Refolding dynamics of stretched biopolymers upon force quench

Single molecule force spectroscopy methods can be used to generate folding trajectories of biopolymers from arbitrary regions of the folding landscape. We illustrate the complexity of the folding kinetics and generic aspects of the collapse of RNA and proteins upon force quench, using simulations of an RNA hairpin and theory based on the de Gennes model for homopolymer collapse. The folding time, $\tau_F$, depends asymmetrically on $\delta f_S = f_S - f_m$ and $\delta f_Q = f_m - f_Q$ where $f_S$ ($f_Q$) is the stretch (quench) force, and $f_m$ is the transition mid-force of the RNA hairpin. In accord with experiments, the relaxation kinetics of the molecular extension, $R(t)$, occurs in three stages: a rapid initial decrease in the extension is followed by a plateau, and finally an abrupt reduction in $R(t)$ that occurs as the native state is approached. The duration of the plateau increases as $\lambda =\tau_Q/\tau_F$ decreases (where $\tau_Q$ is the time in which the force is reduced from $f_S$ to $f_Q$). Variations in the mechanisms of force quench relaxation as $\lambda$ is altered are reflected in the experimentally measurable time-dependent entropy, which is computed directly from the folding trajectories. An analytical solution of the de Gennes model under tension reproduces the multistage stage kinetics in $R(t)$. The prediction that the initial stages of collapse should also be a generic feature of polymers is validated by simulation of the kinetics of toroid (globule) formation in semiflexible (flexible) homopolymers in poor solvents upon quenching the force from a fully stretched state. Our findings give a unified explanation for multiple disparate experimental observations of protein folding.Comment: 31 pages 11 figure

Simulation of Lattice Polymers with Multi-Self-Overlap Ensemble

A novel family of dynamical Monte Carlo algorithms for lattice polymers is proposed. Our central idea is to simulate an extended ensemble in which the self-avoiding condition is systematically weakened. The degree of the self-overlap is controlled in a similar manner as the multicanonical ensemble. As a consequence, the ensemble --the multi-self-overlap ensemble-- contains adequate portions of self-overlapping conformations as well as higher energy ones. It is shown that the multi-self-overlap ensemble algorithm reproduce correctly the canonical averages at finite temperatures of the HP model of lattice proteins. Moreover, it outperforms massively a standard multicanonical algorithm for a difficult example of a polymer with 8-stickers. Alternative algorithm based on exchange Monte Carlo method is also discussed.Comment: 5 Pages, 4 Postscript figures, uses epsf.st

Reply to Comment on "Criterion that Determines the Foldability of Proteins"

We point out that the correlation between folding times and $\sigma = (T_{\theta } - T_{f})/T_{\theta }$ in protein-like heteropolymer models where $T_{\theta }$ and $T_{f}$ are the collapse and folding transition temperatures was already established in 1993 before the other presumed equivalent criterion (folding times correlating with $T_{f}$ alone) was suggested. We argue that the folding times for these models show no useful correlation with the energy gap even if restricted to the ensemble of compact structures as suggested by Karplus and Shakhnovich (cond-mat/9606037).Comment: 6 pages, Latex, 2 Postscript figures. Plots explicitly showing the lack of correlation between folding time and energy gap are adde

Structure-function mapping of a heptameric module in the nuclear pore complex.

The nuclear pore complex (NPC) is a multiprotein assembly that serves as the sole mediator of nucleocytoplasmic exchange in eukaryotic cells. In this paper, we use an integrative approach to determine the structure of an essential component of the yeast NPC, the ~600-kD heptameric Nup84 complex, to a precision of ~1.5 nm. The configuration of the subunit structures was determined by satisfaction of spatial restraints derived from a diverse set of negative-stain electron microscopy and protein domain-mapping data. Phenotypic data were mapped onto the complex, allowing us to identify regions that stabilize the NPC's interaction with the nuclear envelope membrane and connect the complex to the rest of the NPC. Our data allow us to suggest how the Nup84 complex is assembled into the NPC and propose a scenario for the evolution of the Nup84 complex through a series of gene duplication and loss events. This work demonstrates that integrative approaches based on low-resolution data of sufficient quality can generate functionally informative structures at intermediate resolution

Energy landscapes, supergraphs, and "folding funnels" in spin systems

Dynamical connectivity graphs, which describe dynamical transition rates between local energy minima of a system, can be displayed against the background of a disconnectivity graph which represents the energy landscape of the system. The resulting supergraph describes both dynamics and statics of the system in a unified coarse-grained sense. We give examples of the supergraphs for several two dimensional spin and protein-related systems. We demonstrate that disordered ferromagnets have supergraphs akin to those of model proteins whereas spin glasses behave like random sequences of aminoacids which fold badly.Comment: REVTeX, 9 pages, two-column, 13 EPS figures include

Path Integral Approach to Strongly Nonlinear Composite

We study strongly nonlinear disordered media using a functional method. We solve exactly the problem of a nonlinear impurity in a linear host and we obtain a Bruggeman-like formula for the effective nonlinear susceptibility. This formula reduces to the usual Bruggeman effective medium approximation in the linear case and has the following features: (i) It reproduces the weak contrast expansion to the second order and (ii) the effective medium exponent near the percolation threshold are $s=1$, $t=1+\kappa$, where $\kappa$ is the nonlinearity exponent. Finally, we give analytical expressions for previously numerically calculated quantities.Comment: 4 pages, 1 figure, to appear in Phys. Rev.

Optimal shapes of compact strings

Optimal geometrical arrangements, such as the stacking of atoms, are of relevance in diverse disciplines. A classic problem is the determination of the optimal arrangement of spheres in three dimensions in order to achieve the highest packing fraction; only recently has it been proved that the answer for infinite systems is a face-centred-cubic lattice. This simply stated problem has had a profound impact in many areas, ranging from the crystallization and melting of atomic systems, to optimal packing of objects and subdivision of space. Here we study an analogous problem--that of determining the optimal shapes of closely packed compact strings. This problem is a mathematical idealization of situations commonly encountered in biology, chemistry and physics, involving the optimal structure of folded polymeric chains. We find that, in cases where boundary effects are not dominant, helices with a particular pitch-radius ratio are selected. Interestingly, the same geometry is observed in helices in naturally-occurring proteins.Comment: 8 pages, 3 composite ps figure

The PDZ domain of the SpoIVB serine peptidase facilitates multiple functions

During spore formation in Bacillus subtilis, the SpoIVB protein is a critical component of the sigma (K) regulatory checkpoint. SpoIVB has been shown to be a serine peptidase that is synthesized in the spore chamber and which self-cleaves, releasing active forms. These forms can signal proteolytic processing of the transcription factor sigma (K) in the outer mother cell chamber of the sporulating cell. This forms the basis of the sigma (K) checkpoint and ensures accurate sigma (K)-controlled gene expression. SpoIVB has also been shown to activate a second distinct process, termed the second function, which is essential for the formation of heat-resistant spores. In addition to the serine peptidase domain, SpoIVB contains a PDZ domain. We have altered a number of conserved residues in the PDZ domain by site-directed mutagenesis and assayed the sporulation phenotype and signaling properties of mutant SpoIVB proteins. Our work has revealed that the SpoIVB PDZ domain could be used for up to four distinct processes, (i) targeting of itself for trans proteolysis, (11) binding to the protease inhibitor BofC, (iii) signaling of pro-sigma (K) processing, and (iv) signaling of the second function of SpoIVB

Protein folding using contact maps

We present the development of the idea to use dynamics in the space of contact maps as a computational approach to the protein folding problem. We first introduce two important technical ingredients, the reconstruction of a three dimensional conformation from a contact map and the Monte Carlo dynamics in contact map space. We then discuss two approximations to the free energy of the contact maps and a method to derive energy parameters based on perceptron learning. Finally we present results, first for predictions based on threading and then for energy minimization of crambin and of a set of 6 immunoglobulins. The main result is that we proved that the two simple approximations we studied for the free energy are not suitable for protein folding. Perspectives are discussed in the last section.Comment: 29 pages, 10 figure

Protein structures and optimal folding emerging from a geometrical variational principle

Novel numerical techniques, validated by an analysis of barnase and chymotrypsin inhibitor, are used to elucidate the paramount role played by the geometry of the protein backbone in steering the folding to the correct native state. It is found that, irrespective of the sequence, the native state of a protein has exceedingly large number of conformations with a given amount of structural overlap compared to other compact artificial backbones; moreover the conformational entropies of unrelated proteins of the same length are nearly equal at any given stage of folding. These results are suggestive of an extremality principle underlying protein evolution, which, in turn, is shown to be associated with the emergence of secondary structures.Comment: Revtex, 5 pages, 5 postscript figure