The Lagrangian Structure of Long-Time Torsional Dynamics Leading to RNA Folding

Within the context of biopolymer renaturation in vitro, a principle of maximization in the economy of the folding process has been previously formulated as the principle of sequential or stepwise minimization of conformational entropy loss (SMEL). When specialized to the RNA folding context, this principle leads to a predictive folding algorithm under the assumption that an adiabatic approximation is valid. This approximation requires that conformational microstates be lumped up into base-pairing patterns (BPPs) which are treated as quasiequilibrium states, while folding pathways are coarsely represented as sequences of BPP transitions. In this work, we develop a semiempirical microscopic treatment aimed at validating the adiabatic approximation and its underlying SMEL principle. We start by coarse-graining the conformation torsional space X = 3N-torus, with N = length of the chain, representing it as the lattice (Z ₂)^3N , where Z ₂ = integers modulo 2. This is done so that each point in the lattice represents a complete set of local torsional isomeric states coarsely specifying the chain conformation. Then, a coarse Lagrangian governing the long-time dynamics of chain torsions is identified as the variational counterpart of the SMEL principle. To prove this statement, the Lagrangian computation of the coarse Shannon information entropy associated to the specific partition of X into BPPs is performed at different times and contrasted with the adiabatic computation, revealing (a) the subordination of torsional microstate dynamics to BPP transitions within time scales relevant to folding and (b) the coincidence of both plots in the range of folding time scales.     

Single-molecule mechanical folding and unfolding kinetics of armless mitochondrial tRNAArg from Romanomermis culicivorax

The mechanical stability of tRNAs contributes to their biological activities. The mitochondrial tRNA^Arg from Romanomermis culicivorax is the shortest tRNA ever known. This tRNA lacks D- and T-arms, represents a stem-bulge-stem architecture but still folds into a stable tertiary structure. Although its structure had been reported, studies on its mechanical folding and unfolding kinetic characteristics are lacking. Here, we directly measured the single-molecule mechanical folding and unfolding kinetics of the armless mt tRNA^Arg by using optical tweezers in different solution conditions. We revealed a two-step reversible unfolding pathway: the first and large transition corresponds to the unfolding of acceptor stem and bulge below 11 pN, and the second and small transition corresponds to the unfolding of anticodon arm at 12 pN-14 pN. Moreover, the free energy landscapes of the unfolding pathways were reconstructed. We also demonstrated that amino acid-chelated Mg²⁺(aaCM), which mimics the intracellular solution condition, stabilizes the bulge of mitochondrial tRNA^Arg possibly by reducing the topological constraints or stabilizing the possible local non-canonical base pairings within the bulge region. Our study revealed the solution-dependent mechanical stability of an armless mt tRNA, which may shed light on future mt tRNA studies.     

Connecting cluster dynamics and protein folding

The relaxation dynamics of clusters can be interpreted in terms of the topographies of their potential surfaces. Systems with short-range potentials have sawtooth-like potential surfaces with small drops in energy from one local minimum to the next and few-body motions as the clusters move from one minimum to another; such systems readily take on amorphous structures. These are called “glass-formers". Systems with long-range forces have potentials whose topographies are like rough staircases, with some large drops in energy from one minimum to the next; their well-to-well passages involve very collective motions and such systems are excellent structure-seekers. They find their way to well-ordered, highly selective structures under almost all circumstances. These characteristics generalize to describe the potential surfaces and folding behavior of polypeptides and proteins. The forces are effective long-range forces due to the polymer chain. Staircase topographies emerge both from direct sampling of potential surfaces and from the inversion of the kinetics generated by a much more aaabstract topological model, from which folding pathways can be inferred. Received 4 December 2000     

Jamming and crystallization in athermal polymer packings

Dense packings of chains of hard spheres possess characteristic features that do not have a counterpart in corresponding packings of monomeric spheres especially near the maximally random jammed (MRJ) state. From the modelling perspective the additional requirement that spheres keep their connectivity while maximizing the occupied volume fraction imposes severe constraints on generation algorithms of dense chain configurations. The extremely sluggish dynamics imposed by the uncrossability of chains precludes the use of deterministic or stochastic dynamics to generate all but dilute polymer packings. As a viable alternative, especially tailored chain-connectivity-altering Monte Carlo (MC) algorithms have been developed that bypass this kinetic hindrance and have actually been able to produce packings of hard-sphere chains in a volume fraction range spanning from infinite dilution up to the MRJ state. Such very dense athermal polymer packings share a number of structural features with packings of monomeric hard spheres, but also display unique characteristics due to the constraints imposed by connectivity. We give an overview of the most relevant results of our recent modeling work on packings of freely-jointed chains of tangent hard spheres about the MRJ state, local structure, chain dimensions and their scaling with density, topological constraints in the form of entanglements and knots, contact network at jamming, and entropically driven crystallization.     

Topology of protein folding

It is shown that the topological concepts developed previously for the analysis of conformations for strips, double-stranded DNA helices, or elastic thin rods can fruitfully be applied to the study of tertiary folding for complicated protein structures. The topological characteristics determine the integral chirality of proteins and play an important role in the mechanisms of folding and molecular recognition.     

Different pathways in mechanical unfolding/folding cycle of a single semiflexible polymer

Kinetics of conformational change of a semiflexible polymer under mechanical external field were investigated with Langevin dynamics simulations. It is found that a semiflexible polymer exhibits large hysteresis in mechanical folding/unfolding cycle even with a slow operation, whereas in a flexible polymer, the hysteresis almost disappears at a sufficiently slow operation. This suggests that the essential features of the structural transition of a semiflexible polymer should be interpreted at least on a two-dimensional phase space. The appearance of such large hysteresis is discussed in relation to different pathways in the loading and unloading processes. By using a minimal two-variable model, the hysteresis loop is described in terms of different pathways on the transition between two stable states.     

On the dynamics of polymers in dense systems – Results of neutron spin echo spectroscopy

One of the basic problems in the dynamics of polymers concerns the importance of geometrical or topological interactions which are directly related to the large scale molecular structures. In the famous reptation model these constraints are pictured in terms of a tube of localization following the average chain profile and confining the chain motion to the curve‐linear tube. Recently studying the dynamic structure factor of a single labeled chain in a polymer melt by means of neutron spin echo spectroscopy (NSE) led to a direct observation of these tube constraints. Here I shall summarize these neutron spin echo experiments. I shall address the NSE technique, present results on the entropy driven segmental chain dynamics, discuss the dynamics of single chains in the melt where the chain length is increased through the transition to “reptation” dynamics and display NSE measurements on long chain systems which revealed the molecular existence of the entanglement distance. Their magnitudes agree very well with tube diameters derived from dynamical mechanical measurements on the basis of the reptation model proving thereby the basic assumption of this Nobel Price winning concept. This revised version was published online in August 2006 with corrections to the Cover Date.     

Folding of lattice protein model chains with fixed ends

Using Monte Carlo simulations, we have studied the folding dynamics and thermodynamics of geometricall yconstrained lattice protein model chains. The constraints are realized by fixing one or both terminals of the chains. By comparing the results with that of the free-end chains, we find that the folding behaviours of the end-constralned chains are not completely similar to that of the free-end chains. Both kinds of constraints on the chain ends affect the folding dynamics of the chains: i.e., the folding rate, but not the thermodynamics. The thermodynamic behaviour of the one-end-fixed chains shows less difference from that of the free-end chains, while the thermodynamic behaviour of the two-end-fixed chains has obvious difference from that of the free-end chains. The origin of these differences comes from the differences of the ergodicitv of the chains in the conformational space.     

Dynamics and Universality of Unimodal Mappings with Infinite Criticality

We consider infinitely renormalizable unimodal mappings with topological type which are periodic under renormalization. We study the limiting behavior of fixed points of the renormalization operator as the order of the critical point increases to infinity. It is shown that a limiting dynamics exists, with a critical point that is flat, but still having a well-behaved analytic continuation to a neighborhood of the real interval pinched at the critical point. We study the dynamics of limiting maps and prove their rigidity. In particular, the sequence of fixed points of renormalization for finite criticalities converges, uniformly on the real domain, to a mapping of the limiting type.Both authors were supported by Grant No. 2002062 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.Partially supported by NSF grant DMS-0245358.     

Topological Invariants in Terms of Green's Function for the Interacting Kitaev Chain

A one-dimensional closed interacting Kitaev chain and the dimerized version are studied. The topological invariants in terms of Green's function are calculated by the density matrix renormalization group method and the exact diagonalization method. For the interacting Kitaev chain, we point out that the calculation of the topological invariant in the charge density wave phase must consider the dimerized configuration of the ground states. The variation of the topological invariant is attributed to the poles of eigenvalues of the zero-frequency Green functions. For the interacting dimerized Kitaev chain, we show that the topological invariant defined by Green's functions can distinguish more topological nonequivalent phases than the fermion parity.     

A measure on the space of RNA folding pathways: towards a new scheme of statistical inference

In this work we define constructively a measure η on the space of sequential folding (SF) pathways for RNA with the aim of making probabilistic inferences on kinetically-controlled folding. This measure will be shown to be adequate for those RNA species known to search for their structure concurrently with their progressive assembling by sequential nucleotide incorporation. We shall validate our approach by showing that the measure is actually concentrated on the SF pathway that has been experimentally confirmed by pulse-chase kinetic probes. The space of SF pathways will be represented as a product of suitably coarse-grained conformation spaces, one for each length of the growing chain. In this context, the measure is induced by a Markovian stochastic process whose specific realizations are precisely the SF pathways. The coarse-graining is necessary to represent only the essential dynamics of SF: We identify rapidly-interconverting structures by regarding the folding dynamics concurrent with chain growth modulo kinetic barriers of order N ( ). Thus, a pathway corresponds to a sequence of clusters of rapidly-equilibrated structures. Within this representation, we show in specific examples that the measure is concentrated solely on the biologically-significant folding pathway whose destination or final structure is functionally-competent, different appreciably from the global free energy minimum.     

Shear banding in mesoscopic dusty plasma liquids

We experimentally demonstrate shear banding and construct a microscopic dynamic picture of a sheared 2D mesoscopic dust Coulomb liquid at the kinetic level. Under the topological constraints from the discreteness and finite boundary, the nonlinear threshold-type response of motion to the local stress induced by thermal and external drives leads to shear thinning and the enhanced avalanche-type local topological transitions with stress relaxation in the form of clusters. It causes the formation of the outer shear bands in which the mean shear rate, the velocity fluctuations, and the structural rearrangement rate are all enhanced, and leaves a weakly perturbed center band. The typical size of the cooperative hopping vortex (about three interparticle distance) sets up a common length scale for the widths of the confinement induced layering and the shear band.     

Structural and thermodynamic properties of a linearly perturbed matrix model for RNA folding

The structural and thermodynamic properties of a matrix model of homo-RNA folding with linear external interaction are studied. The interaction distinguishes paired bases of the homo-RNA chain from the unpaired bases hence dividing the possible RNA structures given by the linear model into two structural regimes. The genus distribution functions show that the total number of structures for any given length of the chain are reduced for the simple linear interaction considered. The partition function of the model exhibits a scaling relation with the matrix model in which the base pairing strength parameter is re-scaled (G. Vernizzi, H. Orland, A. Zee, Phys. Rev. Lett. 94, 168103 (2005)). The thermodynamics of the model are computed for i) largely secondary structures, (with tertiary structures suppressed by a factor 10^-4) and ii) secondary plus tertiary structures. A structural change for large even lengths is observed in the free energy and specific heat. This change with largely secondary structures appears much before (with respect to length of the chain) than when all the structures (secondary and pseudoknots) are considered. The appearance of different structures which dominate the ensemble with varying temperatures is also found as a function of the interaction parameter for different types of structures (given by different numbers of pairings).