Chirality and Protein Folding

There are several simple criteria of folding to a native state in model proteins. One of them involves crossing of a threshold value of the RMSD distance away from the native state. Another checks whether all native contacts are established, i.e. whether the interacting amino acids come closer than some characteristic distance. We use Go-like models of proteins and show that such simple criteria may prompt one to declare folding even though fragments of the resulting conformations have a wrong sense of chirality. We propose that a better condition of folding should augment the simple criteria with the requirement that most of the local values of the chirality should be nearly native. The kinetic discrepancy between the simple and compound criteria can be substantially reduced in the Go-like models by providing the Hamiltonian with a term which favors native values of the local chirality. We study the effects of this term as a function of its amplitude and compare it to other models such as with the side groups and with the angle-dependent potentials.Comment: To be published in a special issue of J. Phys.: Cond. Mat. (Bedlewo Workshop

Delineation of the Native Basin in Continuum Models of Proteins

We propose two approaches for determining the native basins in off-lattice models of proteins. The first of them is based on exploring the saddle points on selected trajectories emerging from the native state. In the second approach, the basin size can be determined by monitoring random distortions in the shape of the protein around the native state. Both techniques yield the similar results. As a byproduct, a simple method to determine the folding temperature is obtained.Comment: REVTeX, 6 pages, 5 EPS figure

Mechanical Stretching of Proteins: Calmodulin and Titin

Mechanical unfolding of several domains of calmodulin and titin is studied using a Go-like model with a realistic contact map and Lennard-Jones contact interactions. It is shown that this simple model captures the experimentally observed difference between the two proteins: titin is a spring that is tough and strong whereas calmodulin acts like a weak spring with featureless force-displacement curves. The difference is related to the dominance of the alpha secondary structures in the native structure of calmodulin. The tandem arrangements of calmodulin unwind simultaneously in each domain whereas the domains in titin unravel in a serial fashion. The sequences of contact events during unraveling are correlated with the contact order, i.e. with the separation between contact making amino acids along the backbone in the native state. Temperature is found to affect stretching in a profound way.Comment: To be published in a special bio-issue of Physica A; 14 figure

Viscosity Dependence of the Folding Rates of Proteins

The viscosity dependence of the folding rates for four sequences (the native state of three sequences is a beta-sheet, while the fourth forms an alpha-helix) is calculated for off-lattice models of proteins. Assuming that the dynamics is given by the Langevin equation we show that the folding rates increase linearly at low viscosities \eta, decrease as 1/\eta at large \eta and have a maximum at intermediate values. The Kramers theory of barrier crossing provides a quantitative fit of the numerical results. By mapping the simulation results to real proteins we estimate that for optimized sequences the time scale for forming a four turn \alpha-helix topology is about 500 nanoseconds, whereas the time scale for forming a beta-sheet topology is about 10 microseconds.Comment: 14 pages, Latex, 3 figures. One figure is also available at http://www.glue.umd.edu/~klimov/seq_I_H.html, to be published in Physical Review Letter

Investigation of routes and funnels in protein folding by free energy functional methods

We use a free energy functional theory to elucidate general properties of heterogeneously ordering, fast folding proteins, and we test our conclusions with lattice simulations. We find that both structural and energetic heterogeneity can lower the free energy barrier to folding. Correlating stronger contact energies with entropically likely contacts of a given native structure lowers the barrier, and anticorrelating the energies has the reverse effect. Designing in relatively mild energetic heterogeneity can eliminate the barrier completely at the transition temperature. Sequences with native energies tuned to fold uniformly, as well as sequences tuned to fold by a single or a few routes, are rare. Sequences with weak native energetic heterogeneity are more common; their folding kinetics is more strongly determined by properties of the native structure. Sequences with different distributions of stability throughout the protein may still be good folders to the same structure. A measure of folding route narrowness is introduced which correlates with rate, and which can give information about the intrinsic biases in ordering due to native topology. This theoretical framework allows us to systematically investigate the coupled effects of energy and topology in protein folding, and to interpret recent experiments which investigate these effects.Comment: 12 pages, 1 figure, to appear in Proc. Natl. Acad. Sc

Thermodynamic stability of folded proteins against mutations

By balancing the average energy gap with its typical change due to mutations for protein-like heteropolymers with M residues, we show that native states are unstable to mutations on a scale M* ~ (lambda/sigma_mu)^(1/zeta_s), where lambda is the dispersion in the interaction free energies and sigma_mu their typical change. Theoretical bounds and numerical estimates (based on complete enumeration on four lattices) of the instability exponent zeta_s are given. Our analysis suggests that a limiting size of single-domain proteins should exist, and leads to the prediction that small proteins are insensitive to random mutations.Comment: 5 pages, 3 figures, to be published in Physical Review Letter

Mechanical unfolding of RNA: From hairpins to structures with internal multiloops

Mechanical unfolding of RNA structures, ranging from hairpins to ribozymes, using laser optical tweezer (LOT) experiments have begun to reveal the features of the energy landscape that cannot be easily explored using conventional experiments. Upon application of constant force ($f$), RNA hairpins undergo cooperative transitions from folded to unfolded states whereas subdomains of ribozymes unravel one at a time. Here, we use a self-organized polymer (SOP) model and Brownian dynamics simulations to probe mechanical unfolding at constant force and constant-loading rate of four RNA structures of varying complexity. Our work shows (i) the response of RNA to force is largely determined by the native structure; (ii) only by probing mechanical unfolding over a wide range of forces can the underlying energy landscape be fully explored.Comment: 26 pages, 6 figures, Biophys. J. (in press

Folding in two-dimenensional off-lattice models of proteins

Model off-lattice sequences in two dimensions are constructed so that their native states are close to an on-lattice target. The Hamiltonian involves the Lennard-Jones and harmonic interactions. The native states of these sequences are determined with a high degree of certainty through Monte Carlo processes. The sequences are characterized thermodynamically and kinetically. It is shown that the rank-ordering-based scheme of the assignment of contact energies typically fails in off-lattice models even though it generates high stability of on-lattice sequences. Similar to the on-lattice case, Go-like modeling, in which the interaction potentials are restricted to the native contacts in a target shape, gives rise to good folding properties. Involving other contacts deteriorates these properties.Comment: REVTeX, 9 pages, 8 EPS figure

Forced-unfolding and force-quench refolding of RNA hairpins

Using coarse-grained model we have explored forced-unfolding of RNA hairpin as a function of $f_S$ and the loading rate ($r_f$). The simulations and theoretical analysis have been done without and with the handles that are explicitly modeled by semiflexible polymer chains. The mechanisms and time scales for denaturation by temperature jump and mechanical unfolding are vastly different. The directed perturbation of the native state by $f_S$ results in a sequential unfolding of the hairpin starting from their ends whereas thermal denaturation occurs stochastically. From the dependence of the unfolding rates on $r_f$ and $f_S$ we show that the position of the unfolding transition state (TS) is not a constant but moves dramatically as either $r_f$ or $f_S$ is changed. The TS movements are interpreted by adopting the Hammond postulate for forced-unfolding. Forced-unfolding simulations of RNA, with handles attached to the two ends, show that the value of the unfolding force increases (especially at high pulling speeds) as the length of the handles increases. The pathways for refolding of RNA from stretched initial conformation, upon quenching $f_S$ to the quench force $f_Q$, are highly heterogeneous. The refolding times, upon force quench, are at least an order of magnitude greater than those obtained by temperature quench. The long $f_Q$-dependent refolding times starting from fully stretched states are analyzed using a model that accounts for the microscopic steps in the rate limiting step which involves the trans to gauche transitions of the dihedral angles in the GAAA tetraloop. The simulations with explicit molecular model for the handles show that the dynamics of force-quench refolding is strongly dependent on the interplay of their contour length and the persistence length, and the RNA persistence length.Comment: 42 pages, 15 figures, Biophys. J. (in press