[Paper Review] Parametrization of Multiple Pathways in Proteins: Fast Folding versus Tight Transitions

Paul Dommersnes, Alex Hansen+2 more|arXiv (Cornell University)|Jun 20, 2000

Advanced Materials and Mechanics3 citations

TL;DR

This paper proposes a parametrization scheme for multiple folding pathways in proteins using effective Hamiltonians derived from sequential, hierarchical energy and entropy reduction. It demonstrates that such models yield sharply defined two-state transitions and realistic folding times, with entropy barriers explaining the non-Arrhenius temperature dependence of folding kinetics.

ABSTRACT

Growing experimental evidence shows that proteins follow one or a few distinct paths when folding. We propose in this paper a procedure to parametrize these observed pathways, and from this parametrization construct effective Hamiltonians for the proteins. We furthermore study the denaturated-native transitions for a wide class of possible effective Hamiltonians based on this scheme, and find that the sharpness (tightness) of the transitions typically are close to their theoretical maximum and thus in quantitative accordance with the sharp folding transition observed for single domain proteins. Finally we demonstrate that realistic folding times are typical for the proposed class of Hamiltonians, and we discuss the implication of the predicted entropy barriers on the temperature dependence of the folding times.

Motivation & Objective

• To develop a parametrization scheme for multiple folding pathways in proteins that captures experimentally observed two-state transitions.
• To construct effective Hamiltonians from this parametrization that reproduce the sharpness of denaturation transitions observed in single-domain proteins.
• To demonstrate that realistic folding times emerge naturally from the proposed class of Hamiltonians.
• To investigate the role of entropy barriers in shaping the temperature dependence of folding rates.
• To reconcile fast folding with the two-state thermodynamics of protein folding by encoding hierarchical, sequential folding pathways.

Proposed method

• Parametrize protein folding as a sequence of partially folded structures that reduce both energy and entropy stepwise, inspired by a zipper-like mechanism.
• Define effective Hamiltonians based on a set of parameters φi representing local energy contributions to folding steps.
• Use a statistical ensemble approach to compute the average partition function ⟨Z⟩ over all possible configurations of φi parameters.
• Derive expressions for the mean partition function and its fluctuations using combinatorial sums over permutations and binomial coefficients.
• Apply the van’t Hoff relation to analyze the sharpness of the denaturation transition in the derived Hamiltonian class.
• Analyze the temperature dependence of folding times by identifying an entropic barrier that suppresses folding at low temperatures.

Experimental results

Research questions

• RQ1 Can a parametrization of multiple folding pathways reproduce the experimentally observed two-state nature of protein folding?
• RQ2 What is the maximum possible sharpness of the denaturation transition in a model with such pathways, and does it match experimental observations?
• RQ3 Do the proposed Hamiltonians yield folding times consistent with experimental measurements?
• RQ4 How does the presence of an entropy barrier affect the temperature dependence of folding kinetics?
• RQ5 Can hierarchical, sequential folding pathways explain both fast folding and sharp transitions simultaneously?

Key findings

• The proposed class of effective Hamiltonians produces denaturation transitions that are typically at or near the theoretical maximum sharpness, consistent with the two-state behavior seen in experiments.
• Folding times computed from the model are realistic and compatible with experimental observations for single-domain proteins.
• The folding transition is sharpened by an entropic barrier, which causes folding times to increase strongly with temperature below the transition temperature Tc.
• The relative fluctuations of the partition function vanish in the thermodynamic limit (N→∞), indicating a well-defined phase transition.
• The model successfully reconciles fast folding with sharp two-state transitions by encoding sequential, hierarchical folding pathways.
• The temperature dependence of folding rates is non-Arrhenius due to the entropic barrier, explaining the observed increase in folding time at lower T.