Kinetics of interior loop formation in semiflexible chains

Myofilament-associated proteins with intrinsic disorder (MAPIDs) and their resolution by computational modeling

The cardiac sarcomere is a cellular structure in the heart that enables muscle cells to contract. Dozens of proteins belong to the cardiac sarcomere, which work in tandem to generate force and adapt to demands on cardiac output. Intriguingly, the majority of these proteins have significant intrinsic disorder that contributes to their functions, yet the biophysics of these intrinsically disordered regions (IDRs) have been characterized in limited detail. In this review, we first enumerate these myofilament-associated proteins with intrinsic disorder (MAPIDs) and recent biophysical studies to characterize their IDRs. We secondly summarize the biophysics governing IDR properties and the state-of-the-art in computational tools toward MAPID identification and characterization of their conformation ensembles. We conclude with an overview of future computational approaches toward broadening the understanding of intrinsic disorder in the cardiac sarcomere.

Experimental study of a nanoscale translocation ratchet

Despite an extensive theoretical and numerical background, the translocation ratchet mechanism, which is fundamental for the transmembrane transport of biomolecules, has never been experimentally reproduced at the nanoscale. Only the Sec61 and bacterial type IV pilus pores were experimentally shown to exhibit a translocation ratchet mechanism. Here we designed a synthetic translocation ratchet and quantified its efficiency as a nanopump. We measured the translocation frequency of DNA molecules through nanoporous membranes and showed that polycations at the trans side accelerated the translocation in a ratchet-like fashion. We investigated the ratchet efficiency according to geometrical and kinetic parameters and observed the ratchet to be only dependent on the size of the DNA molecule with a power law [Formula: see text]. A threshold length of 3 kbp was observed, below which the ratchet did not operate. We interpreted this threshold in a DNA looping model, which quantitatively explained our results.

Theory and simulations of condensin mediated loop extrusion in DNA

Condensation of hundreds of mega-base-pair-long human chromosomes in a small nuclear volume is a spectacular biological phenomenon. This process is driven by the formation of chromosome loops. The ATP consuming motor, condensin, interacts with chromatin segments to actively extrude loops. Motivated by real-time imaging of loop extrusion (LE), we created an analytically solvable model, predicting the LE velocity and step size distribution as a function of external load. The theory fits the available experimental data quantitatively, and suggests that condensin must undergo a large conformational change, induced by ATP binding, bringing distant parts of the motor to proximity. Simulations using a simple model confirm that the motor transitions between an open and a closed state in order to extrude loops by a scrunching mechanism, similar to that proposed in DNA bubble formation during bacterial transcription. Changes in the orientation of the motor domains are transmitted over ~50 nm, connecting the motor head and the hinge, thus providing an allosteric basis for LE.

Proper adsorptive confinement for efficient production of cyclic polymers: a dissipative particle dynamics study

Efficient production of cyclic polymers has been a hot topic in the past few decades. In this work, we found that an adsorptive porous template with an appropriate size has the capability to accelerate the ring closure of a linear polymer chain in a dilute solution with a higher yield. The restricted pore provides a confined space and the effect of its characteristics, such as pore size, shape and adsorption strength on cyclization time, is systematically studied by using dissipative particle dynamics simulations. As a prerequisite of cyclization in confinement, the entry process of linear precursors has been studied as well. Total production time is governed by a tradeoff between the size effect caused by decreasing the size of the pore and the adsorption of the pore. The strong size effect suppresses polymer entry but accelerates cyclization. The stronger adsorption promotes polymer entry but decelerates cyclization. According to our defined total production time, a small spherical confinement with strong adsorption results in a shorter total production time of cyclic polymers compared to that in free solution. If chain cyclization is permitted during its entering the confinement, the interplay between steric hindrance caused by pore size and adsorption provides an additional 'virtual' confinement at the boundary between confinement and free solution. In this case, an optimal cyclization time is observed with an appropriate adsorption strength under small confinement. Our results provide useful guidance for designing suitable porous templates for producing cyclic polymers with high efficiency.

Kinetics of Loop Closure in Disordered Proteins: Theory vs Simulations vs Experiments

We study intrachain dynamics of intrinsically disordered proteins, as manifested by the time scales of loop formation, using atomistic simulations, experiment-parametrized coarse-grained models, and one-dimensional theories assuming Markov or non-Markov dynamics along the reaction coordinate. Despite the generally non-Markov character of monomer dynamics in polymers, we find that the simplest model of one-dimensional diffusion along the reaction coordinate (equated to the distance between the loop-forming monomers) well captures the mean first passage times to loop closure measured in coarse-grained and atomistic simulations, which, in turn, agree with the experimental values. This justifies use of the one-dimensional diffusion model in interpretation of experimental data. At the same time, the transition path times for loop closure in longer polypeptide chains show significant non-Markov effects; at intermediate times, these effects are better captured by the generalized Langevin equation model. At long times, however, atomistic simulations predict long tails in the distributions of transition path times, which are at odds with both the one-dimensional diffusion model and the generalized Langevin equation model.

Thermal fracture kinetics of heterogeneous semiflexible polymers

The fracture and severing of polymer chains plays a critical role in the failure of fibrous materials and the regulated turnover of intracellular filaments. Using continuum wormlike chain models, we investigate the fracture of semiflexible polymers via thermal bending fluctuations, focusing on the role of filament flexibility and dynamics. Our results highlight a previously unappreciated consequence of mechanical heterogeneity in the filament, which enhances the rate of thermal fragmentation particularly in cases where constraints hinder the movement of the chain ends. Although generally applicable to semiflexible chains with regions of different bending stiffness, the model is motivated by a specific biophysical system: the enhanced severing of actin filaments at the boundary between stiff bare regions and mechanically softened regions that are coated with cofilin regulatory proteins. The results presented here point to a potential mechanism for disassembly of polymeric materials in general and cytoskeletal actin networks in particular by the introduction of locally softened chain regions, as occurs with cofilin binding.

Conformational heterogeneity in human interphase chromosome organization reconciles the FISH and Hi-C paradox

Hi-C experiments are used to infer the contact probabilities between loci separated by varying genome lengths. Contact probability should decrease as the spatial distance between two loci increases. However, studies comparing Hi-C and FISH data show that in some cases the distance between one pair of loci, with larger Hi-C readout, is paradoxically larger compared to another pair with a smaller value of the contact probability. Here, we show that the FISH-Hi-C paradox can be resolved using a theory based on a Generalized Rouse Model for Chromosomes (GRMC). The FISH-Hi-C paradox arises because the cell population is highly heterogeneous, which means that a given contact is present in only a fraction of cells. Insights from the GRMC is used to construct a theory, without any adjustable parameters, to extract the distribution of subpopulations from the FISH data, which quantitatively reproduces the Hi-C data. Our results show that heterogeneity is pervasive in genome organization at all length scales, reflecting large cell-to-cell variations.

Studies comparing Hi-C and FISH data show that in some cases the distance between one pair of loci is paradoxically larger compared to another pair with a smaller value of the contact probability. Here the authors use a theory based on a Generalized Rouse Model for Chromosomes to resolve this paradox.

Heterogeneous Loop Model to Infer 3D Chromosome Structures from Hi-C

Adapting a well-established formalism in polymer physics, we develop a minimalist approach to infer three-dimensional folding of chromatin from Hi-C data. The three-dimensional chromosome structures generated from our heterogeneous loop model (HLM) are used to visualize chromosome organizations that can substantiate the measurements from fluorescence in situ hybridization, chromatin interaction analysis by paired-end tag sequencing, and RNA-seq signals. We demonstrate the utility of the HLM with several case studies. Specifically, the HLM-generated chromosome structures, which reproduce the spatial distribution of topologically associated domains from fluorescence in situ hybridization measurement, show the phase segregation between two types of topologically associated domains explicitly. We discuss the origin of cell-type-dependent gene-expression level by modeling the chromatin globules of α-globin and SOX2 gene loci for two different cell lines. We also use the HLM to discuss how the chromatin folding and gene-expression level of Pax6 loci, associated with mouse neural development, are modulated by interactions with two enhancers. Finally, HLM-generated structures of chromosome 19 of mouse embryonic stem cells, based on single-cell Hi-C data collected over each cell-cycle phase, visualize changes in chromosome conformation along the cell-cycle. Given a contact frequency map between chromatic loci supplied from Hi-C, HLM is a computationally efficient and versatile modeling tool to generate chromosome structures that can complement interpreting other experimental data.

The role of solvent quality and chain stiffness on the end-to-end contact kinetics of semiflexible polymers

Motivated by loop closure during protein folding and DNA packing, we systemically studied the effects of the solvent quality and chain stiffness on the thermodynamics and kinetics of the end-to-end contact formation for semiflexible polymer chains with reactive ends by Langevin dynamics simulations. In thermodynamics, a rich variety of products of the end-to-end contact have been discovered, such as loop, hairpin, toroid, and rodlike bundle, the populations of which are dependent on the solvent quality and chain stiffness. In kinetics, the overall pathways to form the end-to-end contact have been identified. The change of solvent quality and chain stiffness can tune the roughness of energy landscape and modulate the kinetic partitioning of the end-to-end contact formation pathways, leading to differing kinetic behaviors. In good or poor solvents, the first end-to-end contact rate k _c decreases with increasing the strength of bending stiffness k _θ monotonically. In very poor solvents, however, the dependence of the logarithm of the first end-to-end contact rate ln k _c on k _θ exhibits erratic behavior, which stems from more rugged energy landscape due to the polymer chain getting trapped into the intermediate state composed of the rodlike bundles with two ends in separation. For semiflexible chains, with increasing chain length N, the rate k _c increases initially and then decreases: in good solvents, the rate k _c exhibits a power-law relationship to chain length N with an exponent of ∼-1.50 in the region of long chains, which is in good agreement with the value derived from the experiment in the asymptotic limit of large N; and in poor solvents, the rate k _c exhibits a significantly stronger chain length dependence than those observed in good solvents in the region of long chains due to frustration to form the end-to-end contact along a specific path, especially the scaling exponent between the rate k _c and chain length N is ∼-3.62 for the case of polymer chains with k _θ = 4 at the solvent quality ε _ij = 1, in accord with the value obtained from the experiments.

Radial distribution function of semiflexible oligomers with stretching flexibility

The radial distribution of the end-to-end distance R_ee is crucial for quantifying the global size and flexibility of a linear polymer. For semiflexible polymers, several analytical formulas have been derived for the radial distribution of R_ee ignoring the stretching flexibility. However, for semiflexible oligomers, such as DNA or RNA, the stretching flexibility can be rather pronounced and can significantly affect the radial distribution of R_ee. In this study, we obtained an extended formula that includes the stretch modulus to describe the distribution of R_ee for semiflexible oligomers on the basis of previous formulas for semiflexible polymers without stretching flexibility. The extended formula was validated by extensive Monte Carlo simulations over wide ranges of the stretch modulus and persistence length, as well as all-atom molecular dynamics simulations of short DNAs and RNAs. Additionally, our analyses showed that the effect of stretching flexibility on the distribution of R_ee becomes negligible for DNAs longer than ∼130 base pairs and RNAs longer than ∼240 base pairs.

Toroidal Condensates by Semiflexible Polymer Chains: Insights into Nucleation, Growth and Packing Defects

Deciphering the principles of DNA condensation is important to understand problems such as genome packing and DNA compaction for delivery in gene therapy. DNA molecules condense into toroids and spindles upon the addition of multivalent ions. Nucleation of a loop in the semiflexible DNA chain is critical for both the toroid and spindle formation. To understand the structural differences in the nucleated loop, which cause bifurcation in the condensation pathways leading to toroid or spindle formation, we performed molecular dynamics simulations using a coarse-grained bead-spring polymer model. We find that the formation of a toroid or a spindle is correlated with the orientation of the chain segments close to the loop closure in the nucleated loop. Simulations show that toroids grow in size when spindles in solution interact with a pre-existing toroid and merge into it by spooling around the circumference of the toroid, forming multimolecular toroidal condensates. The merging of spindles with toroids is facile, indicating that this should be the dominant pathway through which the toroids grow in size. The Steinhardt bond order parameter analysis of the toroid cross section shows that the chains pack in a hexagonal fashion. In agreement with the experiments there are regions in the toroid with good hexagonal packing and also with considerable disorder. The disorder in packing is due to the defects, which are propagated during the growth of toroids. In addition to the well-known crossover defect, we have identified three other forms of defects, which perturb hexagonal packing. The new defects identified in the simulations are amenable to experimental verification.

Analytical expressions for the closure probability of a stiff wormlike chain for finite capture radius

Estimating the probability that two monomers of the same polymer chain are close together is a key ingredient to characterize intramolecular reactions and polymer looping. In the case of stiff wormlike polymers (rigid fluctuating elastic rods), for which end-to-end encounters are rare events, we derive an explicit analytical formula for the probability η(r_{c}) that the distance between the chain extremities is smaller than some capture radius r_{c}. The formula is asymptotically exact in the limit of stiff chains, and it leads to the identification of two distinct scaling regimes for the closure factor, originating from a strong variation of the fluctuations of the chain orientation at closure. Our theory is compatible with existing analytical results from the literature that cover the cases of a vanishing capture radius and of nearly fully extended chains.

DNA looping mediates nucleosome transfer

Proper cell function requires preservation of the spatial organization of chromatin modifications. Maintenance of this epigenetic landscape necessitates the transfer of parental nucleosomes to newly replicated DNA, a process that is stringently regulated and intrinsically linked to replication fork dynamics. This creates a formidable setting from which to isolate the central mechanism of transfer. Here we utilized a minimal experimental system to track the fate of a single nucleosome following its displacement, and examined whether DNA mechanics itself, in the absence of any chaperones or assembly factors, may serve as a platform for the transfer process. We found that the nucleosome is passively transferred to available dsDNA as predicted by a simple physical model of DNA loop formation. These results demonstrate a fundamental role for DNA mechanics in mediating nucleosome transfer and preserving epigenetic integrity during replication.

Single-molecule fluorescence studies on DNA looping

Structure and dynamics of DNA impact how the genetic code is processed and maintained. In addition to its biological importance, DNA has been utilized as building blocks of various nanomachines and nanostructures. Thus, understanding the physical properties of DNA is of fundamental importance to basic sciences and engineering applications. DNA can undergo various physical changes. Among them, DNA looping is unique in that it can bring two distal sites together, and thus can be used to mediate interactions over long distances. In this paper, we introduce a FRET-based experimental tool to study DNA looping at the single molecule level. We explain the connection between experimental measurables and a theoretical concept known as the J factor with the intent of raising awareness of subtle theoretical details that should be considered when drawing conclusions. We also explore DNA looping-assisted protein diffusion mechanism called intersegmental transfer using protein induced fluorescence enhancement (PIFE). We present some preliminary results and future outlooks.

Interplay of Protein Binding Interactions, DNA Mechanics, and Entropy in DNA Looping Kinetics

DNA looping plays a key role in many fundamental biological processes, including gene regulation, recombination, and chromosomal organization. The looping of DNA is often mediated by proteins whose structural features and physical interactions can alter the length scale at which the looping occurs. Looping and unlooping processes are controlled by thermodynamic contributions associated with mechanical deformation of the DNA strand and entropy arising from thermal fluctuations of the conformation. To determine how these confounding effects influence DNA looping and unlooping kinetics, we present a theoretical model that incorporates the role of the protein interactions, DNA mechanics, and conformational entropy. We show that for shorter DNA strands the interaction distance affects the transition state, resulting in a complex relationship between the looped and unlooped state lifetimes and the physical properties of the looped DNA. We explore the range of behaviors that arise with varying interaction distance and DNA length. These results demonstrate how DNA deformation and entropy dictate the scaling of the looping and unlooping kinetics versus the J-factor, establishing the connection between kinetic and equilibrium behaviors. Our results show how the twist-and-bend elasticity of the DNA chain modulates the kinetics and how the influence of the interaction distance fades away at intermediate to longer chain lengths, in agreement with previous scaling predictions.

Unexpected Swelling of Stiff DNA in a Polydisperse Crowded Environment

We investigate the conformations of DNA-like stiff chains, characterized by contour length (L) and persistence length (lp), in a variety of crowded environments containing monodisperse soft spherical (SS) and spherocylindrical (SC) particles, a mixture of SS and SC, and a milieu mimicking the composition of proteins in the Escherichia coli cytoplasm. The stiff chain, whose size modestly increases in SS crowders up to ϕ ≈ 0.1, is considerably more compact at low volume fractions (ϕ ≤ 0.2) in monodisperse SC particles than in a medium containing SS particles. A 1:1 mixture of SS and SC crowders induces greater chain compaction than the pure SS or SC crowders at the same ϕ, with the effect being highly nonadditive. We also discover a counterintuitive result that the polydisperse crowding environment, mimicking the composition of a cell lysate, swells the DNA-like polymer, which is in stark contrast to the size reduction of flexible polymers in the same milieu. Trapping of the stiff chain in a fluctuating tube-like environment created by large-sized crowders explains the dramatic increase in size and persistence length of the stiff chain. In the polydisperse medium, mimicking the cellular environment, the size of the DNA (or related RNA) is determined by L/lp. At low L/lp, the size of the polymer is unaffected, whereas there is a dramatic swelling at an intermediate value of L/lp. We use these results to provide insights into recent experiments on crowding effects on RNA and also make testable predictions.

Modulation of DNA loop lifetimes by the free energy of loop formation

Storage and retrieval of the genetic information in cells is a dynamic process that requires the DNA to undergo dramatic structural rearrangements. DNA looping is a prominent example of such a structural rearrangement that is essential for transcriptional regulation in both prokaryotes and eukaryotes, and the speed of such regulations affects the fitness of individuals. Here, we examine the in vitro looping dynamics of the classic Lac repressor gene-regulatory motif. We show that both loop association and loop dissociation at the DNA-repressor junctions depend on the elastic deformation of the DNA and protein, and that both looping and unlooping rates approximately scale with the looping J factor, which reflects the system's deformation free energy. We explain this observation by transition state theory and model the DNA-protein complex as an effective worm-like chain with twist. We introduce a finite protein-DNA binding interaction length, in competition with the characteristic DNA deformation length scale, as the physical origin of the previously unidentified loop dissociation dynamics observed here, and discuss the robustness of this behavior to perturbations in several polymer parameters.

Cyclization kinetics of Gaussian semiflexible polymer chains

We consider the dynamics and the cyclization kinetics of Gaussian semiflexible chains, in which the interaction potential tends to align successive bonds. We provide asymptotic expressions for the cyclization time, for the eigenvalues and eigenfunctions, and for the mean square displacement at all time and length scales, with explicit dependence on the capture radius, on the positions of the reactive monomers in the chain, and on the finite number of beads. For the cyclization kinetics, we take into account non-Markovian effects by calculating the distribution of reactive conformations of the polymer, which are not taken into account in the classical Wilemski-Fixman theory. Comparison with numerical simulations confirms the accuracy of this non-Markovian theory.

Correlated Local Bending of a DNA Double Helix and Its Effect on DNA Flexibility in the Sub-Persistence-Length Regime

Mechanical characteristics of DNA in the sub-persistence-length (lP ≈ 150 base pairs) regime are vital to many of its biological functions but not well understood. Recent experimental studies in this regime have shown a dramatic departure from the traditional worm-like chain model, which is designed for long DNA chains and predicts a constant flexibility at all length scales. Here, we report an improved model with explicit considerations of a new length scale lD ≈ 10 base pairs, over which DNA local bend angles are correlated. In this correlated worm-like chain model, a finite length correction term is analytically derived, and DNA flexibility is found to be contour-length-dependent. While our model reduces to the traditional worm-like chain model at length scales much larger than lP, it predicts that DNA becomes much more flexible at shorter sizes, in good agreement with recent cyclization measurements of short DNA fragments around 100 base pairs.

Measuring shape-dependent looping probability of DNA

Recently, several studies have shown that short doubled-stranded DNA (dsDNA) loops more readily than the wormlike chain model predicts. In most of these experiments, the intrinsic bendedness of dsDNA, which in theory can dramatically influence looping dynamics, was either avoided or unaccounted for. To investigate the effect of the shape of dsDNA on looping dynamics, we characterized the shapes of several synthetic dsDNA molecules of equal length but different sequences using gel electrophoresis. We then measured their looping rates using a FRET (Förster resonance energy transfer)-based assay and extracted the looping probability density known as the J factor (jM). We also used, for comparison, several dinucleotide angular parameter sets derived from the observed electrophoretic mobility to compute the jM predicted by the wormlike chain model. Although we found a strong correlation between curvature and jM, the measured jM was higher than most dinucleotide model predictions. This result suggests that it is difficult to reconcile the looping probability with the observed gel mobility within the wormlike chain model and underscores the importance of determining the intrinsic shape of dsDNA for proper theoretical analysis.

Kinetics of loop formation in worm-like chain polymers

A common theoretical approach to calculating reaction kinetics is to approximate a high-dimensional conformational search with a one-dimensional diffusion along an effective reaction coordinate. We employed Brownian dynamics simulations to test the validity of this approximation for loop formation kinetics in the worm-like chain polymer model. This model is often used to describe polymers that exhibit backbone stiffness beyond the monomer length scale. We find that one-dimensional diffusion models overestimate the looping time and do not predict the quantitatively correct dependence of looping time on chain length or capture radius. Our findings highlight the difficulty of describing high-dimensional polymers with simple kinetic theories.

Polymer model with long-range interactions: analysis and applications to the chromatin structure

Chromatin inside the cell nucleus consists of the DNA and its hierarchy of interacting molecules, which can be modeled as a complex polymer. To describe the chromatin dynamic, we develop and analyze here a polymer model that accounts for long-range interactions and not just those between the closest neighbors as in the Rouse polymer model. Our construction of the polymer model allows us to recover the local interaction between monomers from the anomalous diffusion exponent, which can be directly measured experimentally. We compute asymptotically for this polymer model the cross-correlation function for a given monomer and the mean time for a loop to be formed. Finally, we discuss some possible applications for interpretation of chromosome capture data.

Design principles governing the motility of myosin V

The molecular motor myosin V (MyoV) exhibits a wide repertoire of pathways during the stepping process, which is intimately connected to its biological function. The best understood of these is the hand-over-hand stepping by a swinging lever arm movement toward the plus end of actin filaments. Single-molecule experiments have also shown that the motor "foot stomps," with one hand detaching and rebinding to the same site, and back-steps under sufficient load. The complete taxonomy of MyoV's load-dependent stepping pathways, and the extent to which these are constrained by motor structure and mechanochemistry, are not understood. Using a polymer model, we develop an analytical theory to describe the minimal physical properties that govern motor dynamics. We solve the first-passage problem of the head reaching the target-binding site, investigating the competing effects of backward load, strain in the leading head biasing the diffusion in the direction of the target, and the possibility of preferential binding to the forward site due to the recovery stroke. The theory reproduces a variety of experimental data, including the power stroke and slow diffusive search regimes in the mean trajectory of the detached head, and the force dependence of the forward-to-backward step ratio, run length, and velocity. We derive a stall force formula, determined by lever arm compliance and chemical cycle rates. By exploring the MyoV design space, we predict that it is a robust motor whose dynamical behavior is not compromised by reasonable perturbations to the reaction cycle and changes in the architecture of the lever arm.

Intrachain contact dynamics in unfolded cytochrome cb562

We have investigated intrachain contact dynamics in unfolded cytochrome cb562 by monitoring heme quenching of excited ruthenium photosensitizers covalently bound to residues along the polypeptide. Intrachain diffusion for chemically denatured proteins proceeds on the microsecond time scale with an upper limit of 0.1 μs. The rate constants exhibit a power-law dependence on the number of peptide bonds between the heme and Ru complex. The power-law exponent of -1.5 is consistent with theoretical models for freely jointed Gaussian chains, but its magnitude is smaller than that reported for several synthetic polypeptides. Contact formation within a stable loop was examined in a His63-heme ligated form of the protein under denaturing conditions. Loop formation accelerated contact kinetics for the Ru66 labeling site, owing to reduction in the length of the peptide separating redox sites. For other labeling sites within the stable loop, quenching rates were modestly reduced compared to the open chain polymer.

Computation of the mean first-encounter time between the ends of a polymer chain

Using a novel theoretical approach, we study the mean first-encounter time (MFET) between the two ends of a polymer. Previous approaches used various simplifications that reduced the complexity of the problem, leading, however, to incompatible results. We construct here for the first time a general theory that allows us to compute the MFET. The method is based on estimating the mean time for a Brownian particle to reach a narrow domain in the polymer configuration space. In dimension two and three, we find that the MFET depends mainly on the first eigenvalue of the associated Fokker-Planck operator and provide precise estimates that are confirmed by Brownian simulations. Interestingly, although many time scales are involved in the encounter process, its distribution can be well approximated by a single exponential, which has several consequences for modeling chromosome dynamics in the nucleus. Another application of our result is computing the mean time for a DNA molecule to form a closed loop (when its two ends meet for the first time).

Length-dependent aggregation of uninterrupted polyalanine peptides

Polyalanine (polyA) is the third-most prevalent homopeptide repeat in eukaryotes, behind polyglutamine and polyasparagine. Abnormal expansion of the polyA repeat is linked to at least nine human diseases, and the disease mechanism likely involves enhanced length-dependent aggregation. Because of the simplicity of its side chain, polyA has been a favorite target of computational studies, and because of their tendency to fold into α-helix, peptides containing polyA-rich domains have been a popular experimental subject. However, experimental studies on uninterrupted polyA are very limited. We synthesized polyA peptides containing uninterrupted sequences of 7 to 25 alanines (A7 to A25) and characterized their length-dependent conformation and aggregation properties. The peptides were primarily disordered, with a modest component of α-helix that increased with increasing length. From measurements of mean distance spanned by the polyA segment, we concluded that physiological buffers are neutral solvents for shorter polyA peptides and poor solvents for longer peptides. At moderate concentration and near-physiological temperature, polyA assembled into soluble oligomers, with a sharp transition in oligomer physical properties between A19 and A25. With A19, oligomers were large, contained only a small fraction of the total peptide mass, and slowly grew into loose clusters, while A25 rapidly and completely assembled into small stable oligomers of ~7 nm radius. At high temperatures, A19 assembled into fibrils, but A25 precipitated as dense, micrometer-sized particles. A comparison of these results to those obtained with polyglutamine peptides of similar design sheds light on the role of the side chain in regulating conformation and aggregation.

Universality in the timescales of internal loop formation in unfolded proteins and single-stranded oligonucleotides

Understanding the rate at which various parts of a molecular chain come together to facilitate the folding of a biopolymer (e.g., a protein or RNA) into its functional form remains an elusive goal. Here we use experiments, simulations, and theory to study the kinetics of internal loop closure in disordered biopolymers such as single-stranded oligonucleotides and unfolded proteins. We present theoretical arguments and computer simulation data to show that the relationship between the timescale of internal loop formation and the positions of the monomers enclosing the loop can be recast in a form of a universal master dependence. We also perform experimental measurements of the loop closure times of single-stranded oligonucleotides and show that both these and previously reported internal loop closure kinetics of unfolded proteins are well described by this theoretically predicted dependence. Finally, we propose that experimental deviations from the master dependence can then be used as a sensitive probe of dynamical and structural order in unfolded proteins and other biopolymers.

Examining polyglutamine peptide length: a connection between collapsed conformations and increased aggregation

Abnormally expanded polyglutamine domains in proteins are associated with several neurodegenerative diseases, of which the best known is Huntington's. Expansion of the polyglutamine domain facilitates aggregation of the affected protein, and several studies directly link aggregation to neurotoxicity. The age of onset of disease is inversely correlated with the length of the polyglutamine domain; this correlation motivates an examination of the role of the length of the domain on aggregation. In this investigation, peptides containing 8 to 24 glutamines were synthesized, and their conformational and aggregation properties were examined. All peptides lacked secondary structure. Fluorescence resonance energy transfer studies revealed that the peptides became increasingly collapsed as the number of glutamine residues increased. The effective persistence length was estimated to decrease from approximately 11 to approximately 7 A as the number of glutamines increased from 8 to 24. A comparison of our data with theoretical results suggests that phosphate-buffered saline is a good solvent for Q8 and Q12, a theta solvent for Q16, and a poor solvent for Q20 and Q24. By dynamic light scattering, we observed that Q16, Q20, and Q24, but not Q8 or Q12, immediately formed soluble aggregates upon dilution into phosphate-buffered saline at 37 degrees C. Thus, Q16 stands at the transition point between good and poor solvent and between stable and aggregation-prone peptide. Examination of aggregates by transmission electron microscopy, along with kinetic assays for sedimentation, provided evidence indicating that soluble aggregates mature into sedimentable aggregates. Together, the data support a mechanism of aggregation in which monomer collapse is accompanied by formation of soluble oligomers; these soluble species lack regular secondary structure but appear morphologically similar to the sedimentable aggregates into which they eventually mature.

Biochemistry on a leash: the roles of tether length and geometry in signal integration proteins

We use statistical mechanics and simple ideas from polymer physics to develop a quantitative model of proteins whose activity is controlled by flexibly tethered ligands and receptors. We predict how the properties of tethers influence the function of these proteins and demonstrate how their tether length dependence can be exploited to construct proteins whose integration of multiple signals can be tuned. One case study to which we apply these ideas is that of the Wiskott-Aldrich Syndrome Proteins as activators of actin polymerization. More generally, tethered ligands competing with those free in solution are common phenomena in biology, making this an important specific example of a widespread biological idea.

Synchronous vs asynchronous chain motion in alpha-synuclein contact dynamics

alpha-Synuclein (alpha-syn) is an intrinsically unstructured 140-residue neuronal protein of uncertain function that is implicated in the etiology of Parkinson's disease. Tertiary contact formation rate constants in alpha-syn, determined from diffusion-limited electron-transfer kinetics measurements, are poorly approximated by simple random polymer theory. One source of the discrepancy between theory and experiment may be that interior-loop formation rates are not well approximated by end-to-end contact dynamics models. We have addressed this issue with Monte Carlo simulations to model asynchronous and synchronous motion of contacting sites in a random polymer. These simulations suggest that a dynamical drag effect may slow interior-loop formation rates by about a factor of 2 in comparison to end-to-end loops of comparable size. The additional deviations from random coil behavior in alpha-syn likely arise from clustering of hydrophobic residues in the disordered polypeptide.

Semiflexible chains in confined spaces

We develop an analytical method for studying the properties of a noninteracting wormlike chain (WLC) in confined geometries. The mean-field-like theory replaces the rigid constraints of confinement with average constraints, thus allowing us to develop a tractable method for treating a WLC wrapped on the surface of a sphere, and fully encapsulated within it. The efficacy of the theory is established by reproducing the exact correlation functions for a WLC confined to the surface of a sphere. In addition, the coefficients in the free energy are exactly calculated. We also describe the behavior of a surface-confined chain under external tension that is relevant for single molecule experiments on histone-DNA complexes. The force-extension curves display spatial oscillations, and the extension of the chain, whose maximum value is bounded by the sphere diameter, scales as f(-1) at large forces, in contrast to the unconfined chain that approaches the contour length as f(-1/2). A WLC encapsulated in a sphere, that is relevant for the study of the viral encapsulation of DNA, can also be treated using the mean-field approach. The predictions of the theory for various correlation functions are in excellent agreement with Langevin simulations. We find that strongly confined chains are highly structured by examining the correlations using a local winding axis. The predicted pressure of the system is in excellent agreement with simulations but, as is known, is significantly lower than the pressures seen for DNA packaged in viral capsids.

Folding and unfolding kinetics of a single semiflexible polymer

We investigate theoretically the kinetics of the folding transition of a single semiflexible polymer. In the folding transition, the growth rate decreases with an increase in the number of monomers in the collapsed domain, suggesting that the main contribution to dissipation is from the motion of the domain. In the unfolding transition, the dynamic scaling exponents 1/8 and 1/4 were determined for the disentanglement and relaxation steps, respectively. We performed Langevin dynamics simulations to test our theory. It is found that our theory is in good agreement with simulations. We also propose the kinetics of the transitions in the presence of a hydrodynamic interaction.

Excluded volume effects on the intrachain reaction kinetics

On the basis of the recently developed optimized Rouse-Zimm theory of chain polymers with excluded volume interactions, we calculate the long-time first-order rate constant k(1) for end-to-end cyclization of linear chain polymers. We first find that the optimized Rouse-Zimm theory provides the longest chain relaxation times tau(1) of excluded volume chains that are in excellent agreement with the available Brownian dynamics simulation results. In the free-draining limit, the cyclization rate is diffusion-controlled and k(1) is inversely proportional to tau(1), and the k(1) values calculated using the Wilemski-Fixman rate theory are in good agreement with Brownian dynamics simulation results. However, when hydrodynamic interactions are included, noticeable deviations are found. The main sources of errors are fluctuating hydrodynamic interaction and correlation hole effects as well as the non-Markovian reaction dynamic effect. The physical natures of these factors are discussed, and estimates for the magnitudes of required corrections are given. When the corrections are included, the present theory allows the prediction of accurate k(1) values for the cyclization of finite-length chains in good solvents as well as the correct scaling exponent in the long-chain limit.

Kinetics of loop formation in polymer chains

We investigate the kinetics of loop formation in ideal flexible polymer chains (the Rouse model), and polymers in good and poor solvents. We show for the Rouse model, using a modification of the theory of Szabo, Schulten, and Schulten, that the time scale for cyclization is tau(c) approximately tau(0)N(2) (where tau(0) is a microscopic time scale and N is the number of monomers), provided the coupling between the relaxation dynamics of the end-to-end vector and the looping dynamics is taken into account. The resulting analytic expression fits the simulation results accurately when a, the capture radius for contact formation, exceeds b, the average distance between two connected beads. Simulations also show that when a < b, tau(c) approximately N(alpha)(tau), where 1.5 < alpha(tau) < or = 2 in the range 7 < N < 200 used in the simulations. By using a diffusion coefficient that is dependent on the length scales a and b (with a < b), which captures the two-stage mechanism by which looping occurs when a < b, we obtain an analytic expression for tauc that fits the simulation results well. The kinetics of contact formation between the ends of the chain are profoundly effected when interactions between monomers are taken into account. Remarkably, for N < 100, the values of tau(c) decrease by more than 2 orders of magnitude when the solvent quality changes from good to poor. Fits of the simulation data for tau(c) to a power law in N (tau(c) approximately N(alpha)(tau)) show that alpha(tau) varies from about 2.4 in a good solvent to about 1.0 in poor solvents. The effective exponent alpha(tau) decreases as the strength of the attractive monomer-monomer interactions increases. Loop formation in poor solvents, in which the polymer adopts dense, compact globular conformations, occurs by a reptation-like mechanism of the ends of the chain. The time for contact formation between beads that are interior to the chain in good solvents changes nonmonotonically as the loop length varies. In contrast, the variation in interior loop closure time is monotonic in poor solvents. The implications of our results for contact formation in polypeptide chains, RNA, and single-stranded DNA are briefly outlined.

Loop dependence of the stability and dynamics of nucleic acid hairpins

Hairpin loops are critical to the formation of nucleic acid secondary structure, and to their function. Previous studies revealed a steep dependence of single-stranded DNA (ssDNA) hairpin stability with length of the loop (L) as approximately L(8.5 +/- 0.5), in 100 mM NaCl, which was attributed to intraloop stacking interactions. In this article, the loop-size dependence of RNA hairpin stabilities and their folding/unfolding kinetics were monitored with laser temperature-jump spectroscopy. Our results suggest that similar mechanisms stabilize small ssDNA and RNA loops, and show that salt contributes significantly to the dependence of hairpin stability on loop size. In 2.5 mM MgCl2, the stabilities of both ssDNA and RNA hairpins scale as approximately L(4 +/- 0.5), indicating that the intraloop interactions are weaker in the presence of Mg2+. Interestingly, the folding times for ssDNA hairpins (in 100 mM NaCl) and RNA hairpins (in 2.5 mM MgCl2) are similar despite differences in the salt conditions and the stem sequence, and increase similarly with loop size, approximately L(2.2 +/- 0.5) and approximately L(2.6 +/- 0.5), respectively. These results suggest that hairpins with small loops may be specifically stabilized by interactions of the Na+ ions with the loops. The results also reinforce the idea that folding times are dominated by an entropic search for the correct nucleating conformation.

Single-label kinase and phosphatase assays for tyrosine phosphorylation using nanosecond time-resolved fluorescence detection

The collision-induced fluorescence quenching of a 2,3-diazabicyclo[2.2.2]oct-2-ene-labeled asparagine (Dbo) by hydrogen atom abstraction from the tyrosine residue in peptide substrates was introduced as a single-labeling strategy to assay the activity of tyrosine kinases and phosphatases. The assays were tested for 12 different combinations of Dbo-labeled substrates and with the enzymes p60c-Src Src kinase, EGFR kinase, YOP protein tyrosine phosphatase, as well as acid and alkaline phosphatases, thereby demonstrating a broad application potential. The steady-state fluorescence changed by a factor of up to 7 in the course of the enzymatic reaction, which allowed for a sufficient sensitivity of continuous monitoring in steady-state experiments. The fluorescence lifetimes (and intensities) were found to be rather constant for the phosphotyrosine peptides (ca. 300 ns in aerated water), while those of the unphosphorylated peptides were as short as 40 ns (at pH 7) and 7 ns (at pH 13) as a result of intramolecular quenching. Owing to the exceptionally long fluorescence lifetime of Dbo, the assays were alternatively performed by using nanosecond time-resolved fluorescence (Nano-TRF) detection, which leads to an improved discrimination of background fluorescence and an increased sensitivity. The potential for inhibitor screening was demonstrated through the inhibition of acid and alkaline phosphatases by molybdate.

DNA looping and translocation provide an optimal cleavage mechanism for the type III restriction enzymes

EcoP15I is a type III restriction enzyme that requires two recognition sites in a defined orientation separated by up to 3.5 kbp to efficiently cleave DNA. The mechanism through which site-bound EcoP15I enzymes communicate between the two sites is unclear. Here, we use atomic force microscopy to study EcoP15I-DNA pre-cleavage complexes. From the number and size distribution of loops formed, we conclude that the loops observed do not result from translocation, but are instead formed by a contact between site-bound EcoP15I and a nonspecific region of DNA. This conclusion is confirmed by a theoretical polymer model. It is further shown that translocation must play some role, because when translocation is blocked by a Lac repressor protein, DNA cleavage is similarly blocked. On the basis of these results, we present a model for restriction by type III restriction enzymes and highlight the similarities between this and other classes of restriction enzymes.

Alpha-synuclein tertiary contact dynamics

Tertiary contact formation rates in alpha-synuclein, an intrinsically disordered polypeptide implicated in Parkinson's disease, have been determined from measurements of diffusion-limited electron-transfer kinetics between triplet-excited tryptophan:3-nitrotyrosine pairs separated by 10, 12, 55, and 90 residues. Calculations based on a Markovian lattice model developed to describe intrachain diffusion dynamics for a disordered polypeptide give contact quenching rates for various loop sizes ranging from 6 to 48 that are in reasonable agreement with experimentally determined values for small loops (10-20 residues). Contrary to expectations, measured contact rates in alpha-synuclein do not continue to decrease as the loop size increases (>/=35 residues), and substantial deviations from calculated rates are found for the pairs W4-Y94, Y39-W94, and W4-Y136. The contact rates for these large loops indicate much shorter average donor-acceptor separations than expected for a random polymer.

Size, shape, and flexibility of RNA structures

Determination of sizes and flexibilities of RNA molecules is important in understanding the nature of packing in folded structures and in elucidating interactions between RNA and DNA or proteins. Using the coordinates of the structures of RNA in the Protein Data Bank we find that the size of the folded RNA structures, measured using the radius of gyration R(G), follows the Flory scaling law, namely, R(G)=5.5N(1/3) A, where N is the number of nucleotides. The shape of RNA molecules is characterized by the asphericity Delta and the shape S parameters that are computed using the eigenvalues of the moment of inertia tensor. From the distribution of Delta, we find that a large fraction of folded RNA structures are aspherical and the distribution of S values shows that RNA molecules are prolate (S>0). The flexibility of folded structures is characterized by the persistence length l(p). By fitting the distance distribution function P(r), that is computed using the coordinates of the folded RNA, to the wormlike chain model we extracted the persistence length l(p). We find that l(p) approximately 1.5N(0.33) A which might reflect the large separation between the free energies that stabilize secondary and tertiary structures. The dependence of l(p) on N implies that the average length of helices should increase as the size of RNA grows. We also analyze packing in the structures of ribosomes (30S, 50S, and 70S) in terms of R(G), Delta, S, and l(p). The 70S and the 50S subunits are more spherical compared to most RNA molecules. The globularity in 50S is due to the presence of an unusually large number (compared to 30S subunit) of small helices that are stitched together by bulges and loops. Comparison of the shapes of the intact 70S ribosome and the constituent particles suggests that folding of the individual molecules might occur prior to assembly.

Kinetics of internal-loop formation in polypeptide chains: a simulation study

The speed of simple diffusional motions, such as the formation of loops in the polypeptide chain, places one physical limit on the speed of protein folding. Many experimental studies have explored the kinetics of formation of end-to-end loops in polypeptide chains; however, protein folding more often requires the formation of contacts between interior points on the chain. One expects that, for loops of fixed contour length, interior loops will form more slowly than end-to-end loops, owing to the additional excluded volume associated with the "tails". We estimate the magnitude of this effect by generating ensembles of randomly coiled, freely jointed chains, and then using the theory of Szabo, Schulten, and Schulten to calculate the corresponding contact formation rates for these ensembles. Adding just a few residues, to convert an end-to-end loop to an internal loop, sharply decreases the contact rate. Surprisingly, the relative change in rate increases for a longer loop; sufficiently long tails, however, actually reverse the effect and accelerate loop formation slightly. Our results show that excluded volume effects in real, full-length polypeptides may cause the rates of loop formation during folding to depart significantly from the values derived from recent loop-formation experiments on short peptides.