An Exact Algorithm for Side-Chain Placement in Protein Design

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1 S. Canzar 1 N. C. Toussaint 2 G. W. Klau 1 An Exact Algorithm for Side-Chain Placement in Protein Design 1 Centrum Wiskunde & Informatica, Amsterdam, The Netherlands 2 University of Tübingen, Center for Bioinformatics, Tübingen, Germany Life sciences seminar, 13 May 2011

2 Proteins key players in virtually all biological processes function mostly determined by its 3D structure S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 2

3 Proteins key players in virtually all biological processes function mostly determined by its 3D structure sequence of amino acids (=residues) on backbone S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 2

4 Proteins key players in virtually all biological processes function mostly determined by its 3D structure sequence of amino acids (=residues) on backbone each amino acid has exible side-chain S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 2

5 The Side-Chain Placement Problem Side-Chain Placement (SCP) Given a xed backbone, place the amino acid side-chains on the backbone in the energetically most favorable conformation. S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 3

6 The Side-Chain Placement Problem Side-Chain Placement (SCP) Given a xed backbone, place the amino acid side-chains on the backbone in the energetically most favorable conformation. Homology Modeling: use known backbone of similar protein to predict protein structure sequence of amino acids known S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 3

7 The Side-Chain Placement Problem Side-Chain Placement (SCP) Given a xed backbone, place the amino acid side-chains on the backbone in the energetically most favorable conformation. Homology Modeling: use known backbone of similar protein to predict protein structure sequence of amino acids known Protein Design nd sequence of amino acids that will fold into given backbone applications in pharmaceutical and biotechnological industry S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 3

8 The Side-Chain Placement Problem Side-Chain Placement (SCP) Given a xed backbone, place the amino acid side-chains on the backbone in the energetically most favorable conformation. Homology Modeling: use known backbone of similar protein to predict protein structure sequence of amino acids known Protein Design nd sequence of amino acids that will fold into given backbone applications in pharmaceutical and biotechnological industry Same mathematical abstraction for both problems! S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 3

9 Discrete Search Space The side-chain conformation of a residue is discretized into a nite number of states. S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 4

10 Discrete Search Space The side-chain conformation of a residue is discretized into a nite number of states. Each rotamer represents a set of similar, statistically preferred, side-chain conformations S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 4

11 Discrete Search Space The side-chain conformation of a residue is discretized into a nite number of states. Each rotamer represents a set of similar, statistically preferred, side-chain conformations Backbone-(in)dependent rotamer library (Dunbrack et al.) (C. Kingsford) S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 4

Each rotamer represents a set of similar, statistically

12 Discrete Search Space The side-chain conformation of a residue is discretized into a nite number of states. Each rotamer represents a set of similar, statistically preferred, side-chain conformations Backbone-(in)dependent rotamer library (Dunbrack et al.) (C. Kingsford) Combinatorial search problem! S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 4

13 Energy Function Quality of rotamer assignment by energy function: Singleton scores: interaction between backbone and chosen rotamer intrinsic energy of rotamer Pairwise scores: van der Waals electrostatic hydrogen bonding... S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 5

14 Energy Function Quality of rotamer assignment by energy function: Singleton scores: interaction between backbone and chosen rotamer intrinsic energy of rotamer Pairwise scores: van der Waals electrostatic hydrogen bonding... Goal: Find minimum energy solution! S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 5

15 Graph-Theoretic Formulation Represent protein with k residues by k-partite graph G = (V, E): part V i for each residue i node v V i for each candidate rotamer of residue i S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 6

16 Graph-Theoretic Formulation Represent protein with k residues by k-partite graph G = (V, E): part V i for each residue i node v V i for each candidate rotamer of residue i edge uv denotes interaction between u and v v u S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 6

17 Graph-Theoretic Formulation Represent protein with k residues by k-partite graph G = (V, E): part V i for each residue i node v V i for each candidate rotamer of residue i edge uv denotes interaction between u and v v u S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 6

18 Graph-Theoretic Formulation Represent protein with k residues by k-partite graph G = (V, E): part V i for each residue i node v V i for each candidate rotamer of residue i edge uv denotes interaction between u and v node costs c v, v V = self-energy of rotamer v c v c u S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 6

19 Graph-Theoretic Formulation Represent protein with k residues by k-partite graph G = (V, E): part V i for each residue i node v V i for each candidate rotamer of residue i edge uv denotes interaction between u and v node costs c v, v V = self-energy of rotamer v edge costs c uv, uv E = interaction energy of u and v c uv c v c u S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 6

20 Problem SCP Side-Chain Placement (SCP) Given a k-partite graph G = (V, E), V = V 1, V k, with node costs c v, v V, and edge costs c uv, uv E, determine an assignment a : [k] V with a(i) V i, such that cost k i=1 k 1 c a(i) + of induced subgraph is minimum. k i=1 j=i+1 c a(i)a(j) N P-hard [Pierce, Winfree, 2002] inapproximable [Chazelle et al., 2004] S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 7

21 Previous Work Heuristic: Simulated Annealing Monte Carlo Belief Propagation S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 8

22 Previous Work Heuristic: Simulated Annealing Monte Carlo Belief Propagation Less accurate with with increasing problem size! [Voigt et al. 2000] S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 8

23 Previous Work Heuristic: Simulated Annealing Monte Carlo Belief Propagation Less accurate with with increasing problem size! [Voigt et al. 2000] Exact: Dead end elimination + A Branch and Bound Tree decomposition Integer linear programming S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 8

24 Overview of the Approach exact approach based on ILP formulation by [Althaus et al.], [Kingsford et al.] Branch & Bound framework Lagrangian relaxation: lower bounds by shortest path computation Lagrangian dual: Subgradient Optimization primal feasible solutions initial primal bound by randomized local search S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 9

25 Overview of the Approach exact approach based on ILP formulation by [Althaus et al.], [Kingsford et al.] Branch & Bound framework Lagrangian relaxation: lower bounds by shortest path computation Lagrangian dual: Subgradient Optimization primal feasible solutions initial primal bound by randomized local search S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 9

26 An ILP formulation Variables: x u {0, 1}, u V i, indicates wheter a(i) = u. y uv {0, 1}: edge uv is contained in induced subgraph S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 10

27 An ILP formulation Variables: x u {0, 1}, u V i, indicates wheter a(i) = u. y uv {0, 1}: edge uv is contained in induced subgraph Constraints: (Let r(v) = i i v V i ) Pick one rotamer per residue: v V i x v = 1 i [k] S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 10

28 An ILP formulation Variables: x u {0, 1}, u V i, indicates wheter a(i) = u. y uv {0, 1}: edge uv is contained in induced subgraph Constraints: (Let r(v) = i i v V i ) Pick one rotamer per residue: v V i x v = 1 i [k] Select induced edges: u V i y uv = x v v V, i r(v) S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 10

29 Lagrangian Relaxation min s.t. c v x v + v V uv E c uv y uv v V i x v = 1 i [k] u V i y uv = x v v V, i r(v) x v, y uv {0, 1} v V, uv E S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 11

30 Lagrangian Relaxation min s.t. c v x v + v V uv E c uv y uv v V i x v = 1 i [k] u V i y uv = x v v V, i < r(v) u V i y uv = x v v V, i > r(v) x v, y uv {0, 1} v V, uv E S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 11

31 Lagrangian Relaxation min s.t. c v x v + v V uv E c uv y uv v V i x v = 1 i [k] u V i y uv = x v v V, i < r(v) u V i y uv = x v v V, i = r(v) + 1 u V i y uv = x v v V, i > r(v) + 1 x v, y uv {0, 1} v V, uv E S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 11

32 Lagrangian Relaxation min s.t. c v x v + v V uv E c uv y uv v V i x v = 1 i [k] u V i y uv = x v v V, i < r(v) u V i y uv = x v v V, i = r(v) + 1 dualize y uv = x v v V, i > r(v) + 1 u V i x v, y uv {0, 1} v V, uv E S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 11

33 Lagrangian Relaxation min s.t. c v x v + v V uv E c uv y uv + λ i v (x v v V i>r(v)+1 u V i v V i x v = 1 i [k] u V i y uv = x v v V, i < r(v) u V i y uv = x v v V, i = r(v) + 1 x v, y uv {0, 1} v V, uv E y uv ) S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 11

34 Lagrangian Subproblem v V i x v = 1 i [k] u V i y uv = x v v V, i = r(v) 1 u V i y uv = x v v V, i = r(v) + 1 v 1 v 2 v 3 v 4 S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 12

35 Lagrangian Subproblem x v = 1 i [k] v V i y uv = x v v V, i < r(v) i = r(v) 1 u V i y uv = x v v V, i = r(v) + 1 u V i v 1 v 2 v 3 v 4 S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 12

36 Lagrangian Subproblem x v = 1 i [k] v V i y uv = x v v V, i < r(v) i = r(v) 1 u V i y uv = x v v V, i = r(v) + 1 u V i v 1 v 2 v 3 v 4 S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 12

37 Lagrangian Subproblem x v = 1 i [k] v V i y uv = x v v V, i < r(v) i = r(v) 1 u V i y uv = x v v V, i = r(v) + 1 u V i v 1 v 2 v 3 v 4 S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 12

38 Solving the Lagrangian Subproblem minimize (c v + v V i>r(v)+1 λ i v)x v + uv E r(u)<r(v) (c uv λ r(v) u )y uv Consider the prot δ of a node v: δ(v) = (c v + i>r(v)+1 r(v) 2 λ i v) + min(c uv λ r(v) u ) u V i i=1 S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 13

39 Solving the Lagrangian Subproblem minimize (c v + v V i>r(v)+1 λ i v)x v + uv E r(u)<r(v) (c uv λ r(v) u )y uv Consider the prot δ of a node v: δ(v) = (c v + i>r(v)+1 r(v) 2 λ i v) + min(c uv λ r(v) u ) u V i i=1 Then the score of a feasible path p = (v 1, v 2,..., v k ) is: k i=1 k 1 δ(v i ) + i=1 c vi v i+1 S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 13

40 Lagrangian Bound by Shortest Path δ(v 1 ) c v1v 2 δ(v 2 ) c v2v 3 c v3v 4 δ(v 3 ) δ(v 4 ) S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 14

41 Lagrangian Bound by Shortest Path δ(v 1 ) c v1v 2 + δ(v 2 ) c v2v 3 + δ(v 3 ) c v3v 4 + δ(v 4 ) v 3 v 4 v 2 S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 14

42 Lagrangian Bound by Shortest Path s δ(v 1 ) v 1 c v1v 2 + δ(v 2 ) v 2 c v2v 3 + δ(v 3 ) c v3v 4 + δ(v 4 ) v 3 v t S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 14

43 Lagrangian Bound by Shortest Path s δ(v 1 ) v 1 c v1v 2 + δ(v 2 ) v 2 c v2v 3 + δ(v 3 ) c v3v 4 + δ(v 4 ) v 3 v t Shortest path in time linear in the number of edges! S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 14

44 Lagrangian Bound by Shortest Path s δ(v 1 ) v 1 c v1v 2 + δ(v 2 ) v 2 c v2v 3 + δ(v 3 ) c v3v 4 + δ(v 4 ) v 3 v t Shortest path in time linear in the number of edges! Optimal solution in time O( V 2 ) S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 14

45 Experimental Setting C++, LEDA, BALL compare to CPLEX [Kingsford et al.] DEE, TreePack, R3 do not allow multiple candidate amino acids treewidth for small instances reduced instances too large 2.26 GHz Intel Quad Core processors, 4 GB RAM, 64 bit Linux time limit 12 hours, memory limit 16 GB suboptimal rotamers eliminated in preprocessing 2 dierent benchmark sets S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 15

46 Experimental Results (I) Protein design energy les by Kingsford et al. 25 proteins, exible residue positions surface residues xed, 6 amino acids at core positions Instance Lagrangian B&B CPLEX Name #res #rot N H time/s time/s S 1c9o cex cz czp d4t mfm plc qj , , qq rcf pth lzt p rsa S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 16

47 Experimental Results (II) Protein design instances from Yanover et al. 97 proteins, exible residue positions at each position all 20 amino acids allowed Rosetta energy function Instance Lagrangian B&B CPLEX Name #res #rot N H time/s time/s S 1brf bx d3b , en ezg g6x gcq , i kth rb , sem , , rxn , S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 17

48 Conclusion and Outlook Combinatorial relaxation outperforms LP relaxation Performance depends on energy function and number of allowed amino acids Large real-world instances solved optimally in reasonable time Strong heuristics on specic problem classes [Sontag et al.] Wide range of applications: image understanding error correcting codes frequency assignment in telecommunication S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 18

Side-Chain Positioning CMSC 423

Side-Chain Positioning CMSC 423 Protein Structure Backbone Protein Structure Backbone Side-chains Side-chain Positioning Given: - amino acid sequence - position of backbone in space Find best 3D positions