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.

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.

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.

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.

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.

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.

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.

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.

Structural stabilities of calcium proteins: Human intelectin-1 and frog lectin XEEL.

We extend our study of the structural stability of helical and nonhelical regions in chain A of human intelectin-1 to include a second human intelectin (4WMY) and the frog protein "Xenopus embryonic epidermal lectin" (XEEL). These unique lectins have been shown to recognize carbohydrate residues found exclusively in microbes, thus they could potentially be developed into novel microbe detection and sequestration tools. We believe that by studying the structural stability of these proteins we can provide insights on their biological role and activities. Using a geometrical model introduced previously, we perform computational analyses of protein crystal structures that quantify the resiliency of the native state to steric perturbations. Based on these analyses, we conclude that differences in the resiliency of the human and frog proteins can be attributed primarily to differences in non-helical regions and to residues near Ca ions. Since these differences are particularly pronounced in the vicinity of the ligand binding site, they provide an explanation for the finding that human intelectin-1 has a higher affinity for a ligand than XEEL. We also present data on conserved and position-equivalent pairs of residues in 4WMY and XEEL. We identify residue pairs as well as regions in which the influence of neighboring residues is nearly uniform as the parent protein denatures. Since the structural signatures are conserved, this identification provides a basis for understanding why both proteins exhibit trimeric structures despite poor sequence conservation at the interface.

Relaxation of structural constraints during Amicyanin unfolding.

We study the thermal unfolding of amicyanin by quantifying the resiliency of the native state to structural perturbations. Three signatures characterizing stages of unfolding are identified. The first signature, lateral extension of the polypeptide chain, is calculated directly from the reported crystallographic data. Two other signatures, the radial displacement of each residue from Cu(II) and the angular spread in the chain as the protein unfolds, are calculated using crystallographic data in concert with a geometrical model we introduced previously (J.J. Kozak, H. B. Gray, R. A. Garza-López, J. Inorg. Biochem. 155(2016) 44-55). Particular attention is paid to the resiliency of the two beta sheets in amicyanin. The resiliency of residues in the near neighborhood of the Cu center to destabilization provides information on the persistence of the entatic state. Similarly, examining the resiliency of residues intercalated between structured regions (beta sheets, the alpha helix) provides a basis for identifying a "hydrophobic core." A principal focus of our study is to compare results obtained using our geometrical model with the experimental results (C. La Rosa, D. Milardi, D. M. Grasso, M. P. Verbeet, G. W. Canters, L. Sportelli, R. Guzzi, Eur. Biophy. J.30(8),(2002) 559-570) on the denaturation of amicyanin, and we show that our results support a classical model proposed by these authors.

Structural Stability of Intelectin-1.

We study the structural stability of helical and nonhelical regions in chain A of human intelectin-1. Using a geometrical model introduced previously, we carried out a computational analysis based on the recently reported crystal structure of this protein by Kiessling et al. to quantify the resiliency of the native state to steric perturbations. Response to these perturbations is characterized by calculating, relative to the native state, the lateral, radial, and angular displacements of n-residue segments of the polypeptide chain centered on each residue. By quantifying the stability of the protein through six stages of unfolding, we are able to identify regions in chain A of intelectin-1 that are markedly affected by structural perturbations versus those that are relatively unaffected, the latter suggesting that the native-state geometry of these regions is essentially conserved. Importantly, residues in the vicinity of calcium ions comprise a conserved region, suggesting that Ca ions play a role not only in the coordination of carbohydrate hydroxyl groups but also in preserving the integrity of the structure.

Cytochrome unfolding pathways from computational analysis of crystal structures.

We have developed a model to study the role of geometrical factors in influencing the early stages of unfolding in three cytochromes: cyt c', cyt c-b562 and cyt c. Each stage in unfolding is quantified by the spatial extension λ̂i of n-residue segments, and by their angular extension 〈ßn〉. Similarities and differences between and among the three cytochromes in the unfolding of helical and non-helical regions can be determined by analyzing the data for each signature separately. Definite conclusions can be drawn when spatial and angular changes are considered in tandem. To facilitate comparisons, we present graphical portraits of the three cytochromes at the same stage of unfolding, and in relation to their native state structures. We also display specific segments at different stages of unfolding to illustrate differences in stability of defined domains thereby allowing us to make specific predictions on the unfolding of corresponding internal and terminal helices in cyt c' and cyt c-b562. Our work accords with an earlier experimental report on the presence and persistence of a hydrophobic core in cyt c.

Lattice statistical theory of random walks on a fractal-like geometry.

We have designed a two-dimensional, fractal-like lattice and explored, both numerically and analytically, the differences between random walks on this lattice and a regular, square-planar Euclidean lattice. We study the efficiency of diffusion-controlled processes for flows from external sites to a centrosymmetric reaction center and, conversely, for flows from a centrosymmetric source to boundary sites. In both cases, we find that analytic expressions derived for the mean walk length on the fractal-like lattice have an algebraic dependence on system size, whereas for regular Euclidean lattices the dependence can be transcendental. These expressions are compared with those derived in the continuum limit using classical diffusion theory. Our analysis and the numerical results quantify the extent to which one paradigmatic class of spatial inhomogeneities can compromise the efficiency of adatom diffusion on solid supports and of surface-assisted self-assembly in metal-organic materials.

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.

Analytic expression for the mean time to absorption for a random walker on the Sierpinski fractal. III. The effect of non-nearest-neighbor jumps.

We present exact, analytic results for the mean time to trapping of a random walker on the class of deterministic Sierpinski graphs embedded in d≥2 Euclidean dimensions, when both nearest-neighbor (NN) and next-nearest-neighbor (NNN) jumps are included. Mean first-passage times are shown to be modified significantly as a consequence of the fact that NNN transitions connect fractals of two consecutive generations.

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.

A Euclidean Perspective on the Unfolding of Azurin: Spatial Correlations.

We investigate the stability to structural perturbation of Pseudomonas aeruginosa azurin using a previously developed geometric model. Our analysis considers Ru(2,2',6',2″-terpyridine)(1,10-phenanthroline)(His83)-labeled wild-type azurin and five variants with mutations to Cu-ligating residues. We find that in the early stages of unfolding, the ß-strands exhibit the most structural stability. The conserved residues comprising the hydrophobic core are dislocated only after nearly complete unfolding of the ß-barrel. Attachment of the Ru-complex at His83 does not destabilize the protein fold, despite causing some degree of structural rearrangement. Notably, replacing the Cys112 and/or Met121 Cu ligands does not affect the conformational integrity of the protein. Notably, these results are in accord with experimental evidence, as well as molecular dynamics simulations of the denaturation of azurin.

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.