## Module 1

Basic quantitative skills for systems biology

Students will be introduced to course software, which will be used to illustrate the implementation of basic operations (initially at the level of preliminary material). Mathematica and MATLAB are excellent packages having rich ranges of palettes, templates etc for inputting expressions, and a wide variety of output capabilities including animated graphics. The use of this package of tools will be developed through hands-on application by the students as the module progresses.

1. Review & expansion of key concepts (functions, graphs, rates of change)

1.1 Dependent and independent variables. What are rates of change? Derivatives as slopes of tangent lines. Exponential growth and decay – constant rate of growth/decay – exponential growth/decay differential equation. Properties of the exponential function. Inverse functions – natural logarithms. Properties of logarithms. Real powers – properties, graphs. Logistic functions – constrained growth. Hill functions – Michaelis-Menten; co-operative activation.

1.2 Finding derivatives using Mathematica or MATLAB. Higher order derivatives. Local maxima and minima of functions of one variable; points of inflexion. Maximum rates of change for logistic function and Hill functions.

1.3 Equilibria and local stability analysis of 1-dimensional ODE systems. Examples using growth equations and chemical reaction equations.

1.4 Describing periodic processes. General sine, cosine and tangent functions and graphs. Circadian rhythms and heart beat examples. Oscillators: Simple Harmonic oscillator (SHO) defined by 2nd order differential equation. Damped and reinforced oscillators.

2. Linear systems: the basics (first steps to working with systems of equations)

There is no doubt that it will be critical for SysMIC graduates to appreciate the inherent and almost universal non-linearity of biological systems at all scales. However to approach this concept sensibly and to appreciate some of the methods and simplifications required to analyse these process mathematically it is necessary to first be familiar with linear methods and systems.

2.1 Systems of linear equations. Examples. Vector and matrix representation. Composition of systems and matrix multiplication. Solving systems of linear equations by Gaussian elimination using mathematical software. Examples using stochiometric matrices of metabolic networks.

2.2 Square matrices. Identity matrices, diagonal matrices, upper and lower triangular matrices. Inverse matrices. When do they exist? Determinant of 2x2 matrices and relation to the existence of inverses. Determinants and inverses for diagonal and triangular matrices. How determinants can be defined for arbitrary square matrices. Finding matrix inverses using Mathematical software.

3. Networks (methods for describing and handling complicated interactions)

Systems biology is about the interaction of parts. Students must understand how these parts and their relations one to another can be described graphically and unambiguously and then analysed.

3.1 What is a network? Biological networks – metabolic, transcriptional, signalling, neural and food webs. Directed and undirected networks. Neighbours. Cluster measures. Paths and path length. Diameter. Random, small world and scale free networks. Simple association rules for constructing networks.

3.2 Introduction to the logic of networks: “and”, “or”, “xor”, feedforward, feedback, functional motifs. Examples from gene transcription networks in yeast and intracellular protein signalling networks.

4. Probability and Statistics (working with data)

4.1. Review of basic material (optional): Discrete probabilities. Simple examples using permutations and combinations. Distributions: binomial, negative binomial, Poisson, hypergeometric. How they arise in biological applications – sampling with and without replacement. Continuous distributions. Density functions. Exponential distribution; Normal distribution; LogNormal distribution; Power law distributions. Simple biologically relevant examples. Review of available Mathematica or MATLAB resources.

4.2 Further probability and statistics: Small course in basic R. Sampling, hypothesis testing, data fitting.

5. Modelling (introducing a systematic approach)

The 5-step modeling cycle – purpose, creation, implementation, interpret results, evaluate outcomes. Input-output models, sensitivity analysis, dimensional consistency.

6. Modelling challenges I: (First steps in model making)

A set of substantial mini-projects in which the student creates a model of a biological system using the ideas and techniques taught in Module 1. Only suitably simplified models can be expected at this stage and they will lack realism though they will be (often very) simplified versions of real systems.