The Evolutionary Origins of Protein Sequence Variation

Transcription

1 Temple University Structural Bioinformatics II The Evolutionary Origins of Protein Sequence Variation 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 Allan Haldane Ron Levy Group

2 Summary (from last time) New versions of a protein arise by gene duplication Many possible sequences lead to same fold Proteins in a common family/fold accumulate substitutions at a constant rate over time (usually same function) (often different function) Fibronectin Type III domain MSA Average sequence identity: 19% WebLogo # of substitutions per site (fossil record) substitutions occur at a constant rate Rate constant hypothesis or Molecular Clock (Zuckerland & Pauling 1965) Seq ID MSA

3 How is sequence diversity generated? Why is there so much variation? How do substitutions happen? Evolutionary forces acting on proteins: Natural Selection Mutation Genetic Drift Selective forces on proteins Proteins must: Carry out a function (enzymes, etc) be stable (fold) not aggregate... Bridging the physical scales in evolutionary biology: from protein sequence space to fitness of organisms and populations COSB 2017

4 How is sequence diversity generated? Why is there so much variation? How do substitutions happen? The protein s sequence determines whether it folds, reacts, or aggregates. How can we quantify this relationship? Selective forces on proteins Proteins must: Carry out a function (enzymes, etc) be stable (fold) Many mutations afect stability not aggregate (Functional site is often only a small % of protein).... Selective pressure to fold is one of the predominant protein selective forces. Bridging the physical scales in evolutionary biology: from protein sequence space to fitness of organisms and populations COSB 2017

5 Protein Folding Biochemistry (understanding selective forces in protein evolution) Proteins need to Fold before they can carry out their function Mutations can cause a protein not to fold. Two-State model of Folding The folding process depends on the Free energy of folding, which is determined by the interactions among the amino acids in the folded conformation (sequence dependent) We can model the folding process using simple thermodynamics

6 Lattice Models Some intuition for the free energy of folding through lattice proteins Idea Represent a protein as covalently-bonded chain of amino acids on a 3d grid (eg, 27 amino acids on a 3x3 grid) The chain cannot self-intersect, so only certain conformations are possible. Using a computer we can enumerate all conformations. There are 103,346 for a 3x3 grid. Neighboring amino acids in the grid (which are not bonded along the chain) interact. We defne a matrix of interaction free energies for every amino-acid pair (eg A-A, A-C, A-R etc. Miyazawa Jernigan potential). If we add up the interaction energies for all neighbor residues for a particular sequence, in a particular conformation, we get the folding energy of that sequence+conformation.

7 Lattice Models Some intuition for the free energy of folding through lattice proteins We can calculate the folding energy of any sequence in any conformation. Constraint: Only one conformation is functional. Given a sequence, we can compute its folding energy for the folded conformation, and compare to the folding energy of unfolded/other conformations. In thermodynamic equilibrium, the probability for that sequence to be in the folded state is Folded Unfolded

8 Lattice Models Some intuition for the free energy of folding through lattice proteins Key point: The folding energy depends on the amino acid sequence, in an intuitive way. Reality is more complicated: disordered state molten globule state native state (folded) decoy state Hyper-stability Lattice models can be used to understand some of these efects too: Model with decoy states:

9 Lattice Models Further analysis of lattice models can give quantitative insights into protein evolution, in an elegant theory known as the Random Energy Model. Result: Poorly-folding sequences greatly outnumber strongly folding sequences: It s easy to fnd mutations which decrease folding probability, hard to fnd those that increase it. Can make a histogram of folding energy for all (random) sequnces Can make histogram of energy of all conformations for a single sequence Can make histogram of energies of evolved sequences Simulations show the degree of sequence variation we should expect for folded sequences (qualitatively matches observations). More careful analysis of the energy gap between the folded conformations and decoy conformations gives insights into constraints on sequence evolution

10 Variations in Stability Distribution of stabilities Different observed sequences have (slightly) different stabilities Observation: Protein are marginally stable Possible explanations: Hyper-stability is penalized? Greater number of stable sequences/mutations? Stability varies as proteins evolve Missense meanderings in sequence space: a biophysical view of protein evolution DePristo, Weinreich, Hartl. Nat Rev

11 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

12 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

13 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 diferent (equally ft) genotypes. (asexual) Next generation formed by random sample (with replacement) of previous generation Simulation 1 Cyan genotype has fixed Simulation 2 Time

14 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 (ftness) to each genotype, and performing a weighted sample to get the next generation. Then mutant s fixation probability = s = selection coefcient 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).

15 Mutation-Selection Balance stabilizing destabilizing X X X Vocabulary: Mutation: An individual mutates to a new variant Substitution: A mutant genotype appears and fxes in the population Most protein mutations slightly decrease stability (deleterious). Most mutations do not fix, though some do. A small number of mutations increase stability (beneficial). These mutations often fix.

16 Mutation-Selection Balance Selection Bias Mutational Bias X X X Mutation-Selection Balance # deleterious substitutions = # beneficial substitutions Population genetics theory can be used to quantitatively understand when/how this balance occurs: (under certain conditions) (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

17 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

18 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?

19 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

20 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

21 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 )

22 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

23 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

24 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)

25 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

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

27 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

28 Form of the Potts Model Fields (L x q) Couplings ( (L x q)2) L = sequence length (eg 200) Note similarity of felds 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

29 Form of the Potts Model Potts Energy Sequence landscape Couplings A Prevalence G A A R G I V F Potts Statistical Energy A A R A A F A Potts probability Given known values for the fields and couplings: Image: Dill P(S) gives us a probability for any sequence Note similarity to lattice models 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)

30 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

31 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.

32 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

33 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

34 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

35 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 ftness Enzyme ftness Potts Hamiltonian models of protein co-variation, free energy landscapes, and evolutionary ftness Levy, Haldane, Flynn. COSB 2017

36 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)

37 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.

38 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

39 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

40 Application 2: Free Energy and Conformational Landscapes Potts Energy E(S) refects experimental mutant-stability measurements and melting temperatures Mutant stability Melting temperature Biased MD/Go simulations using contacts as bias/constraints can uncover conformational landscape Quantifcation of the efect of mutations using a global probability model of natural sequence variation Hopf, Ingraham, Poelwijk, Springer, Sander, Coevolutionary Marks Oct 2015signals 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

41 Application 2: Free Energy and Conformational Landscapes By only adding up the couplings corresponding to particular conformational changes, can even predict the conformational preferences of individual sequences. Predicted prefences match up with experimental measures of conformational penalty.

42 Application 3: Fitness Landscapes Enzyme ftness Viral ftness Potts energy E(S) refects ftness of sequences and mutants Potts model can describe epistatic efects 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

43 Summary Potts models can be inferred from the patterns of evolutionary covariation of residues observed in an MSA. Many applications in predicting structures and ftnesses of protein families and of individial sequences: Can predict sequence-dependence of protein folding.