What changes when an organism evolves and how? At the most basic biological level, the answer is 'the DNA sequence', and at the most functional level the answer is 'Darwinian fitness'. At a mathematical level, information theory deals with codes which map from alphabets (such as the DNA alphabet of 'A', 'C', 'G' and 'T'), sequences and their probability distributions to numerical functions (such as Darwinian fitness). So information theory is a natural framework in which to consider biological evolution. The relationship between DNA sequences and Darwinian fitness is rather poorly served by current theoretical models of 'fitness landscapes', which are simultaneously too complex (the number of possible genetic make-ups is too vast to be experimentally accessible in any real biological system) and too simple - fitness landscapes are static concepts, whereas in reality fitness is relative to an ever-changing environment of physical and other organisms, as popularised in the 'Red Queen' idea - organisms have to 'run' (evolve) to stay in the same place in terms of fitness. Therefore, the most appropriate branch of information theory to develop for understanding biological evolution is in the context of dynamics of information about the environment and fitness. This project will develop cutting-edge information dynamic theory in a way that makes it applicable to biological evolution.

This project aims to make solid connections between state of the art mathematics (information dynamics and geometry) and real biological evolution. However, that link is not straightforward due to the complexity of real organisms, the difficulty to monitor them, let alone control over evolutionary time-scales. This project will therefore use a series of experimentally evolving systems with different levels of experimental control and biological realism that relate to one another as well as to the theory. Thus, we shall consider the evolution of a 'simple' biochemical interaction between DNA and another molecule and different levels of complexity in evolving systems of computer 'organisms' as well as the evolution of complete biological organisms, microbes, in the laboratory. There will be close feedback between mathematics and experiment in each of these systems, not only to validate the theoretical advances, but to test the nature and differences in the information dynamics and geometry in each of the evolving systems.

The primary level at which these mathematical developments will be tested in experimental systems is in terms of the dynamics of evolutionary operators, such as mutation, recombination, selection and so on, the processes by which the transitions from one generation to the next occur and pass the information. In the computational and biochemical systems we have complete control of these operators, and in the biological evolution system we are able to manipulate mutation rates and monitor the effects by sequencing the organisms' DNA. It will therefore be possible to test how manipulation of these operators relates to changes in the fitness scores. At the mathematical level this will require development of novel 'transition kernels' corresponding to different forms of biologically realistic operators, considering various measures of information and the analysis of its dynamics. By constructive interaction between the mathematics and the different levels of abstraction of biological systems, we shall both develop an important field of mathematics and provide a novel, widely applicable and relevant framework for thinking about testing and understanding biological evolution.