The coil–globule transition: Radius of gyration of polystyrene in cyclohexane

The conformational phase diagram of neutral polymers in the presence of attractive crowders

Extensive coarse-grained molecular dynamics simulations are performed to investigate the conformational phase diagram of a neutral polymer in the presence of attractive crowders. We show that, for low crowder densities, the polymer predominantly shows three phases as a function of both intra-polymer and polymer-crowder interactions: (1) weak intra-polymer and weak polymer-crowder attractive interactions induce extended or coil polymer conformations (phase E), (2) strong intra-polymer and relatively weak polymer-crowder attractive interactions induce collapsed or globular conformations (phase CI), and (3) strong polymer-crowder attractive interactions, regardless of intra-polymer interactions, induce a second collapsed or globular conformation that encloses bridging crowders (phase CB). The detailed phase diagram is obtained by determining the phase boundaries delineating the different phases based on an analysis of the radius of gyration as well as bridging crowders. The dependence of the phase diagram on strength of crowder-crowder attractive interactions and crowder density is clarified. We also show that when the crowder density is increased, a third collapsed phase of the polymer emerges for weak intra-polymer attractive interactions. This crowder density-induced compaction is shown to be enhanced by stronger crowder-crowder attraction and is different from the depletion-induced collapse mechanism, which is primarily driven by repulsive interactions. We also provide a unified explanation of the observed re-entrant swollen/extended conformations of the earlier simulations of weak and strongly self-interacting polymers in terms of crowder-crowder attractive interactions.

Effects of alignment activity on the collapse kinetics of a flexible polymer

The dynamics of various biological filaments can be understood within the framework of active polymer models. Here we consider a bead-spring model for a flexible polymer chain in which the active interaction among the beads is introduced via an alignment rule adapted from the Vicsek model. Following quenching from the high-temperature coil phase to a low-temperature state point, we study the coarsening kinetics via molecular dynamics (MD) simulations using the Langevin thermostat. For the passive polymer case the low-temperature equilibrium state is a compact globule. The results from our MD simulations reveal that though the globular state is also the typical final state in the active case, the nonequilibrium pathways to arrive at such a state differ from the picture for the passive case due to the alignment interaction among the beads. We notice that deviations from the intermediate "pearl-necklace"-like arrangement, which is observed in the passive case, and the formation of more elongated dumbbell-like structures increase with increasing activity. Furthermore, it appears that while a small active force on the beads certainly makes the coarsening process much faster, there exists a nonmonotonic dependence of the collapse time on the strength of active interaction. We quantify these observations by comparing the scaling laws for the collapse time and growth of pearls with the passive case.

Influence of lactide vs glycolide composition of poly (lactic-co-glycolic acid) polymers on encapsulation of hydrophobic molecules: molecular dynamics and formulation studies

The work demonstrates the preparation of PLGA (PLGA 50:50, PLGA 75:25) nanoparticles, to encapsulate a hydrophobic molecule (coumarin-6), using the microreactor-based continuous process. The formulations were characterized using dynamic light scattering and transmission electron microscopy to determine their size, homogeneity, zeta potential, and surface morphology. The resulting nanoparticles were safe to the CHO cells (≈80% cell survival), at the concentration of ≤600 µg/mL and were successfully taken up by the cells, as demonstrated using confocal microscopy. Moreover, imaging flow cytometry confirmed that the nanoparticles were internalized in 73.96% of the cells. Furthermore, molecular dynamics simulation and docking studies were carried out to explore the effect of polymer chain length of PLGA and lactide vs glycolide (LA:GA) ratio on their compatibility with the coumarin-6 molecules and to study the coiling and flexibility of PLGA in the presence of coumarin-6 molecules. Flory-Huggins interaction parameter (χ) was calculated for polymer chains of varying lengths and LA:GA ratio, with respect to coumarin-6. χ parameter increased with increase in polymer chain length, which indicated superior interaction of coumarin-6 with the smaller chains. Amongst all the polymeric systems, PLGA55 exhibited the strongest interaction with coumarin-6, for all the chain lengths, possibly because of their homogeneous spatial arrangements and superior binding energy. PLGA27 showed better compatibility compared to PLGA72 and PGA, whereas PLA-based polymers exhibited the least compatibility. Analysis of the radius of gyration of the polymer chains in the polymer-coumarin-6 complexes, at the end of molecular dynamics run, exhibited that the polymer chains displayed varying coiling behavior and flexibility, depending upon the relative concentrations of the polymer and coumarin-6. Factors like intra-chain interactions, spatial arrangement, inter-chain binding energies, and polymer-coumarin-6 compatibility also influenced the coiling and flexibility of polymer chains.

Expansion of Intrinsically Disordered Proteins Increases the Range of Stability of Liquid-Liquid Phase Separation

Proteins containing intrinsically disordered regions (IDRs) are ubiquitous within biomolecular condensates, which are liquid-like compartments within cells formed through liquid-liquid phase separation (LLPS). The sequence of amino acids of a protein encodes its phase behaviour, not only by establishing the patterning and chemical nature (e.g., hydrophobic, polar, charged) of the various binding sites that facilitate multivalent interactions, but also by dictating the protein conformational dynamics. Besides behaving as random coils, IDRs can exhibit a wide-range of structural behaviours, including conformational switching, where they transition between alternate conformational ensembles. Using Molecular Dynamics simulations of a minimal coarse-grained model for IDRs, we show that the role of protein conformation has a non-trivial effect in the liquid-liquid phase behaviour of IDRs. When an IDR transitions to a conformational ensemble enriched in disordered extended states, LLPS is enhanced. In contrast, IDRs that switch to ensembles that preferentially sample more compact and structured states show inhibited LLPS. This occurs because extended and disordered protein conformations facilitate LLPS-stabilising multivalent protein-protein interactions by reducing steric hindrance; thereby, such conformations maximize the molecular connectivity of the condensed liquid network. Extended protein configurations promote phase separation regardless of whether LLPS is driven by homotypic and/or heterotypic protein-protein interactions. This study sheds light on the link between the dynamic conformational plasticity of IDRs and their liquid-liquid phase behaviour.

Surface interaction parameter measurement of solvated polymers via model end-tethered chains

We present a method for the direct measurement of the relative energy of interaction between a solvated polymer and a solid interface. By tethering linear chains covalently to the surface, we ensured the idealized and constant configuration of polymer molecules for measurement, modeling, and parameter estimation. For the case of amine-terminated polystyrene bound to a glycidoxypropyl silane film submerged in cyclohexane-d12, we estimated the χ parameter for the temperature range 10.7 °C to 52.0 °C, and found a downward sloping trend that crosses the χ = 0.5 threshold at 37 °C to 40 °C, in agreement with solution estimates for the same system. We simultaneously estimated the surface interaction parameter χ_s at each temperature, finding a decreasing affinity of the chains for the surface with increasing temperature, consistent with empirical observations. The theoretical model shows some limitations in a stronger solvent (toluene-d8) that prevent rigorous parameter estimation, but we demonstrate a qualitative change in χ and χ_s towards stronger solvency and weaker surface interaction with increasing temperature.

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.

Intrinsic α helix propensities compact hydrodynamic radii in intrinsically disordered proteins

Proteins that lack tertiary stability under normal conditions, known as intrinsically disordered, exhibit a wide range of biological activities. Molecular descriptions for the biology of intrinsically disordered proteins (IDPs) consequently rely on disordered structural models, which in turn require experiments that assess the origins to structural features observed. For example, while hydrodynamic size is mostly insensitive to sequence composition in chemically denatured proteins, IDPs show strong sequence-specific effects in the hydrodynamic radius (R_h ) when measured under normal conditions. To investigate sequence-modulation of IDP R_h , disordered ensembles generated by a hard sphere collision model modified with a structure-based parameterization of the solution energetics were used to parse the contributions of net charge, main chain dihedral angle bias, and excluded volume on hydrodynamic size. Ensembles for polypeptides 10-35 residues in length were then used to establish power-law scaling relationships for comparison to experimental R_h from 26 IDPs. Results showed the expected outcomes of increased hydrodynamic size from increases in excluded volume and net charge, and compaction from chain-solvent interactions. Chain bias representing intrinsic preferences for α helix and polyproline II (PP_II ), however, modulated R_h with intricate dependence on the simulated propensities. PP_II propensities at levels expected in IDPs correlated with heightened R_h sensitivity to even weak α helix propensities, indicating bias for common (φ, ψ) are important determinants of hydrodynamic size. Moreover, data show that IDP R_h can be predicted from sequence with good accuracy from a small set of physicochemical properties, namely intrinsic conformational propensities and net charge. Proteins 2017; 85:296-311. © 2016 Wiley Periodicals, Inc.

Consistent View of Polypeptide Chain Expansion in Chemical Denaturants from Multiple Experimental Methods

There has been a long-standing controversy regarding the effect of chemical denaturants on the dimensions of unfolded and intrinsically disordered proteins: A wide range of experimental techniques suggest that polypeptide chains expand with increasing denaturant concentration, but several studies using small-angle X-ray scattering (SAXS) have reported no such increase of the radius of gyration (Rg). This inconsistency challenges our current understanding of the mechanism of chemical denaturants, which are widely employed to investigate protein folding and stability. Here, we use a combination of single-molecule Förster resonance energy transfer (FRET), SAXS, dynamic light scattering (DLS), and two-focus fluorescence correlation spectroscopy (2f-FCS) to characterize the denaturant dependence of the unfolded state of the spectrin domain R17 and the intrinsically disordered protein ACTR in two different denaturants. Standard analysis of the primary data clearly indicates an expansion of the unfolded state with increasing denaturant concentration irrespective of the protein, denaturant, or experimental method used. This is the first case in which SAXS and FRET have yielded even qualitatively consistent results regarding expansion in denaturant when applied to the same proteins. To more directly illustrate this self-consistency, we used both SAXS and FRET data in a Bayesian procedure to refine structural ensembles representative of the observed unfolded state. This analysis demonstrates that both of these experimental probes are compatible with a common ensemble of protein configurations for each denaturant concentration. Furthermore, the resulting ensembles reproduce the trend of increasing hydrodynamic radius with denaturant concentration obtained by 2f-FCS and DLS. We were thus able to reconcile the results from all four experimental techniques quantitatively, to obtain a comprehensive structural picture of denaturant-induced unfolded state expansion, and to identify the most likely sources of earlier discrepancies.

Single-Molecule FRET Spectroscopy and the Polymer Physics of Unfolded and Intrinsically Disordered Proteins

The properties of unfolded proteins have long been of interest because of their importance to the protein folding process. Recently, the surprising prevalence of unstructured regions or entirely disordered proteins under physiological conditions has led to the realization that such intrinsically disordered proteins can be functional even in the absence of a folded structure. However, owing to their broad conformational distributions, many of the properties of unstructured proteins are difficult to describe with the established concepts of structural biology. We have thus seen a reemergence of polymer physics as a versatile framework for understanding their structure and dynamics. An important driving force for these developments has been single-molecule spectroscopy, as it allows structural heterogeneity, intramolecular distance distributions, and dynamics to be quantified over a wide range of timescales and solution conditions. Polymer concepts provide an important basis for relating the physical properties of unstructured proteins to folding and function.

Origin and prediction of free-solution interaction studies performed label-free

Interaction/reaction assays have led to significant scientific discoveries in the biochemical, medical, and chemical disciplines. Several fundamental driving forces form the basis of intermolecular and intramolecular interactions in chemical and biochemical systems (London dispersion, hydrogen bonding, hydrophobic, and electrostatic), and in the past three decades the sophistication and power of techniques to interrogate these processes has developed at an unprecedented rate. In particular, label-free methods have flourished, such as NMR, mass spectrometry (MS), surface plasmon resonance (SPR), biolayer interferometry (BLI), and backscattering interferometry (BSI), which can facilitate assays without altering the participating components. The shortcoming of most refractive index (RI)-based label-free methods such as BLI and SPR is the requirement to tether one of the interaction entities to a sensor surface. This is not the case for BSI. Here, our hypothesis is that the signal origin for free-solution, label-free determinations can be attributed to conformation and hydration-induced changes in the solution RI. We propose a model for the free-solution response function (FreeSRF) and show that, when quality bound and unbound structural data are available, FreeSRF correlates well with the experiment (R(2)> 0.99, Spearman rank correlation coefficients >0.9) and the model is predictive within ∼15% of the experimental binding signal. It is also demonstrated that a simple mass-weighted dη/dC response function is the incorrect equation to determine that the change in RI is produced by binding or folding event in free solution.

Marked difference in conformational fluctuation between giant DNA molecules in circular and linear forms

We performed monomolecular observations on linear and circular giant DNAs (208 kbp) in an aqueous solution by the use of fluorescence microscopy. The results showed that the degree of conformational fluctuation in circular DNA was ca. 40% less than that in linear DNA, although the long-axis length of circular DNA was only 10% smaller than that of linear DNA. Additionally, the relaxation time of a circular chain was shorter than that of a linear chain by at least one order of magnitude. The essential features of this marked difference between linear and circular DNAs were reproduced by numerical simulations on a ribbon-like macromolecule as a coarse-grained model of a long semiflexible, double-helical DNA molecule. In addition, we calculated the radius of gyration of an interacting chain in a circular form on the basis of the mean field model, which provides a better understanding of the present experimental trend than a traditional theoretical equation.

Single-molecule spectroscopy of the unexpected collapse of an unfolded protein at low pH

The dimensions of intrinsically disordered and unfolded proteins critically depend on the solution conditions, such as temperature, pH, ionic strength, and osmolyte or denarurant concentration. However, a quantitative understanding of how the complex combination of chain-chain and chain-solvent interactions is affected by the solvent is still missing. Here, we take a step towards this goal by investigating the combined effect of pH and denaturants on the dimensions of an unfolded protein. We use single-molecule fluorescence spectroscopy to extract the dimensions of unfolded cold shock protein (CspTm) in mixtures of the denaturants urea and guanidinium chloride (GdmCl) at neutral and acidic pH. Surprisingly, even though a change in pH from 7 to 2.9 increases the net charge of CspTm from -3.8 to +10.2, the radius of gyration of the chain is very similar under both conditions, indicating that protonation of acidic side chains at low pH results in additional hydrophobic interactions. We use a simple shared binding site model that describes the joint effect of urea and GdmCl, together with polyampholyte theory and an ion cloud model that includes the chemical free energy of counterion interactions and side chain protonation, to quantify this effect.

Temperature-dependent solvation modulates the dimensions of disordered proteins

For disordered proteins, the dimensions of the chain are an important property that is sensitive to environmental conditions. We have used single-molecule Förster resonance energy transfer to probe the temperature-induced chain collapse of five unfolded or intrinsically disordered proteins. Because this behavior is sensitive to the details of intrachain and chain-solvent interactions, the collapse allows us to probe the physical interactions governing the dimensions of disordered proteins. We find that each of the proteins undergoes a collapse with increasing temperature, with the most hydrophobic one, λ-repressor, undergoing a reexpansion at the highest temperatures. Although such a collapse might be expected due to the temperature dependence of the classical "hydrophobic effect," remarkably we find that the largest collapse occurs for the most hydrophilic, charged sequences. Using a combination of theory and simulation, we show that this result can be rationalized in terms of the temperature-dependent solvation free energies of the constituent amino acids, with the solvation properties of the most hydrophilic residues playing a large part in determining the collapse.

Atomistic picture of isothermal volume relaxation behavior of atactic polystyrene glass provided by a molecular dynamics simulation

Using an atomistic molecular dynamics (MD) simulation, we study the volume relaxation behavior of atactic polystyrene and attempt to correlate this macroscopic behavior with certain microscopic aspects. To this end, the gyration radius, the dimensionless relative shape anisotropy, the mean-squared displacement, and the non-Gaussian parameter are examined simultaneously. Our result shows that the structures characterized at different length scales change in a self-similar way and these changes are intimately correlated to the translational mobility of atoms. The initial incubation of structural changes at the different scales originates from the restriction of mobility due to the cage effect. The applicability of the MD simulation to the investigation of the bulk properties is discussed.

How, when and why proteins collapse: the relation to folding

Unfolded proteins under strongly denaturing conditions are highly expanded. However, when the conditions are more close to native, an unfolded protein may collapse to a compact globular structure distinct from the folded state. This transition is akin to the coil-globule transition of homopolymers. Single-molecule FRET experiments have been particularly conducive in revealing the collapsed state under conditions of coexistence with the folded state. The collapse can be even more readily observed in natively unfolded proteins. Time-resolved studies, using FRET and small-angle scattering, have shown that the collapse transition is a very fast event, probably occurring on the submicrosecond time scale. The forces driving collapse are likely to involve both hydrophobic and backbone interactions. The loss of configurational entropy during collapse makes the unfolded state less stable compared to the folded state, thus facilitating folding.

Role of solvation effects in protein denaturation: from thermodynamics to single molecules and back

Protein stability often is studied in vitro through the use of urea and guanidinium chloride, chemical cosolvents that disrupt protein native structure. Much controversy still surrounds the underlying mechanism by which these molecules denature proteins. Here we review current thinking on various aspects of chemical denaturation. We begin by discussing classic models of protein folding and how the effects of denaturants may fit into this picture through their modulation of the collapse, or coil-globule transition, which typically precedes folding. Subsequently, we examine recent molecular dynamics simulations that have shed new light on the possible microscopic origins of the solvation effects brought on by denaturants. It seems likely that both denaturants operate by facilitating solvation of hydrophobic regions of proteins. Finally, we present recent single-molecule fluorescence studies of denatured proteins, the analysis of which corroborates the role of denaturants in shifting the equilibrium of the coil-globule transition.

Aging of thin polymer films cast from a near-theta solvent

We have examined the aging behavior of spin-cast thin polymer films as a function of their processing history. Films prepared from solutions close to the Θ temperature were aged for varying times at room temperature, followed by a dewetting experiment above the glass transition temperature of the polymer. The characteristic aging time varied strongly with the quality of the solvent, which is attributed to distorted chain conformations in the as-cast films. This is an indication for the nonequilibrium nature of thin polymer films, possibly causing some of their unexplained properties.

Hydrodynamic selection of the kinetic pathway of a polymer coil-globule transition

Recently, the role of hydrodynamic interactions in the selection of a kinetic pathway for phase transitions has attracted considerable attention. Here we study this problem numerically by taking as an example a coil-globule transition of a single polymer, which is a prototype model of protein folding. When a swollen polymer collapses into a globule state, hydrodynamic interactions accelerate the transition. We find, on the other hand, that when a rather compact polymer collapses into the same final state, hydrodynamic interactions decelerate the transition due to a slow squeezing process of the solvent. We reveal that the degree of the initial enhancement of anisotropy of the polymer configuration determines whether hydrodynamic interactions accelerate or decelerate the collapsing dynamics. We also discuss the possible relevance of squeezing flow effects in protein folding.

Collapse transition in proteins

The coil-globule transition, a tenet of the physics of polymers, has been identified in recent years as an important unresolved aspect of the initial stages of the folding of proteins. We describe the basics of the collapse transition, starting with homopolymers and continuing with proteins. Studies of denatured-state collapse under equilibrium are then presented. An emphasis is placed on single-molecule fluorescence experiments, which are particularly useful for measuring properties of the denatured state even under conditions of coexistence with the folded state. Attempts to understand the dynamics of collapse, both theoretically and experimentally, are then described. Only an upper limit for the rate of collapse has been obtained so far. Improvements in experimental and theoretical methodology are likely to continue to push our understanding of the importance of the denatured-state thermodynamics and dynamics for protein folding in the coming years.

Atomistic simulations of the effects of polyglutamine chain length and solvent quality on conformational equilibria and spontaneous homodimerization

Aggregation of expanded polyglutamine tracts is associated with nine different neurodegenerative diseases, including Huntington's disease. Experiments and computer simulations have demonstrated that monomeric forms of polyglutamine molecules sample heterogeneous sets of collapsed structures in water. The current work focuses on a mechanistic characterization of polyglutamine homodimerization as a function of chain length and temperature. These studies were carried out using molecular simulations based on a recently developed continuum solvation model that was designed for studying conformational and binding equilibria of intrinsically disordered molecules such as polyglutamine systems. The main results are as follows: Polyglutamine molecules form disordered, collapsed globules in aqueous solution. These molecules spontaneously associate at conditions approaching those of typical in vitro experiments for chains of length N>/=15. The spontaneity of these homotypic associations increases with increasing chain length and decreases with increasing temperature. Similar and generic driving forces govern both collapse and spontaneous homodimerization of polyglutamine in aqueous milieus. Collapse and dimerization maximize self-interactions and reduce the interface between polyglutamine molecules and the surrounding solvent. Other than these generic considerations, there do not appear to be any specific structural requirements for either chain collapse or chain dimerization; that is, both collapse and dimerization are nonspecific in that disordered globules form disordered dimers. In fact, it is shown that the driving force for intermolecular associations is governed by spontaneous conformational fluctuations within monomeric polyglutamine. These results suggest that polyglutamine aggregation is unlikely to follow a homogeneous nucleation mechanism with the monomer as the critical nucleus. Instead, the results support the formation of disordered, non-beta-sheet-like soluble molten oligomers as early intermediates--a proposal that is congruent with recent experimental data.