Entanglement Complexity Research Articles

Scaling regimes of molecular size and self-entanglements in very compact proteins.

Establishing interrelations between size, compactness, and three-dimensional shape in biomolecules is important for a better understanding of the factors governing their folding features and their biological function. Whereas size and compactness can be characterized by parameters such as the radius of gyration, the description of folding features is less well defined. Recently, we have introduced the probability of overcrossings in two-dimensional projections of a rigid backbone as a descriptor of self-entanglements. This function provides a simple and intuitive characterization: the more complex the entanglements, the larger the mean number of overcrossings. In this work, we study relationships between size and entanglements on a special subclass of biomolecules with a global structural constraint: the family of native protein conformations which are the most compact within a range of amino acid residue numbers. Initially, we use a set of 373 experimental protein backbones exhibiting very diverse lengths, composition, and structural features. Within this set, we have located the proteins with the smallest radii of gyration for fixed ranges of monomer numbers. For this class of proteins, we observe power-law scaling behavior in size and entanglement complexity in terms of the residue number. The results suggest that there are two distinct regimes of scaling characterizing short compact proteins and long compact proteins, respectively. The change of regime appears to be localized roughly around 300 amino acid residues. We propose that this difference correlates with a change in the content of secondary structure for compact proteins: the content of \ensuremath{\beta} strands for short chains is almost twice as large as that of longer chains, the latter in turn being much richer in \ensuremath{\alpha} helices. In summary, the work establishes that in forming a very compact polypeptide there are some constraints among the number of residues, the radius of gyration, the entanglement complexity, and the content of secondary structure of the molecular chain. Thus the degree of compactness in heteropolymers appears to exhibit more complex features than those found in homopolymers.

ON RANDOM KNOTS

In this paper, we consider knotting of Gaussian random polygons in 3-space. A Gaussian random polygon is a piecewise linear circle with n edges in which the length of the edges follows a Gaussian distribution. We prove a continuum version of Kesten's Pattern Theorem for these polygons, and use this to prove that the probability that a Gaussian random polygon of n edges in 3-space is knotted tends to one exponentially rapidly as n tends to infinity. We study the properties of Gaussian random knots, and prove that the entanglement complexity of Gaussian random knots gets arbitrarily large as n tends to infinity. We also prove that almost all Gaussian random knots are chiral.

Scaling behavior of some molecular shape descriptors of polymer chains and protein backbones.

Some global folding features of macromolecular chains are characterized using a family of molecular shape descriptors. These descriptors are derived from the following basic notion: the probability of observing N ``crossings'' between bonds (i.e., double points or overcrossings) when a rigid placement of a polymer backbone is projected onto two dimensions. The approach combines simple elements of geometry and topology of linear polymers, and it quantifies the compactness and the complexity of chain entanglements in three-space. The asymptotic behavior of the shape descriptors has been determined as a function of the chain length. It is found that the configurational averages of the most probable number of overcrossings ${\mathit{N}}^{\mathrm{*}}$, the mean number of overcrossings N\ifmmode\bar\else\textasciimacron\fi{}, and the largest probability of overcrossings ${\mathit{A}}^{\mathrm{*}}$, obey power laws in terms of the number of monomers. The critical exponents have been estimated numerically for random-walk polymers with excluded-volume, as well as for a large number of experimental protein backbones. The results indicate that the scaling behavior is little affected by the configurational state of the polymers, since virtually the same exponents are obtained for both ``swollen'' and ``compact'' structures. The same scaling behavior is found in polymers with various excluded-volume interactions and in a set of 197 proteins. The mean number of overcrossings in proteins is well described by a simple law: N\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\approxeq}0.045${\mathit{n}}^{1.4}$, where n is the number of amino acid residues. The shape descriptors for proteins show little dispersion away from the asymptotic regime, whereas a less uniform behavior is found in the radius of gyration. The results complement the analyses based on other more familiar (geometrical) descriptors, and provide some insights into the large-scale folding structure of polymer chains and proteins.

Complexity of entanglements and degree of folding in branched polymers with excluded-volume interaction

A methodology to characterize large-scale shape of macromolecular conformations is generalized and applied to a simple model of branched polymers. The standard global analysis of polymer configurations uses geometric shape descriptors, such as the mean radius of gyration. However, geometric descriptors alone do not depict well the intrincate folding features found in many macromolecules. For a more complete characterization, we employ here a new class of shape descriptors which convey the degree and complexity of the “entanglements” in a branched chain. Recently, the methodology has been applied to study statistical properties of linear polymer chains. In the present work, we extend the technique to branched polymers and analyze the configurational and long-chain behavior of their entanglement descriptors. A family of “stars” (polymers with a single branchpoint) with excluded volume interaction is studied, and the results are contrasted with those for linear chains. The relation between changes in geometric and shape descriptors within the polymer conformational space is presented. The complexity of entanglements for large chains and large stars are compared. The results suggest a similar scaling behavior of the entanglement descriptors for the two polymer architectures. © 1994 John Wiley & Sons, Inc.

The writhe of a self-avoiding polygon

We discuss the writhe of a self-avoiding polygon on a lattice, as a geometrical measure of its entanglement complexity. We prove a rigorous result about the dependence of the absolute value of the writhe on the number n of edges in the polygon, and use Monte Carlo methods to estimate the distribution of the writhe both for all polygons with n edges and for the subset of polygons that are trefoils.

Assessment of molecular shape fluctuations along dynamic trajectories

To assess the flexibility of a molecule (and to establish how it is affected by external conditions), one can analyze the nature of the configurations encountered along molecular dynamics (MD) trajectories. If the molecule is initially at a rigid conformation, then it will mostly undergo deformations which conserve the initial fold or shape. That is, a rigid molecule will exhibit very small shape fluctuations about the initial shape, whereas large fluctuations will be characteristic of flexible molecules. Establishing the persistence of molecular shape features along MD trajectories is a valuable piece of information in the analysis of dynamic phase transitions in large biomolecules, including protein folding and polymer melting. In this work, we discuss a methodology to monitor macromolecular shape along dynamic trajectories. The procedure uses descriptors which convey some global shape features, including the compactness and degree (and complexity) of entanglement in a backbone. The method is used to follow shape fluctuations in chain molecules (carboxylic acids) and cyclic molecules (cycloalkanes), with 6 up to 12 carbon atoms. The role of variable temperature and number of atoms on these fluctuations is analyzed. The results indicate a relatively constant average shape in the cyclic molecules, although the amplitude of the shape fluctuations increases with the temperature. In contrast, the shape of the chain molecules is affected largely at high temperature, where folding or “melting” becomes dominant. At low temperature, the chains are very rigid. In the high temperature regime, the shape fluctuations in chains are large and very similar to those in cycles. In other words, the distinction between configurations of cycles and chains starts to blur at high temperature, thus suggesting that chains are mostly folded into turns. The results would appear not to depend very strongly on the functional group at the end of the chain. © 1993 John Wiley & Sons, Inc.

Entanglement complexity of self-avoiding walks

Self-avoiding walks on three-dimensional lattices are flexible linear objects which can be self-entangled. The authors discuss several ways to measure entanglement complexity for n-step walks, and prove that these complexity measures tend to infinity with n. For small n, they use Monte Carlo methods to estimate and compare the n-dependence of two of these complexity measures.

Nature of interfacial interaction mechanisms between polyacrylic acid macromolecules and oxide metal surfaces

The mechanism for the adhesion of polyacrylic acid (PAA) coatings to oxidized metal surfaces has been studied. The work entailed studies of the mechanical and chemical interactions occurring at the interfaces between PAA polyelectrolyte macromolecules and iron (III) orthophosphate dihydrate or zinc phosphate hydrate (hopeite) crystalline films that were deposited on the metal surfaces. With respect to mechanical interactions, it was determined that the surface topography of the highly crystallized hopeite layers consisted of an open microstructure. This resulted in enhanced wettability of the oxide film by the polyelectrolyte macromolecules, thereby increasing the mechanical interlocking bond forces. Studies of the interfacial chemical reactions indicated that the conformation changes in the PAA macromolecules relate directly to the frequency of the magnitude of acid-base and divalent metallic ion crosslinking interactions between the proton-donating pendent COOH groups in PAA molecules and polar OH groups at hydrated oxide surface sites. Namely, the presence of numerous free nucleophilic ions existing on the deposited oxide film leads to a substantial increase in the coil-up and entanglement macromolecule density. These entangled complex macromolecules at the interfaces result in a decrease in the degree of chemisorption at the oxide film surfaces, whereas regularly oriented COOH groups produce strong interfacial chemisorption with the polar OH groups. Since the polyelectrolyte macromolecules have hydrophilic pendent COOH groups, the polymer structure which appears best for use as an adhesive and coating should have only enough hydrophilic COOH groups to occupy all available polar OH groups at the oxide metal surface sites.

The Effects of Mechanical Degradation on Rheological Properties of Elastomers

Abstract Several anomalies are observed in the flow of the melts of many elastomers which are not commonly observed with melts of plastics. A stress-oscillation at high rates of shear and a tendency to reach a constant level of stress at lower rates of shear are two of the more commonly encountered anomalies. Both anomalies occasionally occur in the flow of a single melt. The role of molecular weight, molecular weight distribution and branching in producing anomalous flow is demonstrated. The presence of supermolecular flow units is invoked to explain certain anomalies and supporting evidence is presented. The data strongly suggest changes in the rheological flow units take place as the conditions of shear are changed. The flow properties of many elastomers are complicated by the severe mechanical degradation taking place under these conditions of shear. Anomalous flow patterns can be either produced or removed by mechanical degradation depending upon the specific elastomer system. The extent of degradation appears to be dependent upon the flow unit governing the flow process and independent of the shearing stress or work input per unit volume of polymer sheared. The complexity and distribution of entanglements play a major role in the flow processes as well as in the mechanical degradation process of certain elastomers.