Protein Docking by Exploiting Multi-dimensional Energy Funnels

Transcription

1 Protein Docking by Exploiting Multi-dimensional Energy Funnels Ioannis Ch. Paschalidis 1 Yang Shen 1 Pirooz Vakili 1 Sandor Vajda 2 1 Center for Information and Systems Engineering and Department of Manufacturing Engineering 2 Department of Biomedical Engineering Boston Univeristy IEEE 2006 International Conference of EMBS Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

2 Outline 1 Background Protein Docking Problem Computational Approach 2 Our Work SDU: Semi-Definite Underestimator method Exploiting Energy Funnels for Conformational Search Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

3 Outline 1 Background Protein Docking Problem Computational Approach 2 Our Work SDU: Semi-Definite Underestimator method Exploiting Energy Funnels for Conformational Search Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

4 Protein Buidling block of life Linear chain of amino acids in sequence. Unique 3D structure. Biological function. figures by D. Leja, NHGRI and Alberts et.al, Garland Science Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

5 Protein Docking Protein-involved molecular interactions mediate many important biological processes. Targets of many drugs are to enhance or suppress protein functions. Protein docking seeks to computationally determine how proteins interact, given independent component structures. Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

6 Protein Docking Protein-involved molecular interactions mediate many important biological processes. Targets of many drugs are to enhance or suppress protein functions. Protein docking seeks to computationally determine how proteins interact, given independent component structures. Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

7 Outline 1 Background Protein Docking Problem Computational Approach 2 Our Work SDU: Semi-Definite Underestimator method Exploiting Energy Funnels for Conformational Search Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

8 Free Energy Minimization Energy minimum principle The native protein complex has the lowest free energy. Free energy model G = G s + E int + E vdw G s = E elec + G des T S sc + G other Characteristics Multi-frequency terms Funnel-like shape C. J. Camcacho and S. Vajda, PNAS, 2001 Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

9 Current Methods Monte Carlo minimization based approaches Low resolution search High resolution refinement Multi-stage approaches Rigid grid-based global search by FFT Filtering and clustering Refinement D. Baker et. al, JMB, 2003; D. Kozakov, K. H. Clodfelter, S. Vajda and C. J. Camacho, Biophysical Journal, 2005 Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

10 Outline 1 Background Protein Docking Problem Computational Approach 2 Our Work SDU: Semi-Definite Underestimator method Exploiting Energy Funnels for Conformational Search Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

11 Constructing Semi-Definite Underestimator Mathematical formulation minimize K j=1 (f (φj ) U(φ j ) subject to f (φ j ) U(φ j ), j = 1,..., K, U(φ) = φ Qφ + b φ + c Q 0 U max g ( x ) (scaled) f ( x ) U ( x ) Theorem Formulation above can be reformulated as a Semi-Definite Programming (SDP) problem and solved in polynomial time. Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

12 Biased Sampling Probability density function (p.d.f) g(φ) = U(φ) U max B (U(φ) U max) dφ = U(φ) U max, φ B. A g ( x ) (scaled) U max f ( x ) U ( x ) Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

13 SDU Algorithm 1 Initialization: Obtain the set of K random local minima. 2 Underestimation: Solve SDP to obtain the underestimator U(φ) and predictive conformation φ P. 3 Elimination: Discard unfavorable local minima. 4 Focalization: Shrink search region B to the neighborhood of φ P. 5 Exploration: Add local minimum starting from φ P. Add m local minima starting from biased samples. 6 Termination: If convergence condition is statisfied, then stop; otherwise go to Step 2. Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

14 Convergence Theorem The predictive conformation φ P converges to the global minimum φ with probability 1 (w.p.1) as sample size K goes to infinity. lim K P[φP = φ ] = 1. I. Ch. Paschalidis, Y. Shen, S. Vajda and P. Vakili, IEEE Transactions on Automatic Control, in print Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

15 Outline 1 Background Protein Docking Problem Computational Approach 2 Our Work SDU: Semi-Definite Underestimator method Exploiting Energy Funnels for Conformational Search Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

16 Exploring Energy Funnels in Conformational Space Motivation Protein binding energy funnels were computationally explored in 1D dissimilarity variables instead of conformational space. Reduced Gō-type potential Residue-level Bound structures with C β side-chain representations Native contacts set N f (x) = (i,j) N (d ij(x) d u ) 1{d ij (x) > d u } + a (i,j) N E ij 1{d l d ij (x) d u } + b (i,j) 1{d ij(x) < d l } Conformational space SE(3) = R 3 SO(3) R 3 : Translational subspace (Euclidean) SO(3): Rotational group (manifold) Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

17 Exploring Energy Funnels in Conformational Space Motivation Protein binding energy funnels were computationally explored in 1D dissimilarity variables instead of conformational space. Reduced Gō-type potential Residue-level Bound structures with C β side-chain representations Native contacts set N f (x) = (i,j) N (d ij(x) d u ) 1{d ij (x) > d u } + a (i,j) N E ij 1{d l d ij (x) d u } + b (i,j) 1{d ij(x) < d l } Conformational space SE(3) = R 3 SO(3) R 3 : Translational subspace (Euclidean) SO(3): Rotational group (manifold) Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

18 Exploring Energy Funnels in Conformational Space Motivation Protein binding energy funnels were computationally explored in 1D dissimilarity variables instead of conformational space. Reduced Gō-type potential Residue-level Bound structures with C β side-chain representations Native contacts set N f (x) = (i,j) N (d ij(x) d u ) 1{d ij (x) > d u } + a (i,j) N E ij 1{d l d ij (x) d u } + b (i,j) 1{d ij(x) < d l } Conformational space SE(3) = R 3 SO(3) R 3 : Translational subspace (Euclidean) SO(3): Rotational group (manifold) Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

19 Energy Funnels in SE(3) Translational funnels Displacement coordinates r Rotational funnels Exponential parametrization ω R 3 R = R 0 e Ω where R rotation matrix R 0 initial rotation matrix Ω skew-symmetric matrix formed by ω Euclidean distance ln(r 1 R 1 2 ) = ln(eω 1 Ω 2 ) = ω 1 ω 2 ( ) Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

20 Energy Funnels in SE(3) Translational funnels Displacement coordinates r Rotational funnels Exponential parametrization ω R 3 R = R 0 e Ω where R rotation matrix R 0 initial rotation matrix Ω skew-symmetric matrix formed by ω Euclidean distance ln(r 1 R 1 2 ) = ln(eω 1 Ω 2 ) = ω 1 ω 2 ( ) Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

21 Search Strategy and Settings Computational search Iterative orientational and translational adjustment with SDU Test settings 10 bound protein complexes of different types Start from Å away Average translational displacement 10 Å Average orientational displacement 0.8 radian Initial search region doesn t include the native structure 12 Å cube in the translational subspace 0.3 radian cube in the orientational subspace Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

22 Comparison with Monte Carlo Measures η: success rate (RMSD 3Å) K : required number of energy evaluations to guarantee a success with a probability of 95% (K = max{log (1 η) (1 P), 1}) Algorithm MC SDU Measure η(%) K (K) η(%) K (K) Enzyme-Inhibitor Complexes 1AVW BRC CSE KAI PTC Antigen-Antibody or Other Complexes 1A0O AHW AVZ MLC JEL Average Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

23 Comparison with Monte Carlo Measures η: success rate (RMSD 3Å) K : required number of energy evaluations to guarantee a success with a probability of 95% (K = max{log (1 η) (1 P), 1}) Algorithm MC SDU Measure η(%) K (K) η(%) K (K) Enzyme-Inhibitor Complexes 1AVW BRC CSE KAI PTC Antigen-Antibody or Other Complexes 1A0O AHW AVZ MLC JEL Average Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

24 Sample Trajectory Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

25 Sample Trajectory Click for the movie* *This page is the web version of the talk. For the movie-embedded talk please download the compressed file at Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

26 Applicability in Real World Binding energy model E = E ACP desolvation + E elec electrostaics + E vdw van der Waals Molecular system settings Atom level resolution Unbound structures with flexible side-chains Test settings Benchmark set of 57 protein complexes Start from cluster centers of initial stage Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22

27 Summary A novel global optimization method (SDU) tailored to funnel-like functions was introduced. Reduced potential helped understand and exploit the translational and orienntational energy funnels of protein binding energy. To achieve the same level of accuracy, our method reduced energy evaluations by more than a factor of two on average, compared to MC. Future Directions More accurate energy models. Side-chain packing. Paschalidis, Shen, Vakili, Vajda (BU) Protein Docking EMBC / 22