Stereochemistry of residues in turning regions of helical proteins.

We have developed a geometrical approach to quantify differences in the stereochemistry of α-helical and turning regions in four iron proteins. Two spatial signatures are used to analyze residue coordinate data for each protein; and a third is employed to analyze amino-acid molecular volume data. The residue-by-residue analysis of the results, taken together with the finding that two major factors stabilize an α-helix (minimization of side-chain steric interference and intrachain H-bonding), lead to the conclusion that certain residues are preferentially selected for α-helix formation. In the sequential, de novo synthesis of a turning region, residues are preferentially selected such that the overall molecular volume profile (representing purely repulsive, excluded-volume effects) spans a small range Δ of values (Δ = 39.1 Å3) relative to the total range that could be spanned (Δ = 167.7 Å3). It follows that excluded-volume effects are of enormous importance for residues in helical regions as well as those in adjacent turning regions. Once steric effects are taken into account, down-range attractive interactions between residues come into play in the formation of α-helical regions. The geometry of α-helices can be accommodated by conformational changes in less-structured turning regions of a polypeptide, thereby producing a globally optimized (native) protein structure.

Density-dependent finite system-size effects in equilibrium molecular dynamics estimation of shear viscosity: Hydrodynamic and configurational study.

We study the intrinsic nature of the finite system-size effect in estimating shear viscosity of dilute and dense fluids within the framework of the Green-Kubo approach. From extensive molecular dynamics simulations, we observe that the size effect on shear viscosity is characterized by an oscillatory behavior with respect to system size L at high density and by a scaling behavior with an L-1 correction term at low density. Analysis of the potential contribution in the shear-stress autocorrelation function reveals that the former is configurational and is attributed to the inaccurate description of the long-range spatial correlations in finite systems. Observation of the long-time inverse-power decay in the kinetic contribution confirms its hydrodynamic nature. The L-1 correction term of shear viscosity is explained by the sensitive change in the long-time tail obtained from a finite system.

A Euclidean perspective on the unfolding of azurin: chain motion.

We present a new approach to visualizing and quantifying the displacement of segments of Pseudomonas aeruginosa azurin in the early stages of denaturation. Our method is based on a geometrical method developed previously by the authors, and elaborated extensively for azurin. In this study, we quantify directional changes in three α-helical regions, two regions having ß-strand residues, and three unstructured regions of azurin. Snapshots of these changes as the protein unfolds are displayed and described quantitatively by introducing a scaling diagnostic. In accord with molecular dynamics simulations, we show that the long α-helix in azurin (residues 54-67) is displaced from the polypeptide scaffolding and then pivots first in one direction, and then in the opposite direction as the protein continues to unfold. The two ß-strand chains remain essentially intact and, except in the earliest stages, move in tandem. We show that unstructured regions 72-81 and 84-91, hinged by ß-strand residues 82-83, pivot oppositely. The region comprising residues 72-91 (40 % hydrophobic and 16 % of the 128 total residues) forms an effectively stationary region that persists as the protein unfolds. This static behavior is a consequence of a dynamic balance between the competing motion of two segments, residues 72-81 and 84-91.

Stability of phases of a square-well fluid within superposition approximation.

The analytic and numerical methods introduced previously to study the phase behavior of hard sphere fluids starting from the Yvon-Born-Green (YBG) equation under the Kirkwood superposition approximation (KSA) are adapted to the square-well fluid. We are able to show conclusively that the YBG equation under the KSA closure when applied to the square-well fluid: (i) predicts the existence of an absolute stability limit corresponding to freezing where undamped oscillations appear in the long-distance behavior of correlations, (ii) in accordance with earlier studies reveals the existence of a liquid-vapor transition by the appearance of a "near-critical region" where monotonically decaying correlations acquire very long range, although the system never loses stability.

Communication: Nonexistence of a critical point within the Kirkwood superposition approximation.

An analytic argument is given to show that the application of the Kirkwood superposition approximation to the description of fluid correlation functions precludes the existence of a critical point. The argument holds irrespective of the dimension of the system and the specific form of the interaction potential and settles a long-standing controversy surrounding the nature of the critical behavior predicted within the approximation.

Modeling the early stages of self-assembly in nanophase materials. II. Role of symmetry and dimensionality.

We study the early stages of self-assembly of elementary building blocks of nanophase materials, considering explicitly their structure and the symmetry and the dimensionality of the reaction space. Previous work [Kozak et al., J. Chem. Phys. 134, 154701 (2007)] focused on characterizing self-assembly on small square-planar templates. Here we consider larger lattices of square-planar symmetry having N = 255 sites, and both hexagonal and triangular lattices of N = 256 sites. Furthermore, to assess the consequences of a depletion zone above a basal layer (λ = 1), we study self-assembly on an augmented diffusion space defined by λ = 2 and λ = 5 stacked layers having the same characteristics as the basal plane. The effective decrease in the efficiency of self-assembly of individual nanophase units when the diffusion space is expanded, by increasing the template size and/or by enlarging the depletion zone, is then quantified. The results obtained reinforce our earlier conclusion that the most significant factor influencing the kinetics of formation of a final self-assembled unit is the number of reaction pathways from one or more precursor states. We draw attention to the relevance of these results to zeolite synthesis and reactions within pillared clays.

Structural transitions in hypersphere fluids: predictions of Kirkwood's approximation.

We use an analytic criterion for vanishing of exponential damping of correlations developed previously [J. Piasecki et al., J. Chem. Phys. 133, 164507 (2010)] to determine the threshold volume fractions for structural transitions in hard sphere systems in dimensions D = 3, 4, 5, and 6, proceeding from the Yvon-Born-Green hierarchy and using the Kirkwood superposition approximation. We conclude that the theory does predict phase transitions in qualitative agreement with numerical studies. We also derive, within the superposition approximation, the asymptotic form of the analytic condition for occurrence of a structural transition in the D → ∞ limit.

Prediction of a structural transition in the hard disk fluid.

Starting from the second equilibrium equation in the BBGKY hierarchy under the Kirkwood superposition closure, we implement a new method for studying the asymptotic decay of correlations in the hard disk fluid in the high density regime. From our analysis and complementary numerical studies, we find that exponentially damped oscillations can occur only up to a packing fraction η(∗)∼0.718, a value that is in substantial agreement with the packing fraction, η∼0.723, believed to characterize the transition from the ordered solid phase to a dense fluid phase, as inferred from Mak's Monte Carlo simulations [Phys. Rev. E 73, 065104 (2006)]. Next, we show that the same method of analysis predicts that the exponential damping of oscillations in the hard sphere fluid becomes impossible when λ=4nπσ(3)[1+H(1)]≥34.81, where H(1) is the contact value of the correlation function, n is the number density, and σ is the sphere diameter in exact agreement with the condition, λ≥34.8, which is first reported in a numerical study of the Kirkwood equation by Kirkwood et al. [J. Chem. Phys. 18, 1040 (1950)]. Finally, we show that our method confirms the absence of any structural transition in hard rods for the entire range of densities below close packing.

Kinetic rougheninglike transition with finite nucleation barrier.

Recent observations of the growth of protein crystals have identified two different growth regimes. At low supersaturation, the surface of the crystal is smooth and increasing in size due to the nucleation of steps at defects and the subsequent growth of the steps. At high supersaturation, nucleation occurs at many places simultaneously, the crystal surface becomes rough, and the growth velocity increases more rapidly with increasing supersaturation than in the smooth regime. Kinetic roughening transitions are typically assumed to be due to the vanishing of the barrier for two-dimension nucleation on the surface of the crystal. We show here, by means of both analytic mean-field models and kinetic Monte Carlo simulations, that a transition between different growth modes reminiscent of kinetic roughening can also arise as a kinetic effect occurring at finite nucleation barriers.

Conjecture on the Design of Helical Proteins.

In an important advance in our understanding of protein folding, Wolynes and Onuchic found that the frustration ratio, Tf/Ts, for funneled energy landscapes is Tf/Ts ∼1.6. In our recent work on four heme proteins, we showed that when a protein unfolds from the native state to an early unfolded state, the degree of departure is characterized by a ratio f ∼1.6, where f is a measure of the elongation of n-residue segments of the polypeptide chain. Our analysis, which accounts for this apparent similarity in calculated signatures, is based on a logistic-map model of unfolding. We offer an important take home for the de novo protein synthesis community: in order to increase the probability of obtaining good quality crystals, nearest-neighbor repulsive interactions between adjacent residues (or sequences of residues) in the polypeptide chain must be propagated correctly.

Unfolding cytochromes c-b562 and Rd apo b562.

We have analyzed the early stages of unfolding of cytochromes c-b562 (PDB ID: 2BC5) and Rd apo b562 (PDB ID: 1YYJ). Our geometrical approach proceeds from an analysis of the crystal structure reported for each protein. We quantify, residue-by-residue and region-by-region, the spatial and angular changes in the structure as the protein denatures, and quantify differences that result from the seven residues that differ in the two proteins. Using two independent analyses, one based on spatial metrics and the second on angular metrics, we establish the order of unfolding of the five helices in cyt c-b562 and the four helices in the apo protein. For the two helices nearest the N-terminal end of both proteins, the ones in the apo protein unfold first. For the two helices nearest the C-terminal end, the interior helix of the apo protein unfolds first, whereas the terminal helix of the holo protein unfolds first. Excluded-volume effects (repulsive interactions) are minimized in turning regions; the overall range in Δ values is Δ = 36.3 Å3 for cyt c-b562 and Δ = 36.6 Å3 for the apo protein, whereas the span for all 20 amino acids is Δ = 167.7 Å3. As our work indicates that the interior helix of cytochrome c-b562 is the first to fold, we suggest that this helix protects the heme from misligation, consistent with ultrafast folding over a minimally frustrated funneled landscape.

Structural stability of the SARS-CoV-2 main protease: Can metal ions affect function?

We have investigated the structural stability of the SARS (Severe acute respiratory syndrome)-CoV-2 main protease monomer (Mpro). We quantified the spatial and angular changes in the structure using two independent analyses, one based on a spatial metrics (δ, ratio), the second on angular metrics. The order of unfolding of the 10 helices in Mpro is characterized by beta vs alpha plots similar to those of cytochromes and globins. The longest turning region is anomalous in the earliest stage of unfolding. In an investigation of excluded-volume effects, we found that the maximum spread in average molecular-volume values for Mpro, cytochrome c-b562, cytochrome c', myoglobin, and cytoglobin is ~10 Å3. This apparent universality is a consequence of the dominant contributions from six residues: ALA, ASP, GLU, LEU, LYS and VAL. Of the seven Mpro histidines, residues 41, 163, 164, and 246 are in stable H-bonded regions; metal ion binding to one or more of these residues could break up the H-bond network, thereby affecting protease function. Our analysis also indicated that metal binding to cysteine residues 44 and 145 could disable the enzyme.

Copper(II) Inhibition of the SARS-CoV-2 Main Protease.

In an analysis of the structural stability of the coronavirus main protease (Mpro), we identified regions of the protein that could be disabled by cobalt(III)-cation binding to histidines and cysteines [1]. Here we have extended our work to include copper(II) chelates, which we have docked to HIS 41 and CYS 145 in the Mpro active-site region. We have found stable docked structures where Cu(II) could readily bond to the CYS 145 thiolate, which would be lethal to the enzyme.

Funneled angle landscapes for helical proteins.

We use crystallographic data for four helical iron proteins (cytochrome c-b562, cytochrome c', sperm whale myoglobin, human cytoglobin) to calculate radial and angular signatures as each unfolds from the native state stepwise though four unfolded states. From these data we construct an angle phase diagram to display the evolution of each protein from its native state; and, in turn, the phase diagram is used to construct a funneled angle landscape for comparison with the topography of its folding energy landscape. We quantify the departure of individual helical and turning regions from the areal, angular profile of corresponding regions of the native state. This procedure allows us to identify the similarities and differences among individual helical and turning regions in the early stages of unfolding of the four helical heme proteins.

Theoretical study of the microscopic origin of magnetocrystalline anisotropy in Fe16N2 and its alloys: comparison with the other L10 alloys.

We study the magnetocrystalline anisotropy (MCA) energy of Fe16-n X n N2 ([Formula: see text]), where X = Ti, V, Cr, Mn, Co, Ni, Cu. To understand the microscopic origin and basic mechanism controlling the MCA energy of Fe16-n X n N2, we first examined the behavior of the MCA energy of Fe16N2, focusing on the spin-orbit coupling (SOC), and compared the behavior with other alloy systems (FeCo, FePt and CoPt) with L10 structure. We find that whereas the MCA energy of FeCo is determined by the spin-conserved terms of the SOC energy, the MCA energy of Fe16N2 is determined by mutual competition between spin-conserved and spin-flip terms. We then studied the effect of the transition element X on the phase stability and MCA of Fe16-n X n N2. The MCA energy and cohesive energy are calculated to determine the most stable configuration for each choice of X and n, and compared with those of Fe16N2. For X = V and Cu, both the MCA and phase stability improved noticeably. For X = Co, the MCA energy improves, but Fe16-n Co n N2 is less stable than Fe16N2. The microscopic mechanism underlying the MCA energy enhancement due to X = V, Cu and Co in Fe16-n X n N2 was studied by examining the data for spin- and site-resolved projected density of states (PDOS), as well as each spin-conserved and spin-flip terms contributing to the SOC energy.

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.

Reaction efficiency of diffusion-controlled processes on finite, planar arrays. II. Crystal surfaces.

We consider 36 planar nets identified by O'Keeffe and Hyde and calculate for each, using the theory of finite Markovian processes, the overall mean walk length n (first passage time) of a reactant diffusing randomly on a finite platelet before being trapped at a reaction center; the results are analyzed in terms of the total number N of lattice sites, the number N(b) of boundary sites, the average valence nu, and the bond orientation function Psi. We establish that crystalline platelets that are members of the same compatible class are characterized by very comparable catalytic efficiencies. The results obtained are also linked to an analysis of the kinetics of docking in postnucleation stages of protein self-assembly and to a recent conjecture on the symmetries of planar nets and the hard disk freezing transition.

First-passage properties of mortal random walks: Ballistic behavior, effective reduction of dimensionality, and scaling functions for hierarchical graphs.

We consider a mortal random walker on a family of hierarchical graphs in the presence of some trap sites. The configuration comprising the graph, the starting point of the walk, and the locations of the trap sites is taken to be exactly self-similar as one goes from one generation of the family to the next. Under these circumstances, the total probability that the walker hits a trap is determined exactly as a function of the single-step survival probability q of the mortal walker. On the nth generation graph of the family, this probability is shown to be given by the nth iterate of a certain scaling function or map q→f(q). The properties of the map then determine, in each case, the behavior of the trapping probability, the mean time to trapping, the temporal scaling factor governing the random walk dimension on the graph, and other related properties. The formalism is illustrated for the cases of a linear hierarchical lattice and the Sierpinski graphs in two and three Euclidean dimensions. We find an effective reduction of the random walk dimensionality due to the ballistic behavior of the surviving particles induced by the mortality constraint. The relevance of this finding for experiments involving travel times of particles in diffusion-decay systems is discussed.

A conjecture concerning the symmetries of planar nets and the hard disk freezing transition.

We examine the conjecture that in a 2D system of hard disks the packing fraction at which the continuous transition from the ordered 2D solid to the hexatic phase occurs, and that at which the very weak first-order or continuous transition from the hexatic to the fluid phase occurs, can be correlated with the packing fractions of patterned networks (tessellations) of disk positions that span the 2D space. We identify three tessellations that have less than close packed density, span 2D space, and have percolated continuity of disk-disk contact. One has a packing fraction of eta = 0.729, very slightly larger than the estimated packing fraction at the ordered solid-to-hexatic transition, eta = 0.723, and the other two have packing fractions of approximately 0.680, slightly smaller than that identified as the upper end of the stability range of the liquid phase, eta = 0.699. The region 0.680 < eta < 0.729 is identified with the hexatic domain. The end points of this region can be placed in correspondence with nets for which the defining unit structures are regular polygons, but not the hexatic domain, in which there are randomly dispersed clusters that need not be regular polygons. The densities at which the percolated tessellations span the 2D space are regarded as special points along the density axis. We suggest that the possibility of forming different symmetry nets with sensibly the same packing fraction is a geometric analogue of a bifurcation condition that divides the configuration space into qualitatively different domains, and that the onset and end of the hexatic region are correlated with such divisions of the configuration space.

Geometrical Description of Protein Structural Motifs.

We present a geometrical method that can identify secondary structural motifs in proteins via angular correlations. The method uses crystal structure coordinates to calculate angular and radial signatures of each residue relative to an external reference point as the number of nearest-neighbor residues increases. We apply our approach to the blue copper protein amicyanin using the copper cofactor as the external reference point. We define a signature termed Δß which describes the change in angular correlation as the span of nearest neighbor residues increases. We find that three turn regions of amicyanin harbor residues with Δß near zero, while residues in other secondary structures have Δß greater than zero: for ß-strands, Δß changes gradually residue by residue along the strand. Extension of our analysis to other blue copper proteins demonstrated that the noted structural trends are general. Importantly, our geometrical description of the folded protein accounts for all forces holding the structure together. Through this analysis, we identified some of the turns in amicyanin as symmetrical anchor points.