Canzar, S. (Stefan)

Klau, G.W. (Gunnar)

Toussaint, N. (Nora)

English

CWI's Institutional Repository

S. Canzar1 N. C. Toussaint2 G. W. Klau1An Exact Algorithm for Side-ChainPlacement in Protein Design1Centrum Wiskunde & Informatica, Amsterdam, The Netherlands2University of Tübingen, Center for Bioinformatics, Tübingen, GermanySEA'11Proteinskey players in virtually all biological processesfunction mostly determined by its 3D structuresequence of amino acids(=residues) on backboneeach amino acid has flexibleside-chainS. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 2Proteinskey players in virtually all biological processesfunction mostly determined by its 3D structuresequence of amino acids(=residues) on backboneeach amino acid has flexibleside-chainS. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 2Proteinskey players in virtually all biological processesfunction mostly determined by its 3D structuresequence of amino acids(=residues) on backboneeach amino acid has flexibleside-chainS. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 2The Side-Chain Placement ProblemSide-Chain Placement (SCP)Given a fixed backbone, place the amino acid side-chains on thebackbone in the energetically most favorable conformation.S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 3Discrete Search SpaceThe side-chain conformation of a residue is discretized into afinite number of states.Each rotamer represents a set of similar, statistically preferred,side-chain conformationsBackbone-(in)dependent rotamer library (Dunbrack et al.)(C. Kingsford)⇒ Combinatorial search problem!S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 4Discrete Search SpaceThe side-chain conformation of a residue is discretized into afinite number of states.Each rotamer represents a set of similar, statistically preferred,side-chain conformationsBackbone-(in)dependent rotamer library (Dunbrack et al.)(C. Kingsford)⇒ Combinatorial search problem!S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 4Discrete Search SpaceThe side-chain conformation of a residue is discretized into afinite number of states.Each rotamer represents a set of similar, statistically preferred,side-chain conformationsBackbone-(in)dependent rotamer library (Dunbrack et al.)(C. Kingsford)⇒ Combinatorial search problem!S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 4Discrete Search SpaceThe side-chain conformation of a residue is discretized into afinite number of states.Each rotamer represents a set of similar, statistically preferred,side-chain conformationsBackbone-(in)dependent rotamer library (Dunbrack et al.)(C. Kingsford)⇒ Combinatorial search problem!S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 4Energy FunctionQuality of rotamer assignment by energy function:Singleton scores: interaction between backbone and chosen rotamer intrinsic energy of rotamerPairwise scores: van der Waals electrostatic hydrogen bonding ...Goal: Find minimum energy solution!S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 5Energy FunctionQuality of rotamer assignment by energy function:Singleton scores: interaction between backbone and chosen rotamer intrinsic energy of rotamerPairwise scores: van der Waals electrostatic hydrogen bonding ...Goal: Find minimum energy solution!S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 5Graph-Theoretic FormulationRepresent protein with k residues by k-partite graph G = (V ,E ):part Vi for each residue inode v ∈ Vi for each candidate rotamer of residue iedge uv denotes interaction between u and vnode costs cv , v ∈ V = self-energy of rotamer vedge costs cuv , uv ∈ E = interaction energy of u and vS. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 6Graph-Theoretic FormulationRepresent protein with k residues by k-partite graph G = (V ,E ):part Vi for each residue inode v ∈ Vi for each candidate rotamer of residue iedge uv denotes interaction between u and vnode costs cv , v ∈ V = self-energy of rotamer vedge costs cuv , uv ∈ E = interaction energy of u and vuvS. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 6Graph-Theoretic FormulationRepresent protein with k residues by k-partite graph G = (V ,E ):part Vi for each residue inode v ∈ Vi for each candidate rotamer of residue iedge uv denotes interaction between u and vnode costs cv , v ∈ V = self-energy of rotamer vedge costs cuv , uv ∈ E = interaction energy of u and vuvS. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 6Graph-Theoretic FormulationRepresent protein with k residues by k-partite graph G = (V ,E ):part Vi for each residue inode v ∈ Vi for each candidate rotamer of residue iedge uv denotes interaction between u and vnode costs cv , v ∈ V = self-energy of rotamer vedge costs cuv , uv ∈ E = interaction energy of u and vcucvS. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 6Graph-Theoretic FormulationRepresent protein with k residues by k-partite graph G = (V ,E ):part Vi for each residue inode v ∈ Vi for each candidate rotamer of residue iedge uv denotes interaction between u and vnode costs cv , v ∈ V = self-energy of rotamer vedge costs cuv , uv ∈ E = interaction energy of u and vcucvcuvS. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 6Problem SCPSide-Chain Placement (SCP)Given a k-partite graph G = (V ,E ), V = V1,∪ · · · ∪ Vk , with nodecosts cv , v ∈ V , and edge costs cuv , uv ∈ E , determine anassignment a : [k] 7→ V with a(i) ∈ Vi , such that costk∑i=1ca(i) +k−1∑i=1k∑j=i+1ca(i)a(j)of induced subgraph is minimum.NP-hard [Pierce, Winfree, 2002]inapproximable [Chazelle et al., 2004]S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 7Previous WorkHeuristic:Simulated AnnealingMonte CarloBelief PropagationLess accurate with with increasing problem size! [Voigt et al. 2000]Exact:Dead end elimination + A∗Branch and BoundTree decompositionInteger linear programmingS. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 8Previous WorkHeuristic:Simulated AnnealingMonte CarloBelief PropagationLess accurate with with increasing problem size! [Voigt et al. 2000]Exact:Dead end elimination + A∗Branch and BoundTree decompositionInteger linear programmingS. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 8Previous WorkHeuristic:Simulated AnnealingMonte CarloBelief PropagationLess accurate with with increasing problem size! [Voigt et al. 2000]Exact:Dead end elimination + A∗Branch and BoundTree decompositionInteger linear programmingS. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 8Overview of the Approachexact approachbased on ILP formulation by [Althaus et al.], [Kingsford et al.]Branch & Bound frameworkLagrangian relaxation: lower bounds by shortest path computation Lagrangian dual: Subgradient Optimization primal feasible solutionsinitial primal bound by randomized local searchS. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 9Overview of the Approachexact approachbased on ILP formulation by [Althaus et al.], [Kingsford et al.]Branch & Bound frameworkLagrangian relaxation: lower bounds by shortest path computation Lagrangian dual: Subgradient Optimization primal feasible solutionsinitial primal bound by randomized local searchS. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 9An ILP formulationVariables:xu ∈ {0, 1}, u ∈ Vi , indicates wheter a(i) = u.yuv ∈ {0, 1}: edge uv is contained in induced subgraphConstraints: (Let r(v) = i iff v ∈ Vi )Pick one rotamer per residue:∑v∈Vixv = 1 ∀i ∈ [k]Select induced edges:∑u∈Viyuv = xv ∀v ∈ V , i 6= r(v)S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 10An ILP formulationVariables:xu ∈ {0, 1}, u ∈ Vi , indicates wheter a(i) = u.yuv ∈ {0, 1}: edge uv is contained in induced subgraphConstraints: (Let r(v) = i iff v ∈ Vi )Pick one rotamer per residue:∑v∈Vixv = 1 ∀i ∈ [k]Select induced edges:∑u∈Viyuv = xv ∀v ∈ V , i 6= r(v)S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 10An ILP formulationVariables:xu ∈ {0, 1}, u ∈ Vi , indicates wheter a(i) = u.yuv ∈ {0, 1}: edge uv is contained in induced subgraphConstraints: (Let r(v) = i iff v ∈ Vi )Pick one rotamer per residue:∑v∈Vixv = 1 ∀i ∈ [k]Select induced edges:∑u∈Viyuv = xv ∀v ∈ V , i 6= r(v)S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 10Lagrangian Relaxationmin∑v∈Vcvxv +∑uv∈Ecuvyuv+∑v∈V∑i>r(v)+1λiv · (xv −∑u∈Viyuv )s.t.∑v∈Vixv = 1 ∀i ∈ [k]∑u∈Viyuv = xv ∀v ∈ V , i 6= r(v)xv , yuv ∈ {0, 1} ∀v ∈ V , uv ∈ ES. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 11Lagrangian Relaxationmin∑v∈Vcvxv +∑uv∈Ecuvyuv+∑v∈V∑i>r(v)+1λiv · (xv −∑u∈Viyuv )s.t.∑v∈Vixv = 1 ∀i ∈ [k]∑u∈Viyuv = xv ∀v ∈ V , i < r(v)∑u∈Viyuv = xv ∀v ∈ V , i > r(v)xv , yuv ∈ {0, 1} ∀v ∈ V , uv ∈ ES. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 11Lagrangian Relaxationmin∑v∈Vcvxv +∑uv∈Ecuvyuv+∑v∈V∑i>r(v)+1λiv · (xv −∑u∈Viyuv )s.t.∑v∈Vixv = 1 ∀i ∈ [k]∑u∈Viyuv = xv ∀v ∈ V , i < r(v)∑u∈Viyuv = xv ∀v ∈ V , i = r(v) + 1∑u∈Viyuv = xv ∀v ∈ V , i > r(v) + 1xv , yuv ∈ {0, 1} ∀v ∈ V , uv ∈ ES. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 11Lagrangian Relaxationmin∑v∈Vcvxv +∑uv∈Ecuvyuv+∑v∈V∑i>r(v)+1λiv · (xv −∑u∈Viyuv )s.t.∑v∈Vixv = 1 ∀i ∈ [k]∑u∈Viyuv = xv ∀v ∈ V , i < r(v)∑u∈Viyuv = xv ∀v ∈ V , i = r(v) + 1*dualize∑u∈Viyuv = xv ∀v ∈ V , i > r(v) + 1xv , yuv ∈ {0, 1} ∀v ∈ V , uv ∈ ES. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 11Lagrangian Relaxationmin∑v∈Vcvxv +∑uv∈Ecuvyuv +∑v∈V∑i>r(v)+1λiv · (xv −∑u∈Viyuv )s.t.∑v∈Vixv = 1 ∀i ∈ [k]∑u∈Viyuv = xv ∀v ∈ V , i < r(v)∑u∈Viyuv = xv ∀v ∈ V , i = r(v) + 1xv , yuv ∈ {0, 1} ∀v ∈ V , uv ∈ ES. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 11Lagrangian Subproblem∑v∈Vixv = 1 ∀i ∈ [k]∑u∈Viyuv = xv ∀v ∈ V , i = r(v)− 1∑u∈Viyuv = xv ∀v ∈ V , i = r(v) + 1v1v2v3 v4S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 12Lagrangian Subproblem∑v∈Vixv = 1 ∀i ∈ [k]∑u∈Viyuv = xv ∀v ∈ V ,:i < r(v)i = r(v)− 1∑u∈Viyuv = xv ∀v ∈ V , i = r(v) + 1v1v2v3 v4S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 12Lagrangian Subproblem∑v∈Vixv = 1 ∀i ∈ [k]∑u∈Viyuv = xv ∀v ∈ V ,:i < r(v)i = r(v)− 1∑u∈Viyuv = xv ∀v ∈ V , i = r(v) + 1v1v2v3 v4S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 12Lagrangian Subproblem∑v∈Vixv = 1 ∀i ∈ [k]∑u∈Viyuv = xv ∀v ∈ V ,:i < r(v)i = r(v)− 1∑u∈Viyuv = xv ∀v ∈ V , i = r(v) + 1v1v2v3 v4S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 12Solving the Lagrangian Subproblemminimize∑v∈V(cv +∑i>r(v)+1λiv )xv +∑uv∈Er(u)<r(v)(cuv − λr(v)u )yuvConsider the profit δ of a node v :δ(v) = (cv +∑i>r(v)+1λiv ) +r(v)−2∑i=1minu∈Vi(cuv − λr(v)u )Then the score of a feasible path p = (v1, v2, . . . , vk) is:k∑i=1δ(vi ) +k−1∑i=1cvivi+1S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 13Solving the Lagrangian Subproblemminimize∑v∈V(cv +∑i>r(v)+1λiv )xv +∑uv∈Er(u)<r(v)(cuv − λr(v)u )yuvConsider the profit δ of a node v :δ(v) = (cv +∑i>r(v)+1λiv ) +r(v)−2∑i=1minu∈Vi(cuv − λr(v)u )Then the score of a feasible path p = (v1, v2, . . . , vk) is:k∑i=1δ(vi ) +k−1∑i=1cvivi+1S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 13Lagrangian Bound by Shortest Pathδ(v1)δ(v2)δ(v3) δ(v4)cv1v2 cv2v3cv3v4⇒ 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 14Lagrangian Bound by Shortest Pathδ(v1)cv1v2 + δ(v2)v2v3 v4cv2v3+ δ(v3)cv3v4 + δ(v4)⇒ 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 14Lagrangian Bound by Shortest Pathcv1v2 + δ(v2)v2v3 v4cv2v3+ δ(v3)cv3v4 + δ(v4)00δ(v1) v1s 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 14Lagrangian Bound by Shortest Pathcv1v2 + δ(v2)v2v3 v4cv2v3+ δ(v3)cv3v4 + δ(v4)00δ(v1) v1s 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 14Lagrangian Bound by Shortest Pathcv1v2 + δ(v2)v2v3 v4cv2v3+ δ(v3)cv3v4 + δ(v4)00δ(v1) v1s 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 14Experimental SettingC++, LEDA, BALLcompare to CPLEX [Kingsford et al.] DEE, TreePack, R3 do not allow multiple candidate amino acids treewidth ≈ 10− 20 for small instances reduced instances too large2.26 GHz Intel Quad Core processors, 4 GB RAM, 64 bit Linuxtime limit 12 hours, memory limit 16 GBsuboptimal rotamers eliminated in preprocessing2 different benchmark setsS. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 15Experimental ResultsProtein design instances from Yanover et al.97 proteins, 40-180 flexible residue positionsat each position all 20 amino acids allowedRosetta energy functionInstance Lagrangian B&B CPLEXName #res #rot N H time/s time/s S1brf 44 3524 9 4 293.97 469.87 1.61bx7 25 1048 1 0 0.54 5.77 10.71d3b 66 5732 1 0 530.37 9,577.68 18.11en2 59 2689 1 0 19.41 39.94 2.11ezg 58 1653 2 1 185.11 441.23 2.41g6x 51 3190 1 0 23.96 160.64 6.71gcq 65 5442 4 2 903.82 5,270.08 9.81i07 52 3186 4 1 187.45 166.20 0.91kth 49 3330 18 4 798.57 642.42 0.81rb9 43 3307 7 2 127.93 9,535.72 74.51sem 54 4348 192 8 5,020.55 6,470.37 1.34rxn 45 3636 1 0 220.33 3,034.57 13.8S. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 16Conclusion and OutlookCombinatorial relaxation outperforms LP relaxationPerformance depends on energy function and number of allowedamino acidsLarge real-world instances solved optimally in reasonable timeStrong heuristics on specific problem classes [Sontag et al.]Wide range of applications: image understanding error correcting codes frequency assignment in telecommunicationS. Canzar An Exact Algorithm for Side-Chain Placement in Protein Design 17

An Exact Algorithm for Side-Chain Placement in Protein Design

https://ir.cwi.nl/pub/18036/sea11_canzar.pdf

An Exact Algorithm for Side-Chain Placement in Protein Design

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