Learning To Fold Proteins Using Energy Landscape Theory.

This review is a tutorial for scientists interested in the problem of protein structure prediction, particularly those interested in using coarse-grained molecular dynamics models that are optimized using lessons learned from the energy landscape theory of protein folding. We also present a review of the results of the AMH/AMC/AMW/AWSEM family of coarse-grained molecular dynamics protein folding models to illustrate the points covered in the first part of the article. Accurate coarse-grained structure prediction models can be used to investigate a wide range of conceptual and mechanistic issues outside of protein structure prediction; specifically, the paper concludes by reviewing how AWSEM has in recent years been able to elucidate questions related to the unusual kinetic behavior of artificially designed proteins, multidomain protein misfolding, and the initial stages of protein aggregation.

Confinement effects on the kinetics and thermodynamics of protein dimerization.

In the cell, protein complexes form by relying on specific interactions between their monomers. Excluded volume effects due to molecular crowding would lead to correlations between molecules even without specific interactions. What is the interplay of these effects in the crowded cellular environment? We study dimerization of a model homodimer when the mondimers are free and when they are tethered to each other. We consider a structured environment: Two monomers first diffuse into a cavity of size L and then fold and bind within the cavity. The folding and binding are simulated by using molecular dynamics based on a simplified topology based model. The confinement in the cell is described by an effective molecular concentration C approximately L(-3). A two-state coupled folding and binding behavior is found. We show the maximal rate of dimerization occurred at an effective molecular concentration C(op) approximately = 1 mM, which is a relevant cellular concentration. In contrast, for tethered chains the rate keeps at a plateau when C < C(op) but then decreases sharply when C > C(op). For both the free and tethered cases, the simulated variation of the rate of dimerization and thermodynamic stability with effective molecular concentration agrees well with experimental observations. In addition, a theoretical argument for the effects of confinement on dimerization is also made.

Electron tunneling paths in proteins.

One of the crucial issues in biological electron transfer is the determination of the role of spatially intermediate amino acid residues in controlling or directing the electronic tunneling interaction between redox sites. A quantum path integral Monte Carlo method is developed for the analysis of electronic tunneling pathways in a highly structured environment. This path integral method is applied to intramolecular electron transfer in a ruthenium-modified myoglobin that contains a tryptophan in the "line-of-flight." A principal result is the identification of the relevant cylindrical zone swept out by the tunneling electron.

Rate theories and puzzles of hemeprotein kinetics.

The binding of dioxygen and carbon monoxide to heme proteins such as myoglobin and hemoglobin has been studied with flash photolysis. At temperatures below 200 K, binding occurs from within the heme pocket and, contrary to expectation, with nearly equal rates for both ligands. This observation has led to a reexamination of the theory of the association reaction taking into account friction, protein structure, and the nature of electronic transitions. The rate coefficients for the limiting cases of large and small friction are found with simple arguments that use characteristic lengths and times. The arguments indicate how transition state theory as well as calculations based on nonadiabatic perturbation theory, which is called the Golden Rule, may fail. For ligand-binding reactions the data suggest the existence of intermediate states not directly observed so far. The general considerations may also apply to other biomolecular processes such as electron transport.

The energy landscapes and motions of proteins.

Recent experiments, advances in theory, and analogies to other complex systems such as glasses and spin glasses yield insight into protein dynamics. The basis of the understanding is the observation that the energy landscape is complex: Proteins can assume a large number of nearly isoenergetic conformations (conformational substates). The concepts that emerge from studies of the conformational substates and the motions between them permit a quantitative discussion of one simple reaction, the binding of small ligands such as carbon monoxide to myoglobin.

Toward protein tertiary structure recognition by means of associative memory hamiltonians.

The statistical mechanics of associative memories and spin glasses suggests ways to design Hamiltonians for protein folding. An associative memory Hamiltonian based on hydrophobicity patterns is shown to have a large capacity for recall and to be capable of recognizing tertiary structure for moderately variant sequences.

Solvophobically driven folding of nonbiological oligomers.

In solution, biopolymers commonly fold into well-defined three-dimensional structures, but only recently has analogous behavior been explored in synthetic chain molecules. An aromatic hydrocarbon backbone is described that spontaneously acquires a stable helical conformation having a large cavity. The chain does not form intramolecular hydrogen bonds, and solvophobic interactions drive the folding transition, which is sensitive to chain length, solvent quality, and temperature.

Exploring structures in protein folding funnels with free energy functionals: the denatured ensemble.

We discuss the formulation of free energy functionals that describe the formation of structure in partially folded proteins. These free energy functionals take into account the inhomogeneous nature of contact energies, chain entropy and cooperative contributions reflecting the many body character of some folding forces like hydrophobicity, but do not directly account for non-native contacts because they assume the validity of the minimal frustration principle. We show how the free energy functionals can be used to interpret experiments on partially folded proteins that probe the fractional occupancy of specific local structures. In particular, we study the hydrogen protection factors in lysozyme studied in transient experiments by Gladwin and Evans and by Nash and Jonas using equilibrium pressure denaturation and the NMR order parameters measured by Dobson and Kim for the homologous protein alpha-lactalbumin.

Local conformational signals and the statistical thermodynamics of collapsed helical proteins.

We investigate the role that local conformational tendencies can have in guiding the folding of helical proteins, using simple statistical mechanical models. The theory provides a synthesis of classical models of the helix-coil transition in polymers with an approximate treatment of the effects of excluded volume, confinement, and packing alignment of the helices based on a free energy function. The theory studies the consequences of signals encoded locally in the sequence as stabilization energies associated with three types of local structure: native helical conformations, native non-helical conformations, and native helix caps or start-stop signals. The role of randomness in the energies of conformations due to tertiary interactions is also studied vis-à-vis the difficulty of conformational search. The thermal behavior of the model is presented for realistic values of the conformational signal energies, which can be estimated from experimental studies on peptide fragments. Estimates are made for the relative contribution of local signals and specific tertiary interactions to the folding stability gap.

Exploring structures in protein folding funnels with free energy functionals: the transition state ensemble.

We use free energy functionals that account for the partial ordering of residues in the transition state ensemble to characterize the free energy surfaces for fast folding proteins. We concentrate on chymotrypsin inhibitor and lambda-repressor. We show how the explicit cooperativity that can arise from many body forces, such as side-chain ordering or hydrophobic surface burial, determines the crossover from folding with a large delocalized nucleus and the specific small classical nucleus of the type envisioned in nucleation growth scenarios. We compare the structural correlations present in the transition state ensemble obtained from free energy functionals with those inferred from experiment using extrathermodynamic free energy relations for folding time obtained via protein engineering kinetics experiments. We also use the free energy functionals to examine both the size of barriers and multidimensional representations of the free energy profiles in order to address the question of appropriate reaction coordinates for folding.

Generalized protein tertiary structure recognition using associative memory Hamiltonians.

In previous papers, a method of protein tertiary structure recognition was described based on the construction of an associative memory Hamiltonian, which encoded the amino acid sequence and the C alpha co-ordinates of a set of database proteins. Using molecular dynamics with simulated annealing, the ability of the Hamiltonian to successfully recall the structure of a protein in the memory database was successfully demonstrated, as long as the total number of database proteins did not exceed a characteristic value, called the capacity of the Hamiltonian, equal to 0.5N to 0.7N, where N is the number of amino acid residues in the protein to be recalled. In this paper, we describe the development of additional methods to increase the capacity of the Hamiltonian, including use of a more complete representation of the protein backbone and the incorporation of contextual information into the Hamiltonian through the use of secondary structure prediction. In addition, we further extend the ability of associative memory models to predict the tertiary structures of proteins not present in the protein data set, by making the Hamiltonian invariant with respect to biological symmetries that represent site mutations and insertions and deletions. The ability of the Hamiltonian to generalize from homologous proteins to an unknown protein in the presence of other unrelated proteins in the data set is demonstrated.

The foldon universe: a survey of structural similarity and self-recognition of independently folding units.

We have identified independently folding units, so called "foldons", from non-homologous proteins representing different folds. We applied simple statistical arguments in order to estimate the size of the foldon universe required to construct all foldable proteins. Various alignment procedures yield about 2600 foldons in the natural protein universe but this estimate is shown to be rather sensitive to the chosen cut-off value for structural similarity. We showed that foldon matching-modelling can reproduce the major part of the main chain of several proteins with a structural similarity measure Q-score of about 0.4 and an r.m.s. error of about 5 A, although the accuracy of structure prediction has been limited so far by the small size of foldon data set. The prediction score may be increased if one uses the set of protein fragments with optimized sequence-structure relationships, in other works, minimally frustrated segments. To quantify the degree of frustration of the structures of foldons from our database, we searched for those foldons which recognize their own sequence and structure upon threading. As a result we found that about half of the foldons from our data set recognize themselves as the best choice upon threading and therefore are individually minimally frustrated. We showed that there is a close connection between the Q-score of self recognition and the relative foldability (Theta) of the folding units. Foldons having high Q-score and Theta values are expected to be formed in the early phase of the folding process and be observed as stable intermediates under appropriate experimental conditions.

Self-regulating gene: an exact solution.

An exact steady-state solution of the stochastic equations governing the behavior of a gene regulated by a self-generated proteomic atmosphere is presented. The solutions depend on an adiabaticity parameter measuring the relative rate of DNA-protein unbinding and protein degradation. The steady-state solution reveals deviations from the commonly used Ackers et al approximation based on the equilibrium law of mass action, allowing anticooperative behavior in the "nonadiabatic" limit of slow binding and unbinding rates. Noise from binding and unbinding events dominates the shot noise of protein synthesis and degradation up to quite high values of the adiabaticity parameter.

Self-consistently optimized statistical mechanical energy functions for sequence structure alignment.

A quantitative form of the principle of minimal frustration is used to obtain from a database analysis statistical mechanical energy functions and gap parameters for aligning sequences to three-dimensional structures. The analysis that partially takes into account correlations in the energy landscape improves upon the previous approximations of Goldstein et al. (1994, 1995) (Goldstein R, Luthey-Schulten Z, Wolynes P, 1994, Proceedings of the 27th Hawaii International Conference on System Sciences. Los Alamitos, California: IEEE Computer Society Press. pp 306-315; Goldstein R, Luthey-Schulten Z, Wolynes P, 1995, In: Elber R, ed. New developments in theoretical studies of proteins. Singapore: World Scientific). The energy function allows for ordering of alignments based on the compatibility of a sequence to be in a given structure (i.e., lowest energy) and therefore removes the necessity of using percent identity or similarity as scoring parameters. The alignments produced by the energy function on distant homologues with low percent identity (less than 21%) are generally better than those generated with evolutionary information. The lowest energy alignment generated with the energy function for sequences containing prosite signatures but unknown structures is a structure containing the same prosite signature, providing a check on the robustness of the algorithm. Finally, the energy function can make use of known experimental evidence as constraints within the alignment algorithm to aid in finding the correct structural alignment.

Quantizing Ulam's control conjecture.

Like classically chaotic systems, can quantum systems be nudged from a given initial state to any other by arbitrarily weak control fields? We explore a scaling theory that sets fundamental limits on the time required for population transfer with weak control fields. Our results depend on the size of the quantum state space and on the rate and extent of energy flow within the system. When the unperturbed dynamics is quantum localized, the total distance in state space over which control can be achieved grows linearly with the number of photons that are expended.

Recent successes of the energy landscape theory of protein folding and function.

Protein folding and binding can be understood using energy landscape theory. When seeming deviations from the predictions of the funnel hypothesis are found, landscape theory helps us locate the cause. Sometimes the deviation reflects symmetry effects, allowing extra degeneracies to occur. Such effects seem to explain some kinetic anomalies in helical bundles. When binding processes were found to use apparently non-funneled landscapes this was traced to an inadequate understanding of biomolecular forces. The discrepancy allowed the discovery of new water-mediated forces - some of which act between hydrophilic residues. Introducing such forces into the algorithms greatly improves the quality of structure predictions.

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

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 free-energy 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 fluctuational fine structure both to the free-energy profile and to the glassy dynamics is highlighted. The fluctuation 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.

Symmetry and the energy landscapes of biomolecules.

The role of symmetry in the folding of proteins is discussed using energy landscape theory. An analytical argument shows it is much easier to find sequences with funneled energy landscape capable of fast folding if the structure is symmetric. The analogy with phase transitions of small clusters with magic numbers is discussed.