A Constraint-Based Approach to Structure Prediction for Simplified Protein Models that Outperforms Other Existing Methods

Transcription

1 1 A Constraint-Based Approach to Structure Prediction for Simplified Protein Models that Outperforms Other Existing Methods Rolf Backofen and Sebastian Will Chair for Bioinformatics Albert-Ludiwgs-Universität Freiburg

2 Overview computer scientist are interested in methods method: constraint-based structure prediction lattice models basic model of HP-type models subproblems: bounds, hydropbic cores, threading bioinformatics are interested in applications as well results and applications degeneracy of sequences finding protein-likes sequences with unique ground state comparing different models (cubic/fcc, HP-model with HPNX)

3 Overview computer scientist are interested in methods method: constraint-based structure prediction lattice models basic model of HP-type models subproblems: bounds, hydropbic cores, threading bioinformatics are interested in applications as well results and applications degeneracy of sequences finding protein-likes sequences with unique ground state comparing different models (cubic/fcc, HP-model with HPNX)

4 Structure Prediction as Optimization Problem searched: structure (conformation) of minimal (free) energy huge search space hence: only parts of the search space considered generate-and-test generate approximation here: broad exploration of search space starting points for fine-tuning hierarchical approaches GPSQPTYPG DDAPVEDLI RFYDNLQQY LNVVTRHRY search in low resolution model often: low-resolution model = lattice model improvement: biolog. knowledge, molecular dynamics

5 Structure Prediction as Optimization Problem searched: structure (conformation) of minimal (free) energy huge search space hence: only parts of the search space considered generate-and-test generate approximation here: broad exploration of search space starting points for fine-tuning hierarchical approaches GPSQPTYPG DDAPVEDLI RFYDNLQQY LNVVTRHRY search in low resolution model often: low-resolution model = lattice model improvement: biolog. knowledge, molecular dynamics

6 Structure Prediction as Optimization Problem searched: structure (conformation) of minimal (free) energy huge search space hence: only parts of the search space considered generate-and-test generate approximation here: broad exploration of search space starting points for fine-tuning hierarchical approaches GPSQPTYPG DDAPVEDLI RFYDNLQQY LNVVTRHRY search in low resolution model often: low-resolution model = lattice model improvement: biolog. knowledge, molecular dynamics

7 4 Previous Prediction Approaches for Latttice Models sometimes: heuristic approaches advantages: disadvantages: chain growth algorithms genetic algorithms... fast only for structure prediction mostly: monte-carlo/simulated annealing advantages: easy to adapt if ergodic, then known distribution disadvantages: also: complete enumeration advantages: disadvantages: for HP-model, optimal solution nearly never found most approaches are not ergodic direct exploration of landscape very short sequences, only D

8 5 Idea of Lattice Models trade-off: choose between models, that closely resembles proteins structure BUT no hope of ever (algorithmically) finding the native structure models, that crudely resembles proteins structure BUT we can find the native structure BUT SO FAR: we cannot find the native structure either here: BUT we can find the native structure using constraint programming

9 5 Idea of Lattice Models trade-off: choose between models, that closely resembles proteins structure BUT no hope of ever (algorithmically) finding the native structure models, that crudely resembles proteins structure BUT we can find the native structure BUT SO FAR: we cannot find the native structure either here: BUT we can find the native structure using constraint programming

10 5 Idea of Lattice Models trade-off: choose between models, that closely resembles proteins structure BUT no hope of ever (algorithmically) finding the native structure models, that crudely resembles proteins structure BUT we can find the native structure BUT SO FAR: we cannot find the native structure either here: BUT we can find the native structure using constraint programming

11 5 Idea of Lattice Models trade-off: choose between models, that closely resembles proteins structure BUT no hope of ever (algorithmically) finding the native structure models, that crudely resembles proteins structure BUT we can find the native structure BUT SO FAR: we cannot find the native structure either here: BUT we can find the native structure using constraint programming

12 5 Idea of Lattice Models trade-off: choose between models, that closely resembles proteins structure BUT no hope of ever (algorithmically) finding the native structure models, that crudely resembles proteins structure BUT we can find the native structure BUT SO FAR: we cannot find the native structure either here: BUT we can find the native structure using constraint programming

13 5 Idea of Lattice Models trade-off: choose between models, that closely resembles proteins structure BUT no hope of ever (algorithmically) finding the native structure models, that crudely resembles proteins structure BUT we can find the native structure BUT SO FAR: we cannot find the native structure either here: BUT we can find the native structure using constraint programming

14 6 Lattice Models lattice models: usually only backbone positions = positions on lattice self-avoiding: no steric conflicts often used lattices: cubic face-centered-cubic BUT: search for native conformation = NP-complete which lattice should be used?

15 The FCC FCC = face-centered cubic lattice Kepler s conjecture: FCC=densest packing of balls proved just recently (after 400 years) [Bagci,Jernigan,Bahar 00]: clusters of near neighbours in proteins... the neighbours are not distributed in a uniform, less dense way, but rather in a clustered dense way, occupying positions that closely approximate those of a distorted FCC packing We confirm that lattices with large coordination numbers provide better 7 fits to protein structures...

16 The FCC FCC = face-centered cubic lattice Kepler s conjecture: FCC=densest packing of balls proved just recently (after 400 years) [Bagci,Jernigan,Bahar 00]: clusters of near neighbours in proteins... the neighbours are not distributed in a uniform, less dense way, but rather in a clustered dense way, occupying positions that closely approximate those of a distorted FCC packing We confirm that lattices with large coordination numbers provide better 7 fits to protein structures...

17 The FCC FCC = face-centered cubic lattice Kepler s conjecture: FCC=densest packing of balls proved just recently (after 400 years) [Bagci,Jernigan,Bahar 00]: clusters of near neighbours in proteins... the neighbours are not distributed in a uniform, less dense way, but rather in a clustered dense way, occupying positions that closely approximate those of a distorted FCC packing We confirm that lattices with large coordination numbers provide better 7 fits to protein structures...

18 The FCC FCC = face-centered cubic lattice Kepler s conjecture: FCC=densest packing of balls proved just recently (after 400 years) [Bagci,Jernigan,Bahar 00]: clusters of near neighbours in proteins... the neighbours are not distributed in a uniform, less dense way, but rather in a clustered dense way, occupying positions that closely approximate those of a distorted FCC packing We confirm that lattices with large coordination numbers provide better 7 fits to protein structures...

19 The FCC FCC = face-centered cubic lattice Kepler s conjecture: FCC=densest packing of balls proved just recently (after 400 years) [Bagci,Jernigan,Bahar 00]: clusters of near neighbours in proteins... the neighbours are not distributed in a uniform, less dense way, but rather in a clustered dense way, occupying positions that closely approximate those of a distorted FCC packing We confirm that lattices with large coordination numbers provide better 7 fits to protein structures...

20 8 Relation to Proteins hydrophob (H) polar (P) how does it related to proteins? hydrophobic and polar (hydrophilic) amino acids hydrophobic are densely packed alphabet: H = Hydrophobic P = Polar (hydrophilic) in the following: search for conformation with densest hydrophobic packing = max. number of contacts between hydrophobic AA (green) HP-model of Ken Dill: folding of sequences consisting of H and P

21 8 Relation to Proteins hydrophob (H) polar (P) how does it related to proteins? hydrophobic and polar (hydrophilic) amino acids hydrophobic are densely packed alphabet: H = Hydrophobic P = Polar (hydrophilic) in the following: search for conformation with densest hydrophobic packing = max. number of contacts between hydrophobic AA (green) HP-model of Ken Dill: folding of sequences consisting of H and P

22 8 Relation to Proteins hydrophob (H) polar (P) how does it related to proteins? hydrophobic and polar (hydrophilic) amino acids hydrophobic are densely packed alphabet: H = Hydrophobic P = Polar (hydrophilic) in the following: search for conformation with densest hydrophobic packing = max. number of contacts between hydrophobic AA (green) HP-model of Ken Dill: folding of sequences consisting of H and P

23 8 Relation to Proteins hydrophob (H) polar (P) how does it related to proteins? hydrophobic and polar (hydrophilic) amino acids hydrophobic are densely packed alphabet: H = Hydrophobic P = Polar (hydrophilic) in the following: search for conformation with densest hydrophobic packing = max. number of contacts between hydrophobic AA (green) HP-model of Ken Dill: folding of sequences consisting of H and P

24 8 Relation to Proteins hydrophob (H) polar (P) how does it related to proteins? hydrophobic and polar (hydrophilic) amino acids hydrophobic are densely packed alphabet: H = Hydrophobic P = Polar (hydrophilic) in the following: search for conformation with densest hydrophobic packing = max. number of contacts between hydrophobic AA (green) HP-model of Ken Dill: folding of sequences consisting of H and P

25 9 General Approach Algorithm consist of three steps: Step 1 and are precomputation steps Step 1: compute lower energy bounds estimate contacts (within layers, between layers) Step : construct hydrophobic cores use bounds from last step, precomputed Step 3: thread sequence to hydrophobic cores of size n. using constraint propagation Step 1 Step Step 3

26 9 General Approach Algorithm consist of three steps: Step 1 and are precomputation steps Step 1: compute lower energy bounds estimate contacts (within layers, between layers) Step : construct hydrophobic cores use bounds from last step, precomputed Step 3: thread sequence to hydrophobic cores of size n. using constraint propagation Step 1 Step Step 3

27 9 General Approach Algorithm consist of three steps: Step 1 and are precomputation steps Step 1: compute lower energy bounds estimate contacts (within layers, between layers) Step : construct hydrophobic cores use bounds from last step, precomputed Step 3: thread sequence to hydrophobic cores of size n. using constraint propagation Step 1 Step Step 3

28 9 General Approach Algorithm consist of three steps: Step 1 and are precomputation steps Step 1: compute lower energy bounds estimate contacts (within layers, between layers) Step : construct hydrophobic cores use bounds from last step, precomputed Step 3: thread sequence to hydrophobic cores of size n. using constraint propagation Step 1 Step Step 3

29 9 General Approach Algorithm consist of three steps: Step 1 and are precomputation steps Step 1: compute lower energy bounds estimate contacts (within layers, between layers) Step : construct hydrophobic cores use bounds from last step, precomputed Step 3: thread sequence to hydrophobic cores of size n. using constraint propagation Step 1 Step Step 3

30 9 General Approach Algorithm consist of three steps: Step 1 and are precomputation steps Step 1: compute lower energy bounds estimate contacts (within layers, between layers) Step : construct hydrophobic cores use bounds from last step, precomputed Step 3: thread sequence to hydrophobic cores of size n. using constraint propagation Step 1 Step Step 3

31 10 Example for a Constraint-Problem: Sudoku every number from exactly once in every row every column every block

32 10 Example for a Constraint-Problem: Sudoku every number from exactly once in every row every column every block

33 Constraints-Formulation Variablen X n k,l Constraints {0, 1} for every row k, column l and number n {1... 9}. 9 X = 1 1, für Reihe 5 1 l in Reihe 5 X = 1 5,l 9 l in Reihe 5 X = 1 5,l 11 similar for all columns and blocks

34 Constraints-Formulation Variablen X n k,l Constraints {0, 1} for every row k, column l and number n {1... 9}. 9 X = 1 1, für Reihe 5 1 l in Reihe 5 X = 1 5,l 9 l in Reihe 5 X = 1 5,l 11 similar for all columns and blocks

35 Constraint-Propagation k = 1..9 X = 1 k, new values are provable correct iteration till stable state if not solved: Search ?1? 3 for one variable: split over all possible values followed by Propagation eine 1 in keine 1 in Reihe 1,Spalte 8 Reihe 1, Spalte 8 1 (X = 1) 1, naive search (generate-and-test): 9 53 steps 1 Constraint-Programming: automatisation of progagation and search

36 Constraint-Propagation k = 1..9 X = 1 k, new values are provable correct iteration till stable state if not solved: Search ?1? 3 for one variable: split over all possible values followed by Propagation eine 1 in keine 1 in Reihe 1,Spalte 8 Reihe 1, Spalte 8 1 (X = 1) 1, naive search (generate-and-test): 9 53 steps 1 Constraint-Programming: automatisation of progagation and search

37 Constraint-Propagation k = 1..9 X = 1 k, new values are provable correct iteration till stable state if not solved: Search ?1? 3 for one variable: split over all possible values followed by Propagation eine 1 in keine 1 in Reihe 1,Spalte 8 Reihe 1, Spalte 8 1 (X = 1) 1, naive search (generate-and-test): 9 53 steps 1 Constraint-Programming: automatisation of progagation and search

38 Constraint-Propagation k = 1..9 X = 1 k, new values are provable correct iteration till stable state if not solved: Search ?1? 3 for one variable: split over all possible values followed by Propagation eine 1 in keine 1 in Reihe 1,Spalte 8 Reihe 1, Spalte 8 1 (X = 1) 1, naive search (generate-and-test): 9 53 steps 1 Constraint-Programming: automatisation of progagation and search

39 Constraint-Propagation k = 1..9 X = 1 k, new values are provable correct iteration till stable state if not solved: Search ?1? 3 for one variable: split over all possible values followed by Propagation eine 1 in keine 1 in Reihe 1,Spalte 8 Reihe 1, Spalte 8 1 (X = 1) 1, naive search (generate-and-test): 9 53 steps 1 Constraint-Programming: automatisation of progagation and search

40 Constraint-Propagation k = 1..9 X = 1 k, new values are provable correct iteration till stable state if not solved: Search ?1? 3 for one variable: split over all possible values followed by Propagation eine 1 in keine 1 in Reihe 1,Spalte 8 Reihe 1, Spalte 8 1 (X = 1) 1, naive search (generate-and-test): 9 53 steps 1 Constraint-Programming: automatisation of progagation and search

41 13 Constraint-based Formulation constraint problem C Pr : position of i-th amino acid: X i, Y i, Z i [1... n] constraints describe Self-Avoiding Walks (X i, Y i, Z i ) (X j, Y j, Z j ) and (X i, Y i, Z i ) (X i+1, Y i+1, Z i+1 ) = 1 constraint-based optimization: distributing over aminoacid positions C Pr X 1 = X 1 X 4 = 3 C Pr & X 1 = problems redundant constraints and search strategy [Backofen:98] symmetry breaking [Backofen&Will:99] bound for number of of HH-contacts [Backofen:00a,03] new constraints, propagation [Backofen:Will:01]

42 13 Constraint-based Formulation constraint problem C Pr : position of i-th amino acid: X i, Y i, Z i [1... n] constraints describe Self-Avoiding Walks (X i, Y i, Z i ) (X j, Y j, Z j ) and (X i, Y i, Z i ) (X i+1, Y i+1, Z i+1 ) = 1 constraint-based optimization: distributing over aminoacid positions C Pr X 1 = X 1 X 4 = 3 C Pr & X 1 = problems redundant constraints and search strategy [Backofen:98] symmetry breaking [Backofen&Will:99] bound for number of of HH-contacts [Backofen:00a,03] new constraints, propagation [Backofen:Will:01]

43 13 Constraint-based Formulation constraint problem C Pr : position of i-th amino acid: X i, Y i, Z i [1... n] constraints describe Self-Avoiding Walks (X i, Y i, Z i ) (X j, Y j, Z j ) and (X i, Y i, Z i ) (X i+1, Y i+1, Z i+1 ) = 1 constraint-based optimization: distributing over aminoacid positions C Pr X 1 = X 1 X 4 = 3 C Pr & X 1 = problems redundant constraints and search strategy [Backofen:98] symmetry breaking [Backofen&Will:99] bound for number of of HH-contacts [Backofen:00a,03] new constraints, propagation [Backofen:Will:01]

44 13 Constraint-based Formulation constraint problem C Pr : position of i-th amino acid: X i, Y i, Z i [1... n] constraints describe Self-Avoiding Walks (X i, Y i, Z i ) (X j, Y j, Z j ) and (X i, Y i, Z i ) (X i+1, Y i+1, Z i+1 ) = 1 constraint-based optimization: distributing over aminoacid positions C Pr X 1 = X 1 X 4 = 3 C Pr & X 1 = problems redundant constraints and search strategy [Backofen:98] symmetry breaking [Backofen&Will:99] bound for number of of HH-contacts [Backofen:00a,03] new constraints, propagation [Backofen:Will:01]

45 13 Constraint-based Formulation constraint problem C Pr : position of i-th amino acid: X i, Y i, Z i [1... n] constraints describe Self-Avoiding Walks (X i, Y i, Z i ) (X j, Y j, Z j ) and (X i, Y i, Z i ) (X i+1, Y i+1, Z i+1 ) = 1 constraint-based optimization: distributing over aminoacid positions C Pr X 1 = X 1 X 4 = 3 C Pr & X 1 = problems redundant constraints and search strategy [Backofen:98] symmetry breaking [Backofen&Will:99] bound for number of of HH-contacts [Backofen:00a,03] new constraints, propagation [Backofen:Will:01]

46 13 Constraint-based Formulation constraint problem C Pr : position of i-th amino acid: X i, Y i, Z i [1... n] constraints describe Self-Avoiding Walks (X i, Y i, Z i ) (X j, Y j, Z j ) and (X i, Y i, Z i ) (X i+1, Y i+1, Z i+1 ) = 1 constraint-based optimization: distributing over aminoacid positions C Pr X 1 = X 1 X 4 = 3 C Pr & X 1 = problems redundant constraints and search strategy [Backofen:98] symmetry breaking [Backofen&Will:99] bound for number of of HH-contacts [Backofen:00a,03] new constraints, propagation [Backofen:Will:01]

47 14 Problem 1: Frame Sequences

48 15 Bounds for FCC FCC models proteins better: 1.5 Å RMSD [Park&Levitt95] BUT: almost nothing was known approximation: 60% of optimum [Agarwala et al.98] only trivial bounds: 6 number of H-amino acids. approach: layer contacts interlayer contacts 4-point 3-point x=1 x= x=3 n 1 =1 n =5 n 3 =

49 15 Bounds for FCC FCC models proteins better: 1.5 Å RMSD [Park&Levitt95] BUT: almost nothing was known approximation: 60% of optimum [Agarwala et al.98] only trivial bounds: 6 number of H-amino acids. approach: layer contacts interlayer contacts 4-point 3-point x=1 x= x=3 n 1 =1 n =5 n 3 =

50 15 Bounds for FCC FCC models proteins better: 1.5 Å RMSD [Park&Levitt95] BUT: almost nothing was known approximation: 60% of optimum [Agarwala et al.98] only trivial bounds: 6 number of H-amino acids. approach: layer contacts interlayer contacts 4-point 3-point x=1 x= x=3 n 1 =1 n =5 n 3 =

51 16 Bound on Layer Contact relation between surface and contacts 4n = H-contacts + H-surface number of H-neighbours contacts to Ps or solution positions relation to frame H-surface = a + b (a, b)= (height,width) horizontal surface 1 3 vertical surface 1 minimal surface for n Hs = minimal frame (a, b) around n point a = n b = n a

52 16 Bound on Layer Contact relation between surface and contacts 4n = H-contacts + H-surface number of H-neighbours contacts to Ps or solution positions relation to frame H-surface = a + b (a, b)= (height,width) horizontal surface 1 3 vertical surface 1 minimal surface for n Hs = minimal frame (a, b) around n point a = n b = n a

53 16 Bound on Layer Contact relation between surface and contacts 4n = H-contacts + H-surface number of H-neighbours contacts to Ps or solution positions relation to frame H-surface = a + b (a, b)= (height,width) horizontal surface 1 3 vertical surface 1 minimal surface for n Hs = minimal frame (a, b) around n point a = n b = n a

54 16 Bound on Layer Contact relation between surface and contacts 4n = H-contacts + H-surface number of H-neighbours contacts to Ps or solution positions relation to frame H-surface = a + b (a, b)= (height,width) horizontal surface 1 3 vertical surface 1 minimal surface for n Hs = minimal frame (a, b) around n point a = n b = n a

55 16 Bound on Layer Contact relation between surface and contacts 4n = H-contacts + H-surface number of H-neighbours contacts to Ps or solution positions relation to frame H-surface = a + b (a, b)= (height,width) horizontal surface 1 3 vertical surface 1 minimal surface for n Hs = minimal frame (a, b) around n point a = n b = n a

56 16 Bound on Layer Contact relation between surface and contacts 4n = H-contacts + H-surface number of H-neighbours contacts to Ps or solution positions relation to frame H-surface = a + b (a, b)= (height,width) horizontal surface 1 3 vertical surface 1 minimal surface for n Hs = minimal frame (a, b) around n point a = n b = n a

57 16 Bound on Layer Contact relation between surface and contacts 4n = H-contacts + H-surface number of H-neighbours contacts to Ps or solution positions relation to frame H-surface = a + b (a, b)= (height,width) horizontal surface 1 3 vertical surface 1 minimal surface for n Hs = minimal frame (a, b) around n point a = n b = n a

58 16 Bound on Layer Contact relation between surface and contacts 4n = H-contacts + H-surface number of H-neighbours contacts to Ps or solution positions relation to frame H-surface = a + b (a, b)= (height,width) horizontal surface 1 3 vertical surface 1 minimal surface for n Hs = minimal frame (a, b) around n point a = n b = n a

59 17 Recursion for Bound b b b1 a1 n1 n a n = b1 a1 n1 + + n a B (n,n1,a1,b1) C B LC (n1,a1,b1) B ILC ( n1,a1,b1 n,a,b ) 3 4 B C (n n1,n,a,b) B C (n, n 1, a 1, b 1 ): contacts in core with n elements and first layer E 1 : n 1, a 1, b 1 = B LC (n 1, a 1, b 1 ) contacts in layer E 1 + B ILC (n 1, a 1, b 1, n, a, b ) contacts between layers E 1 and E : n, a, b + B C (n n 1, n, a, b ) contacts in core with n n 1 elements and first layer E

60 18 Bound on Interlayer Contacts recall: we need an bound on interlayer contacts point 4 point 3 point x=1 x= n1=5 n=3 but: we are given only frames b1= b= a1=3 a= x=1 n1=5 x= n=3 bound number of 4, 3, and 1 points, given frames

61 19 Bound on Number of 4-, 3-, - and 1-Points problem: number of 4-, 3-, - and 1-points in x = i + 1 depends on exact position of Hs in x = i n i = 8 a i = height b i = width needed: parameters, which determine the number of 4-, 3-, - and 1-points Lemma let l be the number of 3-points. Then: number of 4 = n i + 1 a i b i number of = a i + b i l 4 number of 1 = l + 4. for l, there is an upper bound [Backofen00]

62 19 Bound on Number of 4-, 3-, - and 1-Points problem: number of 4-, 3-, - and 1-points in x = i + 1 depends on exact position of Hs in x = i n i = 8 a i = height b i = width needed: parameters, which determine the number of 4-, 3-, - and 1-points Lemma let l be the number of 3-points. Then: number of 4 = n i + 1 a i b i number of = a i + b i l 4 number of 1 = l + 4. for l, there is an upper bound [Backofen00]

63 19 Bound on Number of 4-, 3-, - and 1-Points problem: number of 4-, 3-, - and 1-points in x = i + 1 depends on exact position of Hs in x = i n i = 8 a i = height b i = width needed: parameters, which determine the number of 4-, 3-, - and 1-points Lemma let l be the number of 3-points. Then: number of 4 = n i + 1 a i b i number of = a i + b i l 4 number of 1 = l + 4. for l, there is an upper bound [Backofen00]

64 19 Bound on Number of 4-, 3-, - and 1-Points problem: number of 4-, 3-, - and 1-points in x = i + 1 depends on exact position of Hs in x = i n i = 8 a i = height b i = width needed: parameters, which determine the number of 4-, 3-, - and 1-points Lemma let l be the number of 3-points. Then: number of 4 = n i + 1 a i b i number of = a i + b i l 4 number of 1 = l + 4. for l, there is an upper bound [Backofen00]

65 19 Bound on Number of 4-, 3-, - and 1-Points problem: number of 4-, 3-, - and 1-points in x = i + 1 depends on exact position of Hs in x = i n i = 8 a i = height b i = width needed: parameters, which determine the number of 4-, 3-, - and 1-points Lemma let l be the number of 3-points. Then: number of 4 = n i + 1 a i b i number of = a i + b i l 4 number of 1 = l + 4. for l, there is an upper bound [Backofen00]

66 0 Bounds on the Number l of 3-Points 3-point: from the side from the top 3 point = 3 point observation: l can also be calculated from the frame i3= i4= i=3 i1=1 3 point (next laxer) bound: calculate max. number of diagonals optimal placement: balance numbers between edges

67 0 Bounds on the Number l of 3-Points 3-point: from the side from the top 3 point = 3 point observation: l can also be calculated from the frame i3= i4= i=3 i1=1 3 point (next laxer) bound: calculate max. number of diagonals optimal placement: balance numbers between edges

68 0 Bounds on the Number l of 3-Points 3-point: from the side from the top 3 point = 3 point observation: l can also be calculated from the frame i3= i4= i=3 i1=1 3 point (next laxer) bound: calculate max. number of diagonals optimal placement: balance numbers between edges

69 0 Bounds on the Number l of 3-Points 3-point: from the side from the top 3 point = 3 point observation: l can also be calculated from the frame i3= i4= i=3 i1=1 3 point (next laxer) bound: calculate max. number of diagonals optimal placement: balance numbers between edges

70 0 Bounds on the Number l of 3-Points 3-point: from the side from the top 3 point = 3 point observation: l can also be calculated from the frame i3= i4= i=3 i1=1 3 point (next laxer) bound: calculate max. number of diagonals optimal placement: balance numbers between edges

71 0 Bounds on the Number l of 3-Points 3-point: from the side from the top 3 point = 3 point observation: l can also be calculated from the frame i3= i4= i=3 i1=1 3 point (next laxer) bound: calculate max. number of diagonals optimal placement: balance numbers between edges

72 0 Bounds on the Number l of 3-Points 3-point: from the side from the top 3 point = 3 point observation: l can also be calculated from the frame i3= i4= i=3 i1=1 3 point (next laxer) bound: calculate max. number of diagonals optimal placement: balance numbers between edges

73 0 Bounds on the Number l of 3-Points 3-point: from the side from the top 3 point = 3 point observation: l can also be calculated from the frame i3= i4= i=3 i1=1 3 point (next laxer) bound: calculate max. number of diagonals optimal placement: balance numbers between edges

74 0 Bounds on the Number l of 3-Points 3-point: from the side from the top 3 point = 3 point observation: l can also be calculated from the frame i3= i4= i=3 i1=1 3 point (next laxer) bound: calculate max. number of diagonals optimal placement: balance numbers between edges

75 1 Problem : Enumerate Hydrophobic Cores =

76 Enumerating Hydrophobic Cores constraint variables: boolean variable for every position ˆ= (pnt( p) = 1) ˆ= (pnt( p) = 0) ˆ= (pnt( p) undef.) contact variable for each neighboring position ˆ= con( p, q) = 1 constraints: pnt( p) = number of Hs p frames if optimal, then no caveats...

77 3 Enumerating Hydrophobic Cores remaining problem: relative positions of frames subproblems: symmetries later many subproblems solved several times do not use fixed frame position global bind frame positions by surrounding cube more pruning: optimal core must have optimal frame-sequence in any direction constructive disjunction

78 3 Enumerating Hydrophobic Cores remaining problem: relative positions of frames subproblems: symmetries later many subproblems solved several times do not use fixed frame position global bind frame positions by surrounding cube more pruning: optimal core must have optimal frame-sequence in any direction constructive disjunction

79 3 Enumerating Hydrophobic Cores remaining problem: relative positions of frames subproblems: symmetries later many subproblems solved several times do not use fixed frame position global bind frame positions by surrounding cube more pruning: optimal core must have optimal frame-sequence in any direction constructive disjunction

80 3 Enumerating Hydrophobic Cores remaining problem: relative positions of frames subproblems: symmetries later many subproblems solved several times do not use fixed frame position global bind frame positions by surrounding cube more pruning: optimal core must have optimal frame-sequence in any direction constructive disjunction

81 3 Enumerating Hydrophobic Cores remaining problem: relative positions of frames subproblems: symmetries later many subproblems solved several times do not use fixed frame position global bind frame positions by surrounding cube more pruning: optimal core must have optimal frame-sequence in any direction constructive disjunction

82 3 Enumerating Hydrophobic Cores remaining problem: relative positions of frames subproblems: symmetries later many subproblems solved several times do not use fixed frame position global bind frame positions by surrounding cube more pruning: optimal core must have optimal frame-sequence in any direction constructive disjunction

83 4 Problem 3: Threading Sequence onto Hydrophobic Cores =

84 5 New Constraints for Threading threading: given core, find a sequence of monomer through it main problem: self-avoiding walks new constraint: SAWalk(x 1,..., x m ) walk self-avoiding walk (SAWalk) Pos. used twice problem: complete handling for SAWalk(x 1,..., x m ) is hard therefore: approximate SAWalks k-avoiding walks 4- but not 5-avoiding

85 5 New Constraints for Threading threading: given core, find a sequence of monomer through it main problem: self-avoiding walks new constraint: SAWalk(x 1,..., x m ) walk self-avoiding walk (SAWalk) Pos. used twice problem: complete handling for SAWalk(x 1,..., x m ) is hard therefore: approximate SAWalks k-avoiding walks 4- but not 5-avoiding

86 5 New Constraints for Threading threading: given core, find a sequence of monomer through it main problem: self-avoiding walks new constraint: SAWalk(x 1,..., x m ) walk self-avoiding walk (SAWalk) Pos. used twice problem: complete handling for SAWalk(x 1,..., x m ) is hard therefore: approximate SAWalks k-avoiding walks 4- but not 5-avoiding

87 5 New Constraints for Threading threading: given core, find a sequence of monomer through it main problem: self-avoiding walks new constraint: SAWalk(x 1,..., x m ) walk self-avoiding walk (SAWalk) Pos. used twice problem: complete handling for SAWalk(x 1,..., x m ) is hard therefore: approximate SAWalks k-avoiding walks 4- but not 5-avoiding

88 5 New Constraints for Threading threading: given core, find a sequence of monomer through it main problem: self-avoiding walks new constraint: SAWalk(x 1,..., x m ) walk self-avoiding walk (SAWalk) Pos. used twice problem: complete handling for SAWalk(x 1,..., x m ) is hard therefore: approximate SAWalks k-avoiding walks 4- but not 5-avoiding

89 6 Example: 3-Avoiding psinglets: HPH-subsequence in cubic lattice: has strong influence on core caveat caveat-freeness by path constraint remaining invalid case excluded by 3-avoidingness

90 6 Example: 3-Avoiding psinglets: HPH-subsequence in cubic lattice: has strong influence on core caveat caveat-freeness by path constraint remaining invalid case excluded by 3-avoidingness

91 6 Example: 3-Avoiding psinglets: HPH-subsequence in cubic lattice: has strong influence on core caveat caveat-freeness by path constraint remaining invalid case excluded by 3-avoidingness

92 6 Example: 3-Avoiding psinglets: HPH-subsequence in cubic lattice: has strong influence on core caveat caveat-freeness by path constraint remaining invalid case excluded by 3-avoidingness

93 6 Example: 3-Avoiding psinglets: HPH-subsequence in cubic lattice: has strong influence on core caveat caveat-freeness by path constraint remaining invalid case excluded by 3-avoidingness

94 7 Problem 4: Symmetry Breaking X 1 = 1 X 1 1 X 1 = 3 solved, but skipped here!

95 Comparison of Results small selection of previous approaches: authors model dim. maxlen algorithm comment [Yue& Dill PhysRevE93] cubic HP 3 36 branch-and-bound optimality proven [Yue&Dill PNAS95] cubic HP 3 88 branch-and-bound optimality proven [Sazhin et al. 01] cubic HP, FCC 3 34 branch-and-bound not always optimal [Cui et al. PNAS0] square HP 18 compl. enum [Hart&Istrail JCB97] FCC side chain 3 approximation 86% of optimum 3 [Agarwala et al. JMB97] FCC HP 3 approximation of optimum 5 our results: native conformation up to length proof of optimality number of conformations of length n: 4.5 n threading on 100-Hs core seq. length runtime S s S s S s S s search space handled bigger 8 only existing non-heuristic algorithm for FCC

96 Comparison of Results small selection of previous approaches: authors model dim. maxlen algorithm comment [Yue& Dill PhysRevE93] cubic HP 3 36 branch-and-bound optimality proven [Yue&Dill PNAS95] cubic HP 3 88 branch-and-bound optimality proven [Sazhin et al. 01] cubic HP, FCC 3 34 branch-and-bound not always optimal [Cui et al. PNAS0] square HP 18 compl. enum [Hart&Istrail JCB97] FCC side chain 3 approximation 86% of optimum 3 [Agarwala et al. JMB97] FCC HP 3 approximation of optimum 5 our results: native conformation up to length proof of optimality number of conformations of length n: 4.5 n threading on 100-Hs core seq. length runtime S s S s S s S s search space handled bigger 8 only existing non-heuristic algorithm for FCC

97 Comparison of Results small selection of previous approaches: authors model dim. maxlen algorithm comment [Yue& Dill PhysRevE93] cubic HP 3 36 branch-and-bound optimality proven [Yue&Dill PNAS95] cubic HP 3 88 branch-and-bound optimality proven [Sazhin et al. 01] cubic HP, FCC 3 34 branch-and-bound not always optimal [Cui et al. PNAS0] square HP 18 compl. enum [Hart&Istrail JCB97] FCC side chain 3 approximation 86% of optimum 3 [Agarwala et al. JMB97] FCC HP 3 approximation of optimum 5 our results: native conformation up to length proof of optimality number of conformations of length n: 4.5 n threading on 100-Hs core seq. length runtime S s S s S s S s search space handled bigger 8 only existing non-heuristic algorithm for FCC

98 Comparison of Results small selection of previous approaches: authors model dim. maxlen algorithm comment [Yue& Dill PhysRevE93] cubic HP 3 36 branch-and-bound optimality proven [Yue&Dill PNAS95] cubic HP 3 88 branch-and-bound optimality proven [Sazhin et al. 01] cubic HP, FCC 3 34 branch-and-bound not always optimal [Cui et al. PNAS0] square HP 18 compl. enum [Hart&Istrail JCB97] FCC side chain 3 approximation 86% of optimum 3 [Agarwala et al. JMB97] FCC HP 3 approximation of optimum 5 our results: native conformation up to length proof of optimality number of conformations of length n: 4.5 n threading on 100-Hs core seq. length runtime S s S s S s S s search space handled bigger 8 only existing non-heuristic algorithm for FCC

99 prediction of one optimal structure Runtimes (sequence length 48, Harvard sequences from [Yue et al., 1995]) Nr. sequence CPSP PERM 1 HPH P H 4 PH 3 P H P HPH 3 PHPH P H P 3 HP 8 H 0,1 s 6,9 min H 4 PH PH 5 P HP H P HP 6 HP HP 3 HP H P H 3 PH 0,1 s 40,5 min 3 PHPH PH 6 P HPHP HPH PHPHP 3 HP H P H P HPHP HP 4,5 s 100, min 4 P HP 3 HPH 4 P H 4 PH PH 3 P HPHPHP HP 6 H PH PH 1,8 s 74,7 min 5 H 3 P 3 H PHPH PH PH PHP 7 HPHP HP 3 HP H 6 PH 1,7 s 59, min 6 PHP 4 HPH 3 PHPH 4 PH PH P 3 HPHP 3 H 3 P H P H P 3 H 1,1 s 144,7 min 7 PHPH P HPH 3 P H PH P 3 H 5 P HPH PHPHP 4 HP HPHP 7,3 s 84,0 min 8 PH PH 3 PH 4 P H 3 P 6 HPH P H PHP 3 H PHPHPH P 3 1,5 s 6,6 min 9 PHPHP 4 HPHPHP HPH 6 P H 3 PHP HPH P HPH 3 P 4 H 0,3 s 140,0 min 10 PH P 6 H P 3 H 3 PHP HPH P HP HP H P H 7 P H 0,1 s 18,3 min 9 CPSP: our approach, constraint-based PERM [Bastolla et al., 1998]: stochastic optimization PERM=pruned-enriched Rosenbluth method

100 30 Applications structure prediction investigation of landscape properties degeneracy of sequences finding protein-likes sequences with unique ground state comparing different models (cubic/fcc, HP-model with HPNX)

101 31 Degeneracy degeneracy (g) of a sequence = number of structures with lowest energy known: HP-model has high degeneracy unknow: how high is it? are there sequences with g=1 (unique ground state, protein-like )? how does it compare to other models (FCC, HPNX)? how do neutral nets look like? degeneracy: can only be tested via two algorithms Sequence degeneracy found by CHCC [Yue et al] our approach HPHHPPHHHHPHHHPPHHPPHPHHHPHPHHPPHHPPPHPPPPPPPPHH 1, 500, , 677, 113 HHHHPHHPHHHHHPPHPPHHPPHPPPPPPHPPHPPPHPPHHPPHHHPH 14, 000 8, 180 PHPHHPHHHHHHPPHPHPPHPHHPHPHPPPHPPHHPPHHPPHPHPPHP 5, 000 5, 090 PHHPPPPPPHHPPPHHHPHPPHPHHPPHPPHPPHHPPHHHHHHHPPHH 188, , 751

102 31 Degeneracy degeneracy (g) of a sequence = number of structures with lowest energy known: HP-model has high degeneracy unknow: how high is it? are there sequences with g=1 (unique ground state, protein-like )? how does it compare to other models (FCC, HPNX)? how do neutral nets look like? degeneracy: can only be tested via two algorithms Sequence degeneracy found by CHCC [Yue et al] our approach HPHHPPHHHHPHHHPPHHPPHPHHHPHPHHPPHHPPPHPPPPPPPPHH 1, 500, , 677, 113 HHHHPHHPHHHHHPPHPPHHPPHPPPPPPHPPHPPPHPPHHPPHHHPH 14, 000 8, 180 PHPHHPHHHHHHPPHPHPPHPHHPHPHPPPHPPHHPPHHPPHPHPPHP 5, 000 5, 090 PHHPPPPPPHHPPPHHHPHPPHPHHPPHPPHPPHHPPHHHHHHHPPHH 188, , 751

103 Application: Design of protein-like Sequences find sequences with exactly one optimal structure stochastic local search node: accepted sequences edges: 8 7 simulation step/mutation Degeneracy Step 1

104 33 Run Time Requirements at every step: calculation/estimation of degeneracy (using our CPFL) but: runtime depends on degeneracy good news: runtime grows only linearly with degeneracy Time/s e+00 e+05 4e+05 6e+05 8e+05 1e+06 Degeneracy

105 Example: Sequences with Unique Ground-State length 64: length 80: 34 Note: previously it was assumed that HP-model has none g=1 sequences

106 Three Typical Runs a) b) c) a) 81 b) 19 c)

107 Degeneracy: FCC vs. Cubic log-degeneracy cubic HP-model: Frequency Log Degeneracy log-degeneracy FCC HP-model: Frequency Log Degeneracy

108 37 Degeneracy: HP-Model vs. HPNX-model HPNX: P=positive N=negative X=neutral should reduce the degeneracy How much? preliminary results HP: approx % of all random sequences are uniquely folding. HPNX: approx..6% of all random sequences are uniquely folding. Note: 50% H monomers example for reduction: sequence S HPNX: HXNNHHHHXHXHHNXNHXHHNHPPXHP corresp. HP: HPPPHHHHPHPHHPPPHPHHPHPPPHP

109 38 S HP-sequence: 4 out of 97

110 39 S HPNX-sequence: the 4 native ones

111 40 Connectivity of Neutral Nets S6 S S13 S4 S1 S1 S33 S36 S3 S3 S0 S31 S18 S5 S7 S8 S17 S9 S35 S4 S1 S7 S30 S9 S34 S S3 S5 S16 S6 S14 S15 S19 S8 S11 S10

112 41 WWW-Page

113 4 Conclusion constraint-based approach to protein folding guaranteed to find optima models: HP-like models: HP, HPNX lattices: cubic, FCC applications: properties of landscape degeneracy neutral nets folding tunnel

114 43 Acknowledgment Sebastian Will Erich Bornberg-Bauer Peter Stadler Michael Wolfinger