Optimisation & Landscape Analysis for Studying Protein Folding. Roy L. Johnston School of Chemistry University of Birmingham

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1 Optimisation & Landscape Analysis for Studying Protein Folding Roy L. Johnston School of Chemistry University of Birmingham Bridging the Gap Workshop: Dynamic Optimisation University of Birmingham, 24 February 2011

2 Overview Introduction The Protein Folding Problem Protein Models Genetic Algorithms HP Lattice Bead Model Dynamical Lattice Model Energy Landscapes for Protein Folding BLN Model Principal Component Analysis Landscape Complexity Conclusions

3 The Protein Folding Problem To predict the 3D local spatial arrangement (secondary structure) and folded conformation (tertiary structure) of a protein from knowledge of its primary structure the 1D sequence of amino acid residues Why Study Protein Folding? To rationalise and predict the relationship between sequence, 3D structure and function. To understand the effect of mutations on protein structure and function. To understand protein folding dynamics e.g. in order to understand protein misfolding diseases (Alzheimers, CJD etc).

4 Search Methods in Protein Folding Even for the minimalist HP lattice bead model, global optimization is NP-hard. Search methods adopted include: Monte Carlo Simulated Annealing Chain Growth Algorithms Genetic Algorithms (Unger & Moult) Ant Colony Optimization (Hoos) Immune Algorithms (Cutello, Timmis et al.)

5 Protein Models Bead Models minimalist models each amino acid is represented by a bead, usually based on their hydrophobic or hydrophilic nature e.g. HP and BLN models beads may be constrained to a lattice or may be off-lattice. United Atom Models with backbone and side chain beads e.g. Dynamical Lattice Model. All-Atom Models full atomistic treatment of protein e.g. CHarMM, AMBER.

6 The HP Lattice Bead Model Amino acids are classed as either Hydrophobic (H) or Polar (P). Each amino acid is represented as a hard sphere ( bead ) on a lattice (e.g. 2-D square and 3-D diamond lattices). Interactions occur between beads which are adjacent on the lattice (topological neighbours) but are not directly bonded (sequence neighbours). P H Standard HP Model HH = 1 HP = PP = 0 E i j ε ij ij ij = 1 i and j are topological neighbours (but not sequence neighbours) ij = 0 otherwise.

7 Protein Folding GA: HP Model Local coordinate scheme: Conformation vector c = { } Sequence vector s = {HPPHHP } Initial valid conformations generated using Recoil Growth Algorithm. Invalid Structure (superimposed beads) Dead End Structure (no further growth possible) Fitness simply related to energy of the conformation: F i = E i Roulette wheel & Brood selection. 1-point crossover. Variety of mutation operators. Monte Carlo local search. Diversity checking no duplicate structures allowed.

8 Crossover 1-point crossover leads to higher GA success rates (and fewer structure evaluations) than 2-pt. 1-pt crossover is better at maintaining schemata (good regions of local structure).

9 Mutation Corrector operator introduced to repair invalid structures generated by mutation. (Sequential 1-bit changes.)

10 Benchmark Sequences Name E (GM) Sequence % Success HP-20 9 HPHP 2 H 2 PHP 2 HPH 2 P 2 HPH 99.5 HP-24 9 H 2 P 2 (HP 2 ) 6 H HP-25 8 P 2 HP 2 (H 2 P 4 ) 3 H HP P 3 H 2 P 2 H 2 P 5 H 7 P 2 H 2 P 4 H 2 P 2 HP HP P 2 H(P 2 H 2 ) 2 P 5 H 10 P 6 (H 2 P 2 ) 2 HP 2 H 5 HP H 2 (PH) 3 PH 4 P(HP 3 ) 3 P(HP 3 ) 2 HPH 4 (PH) 4 H 200 GA runs. Parameters: X-over = 1.0, mutation = 0.5, elitism = 30%. Structures sampled capped at 60,000.

11 Local Search Modified GA Introduce long range Monte Carlo move operator to allow local searching around each offspring and mutant. Conformation c 1 (energy = E 1 ) undergoes random fold mutation (changing one bit in conformation vector): c 1 (E 1 ) c 2 (E 2 ) E 2 < E 1 accept move. E 2 > E 1 accept move with probability p = E 2 /15E 1 30 attempted MC steps = 1 local search. Brood Selection More than 2 offspring generated from a selected pair of parents. The best 2 offspring replace the parents. Allows wider exploration of crossover space around the two parents. Optimum brood size = 5.

12 Comparison with Previous GA This Work Unger & Moult Sequence E(GM) D(GM) %Success N eval E (GM) N eval HP , ,492 HP , ,491 HP , ,400 HP , ,339 HP , ,547 HP , , GA runs. Parameters: X-over = 1.0, mutation = 0.1, elitism = 30%, DPL = 1, local search, brood size = 5. Maximum generations = 100. G.A. Cox, T. V. Mortimer-Jones, R. P. Taylor, RLJ, Theor. Chem. Acc. 112, (2004).

13 Example Global Minima for Benchmark Sequences HP-20 HP-24 HP-25 HP-36 HP-48 HP-50

14 Dynamical Lattice Model* Amino acid residues have preferred conformations determined by backbone angles and. Dynamical lattice = discrete but non-regular grid. From cluster analysis of Ramachandran plots, certain allowed (, ) pairs are defined for each residue. * F. Dressel, S. Kobe, Chem. Phys. Lett. 424, (2006).

15 Dynamical Lattice Model Amino acid residues treated explicitly. Beads for all backbone atoms (N, C, C). R Hard sphere beads for side chains (R). Energy obtained by summing interactions between C beads. E ij e E n T E ij i, j tanh 8. 0 r ij e e i, j e S,i e S, j Graham Cox; S. Kobe, F. Dressel (Dresden)

16 Parameters Dynamical Lattice Model * * From cluster analysis

17 Dynamical Lattice Model Gene coding (e.g. cysteine) Code (, ) GA Parameters Population 200 Structure limit 500,000 X-over 1.0 Mutation 0.1 Elitism 30% Local search 20% No duplicates allowed Repair invalid structures (hard sphere overlap)

18 n Results (400 GA runs) * = GM from Branch & Bound Search * * * * * * * 1AL1 (right-handed -helix, n = 13) XELLKKLLEELKG PDB GA 1A1P (Compstatin, n =14) ICVVQDWGHHRCTX PDB GA GM(B+B) =

19 Energy Energy Landscapes The EL determines kinetics and thermodynamics of e.g. clusters, liquids, glasses and biomolecules. Determines ability of system to find the global minimum energy and of search methods to find the GM. Examples: Potential Energy Surfaces Free Energy Surfaces (as function of T) Multidimensional surfaces are difficult to visualise. Consider connected network of minima and transition states: Eigenvector following, successive confinement,parallel tempering, and nudged elastic band methods.

20 Representing Energy Landscapes Disconnectivity Graph (DG) approach (Hoffman, Sibani, Schoen, Becker & Karplus, Berry et al., Wales et al.) allows visualisation of the connectivity of high-dimensional PES (e.g. for proteins, clusters, spin glasses). E E BUT the x-coordinate has no meaning. Can more physically meaningful coordinates be obtained? Metric Disconnectivity Graphs (MDGs): Reproducible placement of superbasins. Separation of superbasins reflects structural difference. Thickness of line represents size of superbasin.

21 Principal Component Analysis (PCA) Perform a linear transformation of coordinates of energy minima and transition states (1 st rank saddles). z D2 D1 Identify the principal components coordinates that maximise the variance of the system. y x PCA finds orthogonal lines of best fit through a data set. These lines of best fit are used as coordinates to re-plot the data. D2 This analysis can be used to show and visualise trends in multi-dimensional data. D1

22 PCA for Proteins There are several ways to represent the structure of a protein: (,, ) dihedral angles (x,y,z) Cartesian co-ordinates of atoms We have (mostly) used (x,y,z) co-ordinates, with translations and rotations removed.

23 Energy Energy Energy PCA-based Disconnectivity Graphs PCA can be combined with DGs to produce MDGs in which the x and y axes are used to display structural information. Structures are grouped into superbasins that are mutually accessible without passing through a transition state with energy > E max. The MDG is produced using the average coordinate of all members of a superbasin to place the node. The number of superbasins and the connectivity is assessed at intervals E sep. The thickness of the lines can be used to represent the number of structures or the structural diversity within the superbasin. Structural diversity = number of dimensions needed to reproduce (say) 99% (SD 0.99 ) of the information (variance) within a superbasin. V X 1 Q 1

24 The Off-Lattice BLN Bead Model 3 Types of bead: HydrophoBic (B), HydrophiLic (L) and Neutral (N). Off-lattice model has (bond r) stretching, (angle ) bending, torsional ( ) and through space (Lennard- Jones) components. Gauche Anti Gauche

25 46-Bead BLN Model Global minimum is a 4-strand -barrel. Frustrated PES. Inefficient folding. Gō Model All non-native attractive contacts removed. Single-funnel PES with same GM as BLN. Efficient folding.

26 PCA for the 46-bead BLN Model 1 st PC (Q 1 ) contains approx. 30% of total variance. 1 st + 2 nd PCs (Q 1,Q 2 ) contain approx. 45% of total variance. 2D and 3D disconnectivity graphs can be plotted against Q 1 and Q 2 Line thickness related to structural diversity within a superbasin. T. Komatsuzaki, K. Hoshino, Y. Matsunaga, G.J. Rylance, RLJ, D.J. Wales, J. Chem. Phys. 122, (2005).

27 BLN Go 2D 3D

28 3D Disconnectivity Graphs: Go vs. BLN Go BLN

29 Dihedral Angles 46 Bead Go model Disconnectivity graph based on 43 dihedrals A B A B

30 Landscape Complexity How can we quantify the complexity of an energy landscape? Residential probability (p r ): probability of being located in a given superbasin at a certain energy. Branching probability (p b ): probability of taking a particular path to a given superbasin compared to all possible paths, leading from parent node. Landscape complexity (C L ): Shannon entropy of residual probabilities: Path complexity (C P, ): Shannon entropy of branching probabilities: C C L V P, i V i p r p log b p log r p b

31 Complexity: Go vs. BLN Go Landscape Complexity BLN GO BLN C L = C L =

32 Conclusions Nature-inspired Computation has proven to be successful in searching for low energy folds for simple model proteins. Disconnectivity Graphs give information about the connectivity and important energy barriers on energy (and other) landscapes. The Dynamical Lattice Model shows promise as an intermediate between simple bead models and all-atom models. PCA-based visualisation and complexity analysis of protein folding landscapes allows us to explain difficulties encountered by global optimisation algorithms in certain cases and may aid the design of more robust search algorithms.

33 Acknowledgments Birmingham Dr Gareth Rylance Dr Graham Cox Dr Ben Curley Dr Lesley Lloyd Dr Andrew Bennett Dr Jun He (now Aberystwyth) Eleanor Turpin Funding EPSRC The Royal Society Wellcome VIP Scheme JSPS Leverhulme Trust University of Birmingham BlueBEAR External Prof. Sigismund Kobe (Dresden) Prof. Tamiki Komatsuzaki (Kobe) Prof. David Wales (Cambridge) Prof. Martin Karplus & Dr Paul Maragakis (Harvard) Prof. Said Salhi (Kent)

34 Cambridge Daan Frenkel, David Wales & Mark Miller Oxford Jon Doye Birmingham Roy Johnston, Mark Oakley Methods New coarse-grained potentials. Analysis of potential energy, free energy and other landscapes. New hybrid search algorithms. Dynamical and thermodynamic simulations. Investigation of hierarchical selfassembly. Example Systems Proteins; DNA, RNA; Liquid crystals Simulation of Self-Assembly (Programme Grant EP/I001352)