Conformational Entropy Research Articles

Entropic barrier of topologically immobilized DNA in hydrogels

The single most intrinsic property of nonrigid polymer chains is their ability to adopt enormous numbers of chain conformations, resulting in huge conformational entropy. When such macromolecules move in media with restrictive spatial constraints, their trajectories are subjected to reductions in their conformational entropy. The corresponding free energy landscapes are interrupted by entropic barriers separating consecutive spatial domains which function as entropic traps where macromolecules can adopt their conformations more favorably. Movement of macromolecules by negotiating a sequence of entropic barriers is a common paradigm for polymer dynamics in restrictive media. However, if a single chain is simultaneously trapped by many entropic traps, it has recently been suggested that the macromolecule does not undergo diffusion and is localized into a topologically frustrated dynamical state, in apparent violation of Einstein's theorem. Using fluorescently labeled λ-DNA as the guest macromolecule embedded inside a similarly charged hydrogel with more than 95% water content, we present direct evidence for this new state of polymer dynamics at intermediate confinements. Furthermore, using a combination of theory and experiments, we measure the entropic barrier for a single macromolecule as several tens of thermal energy, which is responsible for the extraordinarily long extreme metastability. The combined theory-experiment protocol presented here is a determination of single-molecule entropic barriers in polymer dynamics. Furthermore, this method offers a convenient general procedure to quantify the underlying free energy landscapes behind the ubiquitous phenomenon of movement of single charged macromolecules in crowded environments.

Energy-entropy prediction of octanol\u2013water logP of SAMPL7 N-acyl sulfonamide bioisosters

Partition coefficients quantify a molecule’s distribution between two immiscible liquid phases. While there are many methods to compute them, there is not yet a method based on the free energy of each system in terms of energy and entropy, where entropy depends on the probability distribution of all quantum states of the system. Here we test a method in this class called Energy Entropy Multiscale Cell Correlation (EE-MCC) for the calculation of octanol–water logP values for 22 N-acyl sulfonamides in the SAMPL7 Physical Properties Challenge (Statistical Assessment of the Modelling of Proteins and Ligands). EE-MCC logP values have a mean error of 1.8 logP units versus experiment and a standard error of the mean of 1.0 logP units for three separate calculations. These errors are primarily due to getting sufficiently converged energies to give accurate differences of large numbers, particularly for the large-molecule solvent octanol. However, this is also an issue for entropy, and approximations in the force field and MCC theory also contribute to the error. Unique to MCC is that it explains the entropy contributions over all the degrees of freedom of all molecules in the system. A gain in orientational entropy of water is the main favourable entropic contribution, supported by small gains in solute vibrational and orientational entropy but offset by unfavourable changes in the orientational entropy of octanol, the vibrational entropy of both solvents, and the positional and conformational entropy of the solute.

The second law: information theory and self-assembly

An interplay between algorithmic and mechanistic viewpoints is necessary to quantify the rates of transition between different states in developmental pathways. An ab initio method that uses only information theory and geometry to model the conformational entropy of linkages is discussed. This approach also reveals some ties between biological questions and the foundations of mathematics.

Supramolecular Self-assembly Behaviors of Asymmetric Diblock Copolymer Blends with Hydrogen Bonding Interactions between Shorter Blocks Modelled by Yukawa Potentials

We employed the extended self-consistent field theory to investigate the supramolecular self-assembly behaviors of asymmetric diblock copolymer blends (AB/B’C) with hydrogen bonding interactions between shorter B and B’ blocks. The hydrogen bonding interactions are described by Yukawa potentials, where the hydrogen bonding donors and acceptors were modelled as two blocks smeared with opposite screened charges. The hierarchical microstructures with parallelly packed lamellae-in-lamellae (Lam) and 4.8.8 Archimedean tilting pattern (4.8.8) were observed at lower and higher hydrogen bonding density (θ), respectively. The hierarchy of Lam and 4.8.8 were demonstrated by the one- and two-dimensional density profiles and the underlying order of the large-length-scale and small-length-scale microstructures were also clarified. It was found that the 4.8.8 is favorable to the stronger hydrogen bonding density or interactions. As θ increases, the microphase transition from Lam to 4.8.8 occurs at θ =0.34, which is mainly attributed to the optimization of the electrostatic energy and conformational entropy with sacrificing the interfacial energy. This work can provide a new strategy to understand the supramolecular self-assembly as well as the mechanism behind the formation of complex hierarchical microstructures.

Thermodynamic Compensation in Peptides Following Liquid-Liquid Phase Separation.

Liquid-liquid phase separation of proteins often incorporates intrinsically disordered proteins or those with disordered regions. Examining these processes via the entropy change is desirable for establishing a quantitative foundation with which to probe and understand these phase transitions. Of interest is the effect of residue sequence on the entropy of the peptide backbone. In this work we model these systems via all atom simulations of liquid-liquid phase separation of peptides. Systems of supersaturated pentapeptides separate into a peptide-dense liquid droplet phase as well as a dilute (saturated) aqueous phase. An analysis of the change in backbone conformational entropy associated with the phase transition was performed. We examined systems of four different pentapeptides (GGGGG, GGQGG, GGNGG, and GGVGG) in order to explore the effect of sequence variation on the conformational entropy, as well as the effect of side chain variation on the physical characteristics of the droplet phases. We find that the loss of conformational entropy that accompanies aqueous → droplet transitions is more than compensated by a decrease in interaction enthalpy as contributions to the free energy change for the process.

Symmetry-Broken Patches on Gold Nanoparticles through Deficient Ligand Exchange.

Symmetry-broken nanoparticles (NPs) are important building blocks with directional interparticle interaction as a key to access the precise organization of NPs macroscopically. We report a facile, one-pot synthetic approach to prepare high-quality symmetry-broken plasmonic gold NPs (AuNPs). Symmetry-broken patterning is achieved through deficient ligand exchange of isotropic AuNPs with thiol-terminated polystyrene (PS-SH) in the presence of an amphiphilic polymer surfactant. The concentration of PS-SH plays a dominant role in tuning surface patterning and coverage of AuNPs. The formation of asymmetric surface patches arises from the interplay between the conformational entropy of polymer ligands and the interfacial energy between polymer-grafted AuNPs and the solvent. Our method illustrates new paradises to design asymmetric NPs with directional interparticle interactions to access the precise organization of NPs.

On gyration radius distributions of star-like macromolecules

Using the path integral approach, we obtain the characteristic functions of the gyration radius distributions for Gaussian star and Gaussian rosette macromolecules. We derive the analytical expressions for cumulants of both distributions. Applying the steepest descent method, we estimate the probability distribution functions (PDFs) of the gyration radius in the limit of a large number of star and rosette arms in two limiting regimes: for strongly expanded and strongly collapsed macromolecules. We show that in both cases, in the regime of a large gyration radius relative to its mean-square value, the PDFs can be described by the Gaussian functions. In the shrunk macromolecule regime, both distribution functions tend to zero faster than any power of the gyration radius. Based on the asymptotic behavior of the distribution functions and the behavior of statistical dispersions, we demonstrate that the PDF for the rosette is more densely localized near its maximum than that for the star polymer. We construct the interpolation formula for the gyration radius distribution of the Gaussian star macromolecule which can help to take into account the conformational entropy of the flexible star macromolecules within the Flory-type mean-field theories.

Tying up the Loose Ends: A Mathematically Knotted Protein.

Knots have attracted scientists in mathematics, physics, biology, and engineering. Long flexible thin strings easily knot and tangle as experienced in our daily life. Similarly, long polymer chains inevitably tend to get trapped into knots. Little is known about their formation or function in proteins despite >1,000 knotted proteins identified in nature. However, these protein knots are not mathematical knots with their backbone polypeptide chains because of their open termini, and the presence of a “knot” depends on the algorithm used to create path closure. Furthermore, it is generally not possible to control the topology of the unfolded states of proteins, therefore making it challenging to characterize functional and physicochemical properties of knotting in any polymer. Covalently linking the amino and carboxyl termini of the deeply trefoil-knotted YibK from Pseudomonas aeruginosa allowed us to create the truly backbone knotted protein by enzymatic peptide ligation. Moreover, we produced and investigated backbone cyclized YibK without any knotted structure. Thus, we could directly probe the effect of the backbone knot and the decrease in conformational entropy on protein folding. The backbone cyclization did not perturb the native structure and its cofactor binding affinity, but it substantially increased the thermal stability and reduced the aggregation propensity. The enhanced stability of a backbone knotted YibK could be mainly originated from an increased ruggedness of its free energy landscape and the destabilization of the denatured state by backbone cyclization with little contribution from a knot structure. Despite the heterogeneity in the side-chain compositions, the chemically unfolded cyclized YibK exhibited several macroscopic physico-chemical attributes that agree with theoretical predictions derived from polymer physics.

Study of the capacity of poly(N‐vinylcarbazole) derivatives to form honeycomb‐like patterns

AbstractPoly(N‐vinylcarbazole), two dihalogenated derivatives, and nanocellulose/poly(N‐vinylcarbazole) nanocomposite thin films have been fabricated with micrometric pore topography. This patterned structure has been developed by using breath figure method, which is an easy and cost‐effective method to obtain honeycomb‐like arrays. In addition, the effect of the dihalogenated substitutions and the presence of nanocellulose fillers in the topography and conformational entropy were evaluated. The conformational entropy of each sample was calculated by using Voronoi tessellation. It is important to notice that highly regular patterns have been obtained from this photoconductive polymer, since the conductivity of this kind of material is proportional to the absorbed light, this honeycomb‐like topography could improve the light capture and the efficiency of photoelectric conversion, increasing the potential applicability of these materials.

Protein folding - seeing is deceiving.

This Perspective is intended to raise questions about the conventional interpretation of protein folding. According to the conventional interpretation, developed over many decades, a protein population can visit a vast number of conformations under unfolding conditions, but a single dominant native population emerges under folding conditions. Accordingly, folding comes with a substantial loss of conformational entropy. How is this price paid? The conventional answer is that favorable interactions between and among the side chains can compensate for entropy loss, and moreover, these interactions are responsible for the structural particulars of the native conformation. Challenging this interpretation, the Perspective introduces a proposal that high energy (i.e., unfavorable) excluding interactions winnow the accessible population substantially under physical–chemical conditions that favor folding. Both steric clash and unsatisfied hydrogen bond donors and acceptors are classified as excluding interactions, so called because conformers with such disfavored interactions will be largely excluded from the thermodynamic population. Both excluding interactions and solvent factors that induce compactness are somewhat nonspecific, yet together they promote substantial chain organization. Moreover, proteins are built on a backbone scaffold consisting of α‐helices and strands of β‐sheet, where the number of hydrogen bond donors and acceptors is exactly balanced. These repetitive secondary structural elements are the only two conformers that can be both completely hydrogen‐bond satisfied and extended indefinitely without encountering a steric clash. Consequently, the number of fundamental folds is limited to no more than ~10,000 for a protein domain. Once excluding interactions are taken into account, the issue of “frustration” is largely eliminated and the Levinthal paradox is resolved. Putting the “bottom line” at the top: it is likely that hydrogen‐bond satisfaction represents a largely under‐appreciated parameter in protein folding models.

Dissecting the Conformational Free Energy of a Small Peptide in Solution.

The free energy surface of a small peptide was analyzed based on an unbiased microsecond molecular dynamics simulation. The peptide sampled disordered conformational ensembles of distinct compactness, and its free energy was decomposed into separate contributions from the intramolecular potential energy, conformational entropy, and solvation free energy. The latter was further broken down into enthalpic and entropic contributions due to peptide-water and water-water interactions. This decomposition was enabled by a generalized linear response relation between the peptide-water interaction energy and the solvation free energy, which was empirically parametrized by explicit solvation free energy calculations for representative peptide conformations. This full dissection of the peptide free energy identifies individual contributions that stabilize and destabilize compact and extended peptide conformational ensembles and reveals the origin of a free energy barrier associated with transitions between them.

Implementation of the Freely Jointed Chain Model to Assess Kinetics and Thermodynamics of Thermosensitive Coil-Globule Transition by Markov States.

We revived and implemented a method developed by Kuhn in 1934, originally only published in German, that is, the so-called “freely jointed chain” model. This approach turned out to be surprisingly useful for analyzing state-of-the-art computer simulations of the thermosensitive coil–globule transition of N-Isopropylacrylamide 20-mer. Our atomistic computer simulations are orders of magnitude longer than those of previous studies and lead to a reliable description of thermodynamics and kinetics at many different temperatures. The freely jointed chain model provides a coordinate system, which allows us to construct a Markov state model of the conformational transitions. Furthermore, this guarantees a reliable reconstruction of the kinetics in back-and-forth directions. In addition, we obtain a description of the high diversity and variability of both conformational states. Thus, we gain a detailed understanding of the coil–globule transition. Surprisingly, conformational entropy turns out to play only a minor role in the thermodynamic balance of the process. Moreover, we show that the radius of gyration is an unexpectedly unsuitable coordinate to comprehend the transition kinetics because it does not capture the high conformational diversity within the different states. Consequently, the approach presented here allows for an exhaustive description and resolution of the conformational ensembles of arbitrary linear polymer chains.

Adsorption of Block-Polyelectrolytes on an Oppositely Charged Surface

Electrostatic interaction is widely considered to be the main driving force of the adsorption of polyelectrolytes (PEs) on an oppositely charged surface. Therefore, the alteration of charge distribution as a result of varying PE chain structures will directly affect the performance of the adsorption behaviors in the system at a certain monomer concentration. In this work, we studied the effects of chain structure (chain length, Nₘ) as well as sequence variables including charge fraction and blockiness on the adsorption behaviors of block-PEs on an oppositely charged planar surface in the salt-free system using PE-Poisson–Boltzmann equations. Due to the complicated competitions among the translational entropies of PEs and counterions, the conformational entropy of PEs, and the surface attractive interactions, we find that the adsorption amounts show rich behaviors. First, the adsorption amounts of charged ζc and neutral ζₙ monomers both show two-stage behaviors, that is, ζc and ζₙ drastically increase with Nₘ for short chains but moderately increase to an asymptotic value for relatively long chains. Second, the adsorption amounts of charged monomers show non-monotonic behaviors (increase first then decrease) with charge fraction. However, the adsorption amounts of neutral monomers monotonically decrease with charge fraction. Finally, when chain length and charge fraction are unchanged, the adsorption amounts of both neutral and charged monomers are independent of blockiness. Moreover, we find that overcharging can only appear when the bare surface charge density is low enough, while charge reversal and inversion can be observed in the entire range of the bare surface charge density with appropriate conditions. Additionally, our results indicate that the strength of charge reversal and inversion increases monotonically with chain length and charge fraction, while it changes non-monotonically with blockiness.

Conformationally rigid and flexible surfactant binary (pseudoternary) mixed micelle with mutual synergistic interaction

Molar excess Gibbs energy of a binary mixed micelle with a conformationally flexible and rigid surfactant can be represented by first-order Margules function. Excess entropy comes from the conformational entropy rather than the configurational entropy. Such a binary mixed micelle with the synergistic interactions between micellar building blocks can be considered a pseudoternary mixed micelle from the conformationally rigid surfactant, twisted (globular) conformation of the conformationally flexible surfactant, and elongated conformation of the same surfactant. Excess entropy of such mixed micelles can be approximated with a microcanonical ensemble of the conformationally flexible surfactants. Particles of these surfactants are distributed between the average lower energy (elongated conformation) and the average higher energy (twisted conformation). A new structural parameter of mixed micelles was introduced - the parameter of disorder, which represents the molar fraction of the conformation with an average higher energy (twisted conformation) of the conformationally flexible surfactant in the mixed micellar pseudophase. Influence of the new structural parameter on the conformational rigid surfactant's excess chemical potential is discussed.

Reexamining the role of intercrystalline links in the II-I phase transition of Poly(1-butene)

Despite being studied extensively, the role of intercrystalline links has not been elucidated completely. To clarify its role, two kinds of different lengths of poly (1-butene) chains were blended with different proportions, among which only the long chain was able to form intercrystalline links. Phase transitions of these blends were monitored with time-resolved FTIR. Surprisingly, when the content of long chains reached 30% or 40%, phase transition rates were even slower than they existed alone, indicating phase transition was suppressed greatly by long chains. Adding a minority of short chains actually would also slow down the phase transition, though the effect was not as obvious as long chains. Estimation indicates that the change of enthalpy cannot compensate for the loss in conformational entropy in II-I phase transition. To ensure continuous proceeding of II-I phase transition, the role of amorphous chains is indispensable in the quiescent phase transition. Nevertheless, the role of intercrystalline links should not be undue stressed either. Analysis indicates that a transient amorphous network exists in the blends, consisting of different sizes of amorphous coils. Long chains can form larger coils because of intercrystalline links, while small coils are most likely from individual short chains. Entanglements exist between these amorphous coils, stabilizing the transient network, which would also affect phase transition significantly. When a minority of short chains are added, stress on one side of some big coils reduces. The big coils deviate from previous mass centers, slowing down its internal stress on form II crystal and thus its phase transition rate. Conversely, when a minority of long chains are added, big coils not only would lead small coils deviating from their previous mass centers, but also slow down contraction of the small coils through the entanglement, leading to a larger decrease in form I nuclei. The roles of intercrystalline links and entanglement should be considered simultaneously.

Entropy-Entropy Compensation between the Protein, Ligand, and Solvent Degrees of Freedom Fine-Tunes Affinity in Ligand Binding to Galectin-3C.

Molecular recognition is fundamental to biological signaling. A central question is how individual interactions between molecular moieties affect the thermodynamics of ligand binding to proteins and how these effects might propagate beyond the immediate neighborhood of the binding site. Here, we investigate this question by introducing minor changes in ligand structure and characterizing the effects of these on ligand affinity to the carbohydrate recognition domain of galectin-3, using a combination of isothermal titration calorimetry, X-ray crystallography, NMR relaxation, and computational approaches including molecular dynamics (MD) simulations and grid inhomogeneous solvation theory (GIST). We studied a congeneric series of ligands with a fluorophenyl-triazole moiety, where the fluorine substituent varies between the ortho, meta, and para positions (denoted O, M, and P). The M and P ligands have similar affinities, whereas the O ligand has 3-fold lower affinity, reflecting differences in binding enthalpy and entropy. The results reveal surprising differences in conformational and solvation entropy among the three complexes. NMR backbone order parameters show that the O-bound protein has reduced conformational entropy compared to the M and P complexes. By contrast, the bound ligand is more flexible in the O complex, as determined by 19F NMR relaxation, ensemble-refined X-ray diffraction data, and MD simulations. Furthermore, GIST calculations indicate that the O-bound complex has less unfavorable solvation entropy compared to the other two complexes. Thus, the results indicate compensatory effects from ligand conformational entropy and water entropy, on the one hand, and protein conformational entropy, on the other hand. Taken together, these different contributions amount to entropy–entropy compensation among the system components involved in ligand binding to a target protein.

Understanding Conformational Entropy in Small Molecules.

The calculation of the entropy of flexible molecules can be challenging, since the number of possible conformers can grow exponentially with molecule size and many low-energy conformers may be thermally accessible. Different methods have been proposed to approximate the contribution of conformational entropy to the molecular standard entropy, including performing thermochemistry calculations with all possible stable conformations and developing empirical corrections from experimental data. We have performed conformer sampling on over 120,000 small molecules generating some 12 million conformers, to develop models to predict conformational entropy across a wide range of molecules. Using insight into the nature of conformational disorder, our cross-validated physically motivated statistical model gives a mean absolute error of ∼4.8 J/mol·K or under 0.4 kcal/mol at 300 K. Beyond predicting molecular entropies and free energies, the model implies a high degree of correlation between torsions in most molecules, often assumed to be independent. While individual dihedral rotations may have low energetic barriers, the shape and chemical functionality of most molecules necessarily correlate their torsional degrees of freedom and hence restrict the number of low-energy conformations immensely. Our simple models capture these correlations and advance our understanding of small molecule conformational entropy.

Hydration Thermodynamics of Familial Parkinson's Disease-Linked Mutants of α-Synuclein.

The hydration thermodynamics of different mutants of α-synuclein (α-syn) related to familial Parkinson's disease (PD) is explored using a computational approach that combines both molecular dynamics simulations in water and integral equation theory of molecular liquids. This analysis focuses on the change in conformational entropy, hydration free energy (HFE), and partial molar volume of α-syn upon mutation. The results show that A53T, A30P, E46K, and H50Q mutants aggregate more readily and display increased HFE and less negative interaction volume than the wild-type α-syn. In contrast, an opposite trend is observed for the G51D mutant with a lower experimental aggregation rate. The residuewise decomposition analysis of the HFE highlights that the dehydration/hydration of the hydrophilic residue-rich N- and C-termini of α-syn majorly contributes to the change upon mutation. The hydration shell contributions of different residues to the interaction volume are consistent with its increase/decrease upon mutation. This work shows that both HFE and interaction volume determine the aggregation kinetics of α-syn upon mutation and may serve as an appropriate benchmark for the treatment of PD.

Computational Screening of a Functional Cyclodextrin Derivative for Suppressing a Side Effect of Doxorubicin.

The binding affinity of the beta-cyclodextrin (β-CyD) derivatives with Doxorubicin (Dox) is evaluated by means of the 3D-RISM/KH theory combined with the molecular dynamics simulation in order to screen the compounds for suppressing a side-effect of the cancer drug. A protocol revised for the external and conformational entropies of the host-guest system is employed to calculate the binding free energy. It is found that the direct interactions of CyD with Dox and the desolvation free-energies of the both compounds largely cancel out to leave moderate contributions to the affinity, which are comparable to those from the entropies. The results shed light on the entropy terms for determining the binding affinity, although the external-entropy terms are essentially constant over all the compounds examined and do not affect the screening. The theoretical result is compared with the experimental data of the association constant for a CyD derivative which was predicted to be the best compound by the preliminary calculation without the entropy terms.

Enhanced Heterogeneous Diffusion of Nanoparticles in Semiflexible Networks.

The transport of nanoparticles in semiflexible networks, which form diverse principal structural components throughout living systems, is important in biology and biomedical applications. By combining large-scale molecular simulations as well as theoretical analysis, we demonstrate here that nanoparticles in polymer networks with semiflexible strands possess enhanced heterogeneous diffusion characterized by more evident hopping dynamics. Particularly, the hopping energy barrier approximates to linear dependence on confinement parameters in the regime of moderate rigidity, in contrast to the quadratic dependence of both its soft and hard counterparts. This nonmonotonic feature can be attributed to the competition between the conformation entropy and the bending energy regulated by the chain rigidity, captured by developing an analytical model of a hopping energy barrier. Moreover, these theoretical results agree reasonably well with previous experiments. The findings bear significance in unraveling the fundamental physics of substance transport confined in network-topological environments and would provide an explanation for the dynamics diversity of nanoparticles within various networks, biological or synthetic.