Computational complexity of a problem in molecular structure prediction

Ngo, Jacqueline; Marks, Joe

doi:10.1093/protein/5.4.313

Cited by 92 publications

(44 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For very well designed proteins that can be made to stably fold well below T F but above T G , the asymptotic scaling of folding time is polynomial in chain length. There is no contradiction with well known NP completeness theorems (34) about predicting global minima of a random heteropolymer because well designed proteins are selected, minimally frustrated systems. If still above T G but near T F , the scaling for the time grows exponentially in size but the barrier rises sublinearly as N 2/3 in Finkelstein's (21) calculation.…”

mentioning

confidence: 99%

Folding funnels and energy landscapes of larger proteins within the capillarity approximation

Wolynes

1997

Proc. Natl. Acad. Sci. U.S.A.

208

163

View full text Add to dashboard Cite

The characterization of protein-folding kinetics with increasing chain length under various thermodynamic conditions is addressed using the capillarity picture in which distinct spatial regions of the protein are imagined to be folded or trapped and separated by interfaces. The quantitative capillarity theory is based on the nucleation theory of first-order transitions and the droplet analysis of glasses and random magnets. The concepts of folding funnels and rugged energy landscapes are shown to be applicable in the large size limit just as for smaller proteins. An ideal asymptotic freeenergy profile as a function of a reaction coordinate measuring progress down the funnel is shown to be quite broad. This renders traditional transition state theory generally inapplicable but allows a diffusive picture with a transition-state region to be used. The analysis unifies several scaling arguments proposed earlier. The importance of f luctuational fine structure both to the free-energy profile and to the glassy dynamics is highlighted. The f luctuation effects lead to a very broad trapping-time distribution. Considerations necessary for understanding the crossover between the mean field and capillarity pictures of the energy landscapes are discussed. A variety of mechanisms that may roughen the interfaces and may lead to a complex structure of the transition-state ensemble are proposed.Proteins can be thought of as mesoscopic systems; that is, they are large in atomistic terms but are too small to be completely analyzed by macroscopic reasoning alone. The energy landscape theory of protein folding describes folding kinetics through a statistical characterization of the energies of different conformations (1, 2). This makes a bridge with the theory of phase transitions in disordered systems. The theory suggests that rapidly foldable proteins have an energy landscape dominated by a funnel to a large basin of attraction, the native state, and many small rugged features that give rise to trapping in local minima (3-5). Reliable folding requires the guiding forces of the funnel to be strong enough to overcome the entropy of the unfolded states. This must occur at a temperature such that trapping (also favored at low temperature) is not so strong that flow down the funnel is too slow. Much of the qualitative content of this picture has been confirmed through the computer simulation of small lattice models of proteins (5-8). Quantitatively, the analytical theories of folding landscapes have often made use of mean field descriptions that should be most accurate when the protein is small enough that we can think of each rearranging subunit of the chain as being able to interact with a fair fraction of the others at any time (1, 2, 9-13). This is true of the smallest lattice models. If the rearranging units are appropriately taken to be small segments of helices rather than individual residues, it is also true for the smallest globular proteins in nature, which have about 70 residues (4). This paper discusses how the fol...

show abstract

mentioning

confidence: 99%

Folding funnels and energy landscapes of larger proteins within the capillarity approximation

Wolynes

1997

Proc. Natl. Acad. Sci. U.S.A.

208

163

View full text Add to dashboard Cite

show abstract

“…It is therefore desirable to determine the global minimum of the potential function without recourse to the folding dynamics. It has been argued that the resulting minimization problem is NP-hard [12][13][14], i.e. that the number of low-energy local minima grows exponentially with the number of amino acid residues.…”

mentioning

confidence: 99%

Scaling behavior of stochastic minimization algorithms in a perfect funnel landscape

Hamacher

Wenzel

1999

Phys. Rev. E

View full text Add to dashboard Cite

We determined scaling laws for the numerical effort to find the optimal configurations of a simple model potential energy surface (PES) with a perfect funnel structure that reflects key characteristics of the protein interactions. Generalized Monte-Carlo methods(MCM, STUN) avoid an enumerative search of the PES and thus provide a natural resolution of the Levinthal paradox. We find that the computational effort grows with approximately the eighth power of the system size for MCM and STUN, while a genetic algorithm was found to scale exponentially. The scaling behaviour of a derived lattice model is also rationalized.Despite recent successes in the description of the molecular structure [1,2] and the folding process of small polypeptides [3,8] the ab-initio prediction of the molecular structure for larger proteins remains an elusive goal. Since sequencing techniques presently outperform available experimental techniques for protein structure prediction (PSP) by a wide margin, the reservoir of sequenced proteins of unknown structure represents an ever growing pool of available, but as of yet inaccessible, biological information. These observations motivate the search for ab-initio techniques to predict the molecular structure of proteins from the amino acid sequence alone as one of the outstanding challenges to biological physics.In one widely pursued theoretical approach to PSP, the native structure of the protein is sought as the global minimum of an appropriate potential/free energy-function of the molecule [2,9-11] often including interactions with the solvent in an approximate, implicit fashion. As the folding process in nature takes place on a long time scale (10 −3 − 10 s), its direct simulation cannot be accomplished with the presently available computational resources. It is therefore desirable to determine the global minimum of the potential function without recourse to the folding dynamics. It has been argued that the resulting minimization problem is NP-hard [12-14], i.e. that the number of low-energy local minima grows exponentially with the number of amino acid residues. For this reason stochastic minimization procedures [15] are widely believed to be the most promising avenue to avoid an exponential increase of the numerical effort for the probabilistic "solution" to this problem. Since the available computational resources fall short by orders of magnitude to treat large proteins, it is important to obtain an order-of-magnitude estimation of the numerical effort required. This question can be answered by addressing the scaling laws [16,17]:governing the dependence of the computational effort (n CPU ) on the system size (N ). In this investigation we determined the scaling exponents for four different global minimization methods, for a very simple, idealized a model that reflects some key characteristics of the realistic problem. Our results demonstrate that the Levinthal paradox [18][19][20], which arises from the enourmous number of low-lying conformations of the protein, is naturally resolved in ...

show abstract

“…Moreover, a substantial number of these 8-mer chains would be so heavily dominated by one or other monomer type as to be implausible representations of natural proteins and unlikely to respond to the chaperone-based refolding method, relying as it does on the promotion of an energetically favorable segregation of hydrophobic and hydrophilic monomer types. It was pointed out in the work with the BLN 22-mers 38 that since the "protein folding problem" is in its most general form believed to be NPhard, 48,49 it would be unrealistic to expect any given minimization method to reliably locate nativelike structures for all instances of a heteropolymer model, and so it is reasonable in the present context to concentrate on developing a procedure that handles well those AB model chains that have a more proteinlike ratio of hydrophobic to hydrophilic monomers [real (globular) proteins when compared with the binary-state AB model may be observed to contain around 50 -60% "hydrophilic" residues, 50 the exact proportion depending on how the hydrophobic/hydrophilic residue assignment is made]. The subsets of the full 136-chain 8-mer set chosen here for initial development of the model (parameter optimization) are those with 3, 4, or 5 B-monomers, a total of 94 chains (Table Ia).…”

Section: Choice Of the Ab Model Data Setsmentioning

confidence: 99%

Application of a chaperone‐based refolding method to two‐ and three‐dimensional off‐lattice protein models

Gorse

2002

Biopolymers

View full text Add to dashboard Cite

A model of protein-chaperone interaction as a two-phase (unfolding/refolding) iterative annealing mechanism able to promote structural segregation of hydrophobic and hydrophilic monomers and thereby facilitate access to nativelike states has recently been applied successfully to two 22-mers of the Honeycutt and Thirumalai BLN (hydrophobic, hydrophilic, neutral) heteropolymer model. This technique is here applied to a much wider data set: 94 8-mers of the off-lattice protein model originally presented in two dimensions by Stillinger and Head-Gordon, and later extended into three dimensions by Irbäck and Potthast; the model chaperone is shown to be equally successful, and by progressive elaboration of the chaperone model as in the earlier BLN model work, to be utilizing very similar underlying mechanisms. It is demonstrated that on average, contacts with the model chaperone give rise to a consistent movement in structure space in the direction of more nativelike structures; this method of global minimization does not therefore rely fundamentally on random search. Insofar as the responses to the chaperone of the two- and three-dimensional forms of the substrate model do differ, this can be interpreted as reflecting the different handling of hydrophilic monomers in the models-in particular, whether there is active repulsion between these and monomers of hydrophobic character. The chaperone-induced refolding method is also tested on a set of 220 9-mer chains of each version of the substrate model, where it is seen that the two-dimensional model, with its more clearly distinguished roles for the hydrophobic and hydrophilic monomers, shows a more favorable scaling behavior.

show abstract

Computational complexity of a problem in molecular structure prediction

Cited by 92 publications

References 9 publications

Folding funnels and energy landscapes of larger proteins within the capillarity approximation

Folding funnels and energy landscapes of larger proteins within the capillarity approximation

Scaling behavior of stochastic minimization algorithms in a perfect funnel landscape

Application of a chaperone‐based refolding method to two‐ and three‐dimensional off‐lattice protein models

Contact Info

Product

Resources

About