Algorithmic Generation of Freely Jointed Hard Sphere Chains and Properties of Inertial Tensors

Abstract

A statistical algorithm, capable of generating a large number of freely jointed hard sphere chains, is presented. This is the first of a series of algorithms being developed to model unfolded proteins by different modes of hard sphere chains. The aim of these studies is to systematically investigate the effects of different factors, such as atomic radii, bond angles, torsion angles, chain length, etc., on the conformation of unfolded proteins and other random polymers. As continuous models, various types of hard sphere chains enable one to isolate the aforementioned factors one at a time for investigation and thus are advantageous over discrete lattice models. In particular, the freely jointed hard sphere chain model allows one to evaluate the excluded volume effect. As a first step in this endeavor, the average determinant D(N,r) and the average trace T(N, r) of the inertial tensor A of the random chains were calculated at various sphere radii r and chain lengths N. It is found that both the average determinant D(N,r) and the average trace T(N,r) scale linearly with chain length N. after logarithmic transformation. However, the critical exponent of D(N,r) increases with r faster than that of T(N,r) as a result of the non-commutativity between the det operator and the average operator < >. The significance of the algorithm and the results obtained on understanding random polypeptide chains are discussed.     

A comparison of model linear chain molecules with constrained and flexible bond lengths under planar Couette and extensional flows

We compare directly under flow two commonly used coarse grained models of linear polymers, namely the flexible finitely extensible nonlinear elastic (FENE) chain, and the freely jointed tangent sphere chain, otherwise known as the freely jointed chain. The comparison is based on viscometric, structural and dynamical properties. We use non-equilibrium molecular dynamics to simulate steady-state systems under planar Couette flow and planar extensional flow. Computed properties include shear and elongational viscosities, normal stresses, radius of gyration and end-to-end distances, order parameters, alignment angles and spin angular velocities. In all computed properties we observe very little difference between the two molecular models. Therefore, the choice of either model is suitable, though there is a computational advantage in the use of the FENE model.     

Electrophoretic charge density and persistence length of DNA as measured by fluorescence microscopy

Individual ethidium-stained DNA molecules, embedded in an agarose gel made with electrophoresis buffer (0.05 molar salt), are observed using a fluorescence microscope. In the first experiment, open circular 66 kilobase pair (kbp) plasmids, immobilized by agarose fibers threaded through their centers, display entropic "rubber" elasticity. The charged molecules extend in an electric field of several volts per centimeter and contract to a compact random coil when the field is removed. The extension of the plasmids as a function of field strength is consistent with the freely jointed chain model when the effective electrophoretic charge density is set at 15 e-per persistence length. In a second experiment, stained linear 48.5 kbp DNA molecules are observed as random coils immobilized in agarose. A measure of their size, here named the "maximal-X-extent," is taken for 100 molecules and found to average 1.47 mu. A Monte Carlo computer simulation of random coils (freely jointed chain model) gives the same maximal-X-extent value when the persistence length is set at 0.08 mu.     

Influence of local and residual structures on the scaling behavior and dimensions of unfolded proteins

Although recent spectroscopic studies of chemically denatured proteins hint at significant nonrandom residual structure, the results of extensive small angle X-ray scattering studies suggest random coil behavior, calling for a coherent understanding of these seemingly contradicting observations. Here, we report the results of a Monte Carlo study of the effects of two types of local structures, alpha helix and Polyproline II (PPII) helix, on the dimensions of random coil polyalanine chains viewed as a model of highly denatured proteins. We find that although Flory's power law scaling, long regarded as a signature of random coil behavior, holds for chains containing up to 90% alpha or PPII helix, the absolute magnitude of the chain dimensions is sensitive to helix content. As residual alpha helix content increases, the chain contracts until it reaches a minimum radius at approximately 70% helix, after which the chain dimensions expand rapidly. With an alpha helix content of approximately 20%, corresponding to the Ramachandran probability of being in the helical basin, experimentally observed radii of gyration are recovered. Experimental radii are similarly recovered at an alpha helix content of approximately 87%, providing an explanation for the previously puzzling experimental finding that the dimensions of the highly helical methanol-induced unfolded state are experimentally indistinguishable from those of the helix-poor urea-unfolded state. In contrast, the radius of gyration increases monotonically with increasing PPII content, and is always more expanded than the dimensions observed experimentally. These results suggest that PPII is unlikely the sole, dominant preferred conformation for unfolded proteins.     

Computational simulation of the statistical properties of unfolded proteins

A simple Monte Carlo method was used to generate ensembles of simulated polypeptide conformations that are restricted only by steric repulsion. The models used for these simulations were based on the sequences of four real proteins, ranging in size from 26 to 268 amino acid residues, and included all non-hydrogen atoms. Two sets of calculations were performed, one that included only intra-residue steric repulsion terms and those between adjacent residues, and one that included repulsion terms between all possible atom pairs, so as to explicitly account for the excluded volume effect. Excluded volume was found to increase the average radius of gyration of the chains by 20-40%, with the expansion factor increasing with chain length. Contrary to recent suggestions, however, the excluded volume effect did not greatly restrict the distribution of dihedral angles or favor native-like topologies. The average dimensions of the ensembles calculated with excluded volume were consistent with those measured experimentally for unfolded proteins of similar sizes under denaturing conditions, without introducing any adjustable scaling factor. The simulations also reproduced experimentally determined effective concentrations for the formation of disulfide bonds in reduced and unfolded proteins. The statistically generated ensembles included significant numbers of conformations that were nearly as compact as the corresponding native proteins, as well as many that were as accessible to solvent as a fully extended chain. On the other hand, conformations with as much buried surface area as the native proteins were very rare, as were highly extended conformations. These results suggest that the overall properties of unfolded proteins can be usefully described by a random coil model and that an unfolded polypeptide can undergo significant collapse while losing only a relatively small fraction of its conformational entropy.     

Mechanically probing the folding pathway of single RNA molecules

We study theoretically the denaturation of single RNA molecules by mechanical stretching, focusing on signatures of the (un)folding pathway in molecular fluctuations. Our model describes the interactions between nucleotides by incorporating the experimentally determined free energy rules for RNA secondary structure, whereas exterior single-stranded regions are modeled as freely jointed chains. For exemplary RNA sequences (hairpins and the Tetrahymena thermophila group I intron), we compute the quasiequilibrium fluctuations in the end-to-end distance as the molecule is unfolded by pulling on opposite ends. Unlike the average quasiequilibrium force-extension curves, these fluctuations reveal clear signatures from the unfolding of individual structural elements. We find that the resolution of these signatures depends on the spring constant of the force-measuring device, with an optimal value intermediate between very rigid and very soft. We compare and relate our results to recent experiments by Liphardt et al. (2001).     

New version of Monte Carlo expanded ensemble method for precise calculations of free energy difference

A new version of Monte Carlo (MC) expanded ensemble (EE) method is proposed for the calculations of free energy difference (FED) between two different systems with close values of the free energy. In order to check the method the FED between simple model systems (fluid of hard spheres and freely jointed polymer chain of hard spheres) was calculated. The free energy of the mentioned above systems was also calculated by a standard MC EE method in order to compare the results of two simulations. It was shown that the accuracy of a new algorithm is the same as of a standard one. At the same time new version of EE allows us to obtain FED between two systems having quite different structures, but similar free energies, during one simulation run.     

Kinetics of Contact Formation and End-to-End Distance Distributions of Swollen Disordered Peptides

Unstructured polypeptide chains are subject to various degrees of swelling or compaction depending on the combination of solvent condition and amino acid sequence. Highly denatured proteins generally behave like random-coils with excluded volume repulsion, whereas in aqueous buffer more compact conformations have been observed for the low-populated unfolded state of globular proteins as well as for naturally disordered sequences. To quantitatively account for the different mechanisms inducing the swelling of polypeptides, we have examined three 14-residues peptides in aqueous buffer and in denaturant solutions, including the well characterized AGQ repeat as a reference and two variants, in which we have successively introduced charged side chains and removed the glycines. Quenching of the triplet state of tryptophan by close contact with cysteine has been used in conjunction with Förster resonance energy transfer to study the equilibrium and kinetic properties of the peptide chains. The experiments enable accessing end-to-end root mean-square distance, probability of end-to-end contact formation and intrachain diffusion coefficient. The data can be coherently interpreted on the basis of a simple chain model with backbone angles obtained from a library of coil segments of proteins and hard sphere repulsion at each C_α position. In buffered water, we find that introducing charges in a glycine-rich sequence induces a mild chain swelling and a significant speed-up of the intrachain dynamics, whereas the removal of the glycines results in almost a two-fold increase of the chain volume and a drastic slowing down. In denaturants we observe a pronounced swelling of all the chains, with significant differences between the effect of urea and guanidinium chloride.     

Modeling a self-avoiding chromatin loop: relation to the packing problem, action-at-a-distance, and nuclear context

There is now convincing evidence that genomes are organized into loops, and that looping brings distant genes together so that they can bind to local concentrations of polymerases in "factories" or "hubs." As there remains no systematic analysis of how looping affects the probability that a gene can access binding sites in such factories/hubs, we used an algorithm that we devised and Monte Carlo methods to model a DNA or chromatin loop as a semiflexible (self-avoiding) tube attached to a sphere; we examine how loop thickness, rigidity, and contour length affect where particular segments of the loop lie relative to binding sites on the sphere. Results are compared with those obtained with the traditional model of an (infinitely thin) freely jointed chain. They provide insights into the packing problem (how long genomes are packed into small nuclei), and action-at-a-distance (how firing of one origin or gene can prevent firing of an adjacent one).     

Rapid intrachain binding of histidine-26 and histidine-33 to heme in unfolded ferrocytochrome C.

Time-resolved spectroscopic studies of unfolded horse iron(II) cytochrome c have suggested that the imidazole side chains of His26 and His33 bind transiently to the heme iron on microsecond time scales, after photodissociation of a carbon monoxide ligand from the heme. Our studies of four variants of cytochrome c (horse wild type, horse H33N, horse H33N/H26Q, and tuna wild type), unfolded in guanidine hydrochloride at pH 6.5, demonstrate that these side chains are responsible for the observed microsecond spectral changes. As His33 and then His26 are eliminated from the horse wild-type sequence, transient optical absorption spectra show systematic suppression of a rapid (approximately 10-100 micros) Soret absorbance change that follows photolysis of CO. Transient binding of these histidine side chains to the heme therefore generates one of the fast kinetic phases observed in previous photochemically triggered spectroscopic studies of dynamics in unfolded iron(II) cytochrome c. Furthermore, both His33 and His26 appear to contribute to a similar extent in these early kinetics. Thus, the stiffness of the polypeptide chain creates a deviation from Gaussian chain behavior by impeding, although not preventing, the formation of short (<10 peptide bonds) intrachain loops around the heme group.     

The low-pH unfolded state of the C-terminal domain of the ribosomal protein L9 contains significant secondary structure in the absence of denaturant but is no more compact than the low-pH urea unfolded state

There is considerable interest in the properties of the unfolded states of proteins, particularly unfolded states which can be populated in the absence of high concentrations of denaturants. Interest in the unfolded state ensemble reflects the fact that it is the starting point for protein folding as well as the reference state for protein stability studies and can be the starting state for pathological aggregation. The unfolded state of the C-terminal domain (residues 58-149) of the ribosomal protein L9 (CTL9) can be populated in the absence of denaturant at low pH. CTL9 is a 92-residue globular alpha, beta protein. The low-pH unfolded state contains more secondary structure than the low-pH urea unfolded state, but it is not a molten globule. Backbone ( (1)H, (13)C, and (15)N) NMR assignments as well as side chain (13)C beta and (1)H beta assignments and (15)N R 2 values were obtained for the pH 2.0 unfolded form of CTL9 and for the urea unfolded state at pH 2.5. Analysis of the deviations of the chemical shifts from random coil values indicates that residues that comprise the two helices in the native state show a clear preference for adopting helical phi and psi angles in the pH 2.0 unfolded state. There is a less pronounced but nevertheless clear tendency for residues 107-124 to preferentially populate helical phi and psi values in the unfolded state. The urea unfolded state has no detectable tendency to populate any type of secondary structure even though it is as compact as the pH 2.0 unfolded state. Comparison of the two unfolded forms of CTL9 provides direct experimental evidence that states which differ significantly in their secondary structure can have identical hydrodynamic properties. This in turn demonstrates that global parameters such as R h or R g are very poor indicators of "random coil" behavior.     

An analytic treatment of the condensation of phospholipid chains

A theory relating hydrocarbon chain length to temperature is developed and applied to the liquid crystalline to crystalline phase transiion in phospholipids. Interactions in both the chains and head groups are considered. The hydrocarbon tails are treated as freely jointed chains and the heads as a system of interacting dipoles. Analytic expressions are obtained for the internal energy, entropy, free energy and specific heat of the combined system. The free energy is minimized to obtain a relationship between straightening force and chain length. Agreement is good with experiments involving variation of water content and % cholesterol.Work was supported by a grant from the National Research Council of Canada.     

Theoretical framework for NMR residual dipolar couplings in unfolded proteins

A theoretical framework for the prediction of nuclear magnetic resonance (NMR) residual dipolar couplings (RDCs) in unfolded proteins under weakly aligning conditions is presented. The unfolded polypeptide chain is modeled as a random flight chain while the alignment medium is represented by a set of regularly arranged obstacles. For the case of bicelles oriented perpendicular to the magnetic field, a closed-form analytical result is derived. With the obtained analytical expression the RDCs are readily accessible for any locus along the chain, for chains of differing length, and for varying bicelle concentrations. The two general features predicted by the model are (i) RDCs in the center segments of a polypeptide chain are larger than RDCs in the end segments, resulting in a bell-shaped sequential distribution of RDCs, and (ii) couplings are larger for shorter chains than for longer chains at a given bicelle concentration. Experimental data available from the literature confirm the first prediction of the model, providing a tool for recognizing fully unfolded polypeptide chains. With less certainty experimental data appear to support the second prediction as well. However, more systematic experimental studies are needed in order to validate or disprove the predictions of the model. The presented framework is an important step towards a solid theoretical foundation for the analysis of experimentally measured RDCs in unfolded proteins in the case of alignment media such as polyacrylamide gels and neutral bicelle systems which align biomacromolecules by a steric mechanism. Various improvements and generalizations are possible within the suggested approach.     

Structure of gel phase saturated lecithin bilayers: temperature and chain length dependence.

Systematic low-angle and wide-angle x-ray scattering studies have been performed on fully hydrated unoriented multilamamellar vesicles of saturated lecithins with even chain lengths N = 16, 18, 20, 22, and 24 as a function of temperature T in the normal gel (L beta') phase. For all N, the area per chain Ac increases linearly with T with an average slope dAc/dT = 0.027 A2/degree C, and the lamellar D-spacings also increase linearly with an average slope dD/dT = 0.040 A/degree C. At the same T, longer chain length lecithins have more densely packed chains, i.e., smaller Ac's, than shorter chain lengths. The chain packing of longer chain lengths is found to be more distorted from hexagonal packing than that of smaller N, and the distortion epsilon of all N approaches the same value at the respective transition temperatures. The thermal volume expansion of these lipids is accounted for by the expansion in the hydrocarbon chain region. Electron density profiles are constructed using four orders of low-angle lamellar peaks. These show that most of the increase in D with increasing T is due to thickening of the bilayers that is consistent with a decrease in tilt angle theta and with little change in water spacing with either T or N. Because of the opposing effects of temperature on area per chain Ac and tilt angle 0, the area expansivity alpha A is quite small. A qualitative theoretical model based on competing head and chain interactions accounts for our results.     

The hydrodynamic and conformational properties of denatured proteins in dilute solutions

Published data on the characterization of unfolded proteins in dilute solutions in aqueous guanidine hydrochloride are analyzed to show that the data are not fit by either the random flight or wormlike chain models for linear chains. The analysis includes data on the intrinsic viscosity, root‐mean‐square radius of gyration, from small‐angle X‐ray scattering, and hydrodynamic radius, from the translational diffusion coefficient. It is concluded that residual structure consistent with that deduced from nuclear magnetic resonance on these solutions can explain the dilute solution results in a consistent manner through the presence of ring structures, which otherwise have an essentially flexible coil conformation. The ring structures could be in a state of continual flux and rearrangement. Calculation of the radius of gyration for the random‐flight model gives a similar reduction of this measure for chains joined at their endpoints, or those containing loop with two dangling ends, each one‐fourth the total length of the chain. This relative insensitivity to the details of the ring structure is taken to support the behavior observed across a range of proteins.     

Effects of denaturants on the dynamics of loop formation in polypeptides

Quenching of the triplet state of tryptophan by close contact with cysteine has been used to measure the reaction-limited and diffusion-limited rates of loop formation in disordered polypeptides having the sequence cys-(ala-gly-gln)j-trp (j=1-9). The decrease in the length-dependence of the reaction-limited rate for short chains in aqueous buffer, previously attributed to chain stiffness, is not observed at high concentrations of chemical denaturant (6 M GdmCl and 8 M urea), showing that denaturants increase chain flexibility. For long chains, both reaction-limited and diffusion-limited rates are significantly smaller in denaturant and exhibit a steeper length dependence. The results can be explained using end-to-end distributions from a wormlike chain model in which excluded volume interactions are incorporated by associating a 0.4-0.5 nm diameter hard sphere with the end of each virtual peptide bond. Fitting the data with this model shows that the denaturants reduce the persistence length from approximately 0.6 nm to approximately 0.4 nm, only slightly greater than the length of a peptide bond. The same model also describes the reported length dependence for the radii of gyration of chemically denatured proteins containing 50-400 residues. The end-to-end diffusion coefficients obtained from the diffusion-limited rates are smaller than the sum of the monomer diffusion coefficients and exhibit significant temperature dependence, suggesting that diffusion is slowed by internal friction arising from barriers to backbone conformational changes.     

Salt bridges destabilize a leucine zipper designed for maximized ion pairing between helices

Interhelical salt bridges are common in leucine zippers and are thought to stabilize the coiled coil conformation. Here we present a detailed thermodynamic investigation of the designed, disulfide-linked leucine zipper AB(SS) whose high-resolution NMR structure shows six interhelical ion pairs between heptad positions g of one helix and e' of the other helix but no ion pairing within single helices. The average pK(a) value of the Glu side chain carboxyl groups of AB(SS) is slightly higher than the pK(a) of a freely accessible Glu in an unfolded peptide [Marti, D. N., Jelesarov, I., and Bosshard, H. R. (2000) Biochemistry 39, 12804-12818]. This indicates that the salt bridges are destabilizing, a prediction we now have confirmed by determining the pH +/- stability profile of AB(SS). Circular dichroism-monitored unfolding by urea and by heating and differential scanning calorimetry show that the coiled coil conformation is approximately 5 kJ/mol more stable when salt bridges are broken by protonation of the carboxyl side chains. Using guanidinium chloride as the denaturant, the increase in the free energy of unfolding on protonation of the carboxyl side chains is larger, approximately 17 kJ/mol. The discrepancy between urea and guanidinium chloride unfolding can be ascribed to the ionic nature of guanidinium chloride, which screens charge-charge interactions. This work demonstrates the difficulty of predicting the energetic contribution of salt bridges from structural data alone even in a case where the ion pairs are seen in high-resolution NMR structures. The reason is that the contribution to stability results from a fine balance between energetically favorable Coulombic attractions and unfavorable desolvation of charges and conformational constraints of the residues involved in ion pairing. The apparent discrepancy between the results presented here and mutational studies indicating stabilization by salt bridges is discussed and resolved. An explanation is proposed for why interhelical salt bridges are frequently found in natural coiled coils despite evidence that they do not directly contribute to stability.     

ERdj3, a stress-inducible endoplasmic reticulum DnaJ homologue, serves as a cofactor for BiP's interactions with unfolded substrates

We recently identified ERdj3 as a component of unassembled immunoglobulin (Ig) heavy chain:BiP complexes. ERdj3 also associates with a number of other protein substrates, including unfolded light chains, a nonsecreted Ig light chain mutant, and the VSV-G ts045 mutant at the nonpermissive temperature. We produced an ERdj3 mutant that was unable to stimulate BiP's ATPase activity in vitro or to bind BiP in vivo. This mutant retained the ability to interact with unfolded protein substrates, suggesting that ERdj3 binds directly to proteins instead of via interactions with BiP. BiP remained bound to unfolded light chains longer than ERdj3, which interacted with unfolded light chains initially, but quickly disassociated before protein folding was completed. This suggests that ERdj3 may bind first to substrates and serve to inhibit protein aggregation until BiP joins the complex, whereas BiP remains bound until folding is complete. Moreover, our findings support a model where interactions with BiP help trigger the release of ERdj3 from the substrate:BiP complex.     

Small-angle neutron scattering by a strongly denatured protein: analysis using random polymer theory.

Small-angle neutron scattering profiles are presented from phosphoglycerate kinase, in the native form and strongly denatured in 4 M guanidinium chloride (GdnHCl) solution. The data are interpreted using a model in which the excess scattering density associated with the protein is represented as a finite freely jointed chain of spheres. The similarity of the model-derived scattering function to experiment increases asymptotically with the number of spheres. The improvement of the fit obtained with more than approximately 200 spheres (i.e., two residues per sphere) is insignificant. The effects of finite size of the scattering units and of scattering length variation along the polypeptide chain are examined. Improved agreement with experiment is obtained when these effects are taken into account. A method for rapid calculation of the scattering profile of a full, all-atom configuration is examined. It is found that a representation of the chain containing two scattering units per residue, placed at the backbone and side-chain scattering length centroids, reproduces the full, all-atom profile to within 2%.