Random Copolymer: Gaussian Variational Approach

Moskalenko, A. M.; Kuznetsov, Yu. А.; Dawson, Kenneth A.

doi:10.1051/jp2:1997134

Cited by 11 publications

(19 citation statements)

References 20 publications

(48 reference statements)

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“…Nevertheless, it might be that stiffness plays a similar role as heterogeneity, in which case the model of [2] might catch typical features of real protein folding. Indeed, in a recent treatment of random copolymers [11] the authors found a phase diagram ( fig.3 of [11]) which is surprisingly similar to the one found in [2].…”

Section: Introductionsupporting

confidence: 77%

Phase Transitions of Single Semistiff Polymer Chains

1997

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We study numerically a lattice model of semiflexible homopolymers with nearest neighbor attraction and energetic preference for straight joints between bonded monomers. For this we use a new Monte Carlo algorithm, the 'Pruned-Enriched Rosenbluth Method' (PERM). It is very efficient both for relatively open configurations at high temperatures, and for compact and frozen-in low-T states. This allows us to study in detail the phase diagram as a function of nn attraction ǫ and stiffness x. It shows a θ-collapse line with a transition from open coils (small ǫ) to molten compact globules (large ǫ), and a freezing transition toward a state with orientational global order (large stiffness x). Qualitatively this is similar to a recently studied mean field theory (S. Doniach, T. Garel and H. Orland (1996), J. Chem. Phys. 105 (4), 1601), but there are important differences in details. In contrast to the mean field theory and to naive expectations, the θ-temperature increases with stiffness x. The freezing temperature increases even faster, and reaches the θ-line at a finite value of x. For even stiffer chains, the freezing transition takes place directly, without the formation of an intermediate globular state. Although being in conflict with mean field theory, the latter had been conjectured already by Doniach et al. on the basis of heuristic arguments and of low statistics Monte Carlo simulations.Finally, we discuss the relevance of the present model as a very crude model for protein folding.

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Section: Introductionsupporting

confidence: 77%

Phase Transitions of Single Semistiff Polymer Chains

1997

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“…This approach has the advantage that it does not rely on the "ground state dominance" approximation (see appendix B and below), but it is technically rather heavy. Beside the original work 28 , which delt with a polymer chain in a random medium, the only use of this method, in the context of self-interacting random chains, is, as far as we know, the hydrophilic-hydrophobic chain 30 . This study deals with the three-dimensional case, in the framework of a one step replica symmetry breaking scheme for the Parisi-like kernel g(s, x), x ∈ [0, 1].…”

Section: The Random Hydrophilic-hydrophobic Chainmentioning

confidence: 99%

“…The fact that a replica symmetric mean field theory turns, (for finite dimensions) into a broken replica symmetry theory occurs also in the random field Ising model 34 . Interestingly enough, the random term in equation (30) can be rewritten in a "random density" way since…”

Section: The Random Hydrophilic-hydrophobic Chainmentioning

confidence: 99%

Protein Folding and Heteropolymers

Garel

Orland

Pitard

1997

Series on Directions in Condensed Matter Physics

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We present a statistical mechanics approach to the protein folding problem. We first review some of the basic properties of proteins, and introduce some physical models to describe their thermodynamics. These models rely on a random heteropolymeric description of these non random biomolecules. Various kinds of randomness are investigated, and the connection with disordered systems is discussed. We conclude by a brief study of the dynamics of proteins.Natural proteins have the property of folding into an (almost) unique compact native structure, which is of biological interest 1,2 . The compactness of this unique native state is largely due to the existence of an optimal amount of hydrophobic amino-acid residues 3 , since these biological objects are usually designed to work in water. The task of predicting the conformation of the three-dimensional structure from the linear primary sequence is often referred to as the protein folding problem. Already at this stage, a rigorous analytical theory appears difficult, since it amounts to study a mesoscopic system (most proteins have between 100 and 500 residues), notwithstanding the solvent's properties.This mesoscopic system is of a classical nature; the quantum mechanical valence electrons of the atoms induce interactions between the heavy "nuclei" which can then be treated as classical objects interacting through classical many-body interactions. This observation forms the basis of Molecular Dynamics, Monte Carlo, or Statistical Mechanical models.To further complicate the matter, both the compactness and the chemical heterogeneity of a given protein tend to slow down dynamical processes: the question of the kinetic control of the folding (as opposed to a thermodynamic control) is therefore periodically asked. This remark suggests that the folding process may have something in common with the physics of glassy systems, where competing interactions (frustration) and/or disorder lead to a very rugged phase space, resulting in slow dynamical processes. In this review, we will assume that the folded state of proteins is thermodynamically stable (the folded state is the state of minimal free energy).There has been a considerable amount of numerical simulations of proteins, using molecular dynamics or Monte Carlo calculations (for a review see 2,3,4 ).This promising approach will not be discussed in this review. We will use only analytical approaches throughout. Similarly, models emphasizing the micro-crystalline character of the folded proteins will not be addressed here 5 .The outline of this review is the following. In the first section, we introduce at an elementary level, some notions of the physics, chemistry and biology of proteins. In the second section, we study (using statistical physics methods) some heteropolymer models which are possibly relevant to the protein folding problem. The phase diagrams of these models bear some qualitative resemblance to the real systems.In the last section, we tackle the issues of dynamics. In view of the complexity of 1

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“…We therefore believe that, in the thermodynamic limit (N, p → ∞), one has a discontinuous single block collapse transition; the critical temperature T c (∞) is the same as the collapse temperature of the fully hydrophobic chain (T θ = T θ (∞) ≃ 3.7 [20]). The transition is well characterized by the phase separation order parameter [6,7]:…”

Section: A Numerical Simulations For D=3mentioning

confidence: 99%

Collapse transitions of a periodic hydrophilic hydrophobic chain

Orlandini

Garel

1998

Eur. Phys. J. B

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Short title: Transitions of a periodic hydrophilic hydrophobic chain 1We study a single self avoiding hydrophilic hydrophobic polymer chain, through Monte Carlo lattice simulations. The affinity of monomer i for water is characterized by a (scalar) charge λ i , and the monomer-water interaction is short-ranged. Assuming incompressibility yields an effective short ranged interaction between monomer pairs (i, j), proportional to (λ i + λ j ). In this article, we take λ i = +1 (resp. (λ i = −1)) for hydrophilic (resp. hydrophobic) monomers and consider a chain with (i) an equal number of hydro-philic and -phobic monomers (ii) a periodic distribution of the λ i along the chain, with periodicity 2p. The simulations are done for various chain lengths N , in d = 2 (square lattice) and d = 3 (cubic lattice). There is a critical value p c (d, N ) of the periodicity, which distinguishes between different low temperature structures. For p > p c , the ground state corresponds to a macroscopic phase separation between a dense hydrophobic core and hydrophilic loops.For p < p c (but not too small), one gets a microscopic (finite scale) phase separation, and the ground state corresponds to a chain or network of hydrophobic droplets, coated by hydrophilic monomers. We restrict our study to two extreme cases, p ∼ O(N ) and p ∼ O(1) to illustrate the physics of the various phase transitions. A tentative variational approach is also presented.2

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Random Copolymer: Gaussian Variational Approach

Cited by 11 publications

References 20 publications

Phase Transitions of Single Semistiff Polymer Chains

Phase Transitions of Single Semistiff Polymer Chains

Protein Folding and Heteropolymers

Collapse transitions of a periodic hydrophilic hydrophobic chain

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