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Simulations of Я-Hairpin Folding Confined to Spherical Pores Using

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Simulations of Я-Hairpin Folding Confined to Spherical Pores Using Simulations of ␤-hairpin folding confined to spherical pores using distributed computing D. K. Klimov*†, D. Newfield‡, and D. Thirumalai*† *Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742; and ‡Parabon Computation, 3930 Walnut Street, Suite 100, Fairfax, VA 22030-4738 Communicated by George H. Lorimer, University of Maryland, College Park, MD, April 12, 2002 (received for review December 18, 2001) 3 ϱ We report the thermodynamics and kinetics of an off-lattice Go radius of gyration of a chain Rg at D (the size of a chain in ␤ Ӎ ␯ model -hairpin from Ig-binding protein confined to an inert bulk solution) to N according to Rg aN . If the chain is ideal, ␯ ϭ ⌬ ϭ ͞ 2 spherical pore. Confinement enhances the stability of the hairpin then 0.5 and FU RTN(a D) . Because of the reduction due to the decrease in the entropy of the unfolded state. Compared in the translational entropy, confinement also increases the free ⌬ Ͼ ⌬ ͞⌬ ϽϽ with their values in the bulk, the rates of hairpin formation increase energy of the native state, i.e., FN 0. If FN FU 1, then in the spherical pore. Surprisingly, the dependence of the rates on localization of a protein in a confined space stabilizes the native the pore radius, Rs, is nonmonotonic. The rates reach a maximum state compared with the bulk. It also follows that there is a range ͞ b Ӎ b at Rs Rg,N 1.5, where Rg,N is the radius of gyration of the folded of D values over which stability is maximized. These arguments ␤-hairpin in the bulk. The denatured state ensemble of the encap- and the explicit calculations of polymers confined to slits (11) sulated ␤-hairpin is highly structured even at substantially ele- suggest that excluded volume effects should not qualitatively vated temperatures. Remarkably, a profound effect of confine- alter the conclusions reached by Zhou and Dill (6). Because ment is evident even when the ␤-hairpin occupies less than a 10th confining an unfolded chain with excluded volume leads to of the sphere volume. Our calculations show that the emergence higher entropy costs (provided a Ͻ D Ͻ N␯a), the change in of substantial structure in the denatured state of proteins in inert stability estimated by assuming that the denatured state is ideal pores is a consequence of confinement. In contrast, the structure of should be a lower bound. the bulk denatured state ensemble depends dramatically on the In this article we examine the effects of confinement on the extent of denaturation. nature of the denatured state ensemble (DSE), folding rates, and stability by studying ␤-hairpin formation in spherical pores of olding in cells occurs in the presence of lipids, carbohydrates, different radii Rs (Fig. 1). We find that confinement, which may Fand other biological molecules, i.e., in a crowded environ- be considered as a model for the hydrophilic cavity in GroEL ment (1, 2). Therefore, many proteins spontaneously fold to their (12), leads to enhancement in the folding rates and stability 3 ϱ native states in the environment, which geometrically restricts compared with their values at Rs . Throughout the paper we 3 ϱ their conformational space (3). There are many reasons for refer to the bulk hairpin when Rs and the terms ‘‘confined’’ considering folding in confined spaces. The recognition and and ‘‘encapsulated’’ hairpins are used interchangeably. Super- encapsulation of a substrate protein by the GroEL molecule scripts b and c refer to bulk and confined cases, respectively. places it in a cylindrical cavity. Much of the annealing action Methods takes place in a dynamic cage that is built by the interactions of GroEL with GroES (4). In the course of protein synthesis the Model. To study confinement effects we use a 16-residue C- ␤ nascent polypeptide chain is confined to a narrow ‘‘tube’’ (5). terminal -hairpin from protein G (Fig. 1), for which experi- Such a narrow space subjects the protein to a tensile force (f ϳ mental (13, 14) and theoretical (15) studies have been reported. ͞ kBT D, where D is the diameter of the pore), which may play a We provide a brief overview of the model and simulation role in mechanically squeezing the newly synthesized protein out technique (see ref. 15 for details) focusing on the relevant of the ribosome. These considerations and other studies (6–9) changes caused by confinement. probing the effect of confinement on protein stability have The off-lattice coarse-grained protein model explicitly in- motivated the present work. cludes the backbone C␣ carbons and side chains, which are Recently, Eggers and Valentine examined the structure of represented as spheres of appropriate van der Walls radii four proteins encapsulated in a silica matrix (8). They showed positioned at the centers of mass of amino acids. The model that, in general, confinement leads to enhanced stability of the includes backbone hydrogen bonds (HBs) between NH and CO folded protein. For example, they surmised that the encapsula- groups, which are treated as virtual moieties between adjacent tion of ␣-lactalbumin results in the increase in the melting C␣ atoms (15). The potential energy of a bulk conformation of Ϸ temperature, Tm,by 30°C. However, the relationship between polypeptide chain Vb is given by the sum of bond-length poten- confinement and enhanced stability of a folded state is not tial, side chain–backbone connectivity potential, bond-angle simple, because changes in the water structure in pores could potential, dihedral angle potential, HB and nonbonded long- alter the interactions stabilizing the native state (9). Neverthe- range potentials (15). less, it seems that confinement generally increases the stability Because our goal is to study the effect of confinement on the ␤ of the native structure. -hairpin formation, the Go model, which includes only native Folding in a crowded cellular environment may be modeled by interactions, should suffice. Previous studies have shown that the BIOPHYSICS confining a polypeptide chain to narrow pores. The enhanced Go models provide a qualitative description of two-state folding stability in pores was explained recently by Zhou and Dill using sequences (16–18). Therefore, we consider amino acid- polymer physics concepts (6). If the denatured state of a polymer dependent interactions only for the side chains that are in chain is assumed to be a random coil, then the free energy cost contact in the wild-type native structure of C-terminal hairpin to localize it in a region of size D (e.g., sphere, slit, or tube) is ⌬ Ϸ ͞ 1/␯ Ͻ ␯ FU RTN(a D) (10), provided that D N a, where a may Abbreviations: DSE, denatured state ensemble; HB, hydrogen bond; HC, hydrophobic be taken to be the average distance between the C␣ atoms (Ϸ3.8 cluster. Å), and N is the number of residues. (This estimate is accurate †To whom reprint requests may be addressed. E-mail: [email protected] or up to a logarithmic factor in D.) The Flory exponent ␯ relates the [email protected]. www.pnas.org͞cgi͞doi͞10.1073͞pnas.072220699 PNAS ͉ June 11, 2002 ͉ vol. 99 ͉ no. 12 ͉ 8019–8024 meric brushes in polymer melts (19). The form of the confining potential in Eq. 3 was chosen because it is well defined (no Ͻ singularities) for all values of R Rs. The precise functional form of Vw(R) is not important for the conclusions of this study as long as Vw(R) is a short-ranged repulsive potential. For example, we ϭ tested the case in which Eq. 3 is replaced with Eq. 1 with r Rs Ϫ R and found that the conclusions of the study do not change qualitatively. Furthermore, it has been shown (11) that the size of a homopolymer confined to slits is the same irrespective of the interaction potential (soft or hard) with the wall. In general, a soft wall can always be replaced by a hard wall with a slightly larger range of interactions (20). However, the physics of folding would be altered if the confining potential is long-ranged or contains attractive terms. The potential energy of the confined hairpin is the sum of Vb and the energies Vw for all hairpin atoms. In this study we Ն ϭ ϭ b b consider Rs Rs,0 4.475a 1.40Rg,N, where Rg,N is the radius of gyration of the bulk native conformation. The value of Rs,0 is the smallest radius of the confining sphere, which minimally perturbs the bulk native structure. Numerical Simulations. Langevin simulations based on the velocity form of the Verlet algorithm and the multiple histogram tech- nique were used to compute the equilibrium properties of confined hairpins (15). Folding kinetics in a sphere was obtained by using Langevin dynamics at approximately water viscosity. Hundreds of folding trajectories were generated at each value of ϭ ͞ ͚M ␶Ϫ1 Rs to calculate the folding rate kF 1 M iϭ1 1i , where M is ␶ the total number of trajectories and 1i is the first passage time Fig. 1. Encapsulation of 16-mer C-terminal ␤-hairpin from the protein G in to the native state. To assess the heterogeneity of the folding a spherical pore of the radius Rs is sketched. The surprising outcome of this pathways we computed the fraction of unfolded molecules at ͞ b տ ϭ Ϫ͐t ϭ ͞ ͚M ␦ Ϫ ␶ work is that even confinement to relatively large pores (Rs Rg,N 2.0) has a time tPu(t) 1 0 Pfp(s)ds, where Pfp 1 M iϭ1 (s 1i) profound effect on the hairpin stability and folding rates.
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