Equilibrium self-assembly of small RNA viruses.

We propose a description for the quasiequilibrium self-assembly of small, single-stranded (ss) RNA viruses whose capsid proteins (CPs) have flexible, positively charged, disordered tails that associate with the negatively charged RNA genome molecules. We describe the assembly of such viruses as the interplay between two coupled phase-transition-like events: the formation of the protein shell (the capsid) by CPs and the condensation of a large ss viral RNA molecule. Electrostatic repulsion between the CPs competes with attractive hydrophobic interactions and attractive interaction between neutralized RNA segments mediated by the tail groups. An assembly diagram is derived in terms of the strength of attractive interactions between CPs and between CPs and the RNA molecules. It is compared with the results of recent studies of viral assembly. We demonstrate that the conventional theory of self-assembly, which does describe the assembly of empty capsids, is in general not applicable to the self-assembly of RNA-encapsidating virions.

Nonequilibrium statistical mechanics of mixtures of particles in contact with different thermostats.

We introduce a novel type of locally driven systems made of two types of particles (or a polymer with two types of monomers) subject to a chaotic drive with approximately white noise spectrum, but different intensity; in other words, particles of different types are in contact with thermostats at different temperatures. We present complete systematic statistical mechanics treatment starting from first principles. Although we consider only corrections to the dilute limit due to pairwise collisions between particles, meaning we study a nonequilibrium analog of the second virial approximation, we find that the system exhibits a surprisingly rich behavior. In particular, pair correlation function of particles has an unusual quasi-Boltzmann structure governed by an effective temperature distinct from that of any of the two thermostats. We also show that at sufficiently strong drive the uniformly mixed system becomes unstable with respect to steady states consisting of phases enriched with different types of particles. In the second virial approximation, we define nonequilibrium "chemical potentials" whose gradients govern diffusion fluxes and a nonequilibrium "osmotic pressure," which governs the mechanical stability of the interface.

Network Formation by Cross-Hybridization of Complementary Strands to Grafted ssDNA.

When a low density brush of single-stranded DNA (ssDNA) targets end-grafted to a surface is immersed in a solution of complementary ssDNA probes, a regular brush of DNA duplexes is formed by 1:1 hybridization between probe and target DNA. We suggest that in higher density brushes of ssDNA this process competes with cross-hybridization of a target strand to several neighboring probe strands resulting in the formation of a cross-linked DNA network. We analyze a simple 2D model of a dense DNA brush and use analytic methods and computer simulations to find how the conditions for network formation depend on system size and DNA length. We argue that in 3D brushes cross-hybridization will nearly always lead to network formation and suggest that this may explain some intriguing results on dense DNA brushes. Experiments on DNA monolayers and concentrated DNA solutions that could test our predictions are proposed.

Adsorption kinetics of a single polymer on a solid plane.

We study analytically and by means of an off-lattice bead-spring dynamic Monte Carlo simulation model the adsorption kinetics of a single macromolecule on a structureless flat substrate in the regime of strong physisorption. The underlying notion of a "stem-flower" polymer conformation, and the related mechanism of "zipping" during the adsorption process are shown to lead to a Fokker-Planck equation with reflecting boundary conditions for the time-dependent probability distribution function (PDF) of the number of adsorbed monomers. The theoretical treatment predicts that the mean fraction of adsorbed segments grows with time as a power law with a power of (1+nu)-1, where nu approximately 3/5 is the Flory exponent. The instantaneous distribution of train lengths is predicted to follow an exponential relationship. The corresponding PDFs for loops and tails are also derived. The complete solution for the time-dependent PDF of the number of adsorbed monomers is obtained numerically from the set of discrete coupled differential equations and shown to be in perfect agreement with the Monte Carlo simulation results. In addition to homopolymer adsorption, we also study regular multiblock copolymers and random copolymers, and demonstrate that their adsorption kinetics may be considered within the same theoretical model.

Limits of analogy between self-avoidance and topology-driven swelling of polymer loops.

The work addresses the analogy between trivial knotting and excluded volume in looped polymer chains of moderate length, where the effects of knotting are small. A simple expression for the swelling seen in trivially knotted loops is described and shown to agree with simulation data. Contrast between this expression and the well-known expression for excluded volume polymers leads to a graphical mapping of excluded volume to trivial knots, which may be useful for understanding where the analogy between the two physical forms is valid. The work also includes description of a new method for the computational generation of polymer loops via conditional probability. Although computationally intensive, this method generates loops without statistical bias, and thus is preferable to other loop generation routines in the region of interest.

Primary sequences of proteinlike copolymers: Levy-flight-type long-range correlations.

We consider the statistical properties of primary sequences of two-letter HP copolymers (H for hydrophobic and P for polar) designed to have water soluble globular conformations with H monomers shielded from water inside the shell of P monomers. We show, both by computer simulations and by exact analytical calculation, that for large globules and flexible polymers such sequences exhibit long-range correlations which can be described by Levy-flight statistics.

Free energy self-averaging in protein-sized random heteropolymers.

Current theories of heteropolymers are inherently macroscopic, but are applied to mesoscopic proteins. To compute the free energy over sequences, one assumes self-averaging--a property established only in the macroscopic limit. By enumerating the states and energies of compact 18, 27, and 36mers on a lattice with an ensemble of random sequences, we test the self-averaging approximation. We find that fluctuations in the free energy between sequences are weak, and that self-averaging is valid at the scale of real proteins. The results validate sequence design methods which exponentially speed up computational design and simplify experimental realizations.

Unexpected scenario of glass transition in polymer globules: an exactly enumerable model.

We introduce a lattice model of glass transition in polymer globules. This model exhibits ergodicity breaking in which the disjoint regions of phase space do not arise uniformly, but as small chambers whose number increases exponentially with polymer density. Chamber sizes obey power law distribution, making phase space similar to a fractal foam. This clearly demonstrates the importance of the phase space geometry and topology in describing any glass-forming system, such as semicompact polymers during protein folding.

Random walks in the space of conformations of toy proteins.

Monte Carlo dynamics of the lattice toy protein of 48 monomers is interpreted as a random walk in an abstract (discrete) space of conformations. To test the geometry of this space, we examine the return probability P(T), which is the probability to find the polymer in the native state after T Monte Carlo steps, provided that it starts from the native state at the initial moment. Comparing computational data with the theoretical expressions for P(T) for random walks in a variety of different spaces, we show that conformation spaces of polymer loops may have nontrivial dimensions and exhibit negative curvature characteristics of Lobachevskii (hyperbolic) geometry.

Closed loops of nearly standard size: common basic element of protein structure.

By screening the crystal protein structure database for close Calpha-Calpha contacts, a size distribution of the closed loops is generated. The distribution reveals a maximum at 27+/-5 residues, the same for eukaryotic and prokaryotic proteins. This is apparently a consequence of polymer statistic properties of protein chain trajectory. That is, closure into the loops depends on the flexibility (persistence length) of the chain. The observed preferential loop size is consistent with the theoretical optimal loop closure size. The mapping of the detected unit-size loops on the sequences of major typical folds reveals an almost regular compact consecutive arrangement of the loops. Thus, a novel basic element of protein architecture is discovered; structurally diverse closed loops of the particular size.

Reversible molecular adsorption based on multiple-point interaction by shrinkable gels.

A general approach is presented for creating polymer gels that can recognize and capture a target molecule by multiple-point interaction and that can reversibly change their affinity to the target by more than one order of magnitude. The polymers consist of majority monomers that make the gel reversibly swell and shrink and minority monomers that constitute multiple-point adsorption centers for the target molecule. Multiple-point interaction is experimentally proven by power laws found between the affinity and the concentration of the adsorbing monomers within the gels.

Models of protein interactions: how to choose one.

BACKGROUND: There have been many attempts to approximate realistic protein interaction energies by coarse graining (i.e. considering interactions between amino acids rather than those between atoms). In particular, many 20-letter models have been derived (corresponding to the 20 naturally occurring amino acids). Because such models remain computationally infeasible, many two-letter models have been proposed as further simplifications. The choice of which model to use remains arbitrary, however. In this work, we formulate the framework within which the quality of approximate interaction potentials with respect to folding can be defined explicitly. RESULTS: Using a recently proposed criterion for comparing interaction matrices, we compare various 20 x 20 interaction matrices and obtain the two-letter model that most closely approximates each 20 x 20 matrix. We find that there are considerable differences among the 20 x 20 matrices. In particular, some matrices are much more similar to the hydrophobic model than others. Furthermore, we find that although the best two-letter approximation of a 20-letter model is a significantly better approximation than a random two-letter model, it is still a poor approximation of realistic protein interactions. CONCLUSIONS: The determination of the best two-letter approximations of various 20-letter models of protein interaction energies reveals the degree to which hydrophobic interactions dominate in each of the models and hence in proteins.

Statistical mechanics of simple models of protein folding and design.

It is now believed that the primary equilibrium aspects of simple models of protein folding are understood theoretically. However, current theories often resort to rather heavy mathematics to overcome some technical difficulties inherent in the problem or start from a phenomenological model. To this end, we take a new approach in this pedagogical review of the statistical mechanics of protein folding. The benefit of our approach is a drastic mathematical simplification of the theory, without resort to any new approximations or phenomenological prescriptions. Indeed, the results we obtain agree precisely with previous calculations. Because of this simplification, we are able to present here a thorough and self contained treatment of the problem. Topics discussed include the statistical mechanics of the random energy model (REM), tests of the validity of REM as a model for heteropolymer freezing, freezing transition of random sequences, phase diagram of designed ("minimally frustrated") sequences, and the degree to which errors in the interactions employed in simulations of either folding and design can still lead to correct folding behavior.

Nonrandomness in protein sequences: evidence for a physically driven stage of evolution?

The sequences, or primary structures, of existing biopolymers--in particular, proteins--are believed to be a product of evolution. Are the sequences random? If not, what is the character of this nonrandomness? To explore the statistics of protein sequences, we use the idea of mapping the sequence onto the trajectory of a random walk, originally proposed by Peng et al. [Peng, C.-K., Buldyrev, S. V., Goldberger, A. L., Havlin, S., Sciortino, F., Simons, M. & Stanley, H. E. (1992) Nature (London) 356, 168-170] in their analysis of DNA sequences. Using three different mappings, corresponding to three basic physical interactions between amino acids, we found pronounced deviations from pure randomness, and these deviations seem directed toward minimization of the energy of the three-dimensional structure. We consider this result as evidence for a physically driven stage of evolution.

Thermodynamic procedure to synthesize heteropolymers that can renature to recognize a given target molecule.

We suggest a procedure to synthesize polymers with characteristics similar to those observed in globular proteins: renaturability and the existence of an "active site" capable of specifically recognizing a given target molecule. This procedure is investigated by computer simulation, which finds a yield of up to 65%. We believe that, in principle, this scheme can be realized in vitro. The applicability of this approach as a model of prebiotic synthesis in vivo is also discussed.