Course details

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 analysed.

1.1 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.

1.2 Introduction to the analysis of networks. Feedforward, feedback, functional motifs. Examples from gene transcription networks in yeast and intracellular protein signalling networks.

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 using MATLAB. Examples using stoichiometric 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 2×2 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 MATLAB.

3.1 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.

3.2 Finding derivatives using 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.

3.3 Equilibria and local stability analysis of 1-dimensional ODE systems. Examples using growth equations and chemical reaction equations. Introduction to systems of differential equaitons.

3.4 Describing periodic processes. General sine, cosine and tangent functions and graphs. Oscillators: Simple Harmonic oscillator (SHO) defined by 2nd order differential equation.

The modelling cycle, first round: Inspect experimental data – set up model scheme – write down mathematical equations (example: reaction kinetics) –mathematical analysis of model – convert equations to computer code – run computer simulations – compare computer output to data.

The modelling cycle, second and further rounds: Find discrepancies between experimental data and simulations – improve model – make new predictions – compare computer output to data.

Interpretation of the model: search for critical model features.

Application: use successful model as template for advanced investigations (example: physiological control).

5.1. Short course in basic R. 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 including differential expression analysis. Review of available R resources.

Getting Started in Python;  Variables Types and Operations;  Conditional Statements; Arrays; Loops and Iteration; Associative Arrays and Sets; Strings and Files; Functions; Regular Expressions

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 but will serve to practic basic techniques.

Eigenvectors and eigenvalues of the Leslie matrix. Solving linear models. How to find eigenvalues and eigenvectors for numerical examples using MATLAB.

Quantifying interaction using covariance. Creating multivariate random data sets with specified (co-)variance properties. Using eigenvectors to meaningfully transform Omics data. Understanding and applying principal component analysis.

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. From Michaelis-Menten to more realistic kinetic rate laws.

Continuous time models. Equilibria, local stability and instability. Phase plane analysis: sources, sinks, saddles neutral equilibrium points, spiral sources and sinks, phase plane. Solving systems of nonlinear differential equations in MATLAB.

Continuous time models. Parameterized systems and bifurcations. Solving systems of nonlinear differential equations in MATLAB. Using eigenvalues of the Jacobian matrix to characterise bifurcations. Modelling oscillatory phenomena in biology.

The 1-dimensional logistic map. Periodic orbits. Bifurcations, Period doubling route transition to chaos. The logistic bacterial growth model.

Cellular automata (in one and two dimensions). Agent based models. Setting up rules for discrete-state models. Implementing cellular automata in MATLAB.

Stochastic switching processes in biology. Markov chains to describe transition probability models. 2-state Markov Chains. Generalization to higher state numbers. Generation of nucleotide sequences with Markov models.

What is diffusion? Brownian motion and its continuous limit. Compartments coupled by diffusion. Reaction-diffusion models. Partial differential equations. Spatial instability and Turing patterns in one and two spatial dimensions. Substrate-depletion models. Pattern formation in biology.

ODEs as “mean-field” approximations. Stochastic models for small particle numbers. Simulation of stochastic models (Gillespie algorithm). Stochastic population growth dynamics. Stochastic enzyme kinetics. Stochastic steady states under open-flow conditions. Calculating reaction probabilities.

Estimation of parameters in ODE models from data using classical and Bayesian methods, simulated annealing or other search procedures. Data quality, reliability and processing.

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.

If you want to develop your skills further you are invited to perform individual project work (Module 3).