We present a method to investigate the kinetics of protein folding on a long time-scale and the dynamics underlying the formation of secondary and tertiary structures during the entire reaction. The approach is based on the formal analogy between thermal and quantum diffusion: by writing the solution of the Fokker-Planck equation for the time-evolution of a protein in a viscous heat-bath in terms of a path integral, we derive a Hamilton-Jacobi variational principle from which we are able to compute the most probable pathway of folding. The method is applied to the folding of the Villin Headpiece Subdomain, in the framework of a Go-model. We have found that, in this model, the transition occurs through an initial collapsing phase driven by the starting coil configuration and a later rearrangement phase, in which secondary structures are formed and all computed paths display strong similarities. This method is completely general, does not require the prior knowledge of any reaction coordinate and represents an efficient tool to perfom ab-initio simulations of the entire folding process with available computers.Understanding the kinetics of protein folding and the dynamical mechanisms involved in the formation of their structures in an all-atom approach involves simulating a statistically significant ensemble of folding trajectories for a system of ∼ 10 4 degrees of freedom. Unfortunately, the existence of a huge gap between the microscopic time-scale of the rotational degrees of freedom ∼ 10 −12 s and the macroscopic time scales of the full folding process ∼ 10 −6 − 10 1 s makes it extremely computationally challenging to follow the evolution of a typical ∼ 100-residue protein for a time interval longer than few tens of nanoseconds.Several approaches have been proposed to overcome such computational difficulties and address the problem of identifying the relevant pathways of the folding reaction [2]. Unfortunately these methods are either affected by uncontrolled systematic errors associated to ad-hoc approximations, or can only be applied to small proteins with typical folding time of the order of few nanoseconds (fast folders). In this Letter we present a novel approach to overcome these difficulties: we adopt the Langevin approach and devise a method to rigorously define and practically compute the most statistically relevant protein folding pathway. As a first exploratory application, we have studied the folding transition of the 36-monomer Villin Headpiece Subdomain (PDB code 1VII). This molecule has been extensively studied in the literature because it is the smallest polypeptide that has all of the properties of a single domain protein and in addition, it is one of the fastest folders [3]. The ribbon representation of this system is shown in Fig.1. We analyze the transition from different random selfavoiding coil states to the native state, whose structure was obtained from the Brookhaven Protein Data Bank.Our study is based on the analogy between Langevin diffusion and quantum propagation. Previous studies...

The aggregation behavior of two bile acid salts (i.e., sodium cholate and sodium deoxycholate) has been studied in their aqueous solutions of three different concentrations (i.e., 30, 90,and 300 mM) by means of molecular dynamics computer simulations. To let the systems reach thermodynamic equilibrium, rather long simulations have been performed: the equilibration period, lasting for 20-50 ns, has been followed by a 20 ns long production phase, during which the average size of the bile aggregates (regarded to be the slowest varying observable) has already fluctuated around a constant value. The production phase of the runs has been about an order of magnitude longer than the average lifetime of both the monomeric bile ions and the bonds that link two neighboring bile ions together to be part of the same aggregate. This has allowed the bile ions belonging to various aggregates to be in a dynamic equilibrium with the isolated monomers. The observed aggregation behavior of the studied bile ions has been found to be in good qualitative agreement with experimental findings. The analysis of the results has revealed that, due to their molecular structure, which is markedly different from that of the ordinary aliphatic surfactants, the bile ions form rather different aggregates than the usual spherical micelles. In the lowest concentration solution studied, the bile ions only form small oligomers. In the case of deoxycholate, these oligomers, such as the ordinary micelles, are kept together by hydrophobic interactions, whereas in the sodium cholate system, small hydrogen-bonded aggregates (mostly dimers) are also present. In the highest concentration systems, the bile ions form large secondary micelles, which are kept together both by hydrophobic interactions and by hydrogen bonds. Namely, in these secondary micelles, small hydrophobic primary micelles are linked together via the formation of hydrogen bonds between their hydrophilic outer surfaces.

We present a generalized version of the ITIM algorithm for the identification of interfacial molecules, which is able to treat arbitrarily shaped interfaces. The algorithm exploits the similarities between the concept of probe sphere used in ITIM and the circumsphere criterion used in the α-shapes approach, and can be regarded either as a reference-frame independent version of the former, or as an extended version of the latter that includes the atomic excluded volume. The new algorithm is applied to compute the intrinsic orientational order parameters of water around a dodecylphosphocholine and a cholic acid micelle in aqueous environment, and to the identification of solvent-reachable sites in four model structures for soot. The additional algorithm introduced for the calculation of intrinsic density profiles in arbitrary geometries proved to be extremely useful also for planar interfaces, as it allows to solve the paradox of smeared intrinsic profiles far from the interface.

We develop a theoretical approach to the protein folding problem based on out-of-equilibrium stochastic dynamics. Within this framework, the computational difficulties related to the existence of large time scale gaps in the protein folding problem are removed and simulating the entire reaction in atomistic details using existing computers becomes feasible. In addition, this formalism provides a natural framework to investigate the relationships between thermodynamical and kinetic aspects of the folding. For example, it is possible to show that, in order to have a large probability to remain unchanged under Langevin diffusion, the native state has to be characterized by a small conformational entropy. We discuss how to determine the most probable folding pathway, to identify configurations representative of the transition state and to compute the most probable transition time. We perform an illustrative application of these ideas, studying the conformational evolution of alanine di-peptide, within an all-atom model based on the empiric GROMOS96 force field.A critical part of the protein-folding problem is to understand its kinetics and the underlying physical processes. To this aim, several different theoretical methods have been recently developed, spanning from analytical approaches[1, 2, 3] to detailed computer simulations [4,5,6]. A major problem in simulating the folding process using standard molecular dynamics (MD) is the huge gap between the time scale of "elementary moves", of the order of 10-100 ps, and that of the entire folding process, which ranges from a few microseconds for fast-folders [7], up to several seconds or even minutes for more complex proteins. This peculiarity of the folding process makes the brute-force molecular dynamics approach too demanding, and a substantial part of the efforts in the field of protein folding simulation aims at bridging this gap.In a recent paper [8] we have presented a novel theoretical framework for investigating the folding dynamics, named hereafter Dominant Folding Pathways (DFP), which is based on a reformulation in terms of path integrals of the dynamics described by the Langevin equation. The DFP analysis allows to compute rigorously (i.e. without any assumptions other than the validity of the underlying Langevin equation) the most probable conformational pathway connecting an arbitrary initial conformation to an arbitrary final conformation. The major advantage of the method is the possibility of bypassing the computational difficulties associated with the existence of different time scales in the problem, while retaining the ability to recover information on the time evolution of the system. As we shall see, the resulting computational simplification is dramatic and makes it feasible to study the formation pattern of conformational structures along the entire folding process using realistic all-atom force fields, on available computers.In this Letter we further develop our formalism and we present the first DFP simulation performed in full atomistic deta...

We present the results of a combined metadynamics-umbrella sampling investigation of the puckered conformers of pyranoses described using the gromos 45a4 force field. The free energy landscape of Cremer-Pople puckering coordinates has been calculated for the whole series of α and β aldohexoses, showing that the current force field parameters fail in reproducing proper puckering free energy differences between chair conformers. We suggest a modification to the gromos 45a4 parameter set which improves considerably the agreement of simulation results with theoretical and experimental estimates of puckering free energies. We also report on the experimental measurement of altrose conformers populations by means of NMR spectroscopy, which show good agreement with the predictions of current theoretical models.

Optical Tweezers are employed to study the electrophoretic and the electroosmotic motion of a single colloid immersed in electrolyte solutions of ion concentrations between 10(-5) and 1 mol/l and of different valencies (KCl, CaCl(2), LaCl(3)). The measured particle mobility in monovalent salt is found to be in agreement with computations combining primitive model molecular dynamics simulations of the ionic double layer with the standard electrokinetic model. Mobility reversal of a single colloid-for the first time-is observed in the presence of trivalent ions (LaCl(3)) at ionic strengths larger than 10(-2) mol/l. In this case, our numerical model is in a quantitative agreement with the experiment only when ion specific attractive forces are added to the primitive model, demonstrating that at low colloidal charge densities, ion correlation effects alone do not suffice to produce mobility reversal.

The relative arrangement of the neighboring bile ions and the shape of the hydrophobic and hydrogen-bonded primary micelles as well of the large secondary micelles formed by these ions are analyzed in detail on the basis of molecular dynamics computer simulations of 30 and 300 mM sodium cholate and sodium deoxycholate solutions. In the lower concentration considered, the systems only contain primary micelles, whereas in both of the 300 mM systems secondary micelles are also present. The simulations performed were long enough that the systems reached thermodynamic equilibrium. It is found that the neighboring cholate ions prefer alignments in which their quasi-planar tetracyclic ring systems are parallel with each other, whereas for deoxycholate an opening of the angle between these planes is observed. The shape of the micelles is characterized by the ratio of their three principal moments of inertia. The primary deoxycholate micelles are found to be rather spherical, whereas in the case of cholate somewhat flattened, disklike or oblate shaped ellipsoidal primary micelles are found, irrespective of whether these micelles are kept together by hydrogen bonds or are of hydrophobic origin. Finally, the secondary micelles are found to exhibit a large variety of shapes, ranging from flattened oblates to rodlike objects through various different irregular shapes, characterized by markedly different values of the three principal moments of inertia. The observed preferences of the relative arrangement of the neighboring ions and of the aggregate shapes as well as the differences observed in the behavior of the two bile ions studied in these respects are traced back to the molecular structure of these ions.

Simulating coarse-grained models of charged soft-condensed matter systems in presence of dielectric discontinuities between different media requires an efficient calculation of polarization effects. This is almost always the case if implicit solvent models are used near interfaces or large macromolecules. We present a fast and accurate method (ICC( small star, filled)) that allows to simulate the presence of an arbitrary number of interfaces of arbitrary shape, each characterized by a different dielectric permittivity in one-, two-, and three-dimensional periodic boundary conditions. The scaling behavior and accuracy of the underlying electrostatic algorithms allow to choose the most appropriate scheme for the system under investigation in terms of precision and computational speed. Due to these characteristics the method is particularly suited to include nonplanar dielectric boundaries in coarse-grained molecular dynamics simulations.