Probabilistic structural dynamics of protein folding

doi:10.1016/S0096-3003(99)00071-5

Applied Mathematics and Computation

Volume 113, Issue 1, 1 July 2000, Pages 97-100

https://doi.org/10.1016/S0096-3003(99)00071-5 Get rights and content

Abstract

We consider stochastic modeling of protein folding. The protein folding problem requires understanding the characteristics of the folded shape and of the folding process. Recent advances in experimental study of protein folding lead to measurements of mean-square displacements of atoms in proteins, via X-ray diffraction and elastic incoherent neutron scattering. We propose to use techniques from probabilistic structural dynamics in the understanding of statistical characteristics such as mean-square displacements for protein folding. Stationary mean-square displacements of atoms in a protein is expressed in terms of potential energy. Non-stationary mean-square displacements can be obtained by numerically solving stochastic differential equations for the motion of atoms or the associated Fokker–Planck equations.

Introduction

Proteins are large molecules composed of smaller chemical units called amino acids, stably linked together in an ordered linear sequence. Each amino acid has a certain number of atoms 7, 10. A protein folds into the particular three-dimensional shape (“tertiary structure”, “conformation” or “configuration”) having the lowest energy. This gives the protein its specific biochemical properties or functions. It has become simple to determine the amino acid sequence of large numbers of proteins while it remains difficult to determine the structure of even a single protein.

If a protein sequence contains sufficient information to code for a folded structure, it should be possible to construct a potential energy function that reflects the energetics of the protein [7]. A minimum of this potential function should correspond to the protein's folded shape. But it is a difficult issue to locate the global minimum from a random starting point, because of the multiple local minima for potential functions of complex proteins.

Proteins are dynamic and moving systems 4, 9. A more challenging problem is to understand how a protein folds, i.e., what are the kinetics and mechanisms of protein folding from a nearly random configuration to the unique protein?

Mathematically speaking, the protein folding problem is to find characteristics (such as mean-square displacements of atoms in a protein) of the folded shape and of the folding process.

In this paper, we propose to use techniques from probabilistic structural dynamics in the understanding of statistical characteristics such as mean-square displacements for protein folding. Stationary mean-square displacements of atoms in a protein can be expressed in terms of potential energy. Non-stationary mean-square displacements can be obtained by numerically solving stochastic differential equations for the motion of atoms or the associated Fokker–Planck equations.

Section snippets

Stochastic molecular dynamics of protein folding

The equations governing the motion of the atoms in a protein are [11] $x ̈_{i} +a_{i} x ̇_{i} +b_{i} x_{i} +c_{i} ∂ V ∂ x_{i} =f_{i} (t), for i=1,2,…,N,$ where a_i>0, b_i and c_i are constants, V(x₁,…,x_N) is the potential energy, N is the total number of atoms in the protein and f_i(t)^′s are uncorrelated Gaussian white noise random force satisfying $〈f_{i} (t)〉= 0,$ $〈f_{i} (t)f_{i} (t+s)〉= D_{i} 2 δ(s),$ $〈f_{i} (t)f_{j} (t+s)〉= 0 for i≠j.$ Here D_i>0 is the intensity of the white noise f_i(t).

The potential energy V(x₁,…,x_N) can be measured by experimental methods and has been

Folded shape: stationary mean-square displacements

We can write down the associated Fokker–Planck equation for the stochastic model (1). A stationary solution for the Fokker–Planck equation may be found in the case when $a_{i} =a, c_{i} =c, D_{i} =D$ are all constants and then the stationary mean-square displacements can be expressed as $〈x_{i}^{2} 〉= 14 Dab_{i}^{2} − c b_{i}^{2} x_{i} ∂ V ∂ x_{i} for i=1,2,…,N.$ For more results on solving stationary Fokker–Planck equations, see Refs. 3, 8. Other possible stationary mean-square displacements may exist.

For general case, stationary Fokker–Planck

Folding process: non-stationary mean-square displacements

We can gain the understanding of statistical characteristics of protein folding process by computing non-stationary mean-square displacements, again using experimentally measured potential energy V(x₁,…,x_N). The exact solutions are generally not available. The approximate methods may be used to simplify or simulate Fokker–Planck equations 1, 2, 12, 14, 15. The Monte Carlo method can also be used to compute mean-square displacements 1, 5, 6, 15.

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Cited by (2)

Variational supersymmetric approach to evaluate Fokker-Planck probability
2010, Physica A: Statistical Mechanics and its Applications
Citation Excerpt :
The Fokker–Planck equation (FPE) has wide applications in several branches of physics, chemistry and biology. In particular, processes involving diffusion, transfer of electrons, transfer of protons and protein folding can be treated by using this equation, [1–3]. Historically, one of the first applications of this equation is related to the Brownian motion [4,5].
In this work we introduce a method to determine the time dependent probability density for the one-dimensional Fokker–Planck equation. The treatment is based in an analysis of the Schrödinger equation through the variational method associated to the formalism of supersymmetric quantum mechanics (SQM). The approach uses an ansatz for the superpotential which allows us to obtain the trial functions of the variational method. The hierarchy of effective Hamiltonians permits us to determine the variational eigenfunctions and energies of the excited states to the evaluation of the probability. The symmetric bistable potential is used to illustrate the approach whose results are compared with results obtained by the state-dependent diagonalization method and by direct numerical calculation.
Monitoring the unfolding and refolding of creatine kinase by capillary zone electrophoresis
2003, Journal of Capillary Electrophoresis and Microchip Technology

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Applied Mathematics and Computation

Probabilistic structural dynamics of protein folding

Abstract

Introduction

Section snippets

Stochastic molecular dynamics of protein folding

Folded shape: stationary mean-square displacements

Folding process: non-stationary mean-square displacements

Approximate solutions for non-linear random vibration problems

Probabilistic Engineering Mechanics

Derivation and application of the Fokker–Planck equation to discrete nonlinear dynamic systems subjected to white random excitation

J. Acoustical Soc. Amer.

Dynamics and function of proteins: The search for general concepts

Proc. Natl. Acad. Sci. USA

Variational supersymmetric approach to evaluate Fokker-Planck probability

Monitoring the unfolding and refolding of creatine kinase by capillary zone electrophoresis