David L. Weaver

Application of the diffusion-collision model to the folding of three-helix bundle proteins

The diffusion-collision model has been successful in explaining many features of protein folding kinetics, particularly for helical proteins. In the model the folding reaction is described in terms of coupled chemical kinetic (Master) equations of coarse grained entities, called microdomains. Here, the diffusion-collision model is applied to compute the folding kinetics of four three-helix bundle proteins, all of which fold on a time scale of tens of microseconds and appear to have two-state folding. The native structure and the stability of the helical microdomains are used to determine the parameters of the model. The formulation allows computation of the overall rate and determination of the importance of kinetic intermediates. The proteins considered are the B domain of protein A (1BDC), the Engrailed Homeodomain (1ENH), the peripheral sub-unit-binding domain (1EBD C-chain) and the villin headpiece subdomain (1VII). The results for the folding time of protein A, the Engrailed Homeodomain, and 1EBD C-chain are in agreement with experiment, while 1VII is not stable in the present model. In the three proteins that are stable, two-state folding is predicted by the diffusion-collision model. This disagrees with published assertions that multistate kinetics would be obtained from the model. The contact order prediction agrees with experiment for protein A, but yields values that are a factor of 40, 30 and 15 too slow for 1ENH, 1EBD C-chain and 1VII. The effect of mutants on folding is described for protein A and it is demonstrated that significant intermediate concentrations (i.e. deviation from two-state folding) can occur if the stability of some of the helical microdomains is increased. A linear relationship between folding time and the length of the loop between helices B and C in protein A is demonstrated; this is not evident in the contact order description.

Diffusion-Collision Model for the Folding Kinetics of the λ-Repressor Operator-Binding Domain

Abstract The operator-binding domain of the λ-repressor contains five α-helices and an extended N-terminal arm in the crystal structure determined by Pabo and Lewis reported in Nature 298, 443,1982 (1). The four helices form a “box” enclosing a hydrophobic core, with the fifth helix interacting with the equivalent helix in a dimer. With a small number of well-defined secondary structure elements (microdomains), the repressor is well suited for an analysis of its folding pathways and kinetics by use of the diffusion-collision model. In this paper, the basic elements of the model appropriate to a several microdomain protein are formulated and applied to a set of folding pathways consistent with the crystal structure of the operator- binding domain. The overall kinetics, as well as the time-dependence of intermediate states are determined as a function of the microdomain stability parameter.

Nonequilibrium decay effects in diffusion‐controlled processes

It is shown how to include decay or desorption at a boundary in systems with diffusive kinetics. A quantity, the mean equilibrium time, is defined and calculated for one‐dimensional diffusion. Both finite and infinite systems are considered.

One-dimensional potential barrier model of protein folding with intermediates

Protein folding is modeled as one-dimensional diffusion in a potential with square wells representing folding species and square barriers representing transitions among the species. Within the context of the model, one or more intermediate species can either speed up or slow down folding, depending on their energy and on the potential barrier(s) to the final folded state. Intermediate species in deep potential wells may reduce the probability in the final state, as well as slowing the overall folding process. The potential barrier model is consistent with protein folding taking place by diffusion, collision and coalescence of marginally stable subunits of the protein in a sequential but, in principle, arbitrary order, as in the diffusion-collision model. Using parameters taken from the structures of three-helix bundle proteins the potential barrier model gives folding rates consistent with recent experiments on these proteins.

ELECTROSTATIC MULTIPOLE REPRESENTATION OF A POLYPEPTIDE CHAIN : AN ALGORITHM FOR SIMULATION OF POLYPEPTIDE PROPERTIES

A method for describing a polypeptide chain based on an electrostatic multipole representation is introduced. The main features of the description are outlined. Appropriate energy functions for nonbonded interactions are developed. The full atomic representation may be retrieved from the electrostatic multipole representation at any point in a calculation. The multipole description and the energy functions are tested by calculation of steric maps for different amino acid side‐chain groups. The ability to calculate energetically stable structures is demonstrated by energy conformation maps and the results of energy calculations in optimal secondary structural elements. Results from dynamics simulations of helical chains of polyglycine, polyalanine, polyvaline, and a 21‐residue helix obtained from the crystal structure of sperm whale myoglobin are included to demonstrate the efficiency of the algorithm. It is demonstrated that this description of the polypeptide chain is both simple and complete and will allow for the rapid simulation of chain dynamics without loss of essential information about the chain.

A generator of protein folding kinetics states for the diffusion–collision model

Two separate algorithms for calculating the intermediate states, using cellular automata, and the initial conditions in the rate matrix for the diffusion–collision model are introduced. They enable easy and fast calculations of the folding probabilities of the intermediate states, even for a very large number of microdomains.

Exact diagonalization of relativistic Hamiltonians including a constant magnetic field

It is shown how to exactly diagonalize the Dirac and Sakata–Taketani Hamiltonians, including the effect of a constant, external magnetic field, using a unitary transformation. In the latter case, the magnetic moment coupling must be of the Yang–Mills type in order to perform the transformation.

Diffusion-Collision Model Study of Misfolding in aFour-Helix Bundle Protein

Proteins with complex folding kinetics will be susceptible to misfolding at some stage in the folding process. We simulate this problem by using the diffusion-collision model to study non-native kinetic intermediate misfolding in a four-helix bundle protein. We find a limit on the size of the pairwise hydrophobic area loss in non-native intermediates, such that burying above this limit creates long-lasting non-native kinetic intermediates that would disrupt folding and prevent formation of the native state. Our study of misfolding suggests a method for limiting the production of misfolded kinetic intermediates for helical proteins and could, perhaps, lead to more efficient production of proteins in bulk.

Diffusion-controlled kinetics of the helix–coil transition with square barrier hydrogen bonds

The coil α-helix (and reverse) transition in peptides is modeled as a sequential diffusive kinetic process in which the fundamental event is the diffusion back and forth over a square barrier to propagate or dissolve a single hydrogen bond. The model is solved exactly numerically in one-dimension (the reaction coordinate), for helix and coil probabilities as a function of (1) time, (2) the number of hydrogen bonds, and (3) temperature. In addition, a modified first-passage time is calculated as the time scale of the coil to helix transition. The results of the diffusion model calculations are compared with recent experiments and we show how the model may give insight into protein folding kinetics. The mechanistic diffusion model complements the Master equation model applied previously to the coil–helix folding problem and provides insight into the choice of a useful reaction coordinate for the process.

Connection between back-reaction boundary conditions and approach to equilibrium for double square wells

One-dimensional diffusion with back-reaction at a boundary is calculated and compared with diffusion in an asymmetrical double well. Good agreement is found between exact analytical or numerical results and a suggested mean equilibrium time approximation. The results are applied to a model for the helix-coil transition in linear biopolymers and a reasonable time scale for this process results.