORIALS:

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

- Introduction to systems biology and genetic networks
- Ordinary differential equations and vector fields -- visualizing flows in MATLAB
- Nullclines, equilibria, and numerical root finding
- Numerical solution methods
- Stability and oscillations
- Application: genetic oscillators
- Application: genetic switches
- Stochastic differential equations and numerical methods
- Biological memory models and attractors

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
- High-Dimensional Model Representation review article, Saltelli et al
- Oh et al, Mesoscale brain connectome, Nature 2014
- Some "rough" supplemental (or review) notes on ODEs
- noteset 1 on ODEs
- noteset 2 on numerical methods for ODEs
- noteset 3 on stability and the Jacobian

## Course structure and grading

Course grades are composed of: Problem Sets 40%, Case study presentation 10%, Project presentation 10%, Project paper 40%.

Problem sets will be due on select Wednesdays (roughly every other week) at the start of class.

Important formatting instructions: in your writeup please present all material for a given problem together -- e.g. under "Problem II" you'd have any and all code that you used for that problem, a written answer (i.e., "the dominant eigenvalue is 0.921"), plots that explain and back up your findings and answers, and any analysis. Then we'd go to the next problem. (Not stapling all code for all problems together as an appendix at the end.) You may find the publish(code.m) command in MATLAB helpful. You can print out your codes and plots and intermingle this with handwritten answers and explanations, or, again, some have found the MATLAB "publish" function handy. Late policy: 50% credit if turned in late but within 2 days of deadline; not accepted otherwise.

Case study, project, and presentation guidelines linked here!

Case study: Each group of students 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 will be provided. Guidelines and grading criteria will be distributed and discussed in class in advance.

**Presentation dates: Nov 15 and 20.**

Project: These case studies will be developed into course projects,
consisting of a brief presentation and paper writeup.
**Presentation dates: Dec. 4 and 6**.
Papers due to my mailbox, Lewis Hall: **MON. Dec. 11, 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.

I recommend that you obtain a student copy of MATLAB for use on your own machine. Students can obtain a $30 license that will last until the end of the school year from the UWare website.

There is access to MATLAB at the ICL labs on campus. Computers in B022 (across the hall from our classroom) are available for drop-in use.

You may also use a remote desktop client to access MATLAB on the Mechanical Engineering department's servers (any student, faculty, or staff has free access). Instructions are available here