Bayesian Statistical Approach for Protein Residue-Residue Contact Prediction Susann Vorberg 2017 Dissertation zur Erlangung des Doktorgrades der Fakultät für Chemie und Pharmazie der Ludwig-Maximilians-Universität München Bayesian Statistical Approach for Protein Residue-Residue Contact Prediction Susann Vorberg aus Leipzig, Deutschland 2017 Erklärung Diese Dissertation wurde im Sinne von §7 der Promotionsordnung vom 28. November 2011 von Dr. Johannes Söding betreut. Eidesstattliche Versicherung Diese Dissertation wurde eigenständig und ohne unerlaubte Hilfe erarbeitet. München, 03.11.2017 ........................ Ort, Datum Susann Vorberg Dissertation eingereicht am: 03.11.2017 Erstgutachter: Dr. Johannes Söding Zweitgutachter: Prof. Dr. Julien Gagneur Tag der mündlichen Prüfung: 11.12.2017 Acknowledgements I am very grateful to Johannes Söding, for giving me the opportunity to work in his lab, for his supervision and guidance on this fascinating project. I learned a lot from you, not only analytical and statistical skills, but also about being a scientist and what holds the scientific world together at its core. Your enthusiasm and your convincing attitude always kept me going. I also want to thank Julien Gagneur for supervising this thesis together with the other mem- bers of my examination board: Franz Herzog, Klaus Förstermann, Karl-Peter Hopfner and Oliver Keppler. My thanks goes also to Roland Beckmann who was part of my thesis advisory committee. Special thanks goes to Julien who had great and pragmatic ideas that helped to keep the big picture in focus. My gratitude goes also to the Quantitative Biosciences Munich graduate school. Foremost to Ulrike Gaul and Erwin Frey for installing this great melting pot of science. With QBM’s financial support I could visit inspiring conferences that helped me to grow as scientist and maybe even more as a person. Additionally, I thank the staff Mara Kieke, Julia Schlehe, Filiz Civril, Markus Hohle and Michael Mende who organized so many great lectures, workshops, and events for us and were always ready to help. I want to thank my group, the Söding lab, for their support and distraction in- and outside the lab. You were more than mere colleagues, you became true friends and made me enjoy coming to work every single day. In particular to Anja and Mark who stayed, like me, in Munich until the very end. It will be an honor to shut down the lights in our beloved jungle office together with you. Thanks a lot, Jessica, for finding the time for proofreading, even when there is no time at all. My thanks also go to the Gagneur group with whom we shared the office space at the LMU gene center for many years. Thanks for your open office doors during my Garching visits whenever I felt I needed company. I also want to thank my former coaches, Henrik Lindner and Torsten Kunke, who supported my decision to leave the army and send me on my way to becoming Dr. Susi. Without you I might still be soaring the skies. I want to thank my family for raising me curious and skeptical and therefore having me equipped with fundamental scientific skills. Daniel you are the love of my life. i ii Summary Despite continuous efforts in automating experimental structure determination and system- atic target selection in structural genomics projects, the gap between the number of known amino acid sequences and solved 3D structures for proteins is constantly widening. While DNA sequencing technologies are advancing at an extraordinary pace, thereby constantly in- creasing throughput while at the same time reducing costs, protein structure determination is still labour intensive, time-consuming and expensive. This trend illustrates the essen- tial importance of complementary computational approaches in order to bridge the so called sequence-structure gap. About half of the protein families lack structural annotation and therefore are not amenable to techniques that infer protein structure from homologs. These protein families can be addressed by de novo structure prediction approaches that in practice are often limited by the immense computational costs required to search the conformational space for the lowest- energy conformation. Improved predictions of contacts between amino acid residues have been demonstrated to sufficiently constrain the overall protein fold and thereby extend the appli- cability of de novo methods to larger proteins. Residue-residue contact prediction is based on the idea that selection pressure on protein structure and function can lead to compensatory mutations between spatially close residues. This leaves an echo of correlation signatures that can be traced down from the evolutionary record. Despite the success of contact prediction methods, there are several challenges. The most evident limitation lies in the requirement of deep alignments, which excludes the majority of protein families without associated struc- tural information that are the focus for contact guided de novo structure prediction. The heuristics applied by current contact prediction methods pose another challenge, since they omit available coevolutionary information. This work presents two different approaches for addressing the limitations of contact prediction methods. Instead of inferring evolutionary couplings by maximizing the pseudo-likelihood, I maximize the full likelihood of the statistical model for protein sequence families. This approach performed with comparable precision up to minor improvements over the pseudo- likelihood methods for protein families with few homologous sequences. A Bayesian statistical approach has been developed that provides posterior probability estimates for residue-residue contacts and eradicates the use of heuristics. The full information of coevolutionary signatures is exploited by explicitly modelling the distribution of statistical couplings that reflects the nature of residue-residue interactions. Surprisingly, the posterior probabilities do not directly translate into more precise predictions than obtained by pseudo-likelihood methods combined with prior knowledge. However, the Bayesian framework offers a statistically clean and the- oretically solid treatment for the contact prediction problem. This flexible and transparent framework provides a convenient starting point for further developments, such as integrating more complex prior knowledge. The model can also easily be extended towards the deriva- tion of probability estimates for residue-residue distances to enhance precision of predicted structures. iii iv Table of Contents Acknowledgements i Summary iii Table of Contents v 1 Background 1 1.1 Biological Background ............................... 1 1.2 Introduction to Contact Prediction ........................ 4 1.2.1 Local Statistical Models .......................... 5 1.2.2 Global Statistical Models ......................... 6 1.2.3 Machine Learning Methods and Meta-Predictors ............ 7 1.3 Modelling Protein Families with Potts Model .................. 8 1.3.1 Model Properties .............................. 9 1.3.2 Gauge Invariance .............................. 9 1.3.3 Inferring Parameters of the Potts Model ................. 10 1.3.4 Solving the Inverse Potts Problem .................... 11 1.3.5 Maximum Likelihood Inference for Pseudo-Likelihood ......... 12 1.3.6 Computing Contact Maps ......................... 13 1.4 Applications for Contact Prediction ........................ 14 1.5 Evaluating Contact Prediction Methods ..................... 18 1.5.1 Sequence Separation ............................ 19 1.5.2 Interpretation of Evaluation Results ................... 20 1.6 Challenges for Coevolution Methods ....................... 21 1.6.1 Phylogenetic Effects as a Source of Noise ................. 21 1.6.2 Entropic Effects as a Source of Noise ................... 21 1.6.3 Finite Sampling Effects .......................... 22 1.6.4 Multiple Sequence Alignments ....................... 22 1.6.5 Alternative Sources of Coevolution .................... 23 v 2 Interpretation of Coupling Matrices 25 2.1 Single Coupling Values Carry Evidence of Contacts ............... 25 2.2 Coupling Profiles Vary with Distance ....................... 26 2.3 Physico-Chemical Fingerprints in Coupling Matrices .............. 29 2.4 Higher Order Dependencies Between Couplings ................. 34 2.5 Discussion ...................................... 35 2.6 Methods ....................................... 37 2.6.1 Dataset ................................... 37 2.6.2 Computing Pseudo-Likelihood Couplings ................. 37 2.6.3 Sequence Reweighting ........................... 39 2.6.4 Computing Amino Acid Frequencies ................... 40 2.6.5 Regularization ............................... 40 2.6.6 Correlation of Couplings with Contact Class ............... 41 2.6.7 Coupling Distribution Plots ........................ 41 3 Optimizing the Full Likelihood 43 3.1 Approximating the Gradient of the Full Likelihood with Contrastive Divergence 43 3.2 Optimizing the Full Likelihood .......................... 45 3.2.1 Convergence Criterion for Stochastic Gradient Descent ......... 46 3.2.2 Tuning Hyperparameters of Stochastic Gradient Descent Optimizer .. 48 3.3 Tuning the Regularizer of Coupling Parameters ................. 51 3.4 Modifying the Gibbs Sampling Scheme for Contrastive Divergence ...... 53 3.4.1 Varying the Sample Size .......................... 54 3.4.2 Varying the number of Gibbs Steps .................... 57 3.4.3 Persistent Contrastive Divergence ..................... 58 3.5 Using the ADAM Optimizer