Potts Models and Protein Covariation. Allan Haldane Ron Levy Group

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1 Temple University Structural Bioinformatics II Potts Models and Protein Covariation Allan Haldane Ron Levy Group

2 Outline The Evolutionary Origins of Protein Sequence Variation Contents of the Pfam database Protein Evolution (tour of concepts & current ideas) Protein Fitness, Marginal Stability, Compensatory mutations Using Protein-sequence-variation profitably Correlated Mutations and Structural Contacts Potts Models: Theory Review of Potts Model Results Protein Families and their Evolution A Structural perspective. Annual Review of Biochemistry 2005 Bridging the physical scales in ev. biology: from protein sequence space to fitness of organisms and populations COSB 2017 Potts Hamiltonian models of protein co-variation, free energy landscapes, and evolutionary fitness COSB 2017

3 Pfam Pfam is a collection of HMMs and MSAs of many protein families Uses HMMs to detect new sequences 16,306 curated protein families (folds) 23 million protein sequences 7.6 billion residues collected from all branches of life

4 Pfam website example Fibronectin type III Seed Matches All fibronectin sequences in the alignment have the same beta sandwich fold. ~65,000 different sequences ~100 residues long How similar do you think the sequences are (% identical aa)?

5 Fibronectin Type III domain MSA Average sequence identity: 19% WebLogo Seq. ID MSA

6 Fibronectin has quite typical diversity

7 The Protein Universe Protein sequence structure mapping is highly degenerate Typically ~30-40% identity per fold, as low as 8-9% (equal to random sequences) Space of all possible sequences is Space of all sequences sharing a common folded structure may be (rough intuition/guess) (Poses a challenge for HMM methods: Below 20% identity, HMMs have difficulty predicting whether sequence share a fold. This is known as the twilight zone of sequence similarity) How much of protein sequence space has been explored by life on Earth? Dryden et al. J Royal Society Interface (2008) Twilight zone of protein sequence alignments. Rost, Protein Eng Des Sel (1999)

8 The Protein Universe Many authors suggest there are only 1000 to 10,000 existing protein folds across all life 16,384 families in Pfam 1086 folds, 3464 families in SCOP (compare to 20,000 protein-genes in human genome) Protein Families and their Evolution A Structural perspective. Annual Review of Biochemistry 2005 Estimating the total number of protein folds. Govindarajan et al. Proteins: Struct. Func. Bioinfo Expanding protein universe and its origin from the Biological Big Bang PNAS 2002 The Protein Folds as Platonic Forms. Denton et al. Journal of Theoretical Biology 2002

variants/copies are generated by Tree of 512 kinases in the human genome

9 Evolutionary origins of proteins 4 genomes with multiple variants of the same protein fold: (usually same function) (often different function) New variants/copies are generated by Tree of 512 kinases in the human genome (paralogs) Tree of c-src kinase across species (orthologs) Gene duplication (paralog) Speciation (ortholog)

10 How is sequence diversity generated? Observations More closely related species have fewer differences per (orthologous) protein % difference in Hemoglobin between related species The Neutral Theory of Molecular Evolution. Kimura 1983

11 How is sequence diversity generated? # of substitutions per site Observations (fossil record) substitutions occur at a constant rate Rate constant hypothesis or Molecular Clock (Zuckerland & Pauling 1965) Technical detail (correction for repeated substitutions) p = percent final sequence difference K = number of past substitution events

12 How is sequence diversity generated? Observations Proteins appear to accumulate substitutions at a constant rate, usually about 1 substitution per site per billion years

13 How is sequence diversity generated? Observations Proteins appear to accumulate substitutions at a constant rate, usually about 1 substitution per site per billion years Different proteins have different evolutionary rates

per site per billion years Different proteins have different evolutionary rates Different parts of a

14 How is sequence diversity generated? Observations Proteins appear to accumulate substitutions at a constant rate, usually about 1 substitution per site per billion years Different proteins have different evolutionary rates Different parts of a single protein have different evolutionary rates Why is there variation in evolutionary rate? Causes of evolutionary rate variation among protein sites. Nature Reviews Genetics (2016)

15 Summary Many possible sequences lead to same fold Proteins in a common family/fold accumulate substitutions at a constant rate over time

16 How is sequence diversity generated? Why is there so much variation? How do substitutions happen? Selective pressure (Natural Selection) on protein sequences: function stability (folding) (non-)aggregation... Evolutionary forces acting on proteins: Natural Selection (selective pressures above) Mutation Genetic Drift Bridging the physical scales in evolutionary biology: from protein sequence space to fitness of organisms and populations COSB 2017

17 How is sequence diversity generated? Why is there so much variation? How do substitutions happen? Selective pressure (Natural Selection) on protein sequences: function stability (folding) Many mutations affect stability (non-)aggregation (Functional site is often small % of protein)... Evolutionary forces acting on proteins: Natural Selection (selective pressures above) Mutation Genetic Drift Bridging the physical scales in evolutionary biology: from protein sequence space to fitness of organisms and populations COSB 2017

18 Protein Folding Biochemistry (understanding selective forces in protein evolution) Two-State model of Folding Reality is more complicated: disordered state molten globule state native state (folded) decoy state hyper-stability Folded Unfolded

Variations in Stability Distribution of stabilities Different sequences will have (slightly) different stabilities Protein are marginally stable Possible explanations: Hyper-stability is penalized?

19 Variations in Stability Distribution of stabilities Different sequences will have (slightly) different stabilities Protein are marginally stable Possible explanations: Hyper-stability is penalized? More unstable sequences/mutations? Stability related to overall fitness Stability varies as proteins evolve Missense meanderings in sequence space: a biophysical view of protein evolution DePristo, Weinreich, Hartl. Nat Rev Genet 2005

20 Compensatory Mutations/Substitutions Eg, a destabilizing substitution is compensated for by a stabilizing substitution Epistasis When the effect of a mutation (eg, on stability) depends on the identity of other residues. destabilizing stabilizing CTL escape and viral fitness in HIV/SIV infection Front. Microbiol 2010

21 Protein Evolution Why do deleterious (stability-reducing) mutations occur? Why aren t proteins optimally stable?? Evolutionary forces acting on proteins: Natural Selection (previous few slides) Mutation Genetic Drift (next 2 slides) Recombination (not discussed) Quick intro to the Wright-Fisher Model & Population Genetics

It causes new alleles to fix in the population even without any natural selection.

22 The Wright-Fisher Model (without natural selection) Need to understand how new variants arise at the population level Genetic Drift = fluctuations in allele frequencies. It causes new alleles to fix in the population even without any natural selection. Population of 10 individuals with different (equally fit) genotypes. (asexual) Next generation formed by random sample (with replacement) of previous generation Simulation 1 Cyan genotype has fixed Simulation 2 Time

The Wright-Fisher Model Two allele case, with selection Scenario: All individuals in population have the same protein, but one individual mutates Natural Selection modelled by assigning a weight

23 The Wright-Fisher Model Two allele case, with selection Scenario: All individuals in population have the same protein, but one individual mutates Natural Selection modelled by assigning a weight (fitness) to each genotype, and performing a weighted sample to get the next generation. Then mutant s fixation probability = s = selection coefficient N = population size Neutral Say we assign Old genotype has a weight of 1 Mutant individual has a weight of 1+s Deleterious Beneficial (Kimura s fixation probability) Conclusion: Genetic Drift can cause a new mutant to fix even if it s deleterious (s < 0).

24 Mutation-Selection Balance destabilizing stabilizing Mutation: An individual mutates to a new variant Substitution: A mutant genotype appears and fixes in the population Most protein mutations slightly decrease stability (deleterious). A small number of mutations increase stability. Most deleterious mutations do not fix. Mutation-Selection Balance # deleterious substitutions = # beneficial substitutions (population genetics theory can be used to quantitatively understand when/how this balance occurs) (alternative explanation for why proteins are marginally stable) Why are proteins marginally stable? Taverna, Goldstein. Proteins: Structure, Function, and Bioinformatics 2002 Missense meanderings in sequence space: a biophysical view of protein evolution DePristo, Weinreich, Hartl. Nat Rev Genet 2005 Stability effects of mutations and protein evolvability Tokuriki, Tawfik. Current Opinion in Structural Biology 2009 How Protein Stability and New Functions Trade Off Tokuriki, Stricher, Serrano, Tawfik. PLoS Comput Biol 2008

25 Summary Many possible sequences lead to same fold Proteins in a common family/fold accumulate substitutions at a constant rate over time Most substitutions affect protein stability There is a dynamic balance of slightly deleterious (destabilizing) and slightly beneficial (stabilizing) substitutions over time. Marginal stability is maintained. This dynamic balance also involves: Compensatory mutations Epistatic interactions

26 Part II: Potts Models (Using Protein-sequence-variation to study structure) Outline Motivation and Background Parameterizing a Potts Model Applications of Potts models contacts in protein structure Compensatory mutations correlations in MSA columns?

27 Coevolutionary Analysis and Potts Models Correlated Mutations in a MSA imply Structural Interactions Long history (25 years) of Coevolutionary analysis: Detect Correlated positions, then predict contacts Recent Developments: Instead of modeling each residue pair individually, build a correlated statistical model of the MSA: The Potts model The model can be used for more than contact prediction Lövkvist et al, PRE 87, 2013

28 How to measure correlations in an MSA Positions Residue types Bivariate marginal (frequency) Univariate marginal (frequency) (Example: ) Correlations: Observed pairwise frequency Expected pairwise frequency if positions vary independently if the two positions vary independently

29 Pairwise measures of correlations in an MSA Want a correlation score between position-pairs (sum over Different scoring methods in literature (two shown below) All designed to give a score of 0 for independent variation Mutual Information (MI) Can be interpreted as the neg. log likelihood of generating the distribution when sampling from the distribution Statistical Coupling (SCA) Probability of at i excluding sequences with mutation at j Bar means average over all positions )

30 Relationship to contacts Top-ranking MI (and other) scores finds top contact with about 70% true-positive rate, top 50 at 50%. Can this be improved? Protein 3D Structure Computed from Evolutionary Sequence Variation. Marks et al Plos One 2011 Mutual information without the influence of phylogeny or entropy dramatically improves residue contact prediction Dunn, Wahl, Gloor. Bioinformatics 2008

31 Direct vs Indirect Correlations Problem: MI, SCA, Cij can be though of as local models of the correlation: They look at single pairs at a time However, correlations can be caused by indirect interactions, or correlated networks these local models ignore this Eg: Position 7 interacts with 15, and 15 interacts with 25. Then 7 and 25 will be correlated even though they don t interact. Instead want to make a global model of the correlations, in order to distinguish direct from indirect correlations. Idea: Make a statistical model of the sequence as a whole vs Probability of a pair of residues Probability of a whole sequence

32 Potts Models: Origin & Motivation How to model P(S)? P(S) describes the probability of generating sequce S, where S spans the entire sequence space. Most general form of P(S) is the set of probabilities of all sequences a model with parameters! We can t directly measure P(S) from an MSA (since each sequence appears once). Unlike bivariate marginals, which can be directly measured from an MSA. Solution: Get the least biased distribution subject to constraints: The Maximum Entropy distribution P(S)

33 Maximum Entropy The Maximum Entropy distribution P(S) Maximizing entropy minimizes the amount of prior information built into the distribution Number of model parameters will be equal to number of constraints Entropy of a distribution: In our case, set constraint that P(S) gives the right bivariate marginals (pairwise correlation statistics) Bivariate marginals from P(S): Sum over entire sequence space

34 Maximum Entropy Entropy of a distribution Constraints: Method of Lagrange Multipliers: Lagrange multipliers, one per constraint Maximize by solving for all S

35 Maximum Entropy Solved by: Rearrange to give: (Boltzmann distribution) ( statistical energy ) (normalization) This gives the Potts Model Note: This model is named for its history in physics of magnetic materials, has many other applications

36 Form of the Potts Model Fields (L x q) Couplings ( (L x q)2) L = sequence length (eg 200) Note similarity of fields to PSSMs q = # of residue types (eg 20) A G A A R G I V F A A R A A F A Potts parameters interpretated as energy contributions from each position/pair

Form of the Potts Model Sequence landscape Couplings A G A A R G I V F A A R A A F A Potts Energy Fields Statistical Potts Energy Prevalence Statistical Energy Sequence Prevalence Given known values

37 Form of the Potts Model Sequence landscape Couplings A G A A R G I V F A A R A A F A Potts Energy Fields Statistical Potts Energy Prevalence Statistical Energy Sequence Prevalence Given known values for the fields and couplings: Image: Dill P(S) gives us a probability for any sequence Can compute other statistics E(S) gives us a statistical energy landscape Can model effect of mutations, with epistasis + compensation Coupling values give us info about direct interactions between positions (without indirect interactions)

38 Parameterizing the Model given an MSA Above we found the functional form of the Maximum Entropy distribution, but we did not discuss how to find the values of the parameters This is actually a challenging task. We need to find the set of values which satisfly the constraints on the marginals, but there is no obvious way to do so Non-trivial function of Potts parameters

39 Parameterizing the Model given an MSA A number of different numerical methods and approximations have been developed to find the parameters: Belief Propagation, Susceptibility Propagation Mean Field inference Pseudolikelihood Methods + Conjugate Gradient Descent Cluster Expansion Monte Carlo + Quasi-Newton Optimization This is a computationally intensive task.

40 Parameterizing the Model given an MSA Flavor of the algorithms: Problem can be framed as a Maximum Likelihood inference Define a Likelihood function which has a maximum when the constraints are satisfied. (probability of the MSA according to model) Conjugate Gradient methods, Quasi-Newtons methods: Start with an initial guess for the Parameters Compute local gradient of the Likelihood Take a small step in that direction (update parameters) Repeat

41 Aside: Correction for Phylogeny and Sampling Biases Sequences may be phylogenetically related we may have a biased sample This may give the appearance of correlations even when there are none Eg: wild type: Single mutants Double mutant AAAAAAAAAAAAAAAAAA AAAAABAAAAAAAAAAAA AAAAAAAAAABAAAAAAA AAAAABAAAABAAAAAAA If we oversample dbl mut, overestimate correlation One solution: Weight each sequence by how many sequences are similar to it: (weighted average) weight Effective # of seqs

42 Parameterizing the Model given an MSA Summary of Inference Procedure 1)Obtain an MSA (eg from Pfam) 2)Apply phylogenetic weighting. Need > 1000 effective sequences for precise marginals 3)Compute the bivariate (and univariate) marginals of the data 4)Perform Parameter inference (eg Gradient Descent) given bivariate marginals End up with a set of parameters in a number of ways. which we can use

Potts Model Applications Contact Prediction Mutant stability Contact maps Ab-initio Structure Prediction Free Energy (Conformational) Landscapes Fitness landscapes Melting temperature Seq.

43 Potts Model Applications Contact Prediction Mutant stability Contact maps Ab-initio Structure Prediction Free Energy (Conformational) Landscapes Fitness landscapes Melting temperature Seq. Prevalence Electrostatic coupling Structure prediction Viral fitness Enzyme fitness Potts Hamiltonian models of protein co-variation, free energy landscapes, and evolutionary fitness Levy, Haldane, Flynn. COSB 2017

44 Application 1: Contact Prediction Want to get an interaction score (like MI or SCA) but using Potts model Want to summarize the coupling values for each position pair (sum/average over ) Frobenius Norm of Couplings: APC Correction: (removes 'background') Direct Information Similar to MI, but will exclude indirect interactions since it is computed using direct couplings (some technical details related to gauges not discussed here)

45 Application 1: Contact Prediction Contact Map from Potts Model Contact Map from PDB structures (Protein-Kinase domain) Can achieve 80% True Positive rate for top 200 contacts.

indirect interactions, MI does not Identification of direct residue contacts in

46 Application 1: Contact Prediction Direct Interactions Non-interacting DI gives many more True Positives (red) than MI Indirect Interactions DI distinguishes direct from indirect interactions, MI does not Identification of direct residue contacts in protein protein interaction by message passing Weigt, White, Szurmant, Hoch, Hwa. PNAS 2009

map structure) Go Models (coarse grained MD) Genomics-aided

47 Application 1: Structure Prediction Idea: Use predicted contacts as input to further algorithms: NRM (distance geometry: contact map structure) Go Models (coarse grained MD) Genomics-aided structure prediction Sułkowska, Morcos, Weigt, Hwa, Onuchic. PNAS 2012

48 Application 2: Free Energy and Conformational Landscapes Potts Energy E(S) reflects experimental mutantstability measurements and melting temperatures Mutant stability Melting temperature Biased MD/Go simulations using contacts as bias/constraints can uncover conformational landscape Quantification of the effect of mutations using a global probability model of natural sequence variation Hopf, Ingraham, Poelwijk, Springer, Sander, Marks Oct 2015 Coevolutionary signals across protein lineages help capture multiple protein conformations Morcos, Jana, Hwa, Onuchic. PNAS 2013 Coevolutionary information, protein folding landscapes, and the thermodynamics of natural selection Morcos, Schafer, Cheng, Onuchic, Wolynes. PNAS 2014

49 Application 3: Fitness Landscapes Enzyme fitness Viral fitness Potts energy E(S) reflects fitness of sequences and mutants Potts model can describe epistatic effects and compensatory mutations Coevolutionary landscape inference and the context-dependence of mutations in beta-lactamase TEM-1 Molecular Biology and Evolution 2015 The Fitness Landscape of HIV-1 Gag: Advanced Modeling Approaches and Validation of Model Predictions by in vitro Testing PLoS Comput Biol 2014

The Evolutionary Origins of Protein Sequence Variation

Temple University Structural Bioinformatics II The Evolutionary Origins of Protein Sequence Variation Protein Evolution (tour of concepts & current ideas) Protein Fitness, Marginal Stability, Compensatory