## Module 2

Intermediate skills for systems biology

The teaching of this module will again involve hands-on practical biological examples and implementation using mathematical software. The student’s use of Mathematica or MATLAB will be enhanced appropriately as the module progresses.

1. Functions of more than one variable. Functions of 2 continuous real variables.

Graphs as surfaces. Partial derivatives and the Jacobian matrix. Directional derivatives. Taylor expansions up to first order. Extension of the basic concepts to functions of more than 2 variables. To include a wide range of well established biological examples.

2. More on linear systems.

Eigenvectors and eigenvalues of a square matrix. Real and complex eigenvalues. Meaning of complex eigenvalues as shorthand for rotations. How to find eigenvalues and eigenvectors for numerical examples using Mathermatica or MATLAB.

3. Discrete systems

Systems biology does not rely exclusively on continuous mathematics and SysMIC graduates must appreciate this and be able to construct systems models using discrete approaches.

3.1 Markov chains. Simple account of 2-state Markov Chains. Discussion of how this generalizes to higher state number. Simple biological applications; e.g. cell-surface receptor dynamics. Hidden Markov models.

3.2 Discrete time dynamical systems. The 1-dimensional logistic map. Period doubling bifurcations and transition to chaos. Discrete-time dynamic network models – e.g. gene transcription networks.

3.3 Cellular automata (1 and 2 dimensional). Agent based models (ABO). Simple examples from population dynamics (molecular, cellular and whole organism).

4. ODE models and non-linear systems

Systems of 1st order ODEs. Biological examples to be taken from across the full range of the BBSRC portfolio, from genes to population dynamics. Equilibria, local stability and instability. Phase plane analysis: sources, sinks, saddles neutral equilibrium points, spiral sources and sinks, phase plane. Parameterized systems and bifurcations. Solving numerical ODE systems in Mathematica or MATLAB.

5. Diffusion systems:

What is diffusion? Brownian motion and its continuous limit. Drift vs diffusion. Diffusion-reaction systems. Fisher’s equation in spatial population dynamics (1-dimensional). Simple models of chemotaxis. Activator-inhibitor systems –Turing instabilities in developmental systems.

6. Stochastic systems.

How to reconcile the uncertainty introduced by “random” elements with the well described mathematics of continuous functions. Making the dynamics of few‑component systems computationally and mathematically tractable will be desirable especially for biologists interested in single cell-scale biology and dynamics governed by low copy number genes in diploid organisms for example.

Stochastic models and ODEs as ‘mean-field’ approximations. Simulation of stochastic models (Gillespie algorithm). Stochastic chemical dynamics.

7. Data handling:

7.1 Estimation of parameters in ODE models from data. E.g. using Approximate Bayesian Computation (ABC) methods, genetic algorithms, simulated annealing or other search procedures.

Parameterisation of systems biological models is a key process that bridges the gap between the abstract and ideal world of mathematics and the real world of biology. These methods explored here are very widely used in many aspects of systems biology from highlighting and prioritising parameters in subcellular models to the investigations of SNP distributions and disease in populations. This section develops methods applicable to small well defined systems as well as “big data” biological systems.

7.2 Data quality, reliability and processing.

A major focus of the Module 2 is how to handle core data-related tasks, namely how to assess data quality and reliability, and how to preprocess data (e.g. data normalisation).

8. Modelling challenges II: (Further steps in model making).

A set of more sophisticated mini-projects in which the student is asked to create a model of a particular biological system using the more advanced methods described in this module. The exercises will require the full range of techniques introduced in this module, and so will be more biologically realistic. The student will choose either 2 or 3 mini- projects, and have some discretion in choosing projects in their area(s) of interest.