Stretching of homopolymers and contact order

Recently, there have been many experimental studies of mechanical stretching of single biomolecules, such as strands of DNA [1-3], and of large multidomained proteins [4,5]. The force sFd-displacement sdd curves of proteins are spe-cific, intricate, and reproducible so that they can be thought of as fingerprints of the proteins [6]. In addition, theoretical studies [7-9] show that the stretching scenarios are complex and not directly related to the folding scenarios. These sce-narios can be characterized by diagrams which show an av-erage displacement at which a contact interaction is ruptured (or, in the case of folding, an average time needed to establish a contact) as a function of the contact order, i.e., of the sequential distance between two amino acids that interact in the native state. Even though the F-d patterns and the stretching scenarios show dependence on the pulling speed and on the stiffness of the pulling device [7], they are gov-erned primarily by the specific set of interactions that produce the unique folded structure of the protein.

In this paper, we discuss mechanical stretching of mol-ecules without specific interactions: single self-associating homopolymers whose monomers all interact with each other. One goal is to determine how the behavior of molecules with nonspecific interactions contrasts with that of proteins. In addition, self-interacting homopolymers can be considered as models of DNA strands, which show much less specificity than any protein. They also model polymeric chains, and the mechanical properties of PNIPAM [poly(N-isopropyl acrylamide] and PEO [poly(ethylene oxide)] have been studied recently using atomic force microscopy [10].

Theoretical studies of homopolymer stretching have begun from simple models that are amenable to analytical treatments, such as the freely jointed chain (FJC) and the worm-like chain (WLC) [11]. The formula for the force in the WLC model, derived within the continuous chain approximation [12], has been widely used to fit experimental F-d curves for a variety of biomolecules including DNA [13], RNA [14], and proteins. Recently, a more general solution [15] of theWLC model, which takes into account the discreteness of the chain, has been given. This generalized formula can be used to fit the experimental data on single and double stranded DNA in the intermediate and high force regimes, and at the same time gives extra information on the intrabead distance along the chain. Furthermore, it predicts the existence of a crossover force, above which the WLC force approaches the FJC result [15].

The WLC model ignores self-avoidance and attractive interactions between different segments of the chain. Thus it can be applied only in the case of stretching under good solvent conditions, i.e., above the u transition [11] temperature T_u. Under poor solvent conditions, theoretical studies of homopolymer stretching are usually confined to mean-field [16,17] or lattice [18-20] models. A common characteristic of such studies is that at temperatures below T_u they predict existence of a plateau region associated with a well defined force in the F-d curve. The collapse-coil transition at this force is found to be first [16,20,18] and second order [18,19] in three and two dimensional models, respectively.

The studies described above usually refer only to cases in which the polymer is being pulled with a fixed force. However, in typical experimental pulling setups, like in the atomic force microscope, the conditions correspond to a cantilever that has a certain stifness and that moves with a certain speed. This is neither the fixed force nor the fixed extension situation. Furthermore, the analytically derived results adopt equilibrium conditions whereas the speed with which the cantilever moves may often significantly exceed the relaxation rate of the system, especially at low temperatures. Thus establishing the theoretical pulling rate depen-dence of the stretching process is of interest.

Here, we present results that are based on molecular dynamics simulations of self-attracting Lennard-Jones homopolymers which are stretched by a cantilever moving at a constant speed and at temperatures which can be lower than Tu. The simulations are similar to our previous studies of stretching in a Go-like model of proteins [7-9,21]. We show that the stretching behavior of homopolymers is different from that of proteins in that there are no characteristic large peaks in the F-d curves. Instead, variations in the pattern are more regular, of a smaller scale, and thus more akin to the features found when stretching model secondary structures of proteins [7,21]. This bland behavior arises primarily be-cause, in homopolymers, interactions of all contact orders are present through most of the pulling process. However, the changes brought about by increasing temperature are analogous to those occurring in proteins: the F-d curves gradually lose whatever mild features they had as T rises.

An interesting feature of the force-displacement curves is a pronounced plateau. The plateau grows in extent with increasing chain length and is similar to that observed in an experimental study of PNIPAM [10]. Direct spatial analysis shows that the homopolymer is in a “ball-string” configuration. Such a configuration has already been observed by Maurice and Matthai [22] in molecular-dynamics studies of a homopolymer with the Morse-potential-based interactions under poor solvent conditions and by Kreitmeier et al. [23] in Monte Carlo studies of a Lennard-Jones system. The plateau force corresponds to that needed to pull a monomer out of a compact molten globule into an extended chain. It is thus related to the surface tension of the polymer droplet and decreases slowly as the droplet shrinks. It also decreases with increasing temperature, and disappears above a characteristic unfolding temperature.

We also study the unraveling scenarios by monitoring the distances at which each contact breaks for the last time. We show that the scenario diagrams shift logarithmically with pulling rate and they become increasingly simplified as the temperature is raised. In the entropic limit [9] the rupture distances become strictly monotonic as a function of the con-tact order so that the longest ranged contacts break first. This limiting high temperature behavior is very similar to that found for proteins [9].

The following section describes the model interactions and geometry used in our simulations. Section III presents results for thermal unfolding, force-extension curves, and stretching scenarios. We conclude that despite the lack of specificity, the qualitative stretching behavior of selfinteracting homopolymers is similar to that of proteins.

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