4. lecture ss 2006gk 11261 the development of methods that efficiently determine the global minima...

4. Lecture SS 2006

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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

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Unbiased methods vs. 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

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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

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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

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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)

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Simulated annealing (SA)


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.

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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.

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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

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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

t

tVtV

,

,2 XX

XX VV 0,

0,2 tV X

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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

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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 = t1) and a highly smoothed surface (t = tlarge) 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 = tlarge 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 = t1 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.

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Schematic of a more realistic potential smoothing protocol for molecular search problems. Shown is a crossing between the two surviving minima on the t = t2 surface. A reversing schedule encounters the first bifurcation at t = t2. 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 = t2 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 = t1 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.

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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


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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


2

1

2expi

iigaussLJ rbaUU

Fit ai and bi to match LJ 12-6 function with hard sphere radius and well depth of

one.

Here, a1 and b1 are positive to mimick repulsion at small distances,

a2 and b2 are negative and positive to mimick the attractive part of the LJ potential.

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Transform potential function U(r) to Ud(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

dtU

U


2

1

2

23 41exp

41,

i

ii

t

rb

t

atrU

where t is the deformation parameter that controls the extent of potential

smoothing. For the torsional potential term, the smoothened functional form

becomes:

jj tjjV 2expcos1

2

1

where is the torsional angle value, j is the periodicity, Vj is the half-amplitude, is

a phase factor and t is the deformation parameter.

Initial value of smoothing parameter t = 4.25.

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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


1,1,,1, and kikikiki VVVV

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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


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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


docking of NMR helices

docking of idealized helices:gives similar distribution using NMR helices does notbias result.

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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


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Genetic algorithms


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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)

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(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 < ft < 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:

01 fxfSTUN exf

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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)

XX EE min~

XE~

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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.

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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)

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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. 107 energy evalulations 10 ns of

standard MD simulation. Herges, Wenzel, Phys. Rev. Lett. 94, 018101 (2005)

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Good agreement with NMR


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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.


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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.


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.

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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

4. lecture ss 2006gk 11261 the development of methods that efficiently determine the global minima...

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