The Impact of the Tree Prior and Purifying Selection on Estimating Clock Rates During Viral Epidemics

Research Collection Master Thesis The Impact of the Tree Prior and Purifying Selection on Estimating Clock Rates During Viral Epidemics Author(s): Möller, Simon Publication Date: 2016-08-29 Permanent Link: https://doi.org/10.3929/ethz-b-000184546 Rights / License: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library The impact of the tree prior and purifying selection on estimating clock rates during viral epidemics Master Thesis Simon Moller¨ Monday 29th August, 2016 Advisors: Prof. Dr. Tanja Stadler, Dr. David Alan Rasmussen Department of Biosystems, Science and Engineering, ETH Zurich¨ Abstract This thesis is concerned with Bayesian phylogenetic inference of clock rates for viral epidemics. In this and other areas of application of phylogenetics it has been observed that the inferred rate decreases with an increase in the sampling period. Purifying selection is a likely biological factor that contributes to this phenomenon since it purges slightly deleterious mutations from a population over time thus decreasing the overall genetic diversity that is observed per unit time. However, other factors such as methodological biases also play a role and make a biological interpretation of results difficult. We aim to contribute towards disentangling these different influences. With a simulation study we demonstrate that a misspecified tree prior can upwardly bias the inferred clock rate and that the interplay of the different models involved in the inference can be complex and non-intuitive. We also show that the choice of tree prior can influence the inference of clock rate on a real world Ebola dataset from Sierra Leone, but fail to see such influence for a larger dataset from Guinea. Furthermore, we detect signs of purifying selection in the Guinea dataset by comparing rate estimates on internal and pendant branches. i Acknowledgements Working on my thesis in the past six months has been an interesting and rewarding experience. I would like to thank the entire Computational Evolution group, especially Tanja Stadler and Louis du Plessis, for stimulating discussions and great advice. The time in the group has been great from a personal as well as a scientific perspective. I also want to thank Rudolphus Goclenius, who gave the working title – Rudolphus – to my project since his name sounded pleasant and he was born exactly 468 years before I officially started my thesis. Figure 1: A picture of Rudolphus Goclenius (obtained from Wikipedia, original source: http://www.uni- mannheim.de/mateo/desbillons/aport/seite206.html). ii Contents Contents iii 1 Introduction 1 1.1 Phylogenetics...................................... 1 1.2 Bayesian inference ................................... 3 1.3 Bayesian phylogenetic inference............................ 5 1.4 Clock rate, substitution rate and mutation rate................... 6 1.5 Motivation ....................................... 7 2 The interplay of tree prior and clock rate estimation9 2.1 Introduction....................................... 9 2.2 Methods......................................... 11 2.2.1 Simulation study................................ 11 2.2.2 Inference under different tree priors ..................... 11 2.3 Results.......................................... 12 2.3.1 Simulation study................................ 12 2.3.2 Inference under different tree priors ..................... 13 2.4 Discussion and conclusion............................... 14 3 Purifying selection during the Ebola epidemic in Guinea 19 3.1 Introduction....................................... 19 3.2 Data........................................... 19 3.3 Analysis using a relaxed clock............................. 20 3.4 Analysis using a constrained relaxed clock...................... 21 3.5 Conclusion and further work ............................. 22 4 Discussion 25 A Supplementary material to chapter2 27 A.1 Analysis of influenza dataset ............................. 34 iii Contents B Supplementary material to chapter3 36 C Analytic results for the tree length under a serially sampled constant size coalescent model 39 D Implementation of relaxed clock in Beast2 42 E Reproducibility 44 Bibliography 47 iv Chapter 1 Introduction Determining the depth to cover in the intro- 1.1 Phylogenetics duction to a Master’s thesis is probably always a difficult task. The supervisor should not re- Yang and Rannala [1] recently published an ex- quire any introduction to understand the con- cellent introduction to phylogenetic methods tent, whereas including enough information so including the ones that are based on sequence that the “general audience” can follow would distances or parsimony alone which I do not mean writing a couple of text books. As my cover here. For even more details, Yang’s book target audience I chose an ETH Master’s stu- [2] is a valuable resource. dent from a field other than Computational Biology and Bioinformatics with some back- History Phylogenetic trees as we know and ground in mathematics and statistics. I do use them today originated with Darwin’s con- not cover much of biology and hope that any cept of random variation and natural selection reader is roughly familiar with DNA, Darwin (see Fig. 1.1 (a) for a draft in his original note- and diseases in general. While the results book). These trees represent the evolutionary in this thesis are in some sense quite techni- relationships between species (or, more gener- cal and constrained to a narrow field, I hope ally, organisms) and are based on their differ- that the introduction also conveys some of the ence in heritable traits. For a long time, scien- beauty of the foundations that this work is tists used expert knowledge and morphological built upon. Since everything in this thesis characters to construct them. Usually, by us- has to do with Bayesian phylogenetic infer- ing a parsimony argument stating that the best ence I start by describing phylogenetics (sec- tree is the one that requires the least number tion 1.1) and Bayesian inference (section 1.2) of changes throughout the tree. separately before discussing their conjunction (section 1.3). With the advances in sequencing technology it has now for some decades been possible to con- The second purpose of this chapter is to out- struct phylogenetic trees based on DNA and line the problems that are addressed in the re- RNA sequence data. These are potentially mainder of the thesis. To this end, section 1.4 much more accurate since evolution is actually introduces the clock rate which is the main pa- happening on the nucleotide level. The data rameter we are interested in and section 1.5 also suit themselves for a stringent mathemati- briefly describes our contributions. cal analysis because, instead of having to weigh 1 1. Introduction differences between having four and six legs or 3. Neutrality: Nucleotide substitutions do being herbivore or carnivore, algorithms can not cause a change in fitness. operate on a sequence of letters. In this sec- 4. Reversibility: After correcting for varying tion I will describe the framework used for such nucleotide frequencies, substitutions hap- analyses. pen at the same rate in both directions between any two nucleotides. Models of evolution and the Felsenstein like- The first assumption allows us to compute the lihood We assume that as data we are given likelihood for each site separately and then a multiple sequence alignment, i.e. a set of multiply the outcomes to obtain the final re- sequences in which the positions of the nu- sult. The second assumption allows us to treat cleotides are thought to correspond to one model evolution at each site as a Markov chain. another such that differences can be inter- The third assumption allows us to treat all sub- preted as mutations in the past. Our goal stitutions from one nucleotide to another the is to compute the likelihood, P (D|T , M), of same. This makes the Markov chain time ho- the data, D, given an unrooted phylogenetic mogeneous. The final assumptions makes the tree, T , that describes the evolutionary rela- Markov chain reversible. tionships between the species whose sequences we are comparing and a substitution model, D provides us with the nucleotides at the tips M, which describes the dynamics of nucleotide of the tree. If we, for a moment, assume that changes. The tree consists of a topology and we also know all nucleotides on internal nodes, a set of branch lengths given in evolutionary we can compute the likelihood (for a single distance, i.e. units of expected number of sub- site in the alignment) by simply multiplying stitutions per site. An example is shown in pik,jk (dk), where k enumerates all branches in Fig. 1.1 (b). The concept of a substitution will the tree. Note that it does not matter which be explained further below. Any reader, who end of the branch is i and which one is j be- is not familiar with them, can for now think cause of the reversibility assumption. The chal- of mutations instead. Multiple substitution lenge is to enumerate all possible assignments models with varying degrees of complexity and of internal nodes and then multiply the likeli- number of parameters exist, but they all essen- hoods for each one. This can be done efficiently tially provide us with a transition probability using Felsenstein’s pruning algorithm [3]. matrix, P (d). Its entries, pi,j(d), describe the probability that nucleotide i is substituted by Clock models So far we were concerned with nucleotide j over a distance d. a tree in which branch lengths were given in expected number of substitutions. Usually it Let us start by stating explicitly some of the is more interesting to have branch lengths in assumptions that we make: units of real time. For this we introduce the 1. Independence: Sites (i.e. different posi- clock rate, r, a model parameter which scales tions) in the alignment evolve indepen- the (temporal) branch length to give us ex- dently of each other. pected number of substitutions again. Math- ematically we have dk = tk · r, where tk is the 2.

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