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4. Lecture SS 2006GK 11261 The development of methods that efficiently determine the global minima of complex and rugged energy landscapes remains a challenging.

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Presentation on theme: "4. Lecture SS 2006GK 11261 The development of methods that efficiently determine the global minima of complex and rugged energy landscapes remains a challenging."— Presentation transcript:

1 4. Lecture SS 2006GK 11261 The development of methods that efficiently determine the global minima of complex and rugged energy landscapes remains a challenging problem with applications in many scientific and technological areas. In particular, for NP-hard problems, stochastic methods offer an acceptable compromise between the reliability of the method and its computational cost, which scales only as a power law with the number of variables (for a fixed probability to locate the true minimum). In such techniques the global minimization is performed through the simulation of a dynamical process for a “particle” on the multidimensional potential energy surface. V4: Optimization methods for biomolecular structure modelling

2 4. Lecture SS 2006GK 11262 Unbiased methodsvs.Biased methods (not treated here) e.g. protein structure based on databases of known structures algorithms for clusters that favor compact geometries The most efficient method for any given problem is likely to be system dependent. V4: Global optimisation methods Chapter 6.7

3 4. Lecture SS 2006GK 11263 The effort involved in solving a given global optimisation problem is described by computational complexity theory. Locating the global minimum on a PES belongs to the class of NP-hard problems. However, this may be a worst case scenario. Certainly, there are PESs for crystals, naturally occurring proteins and many clusters that contain exponentially large number of minima. But locating the probable global minimum is not difficult experimentally and is sometimes easy in computer models as well. Looking at the disconnectivity graphs on the following slides suggests that finding the global minimum should be relatively easy if the system corresponds to a ‚palm tree‘, but relatively difficult if it corresponds to the ‚banyan tree‘. Complexity of global search

4 4. Lecture SS 2006GK 11264 Fig. 5.4. One-dimensional potential energy functions (left) and the corresponding disconnectivity graphs (right). The dotted lines indicate the energies at which the superbasin analysis was performed. (a) Low downhill barriers and a well-defined global minimum produced by relatively large systematic changes in potential energy between successive minima (‚palm tree‘); (b) higher downhill barriers (‚willow tree‘); (c) a landscape where the barrier heights are larger than the typical energy difference between successive minima (‚banyan tree‘). Disconnectivity graphs Taken from Wales book

5 4. Lecture SS 2006GK 11265 Lennard-Jones clusters are clusters of particles that interact via simple LJ interactions. Optimal test system for global optimization algorithms. Global minima of Lennard-Jones clusters Wales, Scheraga, Science 285, 1368 (1999)

6 4. Lecture SS 2006GK 11266 Simulated annealing (SA) Taken from Wales book SA offered perhaps the first generally applicable global optimisation algorithm. The system is equilibrated at a high temperature and then cooled, with the objective of maintaining equilibrium until the global free energy minimum and potential energy minimum coincide. A necessary and sufficient condition for a SA run to locate the global minimum is that the temperature decreases logarithmically with time. However, such a schedule is rather slow in practice. The problem with SA occurs when the global free energy minimum changes with temperature. Then, the system may become trapped behind a barrier that will rise compared to kT as the temperature is further decreased.

7 4. Lecture SS 2006GK 11267 Conformational space annealing (CSA) In the CSA approach attention is gradually focused onto smaller regions of conformation space as the search progresses. A set of conformations is continuously updated, as for genetic algorithms, starting from pre-assigned randomly generated and energy-minimized conformations. In each cycle a number of dissimilar conformations are selected from the current set, perturbed and energy minimised. For each of the resulting structures the closest conformation from the current set is identified and the two minima are considered „similar“ if they lie within a cutoff, which decreases with time. For two similar structures, the one with the lower energy is saved in the current set. New structures that are not similar to any of the current set can still replace the highest energy member of that set if they lie lower in energy.

8 4. Lecture SS 2006GK 11268 Hypersurface deformation methods attempt to simplify the global optimisation problem by applying a transformation to the PES that smoothes it and reduces the number of local minima. The global minimum is accordingly easier to find on the transformed surface, but it is then necessary to reverse the transformation and map the global minimum back to the real surface. Since the global minimum may change during the reverse mapping, depending on how the surface was smoothed, an efficient local search procedure must be applied during this process, and more than one minimum must be tracked backwards. Deformation methods

9 4. Lecture SS 2006GK 11269 In the DEM the true PES is used as the initial boundary condition for a multidimensional diffusion equation with the boundary condition As t  we have and for sufficiently large values of t the deformed surface has a single minimum. The diffusion equation

10 4. Lecture SS 2006GK 112610 Model packing of 2 TM helices for glycophorin A dimer. Smoothing: pairwise interactions between point atoms are altered to be interactions between spacially delocalized atoms. Assume that the thermodynamic ground state – the conformation of lowest free energy – is also the one of lowest potential energy. Success story: potential smoothing algorithm accurately predicts TM helix packing

11 4. Lecture SS 2006GK 112611 The diffusion equation Pappu, Marshall, Ponder, Nat. Struct. Biol. 6, 50 (1999) Fig. 1 One-dimensional schematic of the effect of a smoothing protocol on a potential energy surface. The original PES is transformed by successive application of a smoothing operator, where the extent of smoothing is dictated by a control parameter t. The undeformed original surface (t = 0), the surface at an intermediate level of smoothing (t = t 1 ) and a highly smoothed surface (t = t large ) are shown. As the surface is transformed, higher lying minima merge into catchment regions of low lying minima and barriers between minima are progressively lowered. Open circles are starting or intermediate points on each surface. Solid circles are local minima. Dashed arrows show the result of local optimization ending at a local minimum. Solid arrows represent adiabatic movement from a local minimum on one surface to the corresponding starting point on a rougher surface. A simple smoothing protocol consists of repeated cycles of local optimization followed by adiabatic transfer to the next surface. This figure shows an idealized smoothing protocol wherein the unique minimum that remains on the t = t large surface is directly related to the global minimum on the original PES. A series of optimizations followed by a gradual reduction in the level of smoothing will therefore lead back to the global minimum. Note that the reversing protocol depends on a set of discrete  t steps between surfaces. For small  t, vertical transfer to a less smooth surface will result in a point close to a transition state whenever a bifurcation has been introduced, as at t = t 1 in the figure. The simple protocol will only succeed if there is a consistent bias toward the minimum on the broader, deeper side of the bifurcation.

12 4. Lecture SS 2006GK 112612 The diffusion equation Pappu, Marshall, Ponder, Nat. Struct. Biol. 6, 50 (1999) Schematic of a more realistic potential smoothing protocol for molecular search problems. Shown is a crossing between the two surviving minima on the t = t 2 surface. A reversing schedule encounters the first bifurcation at t = t 2. At this level of smoothing the protocol favors basin B over basin A due to a crossing of relative energies, which is an artifact of the averaging process. The reversing protocol from Fig. 1 follows a path it chooses at the first bifurcation. If bifurcations are sampled where the relative energies of the alternative basins are inverted from the t = 0 surface, then the simple method will not converge to the global minimum. Between t = t 2 and t = 0 there exist values of t for which the energy ordering resembles that of the original PES. A local search process coupled to the smoothing schedule can potentially recognize errors due to earlier energy crossings. For example, a local search represented by the dotted arrow on the t = t 1 surface would correctly decide that basin A should be favored over basin B. Local searches are especially efficient when carried out on smoother surfaces since the extent of conformational space sampled is larger than for the original PES. If the global minimum is a very narrow and deep well on the PES then crossings can occur for very small values of the smoothing parameter t. For such problems, smoothing coupled to local search may fail to converge to locate the global minimum due to inadequate local sampling.

13 4. Lecture SS 2006GK 112613 Fig. 3 Schematic of a helix dimer illustrating the method used to compute helix packing parameters. H1 and H2 denote the helix axes for helix 1 and helix 2 respectively. P4 is a point along the line of contact connecting the two helices between points P3 and P5. P1 is the C  position of Phe 78 for helix 1, P2 is a point on the helix axis H1 and C1 is the location of the center of mass of helix 1. Similarly P7 is the C  position of Phe 78 for helix 2, P6 is a point on the helix axis H2, and C2 is the location of the center of mass for helix 2. The crossing angle W is the torsion angle defined by the points P2, P3, P5 and P6. The distance of closest contact d is the distance between the points P3 and P5. The angle a that measures the rotation of helix 1 about its axis H1 is the torsion angle defined by the points P1, P2, P3 and P4. Similarly, the angle b that measures the rotation of helix 2 about its axis H2 is the torsion angle defined by the points P7, P6, P5 and P4. The scalar shift parameter s is defined as |T1-T2|, where T1 is the distance between points P3 and C1 and T2 is the distance between points P5 and C2. Inter-helix degrees of freedom Pappu, Marshall, Ponder, Nat. Struct. Biol. 6, 50 (1999)

14 4. Lecture SS 2006GK 112614 Apply modified OPLS united atom force-field: replace each of the pairwise LJ 12-6 van der Waals terms by a sum of two Gaussians: Force field Pappu, Marshall, Ponder, Nat. Struct. Biol. 6, 50 (1999) Fit a i and b i to match LJ 12-6 function with hard sphere radius  and well depth of one. Here, a 1 and b 1 are positive to mimick repulsion at small distances, a 2 and b 2 are negative and positive to mimick the attractive part of the LJ potential.

15 4. Lecture SS 2006GK 112615 Transform potential function U(r) to U d (r,t) such that where  is a multidimensional diffusion operator. The deformed Gaussian van der Waals function that is an analytical solution to the diffusion equation is of the form: Force field Pappu, Marshall, Ponder, Nat. Struct. Biol. 6, 50 (1999) where t is the deformation parameter that controls the extent of potential smoothing. For the torsional potential term, the smoothened functional form becomes: where  is the torsional angle value, j is the periodicity, V j is the half-amplitude,  is a phase factor and t is the deformation parameter. Initial value of smoothing parameter t = 4.25.

16 4. Lecture SS 2006GK 112616 At some chosen value of t along the reveral protocol the level of smoothing is reduced followed by local minimization. The system is moved out of this local minimum along a set of search directions corresponding to the eigenvectors of the Hessian matrix at the current local minimum. The system is moved along a search direction i and the conformational energy is computed at each equidistant point k along the search direction. If the energy at point k satistifies it is chosed to be a new point from which to start a minimization. Potential smoothing and search (PSS) algorithm Pappu, Marshall, Ponder, Nat. Struct. Biol. 6, 50 (1999)

17 4. Lecture SS 2006GK 112617 This pair of conditions suggests apparent downhill progress on a PES. If the energy of the alternate minimum is lower than the energy of the original minimum, the system is moved to the alternate location and the search process is iterated until no new minima of lower energy can be found at the current level of smoothing. Start with t = 4.25. Local searches along Hessian eigenvector directions are performed for all values of t < 4.0 during the reversing schedule. A typical reversing schedule includes 50 – 100 values of t between 4.25 and 0. During search, alternate low energy minima are found on the t =1,8, 0.54, 0.23 and 0.18 surfaces. Search converges to the same minima irrespective of starting conformations. Potential smoothing and search (PSS) algorithm Pappu, Marshall, Ponder, Nat. Struct. Biol. 6, 50 (1999)

18 4. Lecture SS 2006GK 112618 Distribution of interhelical energies for the 5,834 local minima found from a two-body grid search for helices from the consensus NMR structure for the GpA helix dimer. The global minimum on this grid has an energy of -29.56 kcal mol –1, and is also located by PSS. The low energy conformers obtained from this grid search show dissimilarities from the NMR structure akin to the results for the idealized helices described in Table 1. b, Distribution of interhelical energies for the 4,105 local minima found from a two-body grid search for idealized helices built using ( ,  ) angles for a canonical  -helix and  -angles from a rotamer library. The global minimum on this grid has an energy of -31.84 kcal mol –1. This is also the structure found using the PSS algorithm. Interhelical energies during a grid search Pappu, Marshall, Ponder, Nat. Struct. Biol. 6, 50 (1999) docking of NMR helices docking of idealized helices: gives similar distribution  using NMR helices does not bias result.

19 4. Lecture SS 2006GK 112619 Ribbon drawing derived from the TM helix portion of the experimental NMR structure (PDB file 1AFO, Model 13). b, The corresponding helix backbone and side chains from the global minimum determined by the PSS algorithm. Regions involved in interhelical packing are very similar in the two structures, and the RMSD for superposition over all Ca atoms is 0.59 Å. Comparison of modeled structure with NMR structure Pappu, Marshall, Ponder, Nat. Struct. Biol. 6, 50 (1999)

20 4. Lecture SS 2006GK 112620 Genetic algorithms Taken from Wales book

21 4. Lecture SS 2006GK 112621 The freezing problem in stochastic minimization methods arises when the energy difference between “adjacent” local minima on the PES is much smaller than the energy of intervening transition states separating them. As an example consider the dynamics on the model potential below. At high temperatures a particle can still cross the barriers, but not differentiate between the wells. As the temperature drops, the particle will eventually become trapped with almost equal probability in any of the wells, failing to resolve the energy difference between them. The physical idea behind the stochastic tunneling method (STUN) is to allow the particle to “tunnel” forbidden regions of the PES, once it has been determined that they are irrelevant for the low-energy properties of the problem. Stochastic tunnelling method Wenzel, Hamacher, Phys. Rev. Lett. 82, 3003 (1999)

22 4. Lecture SS 2006GK 112622 (a) Schematic one-dimensional PES and (b) STUN effective potential, where the minimum indicated by the arrow is the best minimum found so far. All wells that lie above the best minimum found are suppressed. If the dynamical process can escape the well around the current ground-state estimate, it will not be trapped by local minima that are higher in energy. Wells with deeper minima are preserved and enhanced. (c) After the next minimum has been located, wells that were still pronounced in (b) are also suppressed. Once the true ground state has been found (not shown), all other wells have been suppressed and will no longer trap the dynamical process. The dotted line in (c) illustrates an energy threshold 0 < f t < 1 to classify the nature of the dynamics. Adjusting the temperature to maintain a particular average effective energy balances the tunneling and the local-search phases of the algorithm. Stochastic tunnelling method Wenzel, Hamacher, Phys. Rev. Lett. 82, 3003 (1999) Apply nonlinear transformation to PES:

23 4. Lecture SS 2006GK 112623 The following transformation of the energy landscape does not change the relative energies of any minima: where X represents the 3D vector of nuclear coordinates and „min“ signifies that an energy minimization is carried out starting from X. The transformed energy at any point, X, becomes the energy of the structure obtained by minimization. Each local minimum is, therefore, surrounded by a catchment basin of constant energy consisting of all the neighboring geometries from which that particular minimum is obtained. The overall energy landscape becomes a set of plateaus, one for each catchment basin, but the energies of the local minima are unaffected by the transformation. Basin-hopping Wales, Scheraga, Science 285, 1368 (1999)

24 4. Lecture SS 2006GK 112624 Basin-hopping Wales, Doye, J Phys Chem A 101, 5111 (1997) Aside from removing all the transition state regions from the surface, the catchment basin transformation also accelerates the dynamics, because the system can psss between basins all along their boundaries. Atoms can even pass through each other without encountering prohibitive energy barriers.

25 4. Lecture SS 2006GK 112625 Success story Ab initio folding of 40-residue headpiece of HIV accessory protein 1F4I-40. Use all-atom force field: atomically resolved electrostatic model, group specific dielectric constants, LJ parametrization adapted to exp. distances in X-ray structures. SASA solvation model fitted to exp. solvation free energies of Gly-X-Gly peptides. Only degrees of freedom that are optimized: rotations of dihedral angles of the backbone and rotations of dihedral angles the side chains Herges, Wenzel, Phys. Rev. Lett. 94, 018101 (2005)

26 4. Lecture SS 2006GK 112626 Optimization strategy: adapted basin-hopping technique Replace single minimization step by SA run with self-adapting cooling cycle and length. At the end of one annealing step, accept new conformation if its energy difference to the current configuration was no higher than a given threshold  T. Start each simulation at nonclashing conformation with random backbone dihedral angles. 3 steps: (a) high-temperature bracket of 800/300 K,  T = 15 K, reduced solvent effect, (b) start from final configuration of (a), same SA, full solvent effect (c) low-temperature bracket of 600/3 K,  T = 1K. Within each annealing run, the temperature is geometrically decreased. The number of steps per annealing run is gradually increased to ensure better convergence. Total effort: each full simulation requies ca. 10 7 energy evalulations  10 ns of standard MD simulation. Herges, Wenzel, Phys. Rev. Lett. 94, 018101 (2005)

27 4. Lecture SS 2006GK 112627 Good agreement with NMR Herges, Wenzel, Phys. Rev. Lett. 94, 018101 (2005)

28 4. Lecture SS 2006GK 112628 Comparison of C  – C  distances Dark diagonal block: intrahelical contacts well resolved (not surprising). Off-diagonal dark blocks indicate that also a large fraction of long-range native contacts is reproduced correctly. Herges, Wenzel, Phys. Rev. Lett. 94, 018101 (2005)

29 4. Lecture SS 2006GK 112629 Classification of decoy structures 60.000 low-energy conformations (decoys) were generated. Group decoys with RMSD < 3.0 Å into families with free-energy brackets of 2 kcal/mol. Construct decoy tree. As soon as two decoys in different branches have an RMSD of < 3.0 Å, the 2 families are merged into one superfamily. Herges, Wenzel, Phys. Rev. Lett. 94, 018101 (2005) Trees with very short stems and many low-energy branches are characteristic of glassy PES = Levinthal paradoxon. Trees with well structured trees with few terminal branches suggest existence of a folding funnel. This tree here is consistent with a broad folding funnel.

30 4. Lecture SS 2006GK 112630 Global optimisation is a great challenge. Efficient schemes exist that may each be most advantageous in a particular class of problems. - Genetic algorithms - Deformation of PES – diffusion equation method, stochastic tunneling, basin hopping Ab initio protein structure prediction of small domains up to 100 amino acids is about to be mastered for „easy“ cases. Remains to be seen if one can distinguish „easy“ and „hard“ cases. Summary

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