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.

https://science.sciencemag.org/content/sci/254/5038/1598.full.pdf

The energy landscapes and motions of proteins.

We describe a global optimization technique using “basin-hopping” in which the potential energy surface is transformed into a collection of interpenetrating staircases. This method has been designed to exploit the features that recent work suggests must be present in an energy landscape for efficient relaxation to the global minimum. The transformation associates any point in configuration space with the local minimum obtained by a geometry optimization started from that point, effectively removing transition state regions from the problem. However, unlike other methods based upon hypersurface deformation, this transformation does not change the global minimum. The lowest known structures are located for all Lennard-Jones clusters up to 110 atoms, including a number that have never been found before in unbiased searches.

/pdf/global-optimization-by-basin-hopping-and-the-lowest-energy-2s1wxtsj18.pdf

Global Optimization by Basin-Hopping and the Lowest Energy Structures of Lennard-Jones Clusters Containing up to 110 Atoms

The understanding, and even the description of protein folding is impeded by the complexity of the process. Much of this complexity can be described and understood by taking a statistical approach to the energetics of protein conformation, that is, to the energy landscape. The statistical energy landscape approach explains when and why unique behaviors, such as specific folding pathways, occur in some proteins and more generally explains the distinction between folding processes common to all sequences and those peculiar to individual sequences. This approach also gives new, quantitative insights into the interpretation of experiments and simulations of protein folding thermodynamics and kinetics. Specifically, the picture provides simple explanations for folding as a two-state first-order phase transition, for the origin of metastable collapsed unfolded states and for the curved Arrhenius plots observed in both laboratory experiments and discrete lattice simulations. The relation of these quantitative ideas to folding pathways, to uniexponential vs. multiexponential behavior in protein folding experiments and to the effect of mutations on folding is also discussed. The success of energy landscape ideas in protein structure prediction is also described. The use of the energy landscape approach for analyzing data is illustrated with a quantitative analysis of some recent simulations, and a qualitative analysis of experiments on the folding of three proteins. The work unifies several previously proposed ideas concerning the mechanism protein folding and delimits the regions of validity of these ideas under different thermodynamic conditions. © 1995 Wiley-Liss, Inc.

Funnels, pathways, and the energy landscape of protein folding: A synthesis

A new view of protein folding kinetics replaces the idea of ‘folding pathways’ with the broader notions of energy landscapes and folding funnels. New experiments are needed to explore them.

From Levinthal to pathways to funnels

Protein structures in the Protein Data Bank provide a wealth of data about the interactions that determine the native states of proteins. Using the probability theory, we derive an atomic distance-dependent statistical potential from a sample of native structures that does not depend on any adjustable parameters (Discrete Optimized Protein Energy, or DOPE). DOPE is based on an improved reference state that corresponds to noninteracting atoms in a homogeneous sphere with the radius dependent on a sample native structure; it thus accounts for the finite and spherical shape of the native structures. The DOPE potential was extracted from a nonredundant set of 1472 crystallographic structures. We tested DOPE and five other scoring functions by the detection of the native state among six multiple target decoy sets, the correlation between the score and model error, and the identification of the most accurate non-native structure in the decoy set. For all decoy sets, DOPE is the best performing function in terms of all criteria, except for a tie in one criterion for one decoy set. To facilitate its use in various applications, such as model assessment, loop modeling, and fitting into cryo-electron microscopy mass density maps combined with comparative protein structure modeling, DOPE was incorporated into the modeling package MODELLER-8.

/pdf/statistical-potential-for-assessment-and-prediction-of-16jn1n7saa.pdf

Statistical potential for assessment and prediction of protein structures

The theory of spin glasses was used to study a simple model of protein folding. The phase diagram of the model was calculated, and the results of dynamics calculations are briefly reported. The relation of these results to folding experiments, the relation of these hypotheses to previous protein folding theories, and the implication of these hypotheses for protein folding prediction schemes are discussed.

https://www.pnas.org/content/pnas/84/21/7524.full.pdf

Spin glasses and the statistical mechanics of protein folding

Utilisation d'un hamiltonien stochastique pour etudier une reaction chimique complexe. Une simple generalisation de la theorie de l'etat de transition est developpee et est utilisee pour estimer le temps de repliement d'une proteine

Intermediates and barrier crossing in a random energy model (with applications to protein folding)

The understanding, and even the description of protein folding is impeded by the complexity of the process. Much of this complexity can be described and understood by taking a statistical approach to the energetics of protein conformation, that is, to the energy landscape. The statistical energy landscape approach explains when and why unique behaviors, such as specific folding pathways, occur in some proteins and more generally explains the distinction between folding processes common to all sequences and those peculiar to individual sequences. This approach also gives new, quantitative insights into the interpretation of experiments and simulations of protein folding thermodynamics and kinetics. Specifically, the picture provides simple explanations for folding as a two-state first-order phase transition, for the origin of metastable collapsed unfolded states and for the curved Arrhenius plots observed in both laboratory experiments and discrete lattice simulations. The relation of these quantitative ideas to folding pathways, to uni-exponential {\em vs.} multi-exponential behavior in protein folding experiments and to the effect of mutations on folding is also discussed. The success of energy landscape ideas in protein structure prediction is also described. The use of the energy landscape approach for analyzing data is illustrated with a quantitative analysis of some recent simulations, and a qualitative analysis of experiments on the folding of three proteins. The work unifies several previously proposed ideas concerning the mechanism protein folding and delimits the regions of validity of these ideas under different thermodynamic conditions.

Funnels, Pathways and the Energy Landscape of Protein Folding: A Synthesis

A simple statistical extension of the Flory theory of excluded volume is developed, and applied to random heteropolymer collapse and a simple model of protein folding. Sequence heterogeneity is found to lead to new effects. The nucleation of protein folding is also briefly considered.

Joseph D. Bryngelson

Papers

Funnels, pathways, and the energy landscape of protein folding: A synthesis

Spin glasses and the statistical mechanics of protein folding

Intermediates and barrier crossing in a random energy model (with applications to protein folding)

Funnels, Pathways and the Energy Landscape of Protein Folding: A Synthesis

A simple statistical field theory of heteropolymer collapse with application to protein folding