Θ-polymers in crowded media under stretching force

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Abstract

We study the peculiarities of stretching of globular polymer macromolecules in a disordered (crowded) environment, using the model of self-attracting self-avoiding walks on site-diluted percolative lattices in space dimensions d=3. Applying the pruned–enriched Rosenbluth chain-growth method (PERM), we construct the phase diagram of collapsed–extended state coexistence when varying temperature and stretching force. The change in shape characteristics of globular polymers under stretching is analyzed as well.

Introduction

Long flexible polymer macromolecules in a good solvent form crumpled coil configurations which are perfectly captured by the model of self-avoiding random walks (SAWs) on a regular lattice [1]. This regime holds at temperatures T well above the so-called Θ-point. When lowering the temperature, the effect of monomer-monomer attraction grows and the polymer radius shrinks. At T=TΘ, a crossover occurs from high-temperature SAW behavior to the Θ-statistics. At this particular temperature polymers in d=3 dimensions behave effectively as simple random walks (RWs). Below the Θ-temperature, the entropic effects, which make the polymer chain swell, are overcome by interaction energy and a collapse to the globule regime occurs. The coil–globule transition is considered to be of second order [1], in the sense that the density of an infinite globule is zero at T=TΘ and increases continuously when further lowering the temperature.

The coil–globule transition is of interest in various respects, being deeply connected with problems like protein folding and DNA condensation. The properties of polymers in the vicinity of the Θ-point can be successfully studied on the basis of self-attracting self-avoiding walks (SASAWs), where a nearest-neighbor interaction is included: an attractive energy −ϵ between two neighbor sites is introduced. The coil–globule transition of flexible polymers has been so far the subject of numerous studies [2], [3], [4], [5], [6], [7]. Recent numerical estimates give for a simple cubic lattice kBTΘ(d=3)/ϵ=3.717(3) [5], where kB is the Boltzmann constant.

In studying the folding dynamics and transport properties of proteins, an important role is played by global shape properties of a typical polymer configuration. The asymmetry of polymer shape can be characterized, e.g., by the so-called averaged asphericity Ad [8], [9], which takes on a maximum value of one for a completely stretched configuration, and equals zero for a spherical form. It was realized experimentally [10], [11] that the majority of globular proteins are characterized by an asphericity value Ad0.1, thus being almost spherical.

In polymer physics, of great importance is the understanding of the behavior of macromolecules in the presence of structural disorder. In particular, related problems have been raised recently in studies of protein folding in the natural cellular environment [12], [13]. Real biological cells can be described as a very crowded environment built of the biochemical species, which occupy a large fraction of the total volume. In the language of lattice models, the crowded environment with structural obstacles can be considered as a disordered lattice, where some amount of randomly chosen sites contains defects. Of particular interest is the case, when the concentration p of lattice sites allowed for the SAWs equals the critical concentration pc(d=3)=0.31160 [14] and the lattice becomes percolative. It is established that the value of the Θ-temperature is lowered due to the presence of disorder [15], [16], [17], [18], numerical estimates give kBTΘpc(d=3)/ϵ=0.71(2) [18].

The recent progress in experimental techniques makes it possible to monitor the behavior of various polymers under tension and stress. In particular, applying a force on an isolated protein, the unfolding of the giant titine protein [19] and stretching of collapsed DNA molecules [20] have been studied. Of special interest in biophysics is the stretching of globular polymers below the Θ-point. Force not only influences the structural properties of polymers, but also may introduce a new completely stretched state which is otherwise not accessible. The properties of force-induced transitions in polymers have been studied intensively [21], [22], [23], [24], [25]. The response of a polymer in crowded media to the stretching force within the SASAW model on a percolative lattice has been analyzed recently in Refs. [26], [18]. The interesting question about how the shape properties of almost spherical polymer globules are influenced by stretching remains, however, completely unresolved.

The aim of the present study is to apply numerical simulations to analyze the properties of SASAWs on site-diluted lattices at the percolation threshold under applied external stretching force in space dimensions d=3. We analyze the effect of applied force on the phase transitions between collapsed, extended and stretched phases and estimate the influence of stretching on the shape parameters of globular proteins in crowded environments.

Section snippets

The method

We consider site percolation on regular lattices of edge lengths up to Lmax=200 in d=3. Each site of the lattice was assigned to be occupied with probability pc and empty otherwise. To obtain the backbone of a percolation cluster on a given disordered lattice, we apply an algorithm explained in detail in our previous papers [27].

To study SASAWs on the backbone of percolation clusters, we apply the pruned–enriched Rosenbluth method (PERM) [5], taking into account that the SASAW can have its

Results

The properties of systems in the vicinity of a second-order phase transition can be studied by analyzing the peak structure of the specific heat CV as a function of temperature indicating crossovers between physically different states. In the case of a polymer system, this corresponds to the transition between globule and coil regimes. CV can be expressed via energy fluctuations as follows:CV(T)=1NT2(E2¯E2¯).

To study the Θ-transition of SASAWs, when the external stretching force is acting

Conclusions

We studied self-attracting self-avoiding walks on disordered lattices in space dimensions d=3, modeling flexible polymer macromolecules in porous environment. We considered the special case, when the concentration of disorder is exactly at the percolation threshold, so that an incipient percolation cluster of sites, allowed for SAWs, emerges on the lattice. Keeping one end of a SASAW trajectory on the backbone of a percolation cluster fixed, we applied a stretching force F, acting in some

Acknowledgement

V.B. is grateful for support through the Sächsische DFG-Forschergruppe FOR877.

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