Diversity in Evolving Systems: Scaling and Dynamics of Genealogical Trees by Erik Rauch B.S., Computer Science and Mathematics Yale University (1996) S.M., Electrical Engineering and Computer Science Massachusetts Institute of Technology (1999) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY F~V~ rt ( j,-1 7 January 2004 @Erik Rauch, 2004. All rights reserved. The aumor hereby gtvlh to Mfr penmtson to reproduce and to duMibute pub* pope and elacwonic copes of this thesis documentin whole or in port A uthor .. ................................ Department of Electrical Engineering and Computer Science January 30, 2004 Certified by. ................ Gerald Jay Sussman Matsushita Profgssor of Electrical Engineering Thsig's Sup ifrisor Accepted by........ Arthur C. Smith Chairman, Department Committee on Graduate Students MASSACHUSETTS INSTIuTE OF TECHNOLOGY APR 15 2004 BARKER LIBRARIES Diversity in Evolving Systems: Scaling and Dynamics of Genealogical Trees by Erik Rauch Submitted to the Department of Electrical Engineering and Computer Science on January 30, 2004 in partial fulfillment of the degree of Doctor of Philosophy Abstract Diversity is a fundamental property of all evolving systems. This thesis examines spa- tial and temporal patterns of diversity. The systems I will study consist of a population of individuals, each with a potentially unique state, together with a dynamics consist- ing of copying or reproduction of individual states with small modifications to them (innovations). I show that properties of diversity can be understood by modelling the evolving genealogical tree of the population. This formulation is general enough that it captures interesting features of a range of natural and artificial systems, though I will pay particular attention to genetic diversity in biological populations, and discuss the implications of the results to conservation. I show that diversity is unevenly distributed in populations, and a disproportionate fraction is found in small sub-populations. The evolution of diversity is a dynamic process, and I show that large fluctuations in diversity can result purely from the inter- nal dynamics of the population, and not necessarily from external causes. I also show how diversity is affected by the structure of the population (spatial or well-mixed), and determine the scaling of diversity with habitat area in spatial systems. Predictions from the model agree with existing experimental genetic data on global populations of bacteria. I then apply the method of modelling the genealogical tree of a population to further questions in evolution. Using a generic model of a pathogen evolving to coexist with a population of hosts, I show that the evolutionary dynamics of the system can be better understood by considering the dynamics of strains (groups of individuals descended from a common ancestor) rather than individuals. A fundamental question in the study of evolution is how selection can operate above the level of the individual, and these results suggests a more general mechanism for such selection. 3 Thesis supervisor: Gerald Jay Sussman Title: Matsushita Professor of Electrical Engineering 4 Acknowledgements Yaneer Bar-Yam acted as a co-supervisor for this work and contributed especially to the presentation of the results and their relevance. Hiroki Sayama contributed to the work on host-pathogen evolution. Gerald Jay Sussman provided valuable guidance and strongly encouraged this work. Stephen Hubbell, James Tiedje, John Wakeley, Simon Levin, Mehran Kardar and Stuart Pimm provided useful comments on the diversity results. Jae-Chang Cho and James Tiedje provided the original figure with data on Pseudomonas bacteria populations. Charles Goodnight provided valuable comments on the host-pathogen evolution papers. Daniel Rothman and Joshua Weitz organized the Theoretical Ecology seminar which led to the genesis of the work on host-pathogen evolution. 5 Table of contents Overview 7 1 Scaling, dynamics and distribution of diversity 8 2 Within-species diversity - analytic and simulation results and 23 comparison with experimental data 3 Details of comparison with experimental results 44 4 Dynamics and genealogy of strains in spatially extended host- 51 pathogen models 5 Long-range interactions and evolutionary stability in a predator- 76 prey system 6 Related work 86 7 Conclusion and future work 94 6 Overview The mechanisms that give rise to the enormous variety found in natural systems are of great inherent interest. What causes the enormous variation we see in nature? Why are some species and environments highly diverse, and others less so? How can we characterize the diversity that exists? How does diversity affect evolution? These questions have been studied since Darwin's Origin of Species and before, but this thesis presents a new approach to aspects of this problem based on analyzing and simulating properties of evolving genealogical trees of populations. Chapter 1 gives an overview of the results on diversity. Chapter 2 presents the results covered in Chapter 1 in detail, and gives additional results. I also compare these results with experimental genetic data on microbial populations, showing that the distribution of diversity within populations and fundamental property relating to the shape of genealogical trees both match the data. Chapter 3 details this comparison. In chapter 4 I apply the method of dynamically tracing the genealogical tree of a population to further questions in evolution. In many systems, organisms modify their environment, which in turn affects the evolution of the organisms, but the effects of this are not yet well understood. I show that such systems can be better understood by considering evolution as the dynamics of strains (groups of genealogically related organisms) rather than individuals. In chapter 5 I use the methods of chapter 4 to explore the effect of local and long-range interactions on such evolutionary systems. Chapter 6 reviews existing work related to this thesis, and chapter 7 concludes and presents potential future work. 7 Chapter 1: Scaling, dynamics and distribution of diversity Abstract Here we introduce the method of modelling the evolving genealogical tree of a population, and show how it can be used to study spatial and temporal patterns of diversity. These results are given in more detail in Chapter 2. Why diversity is important Evolution is the phenomenon of a population of interacting individuals changing over time. We usually think of evolution as happening in biological systems, but the concept of evolution can be applied more broadly to other complex systems as well. These systems have the property that they are made up of individual elements, each with possibly unique characteristics, with new elements arising or replacing old elements in the following way: some other element or elements are copied, but occasionally small changes are introduced into the copy. The small changes can be thought of as potential innovations. The more times an element is copied, the more different it becomes from the original. For example, in a biological context, the small changes are mutations and the copying process is reproduction with inheritance. (The 'copy' does not have to be of a single individual; in sexually reproducing organisms, it is a combination of two individuals). There are several ways for interaction to take place: it can occur directly between individuals (for example, through predatator-prey interaction), but more commonly it takes place through the environment. In the simplest way to account for this, individuals can be thought of as "replacing" others because limited resources support a finite population. A population evolving according to this process explores the space of possible states that the individuals can have. A fundamental way to characterize such a pop- ulation is its diversity - a measure of how much of the state space it covers. Diversity is fundamental to adaptation, which is one of the most important charac- 8 teristics of evolving systems. In complex systems, the individuals are generally coupled to a complex environment. To be suited to a complex environment, the individuals' state space must be large. To discover adaptive parts of this state space, it is more ef- fective if the population explores it in parallel, that is, if it is spread out over a region of the state space rather than be concentrated in one part of it. The more variation exists in a population, the faster it can change[1]. Furthermore, complex environments are usually dynamic, and in order to remain adapted, the population must change in response to changes of the environment. This requires that it be flexible (that is, capable of a wide range of responses), because these new conditions may never have been experienced by the system before. However, populations generally cannot change all at once. Any 'solution' must usually start as an innovation in a single or small number of individuals, and adaptation at the level of the population happens when these innovations spread. Potential solutions could conceivably be generated by modifying existing ones when they are needed; however, when the state space is large, it generally takes a long time to discover an adaptive solution from scratch - possibly too much time for the response to happen quickly enough. Although in the immune system, variation is generated only when it is needed, evidence from biology[2] suggests that response to change is usually more effective when there is an existing pool of variation from which adaptive innovations can be drawn. This implies that effective adaptation to change also depends on diversity. Model of diversity in evolving systems I will now choose a simple state space for individuals in order to illustrate how the properties of diversity can be obtained from the genealogical tree of a population. It is chosen only for concreteness, and this method does not depend strongly on what the state space is as long as it satisfies a certain basic property I will give.