Girsanov Transformation and the Rate of Adaptation

The term natural selection was introduced by Darwin in his 1859 book On the Origins of Species. It is central to the understanding of how species evolve and adapt. Evolution is the product of two opposing forces. Mutations give rise to genetic variation, but natural selection and genetic drift cause these variants to be more or less abundant. Mutations that cause its carrier individual to contribute more offspring to the next generation are referred to as being beneficial, which are made more and more abundant (on average) by the process of natural selection, until they are present in every individual in the population. The presence of genetic drift, however, renders the reproduction process random and thus may cause beneficial mutations to become extinct as well as spread to the entire population. Which of these two scenarios actually happens to a beneficial mutation is further complicated by the fact that in a large population, there will be many beneficial mutations that compete with each other, reducing the probability that each beneficial mutation will spread. In the 1930's, the eminent evolutionary biologist R. A. Fisher raised the following question: how quickly can populations adapt to a novel environment by incorporating beneficial mutations? This is what I call the rate of adaptation problem and has fascinated many biologists, and more recently physicists. Up to now, researchers can only calculate the rate of adaptation approximately for large populations. In the project, we hope to use a technique from probability theory, known as Girsanov transformation, to calculate exactly the rate of adaptation for any population size. Girsanov transformation is a powerful technique that has found wide applications in probability theory, but has so far not been applied to the rate of adaptation problem. We discovered that we could transform a selected model to a non-selected one, which is considerably easier to analyse. Quantities in the selected model have a delicate and complex relationship with quantities in the non-selected model, and we hope to reveal how they relate to each other in more detail in this project. We hope this project will be a vivid illustration of the power of Girsanov transformation in the study of selection.An equally fascinating question is the evolution of sex and recombination. Considering the reduction is the overall number of offspring, known as the two-fold cost of sex, sexual reproduction must confer some benefit, being so prevalent among living organisms. As early as 1889, A. Weissman already understood that the purpose of sex was to generate genetic variation, upon which natural selection act. Thus a sexually reproducing population may adapt faster than an asexually one. This understanding, however, has not been quantified exactly up to now. The methods we have developed for the asexual model, i.e. Girsanov transformation, can also be applied to study the advantage of sex. We should be able to develop exact formulae for the rate of adaptation for any recombination rate, and thus help to quantify the effects of recombination on the rate of adaptation. On top of being interesting to evolutionary biologists, the rate of adaptation problem has practical applications in the study of evolution of viruses and bacteria. For example, the HIV virus eventually evolves drug resistance in individuals undergoing antiretroviral therapy. Being able to predict the rate of adaptation of the HIV virus in this context can help to calibrate drug dosage and combination to maximise their effectiveness, and thus prolong and enhance the quality of life of the patient.

The direct beneficiaries of this research are mainly researchers within academic circles, amongst whom evolutionary biologists, population geneticists and probablists are the most obvious. The project is inspired by a question in the mathematical theory of evolutionary: how quickly can populations adapt to a novel environment by incorporating beneficial mutations? This project will yield an expansion formula for the exact rate of adaptation in both asexual and sexual models, which is a breakthrough since previously only approximate formulae for large populations are available. In addition, evolutionary biologists can compare the rate of adaptation for asexual and sexual models, and quantify the effects of recombination and the advantage of sex in this context exactly. The principal innovation of this project is the application of the powerful technique of Girsanov transformation to the study of the rate of adaptation. Population geneticists will thus have a new tool in their study of selection, which is difficult to handle. This project also opens up new lines of research for probablists. For example, the noise structure revealed by the mean fitness of the population is a fascinating object, with a time dependence structure that has not been encountered before. Academics outside these circles will also benefit from this project. Examples include researchers who study biological organisms that evolve rapidly, such as viruses and bacteria, which have very short life cycles compared to humans. The introduction of antiretroviral therapy has greatly prolonged and enhanced the quality of life for patients who are HIV positive. However, new treatments continue to be developed as HIV viruses continue to evolve drug resistance to existing treatments. Being able to predict the rate of adaptation for HIV viruses will help in the calibration of drug dosage and combination. The general public may thus indirectly benefit from this research due to the more accurate calculation of the rate of adaptation of HIV viruses. Finally, the methodology developed in the project will add an additional piece of understanding to the theory of evolution and is yet another example of how advanced mathematical techniques can be used to shed light on questions about processes in nature. Thus it will be an effective illustration of the power of mathematics, and may inspire young people to take up study of this beautiful subject, which will be important in the new knowledge based economy of this century.