Metal Ion Modeling Using Classical Mechanics

Probing the Suitability of Different Ca2+ Parameters for Long Simulations of Diisopropyl Fluorophosphatase

Organophosphate hydrolases are promising as potential biotherapeutic agents to treat poisoning with pesticides or nerve gases. However, these enzymes often need to be further engineered in order to become useful in practice. One example of such enhancement is the alteration of enantioselectivity of diisopropyl fluorophosphatase (DFPase). Molecular modeling techniques offer a unique opportunity to address this task rationally by providing a physical description of the substrate-binding process. However, DFPase is a metalloenzyme, and correct modeling of metal cations is a challenging task generally coming with a tradeoff between simulation speed and accuracy. Here, we probe several molecular mechanical parameter combinations for their ability to empower long simulations needed to achieve a quantitative description of substrate binding. We demonstrate that a combination of the Amber19sb force field with the recently developed 12-6 Ca2+ models allows us to both correctly model DFPase and obtain new insights into the DFP binding process.

Hybrid polarizable simulations of a conventional hydrophobic polyelectrolyte. Toward a theoretical tool for green science innovation

Hybrid modeling approaches based on all-atom force fields to handle a solute and coarse-grained models to account for the solvent are promising numerical tools that can be used to understand the properties of large and multi-components solutions and thus to speed up the development of new industrial products that obey the standard of green and sustainable chemistry. Here, we discuss the ability of a full polarizable hybrid approach coupled to a standard molecular dynamics scheme to model the behavior in the aqueous phase and at infinite dilution conditions of a standard hydrophobic polyelectrolyte polymer whose charge is neutralized by explicit counterions. Beyond the standard picture of a polyelectrolyte behavior governed by an interplay between opposite intra-polyelectrolyte and inter-polyelectrolyte/counterion Coulombic effects, our simulations show the key role played by both intra-solute polarization effects and long range solute/solvent electrostatics to stabilize compact globular conformations of that polyelectrolyte. Our full polarizable hybrid modeling approach is thus a new theoretical tool well suited to be used in digital strategies for accelerating innovation for green science, for instance.

Coarse-Grained Modeling and Molecular Dynamics Simulations of Ca2+-Calmodulin

Calmodulin (CaM) is a calcium-binding protein that transduces signals to downstream proteins through target binding upon calcium binding in a time-dependent manner. Understanding the target binding process that tunes CaM’s affinity for the calcium ions (Ca²⁺), or vice versa, may provide insight into how Ca²⁺-CaM selects its target binding proteins. However, modeling of Ca²⁺-CaM in molecular simulations is challenging because of the gross structural changes in its central linker regions while the two lobes are relatively rigid due to tight binding of the Ca²⁺ to the calcium-binding loops where the loop forms a pentagonal bipyramidal coordination geometry with Ca²⁺. This feature that underlies the reciprocal relation between Ca²⁺ binding and target binding of CaM, however, has yet to be considered in the structural modeling. Here, we presented a coarse-grained model based on the Associative memory, Water mediated, Structure, and Energy Model (AWSEM) protein force field, to investigate the salient features of CaM. Particularly, we optimized the force field of CaM and that of Ca²⁺ ions by using its coordination chemistry in the calcium-binding loops to match with experimental observations. We presented a “community model” of CaM that is capable of sampling various conformations of CaM, incorporating various calcium-binding states, and carrying the memory of binding with various targets, which sets the foundation of the reciprocal relation of target binding and Ca²⁺ binding in future studies.

The Time Scale of Electronic Resonance in Oxidized DNA as Modulated by Solvent Response: An MD/QM-MM Study

The time needed to establish electronic resonant conditions for charge transfer in oxidized DNA has been evaluated by molecular dynamics simulations followed by QM/MM computations which include counterions and a realistic solvation shell. The solvent response is predicted to take ca. 800-1000 ps to bring two guanine sites into resonance, a range of values in reasonable agreement with the estimate previously obtained by a kinetic model able to correctly reproduce the observed yield ratios of oxidative damage for several sequences of oxidized DNA.

Computational Discovery of Transition-metal Complexes: From High-throughput Screening to Machine Learning

Transition-metal complexes are attractive targets for the design of catalysts and functional materials. The behavior of the metal-organic bond, while very tunable for achieving target properties, is challenging to predict and necessitates searching a wide and complex space to identify needles in haystacks for target applications. This review will focus on the techniques that make high-throughput search of transition-metal chemical space feasible for the discovery of complexes with desirable properties. The review will cover the development, promise, and limitations of "traditional" computational chemistry (i.e., force field, semiempirical, and density functional theory methods) as it pertains to data generation for inorganic molecular discovery. The review will also discuss the opportunities and limitations in leveraging experimental data sources. We will focus on how advances in statistical modeling, artificial intelligence, multiobjective optimization, and automation accelerate discovery of lead compounds and design rules. The overall objective of this review is to showcase how bringing together advances from diverse areas of computational chemistry and computer science have enabled the rapid uncovering of structure-property relationships in transition-metal chemistry. We aim to highlight how unique considerations in motifs of metal-organic bonding (e.g., variable spin and oxidation state, and bonding strength/nature) set them and their discovery apart from more commonly considered organic molecules. We will also highlight how uncertainty and relative data scarcity in transition-metal chemistry motivate specific developments in machine learning representations, model training, and in computational chemistry. Finally, we will conclude with an outlook of areas of opportunity for the accelerated discovery of transition-metal complexes.

A transferable active-learning strategy for reactive molecular force fields

Predictive molecular simulations require fast, accurate and reactive interatomic potentials. Machine learning offers a promising approach to construct such potentials by fitting energies and forces to high-level quantum-mechanical data, but doing so typically requires considerable human intervention and data volume. Here we show that, by leveraging hierarchical and active learning, accurate Gaussian Approximation Potential (GAP) models can be developed for diverse chemical systems in an autonomous manner, requiring only hundreds to a few thousand energy and gradient evaluations on a reference potential-energy surface. The approach uses separate intra- and inter-molecular fits and employs a prospective error metric to assess the accuracy of the potentials. We demonstrate applications to a range of molecular systems with relevance to computational organic chemistry: ranging from bulk solvents, a solvated metal ion and a metallocage onwards to chemical reactivity, including a bifurcating Diels-Alder reaction in the gas phase and non-equilibrium dynamics (a model SN2 reaction) in explicit solvent. The method provides a route to routinely generating machine-learned force fields for reactive molecular systems.

Comparison of different approaches to derive classical bonded force-field parameters for a transition metal cofactor: a case study for non-heme iron site of ectoine synthase

Computational investigations into the structure and function of metalloenzymes with transition metal cofactors require proper preparation of the model, which requires obtaining reliable force field parameters for the cofactor. Here, we present a test case where several methods were used to derive amber force field parameters for a bonded model of the Fe(II) cofactor of ectoine synthase. Moreover, the spin of the ground state of the cofactor was probed by DFT and post-HF methods, which consistently indicated the quintet state is lowest in energy and well separated from triplet and singlet. The performance of the obtained force field parameter sets, derived for the quintet spin state, was scrutinized and compared taking into account metrics focused on geometric features of the models as well as their energetics. The main conclusion of this study is that Hessian-based methods yield parameters which represent the geometry around the metal ion, but poorly reproduce energy variance with geometrical changes. On the other hand, the energy-based method yields parameters accurately reproducing energy-structure relationships, but with bad performance in geometry optimization. Preliminary tests show that admixing geometrical criteria to energy-based methods may allow to derive parameters with acceptable performance for both energy and geometry.

Rational Design of Nonbonded Point Charge Models for Divalent Metal Cations with Lennard-Jones 12-6 Potential

Exploring a metal-involved biochemical process at a molecular level often requires a reliable description of metal properties in aqueous solution by classical nonbonded models. An additional C₄ term for considering ion-induced dipole interactions was previously proposed to supplement the widely used Lennard-Jones 12-6 potential (known as the 12-6-4 LJ-type model) with good accuracy. Here, we demonstrate an alternative to modeling divalent metal cations (M²⁺) with the traditional 12-6 LJ potential by developing nonbonded point charge models for use with 11 water models: TIP3P, SPC/E, SPC/E_b, TIP4P-Ew, TIP4P-D, and TIP4P/2005 and the more recent OPC3, TIP3P-FB, OPC, TIP4P-FB, and a99SB-disp. Our designed models simultaneously reproduce the experimental hydration free energy, ion-oxygen distance, and coordination number in the first hydration shell accurately for most of the metal cations, an accuracy equivalent to that of the complex 12-6-4 LJ-type and double exponential potential models. A systematic comparison with the existing M²⁺ models is presented as well in terms of effective ion radii, diffusion constants, water exchange rates, and ion-water interactions. Molecular dynamics simulations of metal substitution in Escherichia coli glyoxalase I variants show the great potential of our new models for metalloproteins.

Bridging the 12-6-4 Model and the Fluctuating Charge Model

Metal ions play important roles in various biological systems. Molecular dynamics (MD) using classical force field has become a popular research tool to study biological systems at the atomic level. However, meaningful MD simulations require reliable models and parameters. Previously we showed that the 12-6 Lennard-Jones nonbonded model for ions could not reproduce the experimental hydration free energy (HFE) and ion-oxygen distance (IOD) values simultaneously when ion has a charge of +2 or higher. We discussed that this deficiency arises from the overlook of the ion-induced dipole interaction in the 12-6 model, and this term is proportional to 1/r⁴ based on theory. Hence, we developed the 12-6-4 model and showed it could solve this deficiency in a physically meaningful way. However, our previous research also found that the 12-6-4 model overestimated the coordination numbers (CNs) for some highly charged metal ions. And we attributed this artifact to that the current 12-6-4 scheme lacks a correction for the interactions among the first solvation shell water molecules. In the present study, we considered the ion-included dipole interaction by using the 12-6 model with adjusting the atomic charges of the first solvation shell water molecules. This strategy not only considers the ion-induced dipole interaction between ion and the first solvation shell water molecules but also well accounts for the increased repulsion among these water molecules compared to the bulk water molecules. We showed this strategy could well reproduce the experimental HFE and IOD values for Mg²⁺, Zn²⁺, Al³⁺, Fe³⁺, and In³⁺ and solve the CN overestimation issue of the 12-6-4 model for Fe³⁺ and In³⁺. Moreover, our simulation results showed good agreement with previous ab initio MD simulations. In addition, we derived the physical relationship between the C₄ parameter and induced dipole moment, which agreed well with our simulation results. Finally, we discussed the implications of the present work for simulating metalloproteins. Due to the fluctuating charge model uses a similar concept to the 12-6 model with adjusting atomic charges, we believe the present study builds a bridge between the 12-6-4 model and the fluctuating charge model.

Zinc binding alters the conformational dynamics and drives the transport cycle of the cation diffusion facilitator YiiP

YiiP is a secondary transporter that couples Zn2+ transport to the proton motive force. Structural studies of YiiP from prokaryotes and Znt8 from humans have revealed three different Zn2+ sites and a conserved homodimeric architecture. These structures define the inward-facing and outward-facing states that characterize the archetypal alternating access mechanism of transport. To study the effects of Zn2+ binding on the conformational transition, we use cryo-EM together with molecular dynamics simulation to compare structures of YiiP from Shewanella oneidensis in the presence and absence of Zn2+. To enable single-particle cryo-EM, we used a phage-display library to develop a Fab antibody fragment with high affinity for YiiP, thus producing a YiiP/Fab complex. To perform MD simulations, we developed a nonbonded dummy model for Zn2+ and validated its performance with known Zn2+-binding proteins. Using these tools, we find that, in the presence of Zn2+, YiiP adopts an inward-facing conformation consistent with that previously seen in tubular crystals. After removal of Zn2+ with high-affinity chelators, YiiP exhibits enhanced flexibility and adopts a novel conformation that appears to be intermediate between inward-facing and outward-facing states. This conformation involves closure of a hydrophobic gate that has been postulated to control access to the primary transport site. Comparison of several independent cryo-EM maps suggests that the transition from the inward-facing state is controlled by occupancy of a secondary Zn2+ site at the cytoplasmic membrane interface. This work enhances our understanding of individual Zn2+ binding sites and their role in the conformational dynamics that govern the transport cycle.

Conformational Change of H64 and Substrate Transportation: Insight Into a Full Picture of Enzymatic Hydration of CO2 by Carbonic Anhydrase

The enzymatic hydration of CO₂ into HCO₃ ^- by carbonic anhydrase (CA) is highly efficient and environment-friendly measure for CO₂ sequestration. Here extensive MM MD and QM/MM MD simulations were used to explore the whole enzymatic process, and a full picture of the enzymatic hydration of CO₂ by CA was achieved. Prior to CO₂ hydration, the proton transfer from the water molecule (WT1) to H64 is the rate-limiting step with the free energy barrier of 10.4 kcal/mol, which leads to the ready state with the Zn-bound OH^-. The nucleophilic attack of OH^- on CO₂ produces HCO₃ ^- with the free energy barrier of 4.4 kcal/mol and the free energy release of about 8.0 kcal/mol. Q92 as the key residue manipulates both CO₂ transportation to the active site and release of HCO₃ ^-. The unprotonated H64 in CA prefers in an inward orientation, while the outward conformation is favorable energetically for its protonated counterpart. The conformational transition of H64 between inward and outward correlates with its protonation state, which is mediated by the proton transfer and the product release. The whole enzymatic cycle has the free energy span of 10.4 kcal/mol for the initial proton transfer step and the free energy change of -6.5 kcal/mol. The mechanistic details provide a comprehensive understanding of the entire reversible conversion of CO₂ into bicarbonate and roles of key residues in chemical and nonchemical steps for the enzymatic hydration of CO₂.

Development and application of quantum mechanics/molecular mechanics methods with advanced polarizable potentials

Quantum mechanics/molecular mechanics (QM/MM) simulations are a popular approach to study various features of large systems. A common application of QM/MM calculations is in the investigation of reaction mechanisms in condensed-phase and biological systems. The combination of QM and MM methods to represent a system gives rise to several challenges that need to be addressed. The increase in computational speed has allowed the expanded use of more complicated and accurate methods for both QM and MM simulations. Here, we review some approaches that address several common challenges encountered in QM/MM simulations with advanced polarizable potentials, from methods to account for boundary across covalent bonds and long-range effects, to polarization and advanced embedding potentials.

Advanced Electrostatic Model for Monovalent Ions Based on Ab Initio Energy Decomposition

Ions play important roles in the structures and functions of biomolecules. In biomolecular simulations, ions either directly interact with biomolecules or provide an ionic environment that influences electrostatic interactions of solutes. The AMOEBA+ water model has demonstrated significant advancement of the classical force field for describing molecular interactions due to its improvements on the functional forms to account for essential physics. This work expands the applicability of the AMOEBA+ model toward alkali metal (Li, Na, K, Rb, and Cs) and halogen (F, Cl, Br, and I) ions. Various quantum chemical data on ion-ion and ion-water interactions, experimental ion hydration free energies, and lattice energies of salt crystals are used in the parametrization. The final parameters are verified with other properties outside of the parametrization data, including lattice energies of additional salt crystals and ionic activity coefficients in solution. The new model captures a wide range of ion properties from the gas phase to solution phase and crystals. More importantly, AMOEBA+ provides energy components that are consistent with ab initio energy decomposition. Thus, we expect AMOEBA+ to be more general, transferable, and valuable for the interpretation of intermolecular forces in efficient classical simulations.

Automatically Constructed Neural Network Potentials for Molecular Dynamics Simulation of Zinc Proteins

The development of accurate and efficient potential energy functions for the molecular dynamics simulation of metalloproteins has long been a great challenge for the theoretical chemistry community. An artificial neural network provides the possibility to develop potential energy functions with both the efficiency of the classical force fields and the accuracy of the quantum chemical methods. In this work, neural network potentials were automatically constructed by using the ESOINN-DP method for typical zinc proteins. For the four most common zinc coordination modes in proteins, the potential energy, atomic forces, and atomic charges predicted by neural network models show great agreement with quantum mechanics calculations and the neural network potential can maintain the coordination geometry correctly. In addition, MD simulation and energy optimization with the neural network potential can be readily used for structural refinement. The neural network potential is not limited by the function form and complex parameterization process, and important quantum effects such as polarization and charge transfer can be accurately considered. The algorithm proposed in this work can also be directly applied to proteins containing other metal ions.

The Permeation Mechanism of Cisplatin Through a Dioleoylphosphocholine Bilayer*

The investigation of the intermolecular interactions between platinum-based anticancer drugs and lipid bilayers is of special relevance to unveil the mechanisms involved in different steps of the anticancer mode of action of these drugs. We have simulated the permeation of cisplatin through a model membrane composed of 1,2-dioleoyl-sn-glycero-3-phosphocholine lipids by means of umbrella sampling classical molecular dynamics simulations. The initial physisorption of cisplatin into the polar region of the lipid membrane is controlled by long-range electrostatic interactions with the choline groups in a first step and, in a second step, by long-range electrostatic and hydrogen bond interactions with the phosphate groups. The second half of the permeation pathway, in which cisplatin diffuses through the nonpolar region of the bilayer, is characterized by the drop of the interactions with the polar heads and the rise of attractive interactions with the non-polar tails, which are dominated by van der Waals contributions. The permeation free-energy profile is explained by a complex balance between the drug/lipid interactions and the energy and entropy contributions associated with the dehydration of the drug along the permeation pathway and with the decrease and increase of the membrane ordering along the first and second half of the mechanism, respectively.

Multiscale Quantum Refinement Approaches for Metalloproteins

Biomolecules with metal ion(s) (e.g., metalloproteins) play many important biological roles. However, accurate structural determination of metalloproteins, particularly those containing transition metal ion(s), is challenging due to their complicated electronic structure, complex bonding of metal ions, and high number of conformations in biomolecules. Quantum refinement, which was proposed to combine crystallographic data with computational chemistry methods by several groups, can improve the local structures of some proteins. In this study, a quantum refinement method combining several multiscale computational schemes with experimental (X-ray diffraction) information was developed for metalloproteins. Various quantum refinement approaches using different ONIOM (our own N-layered integrated molecular orbital and molecular mechanics) combinations of quantum mechanics (QM), semiempirical (SE), and molecular mechanics (MM) methods were conducted to assess the performance and reliability on the refined local structure in two metalloproteins. The structures for two (Cu- or Zn-containing) metalloproteins were refined by combining two-layer ONIOM2(QM1/QM2) and ONIOM2(QM/MM) and three-layer ONIOM3(QM1/QM2/MM) schemes with experimental data. The accuracy of the quantum-refined metal binding sites was also examined and compared in these multiscale quantum refinement calculations. ONIOM3(QM/SE/MM) schemes were found to give good results with lower computational costs and were proposed to be a good choice for the multiscale computational scheme for quantum refinement calculations of metal binding site(s) in metalloproteins with high efficiency. Additionally, a two-center ONIOM approach was employed to speed up the quantum refinement calculations for the Zn metalloprotein with two remote active sites/ligands. Moreover, a recent quantum-embedding wavefunction-in-density functional theory (WF-in-DFT) method was also adopted as the high-level method in unprecedented ONIOM2(CCSD-in-B3LYP/MM) and ONIOM3(CCSD-in-B3LYP/SE/MM) calculations, which can be regarded as novel pseudo-three- and pseudo-four-layer ONIOM methods, respectively, to refine the key Zn binding site at the coupled-cluster singles and doubles (CCSD) level. These refined results indicate that multiscale quantum refinement schemes can be used to improve the structural accuracy obtained for local metal binding site(s) in metalloproteins with high efficiency.

Developing and Assessing Nonbonded Dummy Models of Magnesium Ion with Different Hydration Free Energy References

A large diversity in the targeted hydration free energies (HFEs) during model parameterization of metal ions was reported in the literature with a difference by dozens of kcal/mol. Here, we developed a series of nonbonded dummy models of the Mg²⁺ ion targeting different HFE references in TIP3P water, followed by assessments of the designed models in the simulations of MgCl₂ solution and biological systems. Together with the comparison of existing models, we conclude that the difference in the targeted HFEs has a limited influence on the model performance, while the usability of these models differs from case to case. The feasibility of reproducing more properties of Mg²⁺ such as diffusion constants and water exchange rates using a nonbonded dummy model is demonstrated. Underestimated activity derivative and osmotic coefficient of MgCl₂ solutions in high concentration reveal a necessity for further optimization of ion-pair interactions. The developed dummy models are applicable to metal coordination with Asp, Glu, and His residues in metalloenzymes, and the performance in predicting monodentate or bidentate binding modes of Asp/Glu residues depends on the complexity of metal centers and the choice of protein force fields. When both the binding modes coexist, the nonbonded dummy models outperform point charge models, probably in need of considering polarization of metal-binding residues by, for instance, charge calibration in classical force fields. This work is valuable for the use and further development of magnesium ion models for simulations of metal-containing systems with good accuracy.

Force Field Parameters for Fe2+4S2-4 Clusters of Dihydropyrimidine Dehydrogenase, the 5-Fluorouracil Cancer Drug Deactivation Protein: A Step towards In Silico Pharmacogenomics Studies

The dimeric dihydropyrimidine dehydrogenase (DPD), metalloenzyme, an adjunct anti-cancer drug target, contains highly specialized 4 × Fe²⁺₄S²⁻₄ clusters per chain. These clusters facilitate the catalysis of the rate-limiting step in the pyrimidine degradation pathway through a harmonized electron transfer cascade that triggers a redox catabolic reaction. In the process, the bulk of the administered 5-fluorouracil (5-FU) cancer drug is inactivated, while a small proportion is activated to nucleic acid antimetabolites. The occurrence of missense mutations in DPD protein within the general population, including those of African descent, has adverse toxicity effects due to altered 5-FU metabolism. Thus, deciphering mutation effects on protein structure and function is vital, especially for precision medicine purposes. We previously proposed combining molecular dynamics (MD) and dynamic residue network (DRN) analysis to decipher the molecular mechanisms of missense mutations in other proteins. However, the presence of Fe²⁺₄S²⁻₄ clusters in DPD poses a challenge for such in silico studies. The existing AMBER force field parameters cannot accurately describe the Fe²⁺ center coordination exhibited by this enzyme. Therefore, this study aimed to derive AMBER force field parameters for DPD enzyme Fe²⁺ centers, using the original Seminario method and the collation features Visual Force Field Derivation Toolkit as a supportive approach. All-atom MD simulations were performed to validate the results. Both approaches generated similar force field parameters, which accurately described the human DPD protein Fe²⁺₄S²⁻₄ cluster architecture. This information is crucial and opens new avenues for in silico cancer pharmacogenomics and drug discovery related research on 5-FU drug efficacy and toxicity issues.

Exploring the Minimum-Energy Pathways and Free-Energy Profiles of Enzymatic Reactions with QM/MM Calculations

Understanding molecular mechanisms of enzymatic reactions is of vital importance in biochemistry and biophysics. Here, we introduce new functions of hybrid quantum mechanical/molecular mechanical (QM/MM) calculations in the GENESIS program to compute the minimum-energy pathways (MEPs) and free-energy profiles of enzymatic reactions. For this purpose, an interface in GENESIS is developed to utilize a highly parallel electronic structure program, QSimulate-QM (https://qsimulate.com), calling it as a shared library from GENESIS. Second, algorithms to search the MEP are implemented, combining the string method (E et al. J. Chem. Phys. 2007, 126, 164103) with the energy minimization of the buffer MM region. The method implemented in GENESIS is applied to an enzyme, triosephosphate isomerase, which converts dihyroxyacetone phosphate to glyceraldehyde 3-phosphate in four proton-transfer processes. QM/MM-molecular dynamics simulations show performances of greater than 1 ns/day with the density functional tight binding (DFTB), and 10-30 ps/day with the hybrid density functional theory, B3LYP-D3. These performances allow us to compute not only MEP but also the potential of mean force (PMF) of the enzymatic reactions using the QM/MM calculations. The barrier height obtained as 13 kcal mol-1 with B3LYP-D3 in the QM/MM calculation is in agreement with the experimental results. The impact of conformational sampling in PMF calculations and the level of electronic structure calculations (DFTB vs B3LYP-D3) suggests reliable computational protocols for enzymatic reactions without high computational costs.

Atomistic Details of Charge/Space Competition in the Ca2+ Selectivity of Ryanodine Receptors

Ryanodine receptors (RyRs) are ion channels responsible for the fast release of Ca²⁺ from the sarco/endoplasmic reticulum to the cytosol and show a selectivity of Ca²⁺ over monovalent cations. By utilizing a recently developed multisite Ca²⁺ model in molecular dynamic simulations, we show that multiple cations accumulate in the upper selectivity filter of RyRs, and the small size and high valence of Ca²⁺ make it preferable to K⁺ in competition for space in this confined region of negative electrostatic potential. The presence of Ca²⁺ in the upper selectivity filter significantly increases the energy barrier of K⁺ permeation, while the presence of K⁺ has little impact on the Ca²⁺ permeation. Our results provide the atomistic details of the charge/space competition mechanism for the ion selectivity of RyRs, which ensures the robustness of their Ca²⁺ release function. The mechanism could be utilized in protein- and nanoengineering for valence selectivity of ion species.

Computing Metal-Binding Proteins for Therapeutic Benefit

Over one third of biomolecules rely on metal ions to exert their cellular functions. Metal ions can play a structural role by stabilizing the structure of biomolecules, a functional role by promoting a wide variety of biochemical reactions, and a regulatory role by acting as messengers upon binding to proteins regulating cellular metal-homeostasis. These diverse roles in biology ascribe critical implications to metal-binding proteins in the onset of many diseases. Hence, it is of utmost importance to exhaustively unlock the different mechanistic facets of metal-binding proteins and to harness this knowledge to rationally devise novel therapeutic strategies to prevent or cure pathological states associated with metal-dependent cellular dysfunctions. In this compendium, we illustrate how the use of a computational arsenal based on docking, classical, and quantum-classical molecular dynamics simulations can contribute to extricate the minutiae of the catalytic, transport, and inhibition mechanisms of metal-binding proteins at the atomic level. This knowledge represents a fertile ground and an essential prerequisite for selectively targeting metal-binding proteins with small-molecule inhibitors aiming to (i) abrogate deregulated metal-dependent (mis)functions or (ii) leverage metal-dyshomeostasis to selectively trigger harmful cells death.

The study of the molecular mechanism of Lactobacillus paracasei clumping via divalent metal ions by electrophoretic separation

In this work, the molecular mechanism of Lactobacillus paracasei bio-colloid clumping under divalent metal ions treatment such as zinc, copper and magnesium at constant concentrations was studied. The work involved experimental (electrophoretic - capillary electrophoresis in pseudo-isotachophoresis mode, spectroscopic and spectrometric - FT-IR and MALDI-TOF-MS, microscopic - fluorescent microscopy, and flow cytometry) and theoretical (DFT calculations of model complex systems) characterization. Electrophoretic results have pointed out the formation of aggregates under the Zn²⁺ and Cu²⁺ modification, whereas the use of the Mg²⁺ allowed focusing the zone of L. paracasei biocolloid. According to the FT-IR analysis, the major functional groups involved in the aggregation are deprotonated carboxyl and amide groups derived from the bacterial surface structure. Nature of the divalent metal ions was shown to be one of the key factors influencing the bacterial aggregation process. Proteomic analysis showed that surface modification had a considerable impact on bacteria molecular profiles and protein expression, mainly linked to the activation of carbohydrate and nucleotides metabolism as well with the transcription regulation and membrane transport. Density-functional theory (DFT) calculations of modeled Cu²⁺, Mg²⁺ and Zn²⁺ coordination complexes support the interaction between the divalent metal ions and bacterial proteins. Consequently, the possible mechanism of the aggregation phenomenon was proposed. Therefore, this comprehensive study could be further applied in evaluation of biocolloid aggregation under different types of metal ions.

Passive Diffusion of Ciprofloxacin and its Metalloantibiotic: A Computational and Experimental study

Fluoroquinolones (FQ) are antibiotics widely used in clinical practise, but the development of bacterial resistance to these drugs is currently a critical public health problem. In this context, ternary copper complexes of FQ (CuFQPhen) have been studied as a potential alternative. In this study, we compared the passive diffusion across the lipid bilayer of one of the most used FQ, ciprofloxacin (Cpx), and its ternary copper complex, CuCpxPhen, that has shown previous promising results regarding antibacterial activity and membrane partition. A combination of spectroscopic studies and molecular dynamics simulations were used and two different model membranes tested: one composed of anionic phospholipids, and the other composed of zwitterionic phospholipids. The obtained results showed a significantly higher membrane permeabilization activity, larger partition, and a more favourable free energy landscape for the permeation of CuCpxPhen across the membrane, when compared to Cpx. Furthermore, the computational results indicated a more favourable translocation of CuCpxPhen across the anionic membrane, when compared to the zwitterionic one, suggesting a higher specificity towards the former. These findings are important to decipher the influx mechanism of CuFQPhen in bacterial cells, which is crucial for the ultimate use of CuFQPhen complexes as an alternative to FQ to tackle multidrug-resistant bacteria.

Parameterization of Monovalent Ions for the OPC3, OPC, TIP3P-FB, and TIP4P-FB Water Models

Monovalent ions play significant roles in various biological and material systems. Recently, four new water models (OPC3, OPC, TIP3P-FB, and TIP4P-FB), with significantly improved descriptions of condensed phase water, have been developed. The pairwise interaction between the metal ion and water necessitates the development of ion parameters specifically for these water models. Herein, we parameterized the 12-6 and the 12-6-4 nonbonded models for 12 monovalent ions with the respective four new water models. These monovalent ions contain eight cations including alkali metal ions (Li⁺, Na⁺, K⁺, Rb⁺, Cs⁺), transition-metal ions (Cu⁺ and Ag⁺), and Tl⁺ from the boron family, along with four halide anions (F^-, Cl^-, Br^-, I^-). Our parameters were designed to reproduce the target hydration free energies (the 12-6 hydration free energy (HFE) set), the ion-oxygen distances (the 12-6 ion-oxygen distance (IOD) set), or both of them (the 12-6-4 set). The 12-6-4 parameter set provides highly accurate structural features overcoming the limitations of the routinely used 12-6 nonbonded model for ions. Specifically, we note that the 12-6-4 parameter set is able to reproduce experimental hydration free energies within 1 kcal/mol and experimental ion-oxygen distances within 0.01 Å simultaneously. We further reproduced the experimentally determined activity derivatives for salt solutions, validating the ion parameters for simulations of ion pairs. The improved performance of the present water models over our previous parameter sets for the TIP3P, TIP4P, and SPC/E water models (Li, P. et al J. Chem. Theory Comput. 2015 11 1645 1657) highlights the importance of the choice of water model in conjunction with the metal ion parameter set.

Navigating Transition-Metal Chemical Space: Artificial Intelligence for First-Principles Design

The variability of chemical bonding in open-shell transition-metal complexes not only motivates their study as functional materials and catalysts but also challenges conventional computational modeling tools. Here, tailoring ligand chemistry can alter preferred spin or oxidation states as well as electronic structure properties and reactivity, creating vast regions of chemical space to explore when designing new materials atom by atom. Although first-principles density functional theory (DFT) remains the workhorse of computational chemistry in mechanism deduction and property prediction, it is of limited use here. DFT is both far too computationally costly for widespread exploration of transition-metal chemical space and also prone to inaccuracies that limit its predictive performance for localized d electrons in transition-metal complexes. These challenges starkly contrast with the well-trodden regions of small-organic-molecule chemical space, where the analytical forms of molecular mechanics force fields and semiempirical theories have for decades accelerated the discovery of new molecules, accurate DFT functional performance has been demonstrated, and gold-standard methods from correlated wavefunction theory can predict experimental results to chemical accuracy.The combined promise of transition-metal chemical space exploration and lack of established tools has mandated a distinct approach. In this Account, we outline the path we charted in exploration of transition-metal chemical space starting from the first machine learning (ML) models (i.e., artificial neural network and kernel ridge regression) and representations for the prediction of open-shell transition-metal complex properties. The distinct importance of the immediate coordination environment of the metal center as well as the lack of low-level methods to accurately predict structural properties in this coordination environment first motivated and then benefited from these ML models and representations. Once developed, the recipe for prediction of geometric, spin state, and redox potential properties was straightforwardly extended to a diverse range of other properties, including in catalysis, computational "feasibility", and the gas separation properties of periodic metal-organic frameworks. Interpretation of selected features most important for model prediction revealed new ways to encapsulate design rules and confirmed that models were robustly mapping essential structure-property relationships. Encountering the special challenge of ensuring that good model performance could generalize to new discovery targets motivated investigation of how to best carry out model uncertainty quantification. Distance-based approaches, whether in model latent space or in carefully engineered feature space, provided intuitive measures of the domain of applicability. With all of these pieces together, ML can be harnessed as an engine to tackle the large-scale exploration of transition-metal chemical space needed to satisfy multiple objectives using efficient global optimization methods. In practical terms, bringing these artificial intelligence tools to bear on the problems of transition-metal chemical space exploration has resulted in ML-model assessments of large, multimillion compound spaces in minutes and validated new design leads in weeks instead of decades.

Molecular dynamics simulation of the Pb(II) coordination in biological media via cationic dummy atom models

The coordination of Pb(II) in aqueous solutions containing thiols is a pivotal topic to the understanding of the pollutant potential of this cation. Based on its hard/soft borderline nature, Pb(II) forms stable hydrated ions as well as stable complexes with the thiol groups of proteins. In this paper, the modeling of Pb(II) coordination via classical molecular dynamics simulations was investigated to assess the possible use of non-bonded potentials for the description of the metal–ligand interaction. In particular, this study aimed at testing the capability of cationic dummy atom schemes—in which part of the mass and charge of the Pb(II) is fractioned in three or four sites anchored to the metal center—in reproducing the correct coordination geometry and, also, in describing the hard/soft borderline character of this cation. Preliminary DFT calculations were used to design two topological schemes, PB3 and PB4, that were subsequently implemented in the Amber force field and employed in molecular dynamics simulation of either pure water or thiol/thiolate-containing aqueous solutions. The PB3 scheme was then tested to model the binding of Pb(II) to the lead-sensing protein pbrR. The potential use of CDA topological schemes in the modeling of Pb(II) coordination was here critically discussed.

Conformational Motion of Ferredoxin Enables Efficient Electron Transfer to Heme in the Full-Length P450TT

Cytochrome P450 monooxygenases (P450s) are versatile biocatalysts used in natural products biosynthesis, xenobiotic metabolisms, and biotechnologies. In P450s, the electrons required for O₂ activation are supplied by NAD(P)H through stepwise electron transfers (ETs) mediated by redox partners. While much is known about the machinery of the catalytic cycle of P450s, the mechanisms of long-range ET are largely unknown. Very recently, the first crystal structure of full-length P450_TT was solved. This enables us to decipher the interdomain ET mechanism between the [2Fe-2S]-containing ferredoxin and the heme, by use of molecular dynamics simulations. In contrast to the "distal" conformation characterized in the crystal structure where the [2Fe-2S] cluster is ∼28 Å away from heme-Fe, our simulations demonstrated a "proximal" conformation of [2Fe-2S] that is ∼17 Å [and 13.7 Å edge-to-edge] away from heme-Fe, which may enable the interdomain ET. Key residues involved in ET pathways and interdomain complexation were identified, some of which have already been verified by recent mutation studies. The conformational transit of ferredoxin between "distal" and "proximal" was found to be controlled mostly by the long-range electrostatic interactions between the ferredoxin domain and the other two domains. Furthermore, our simulations show that the full-length P450_TT utilizes a flexible ET pathway that resembles either P450_Scc or P450_cam. Thus, this study provides a uniform picture of the ET process between reductase domains and heme domain.

The molecular mechanism of P450-catalyzed amination of the pyrrolidine derivative of lidocaine: insights from multiscale simulations

Nitrogen heterocycles are key and prevalent motifs in drugs. Evolved variants of cytochrome P450_BM3 (CYP102A1) from Bacillus megaterium employ high-valent oxo-iron(iv) species to catalyze the synthesis of imidazolidine-4-ones via an intramolecular C–H amination. Herein, we use multi-scale simulations, including classical molecular dynamics (MD) simulations, quantum mechanical/molecular mechanical (QM/MM) calculations and QM calculations, to reveal the molecular mechanism of the intramolecular C–H amination of the pyrrolidine derivative of lidocaine bearing cyclic amino moieties catalyzed by the variant RP/FV/EV of P450_BM3, which bears five mutations compared to wild type. Our calculations show that overall catalysis includes both the enzymatic transformation in P450 and non-enzymatic transformation in water solution. The enzymatic transformation involves the exclusive hydroxylation of the C–H bond of the pyrrolidine derivative of lidocaine, leading to the hydroxylated intermediate, during which the substrate radical would be bypassed. The following dehydration and C–N coupling reactions are found to be much favored in aqueous situation compared to that in the non-polar protein environment. The present findings expand our understanding of the P450-catalyzed C(sp₃)–H amination reaction.

Nitrogen heterocycles are key and prevalent motifs in drugs.

Integrated experimental/computational approaches to characterize the systems formed by vanadium with proteins and enzymes

An integrated instrumental/computational approach to characterize metallodrug–protein adducts at the molecular level is reviewed. A series of applications are described, focusing on potential vanadium drugs with a generalization to other metals.

Cu2+, Ca2+, and methionine oxidation expose the hydrophobic α-synuclein NAC domain

α-Synuclein is an intrinsically disordered protein whose aggregation is related to Parkinson's disease and other neurodegenerative disorders. Metal cations are one of the main factors affecting the propensity of α-synuclein to aggregate, either by directly binding to it or by catalyzing the production of reactive oxygen species that oxidize it. His50, Asp121 and several additional C-terminal α-synuclein residues are binding sites for numerous metal cations, while methionine sulfoxidation occurs readily on this protein under oxidative stress conditions. Molecular dynamics simulations are an excellent tool to obtain a microscopic picture of how metal binding or methionine sulfoxidation alter the conformational preferences of α-synuclein and, hence, its aggregation propensity. In this work, we report the first coarse-grained molecular dynamics study comparing the conformational ensembles of the native protein, the protein bound to either Cu²⁺ or Ca²⁺ at its main binding sites, and the methionine-sulfoxidized protein. Our results suggest that these events alter the transient α-synuclein intramolecular contacts, inducing a greater solvent exposure of its hydrophobic, aggregation-prone NAC domain, in full agreement with a recent experimental study on Ca²⁺ binding. Moreover, metal-binding residues directly participate in the long-range contacts that shield this domain and regulate α-synuclein aggregation. These results provide a molecular-level rationalization of the enhanced fibrillation experimentally observed in the presence of Cu²⁺ or Ca²⁺ and the oligomerization induced by methionine sulfoxidation.

Catalytic Mechanism of Non-Target DNA Cleavage in CRISPR-Cas9 Revealed by Ab Initio Molecular Dynamics

CRISPR-Cas9 is a cutting-edge genome editing technology, which uses the endonuclease Cas9 to introduce mutations at desired sites of the genome. This revolutionary tool is promising to treat a myriad of human genetic diseases. Nevertheless, the molecular basis of DNA cleavage, which is a fundamental step for genome editing, has not been established. Here, quantum-classical molecular dynamics (MD) and free energy methods are used to disclose the two-metal-dependent mechanism of phosphodiester bond cleavage in CRISPR-Cas9. Ab initio MD reveals a conformational rearrangement of the Mg2+-bound RuvC active site, which entails the relocation of H983 to act as a general base. Then, the DNA cleavage proceeds through a concerted associative pathway fundamentally assisted by the joint dynamics of the two Mg2+ ions. This clarifies previous controversial experimental evidence, which could not fully establish the catalytic role of the conserved H983 and the metal cluster conformation. The comparison with other two-metal-dependent enzymes supports the identified mechanism and suggests a common catalytic strategy for genome editing and recombination. Overall, the non-target DNA cleavage catalysis described here resolves a fundamental open question in the CRISPR-Cas9 biology and provides valuable insights for improving the catalytic efficiency and the metal-dependent function of the Cas9 enzyme, which are at the basis of the development of genome editing tools.

Homology modeling in the time of collective and artificial intelligence

Homology modeling is a method for building protein 3D structures using protein primary sequence and utilizing prior knowledge gained from structural similarities with other proteins. The homology modeling process is done in sequential steps where sequence/structure alignment is optimized, then a backbone is built and later, side-chains are added. Once the low-homology loops are modeled, the whole 3D structure is optimized and validated. In the past three decades, a few collective and collaborative initiatives allowed for continuous progress in both homology and ab initio modeling. Critical Assessment of protein Structure Prediction (CASP) is a worldwide community experiment that has historically recorded the progress in this field. Folding@Home and Rosetta@Home are examples of crowd-sourcing initiatives where the community is sharing computational resources, whereas RosettaCommons is an example of an initiative where a community is sharing a codebase for the development of computational algorithms. Foldit is another initiative where participants compete with each other in a protein folding video game to predict 3D structure. In the past few years, contact maps deep machine learning was introduced to the 3D structure prediction process, adding more information and increasing the accuracy of models significantly. In this review, we will take the reader in a journey of exploration from the beginnings to the most recent turnabouts, which have revolutionized the field of homology modeling. Moreover, we discuss the new trends emerging in this rapidly growing field.

Out of Sight, Out of Mind: The Effect of the Equilibration Protocol on the Structural Ensembles of Charged Glycolipid Bilayers

Molecular dynamics (MD) simulations represent an essential tool in the toolbox of modern chemistry, enabling the prediction of experimental observables for a variety of chemical systems and processes and majorly impacting the study of biological membranes. However, the chemical diversity of complex lipids beyond phospholipids brings new challenges to well-established protocols used in MD simulations of soft matter and requires continuous assessment to ensure simulation reproducibility and minimize unphysical behavior. Lipopolysaccharides (LPS) are highly charged glycolipids whose aggregation in a lamellar arrangement requires the binding of numerous cations to oppositely charged groups deep inside the membrane. The delicate balance between the fully hydrated carbohydrate region and the smaller hydrophobic core makes LPS membranes very sensitive to the choice of equilibration protocol. In this work, we show that the protocol successfully used to equilibrate phospholipid bilayers when applied to complex lipopolysaccharide membranes occasionally leads to a small expansion of the simulation box very early in the equilibration phase. Although the use of a barostat algorithm controls the system dimension and particle distances according to the target pressure, fluctuation in the fleeting pressure occasionally enables a few water molecules to trickle into the hydrophobic region of the membrane, with spurious solvent buildup. We show that this effect stems from the initial steps of NPT equilibration, where initial pressure can be fairly high. This can be solved with the use of a stepwise-thermalization NVT/NPT protocol, as demonstrated for atomistic MD simulations of LPS/DPPE and lipid-A membranes in the presence of different salts using an extension of the GROMOS forcefield within the GROMACS software. This equilibration protocol should be standard procedure for the generation of consistent structural ensembles of charged glycolipids starting from atomic coordinates not previously pre-equilibrated. Although different ways to deal with this issue can be envisioned, we investigated one alternative that could be readily available in major MD engines with general users in mind.

Protein Matrix Control of Reaction Center Excitation in Photosystem II

Photosystem II (PSII) is a multisubunit pigment-protein complex that uses light-induced charge separation to power oxygenic photosynthesis. Its reaction center chromophores, where the charge transfer cascade is initiated, are arranged symmetrically along the D1 and D2 core polypeptides and comprise four chlorophyll (PD1, PD2, ChlD1, ChlD2) and two pheophytin molecules (PheoD1 and PheoD2). Evolution favored productive electron transfer only via the D1 branch, with the precise nature of primary excitation and the factors that control asymmetric charge transfer remaining under investigation. Here we present a detailed atomistic description for both. We combine large-scale simulations of membrane-embedded PSII with high-level quantum-mechanics/molecular-mechanics (QM/MM) calculations of individual and coupled reaction center chromophores to describe reaction center excited states. We employ both range-separated time-dependent density functional theory and the recently developed domain based local pair natural orbital (DLPNO) implementation of the similarity transformed equation of motion coupled cluster theory with single and double excitations (STEOM-CCSD), the first coupled cluster QM/MM calculations of the reaction center. We find that the protein matrix is exclusively responsible for both transverse (chlorophylls versus pheophytins) and lateral (D1 versus D2 branch) excitation asymmetry, making ChlD1 the chromophore with the lowest site energy. Multipigment calculations show that the protein matrix renders the ChlD1 → PheoD1 charge-transfer the lowest energy excitation globally within the reaction center, lower than any pigment-centered local excitation. Remarkably, no low-energy charge transfer states are located within the "special pair" PD1-PD2, which is therefore excluded as the site of initial charge separation in PSII. Finally, molecular dynamics simulations suggest that modulation of the electrostatic environment due to protein conformational flexibility enables direct excitation of low-lying charge transfer states by far-red light.

Explicit Representation of Cation-π Interactions in Force Fields with 1/r4 Nonbonded Terms

The binding energies for cation-π complexation are underestimated by traditional fixed-charge force fields owing to their lack of explicit treatment of ion-induced dipole interactions. To address this deficiency, an explicit treatment of cation-π interactions has been introduced into the OPLS-AA force field. Following prior work with atomic cations, it is found that cation-π interactions can be handled efficiently by augmenting the usual 12-6 Lennard-Jones potentials with 1/r⁴ terms. Results are provided for prototypical complexes as well as protein-ligand systems of relevance for drug design. Alkali cation, ammonium, guanidinium, and tetramethylammonium were chosen for the representative cations, while benzene and six heteroaromatic molecules were used as the π systems. The required nonbonded parameters were fit to reproduce structure and interaction energies for gas-phase complexes from density functional theory (DFT) calculations at the ωB97X-D/6-311++G(d,p) level. The impact of the solvent was then examined by computing potentials of mean force (pmfs) in both aqueous and tetrahydrofuran (THF) solutions using the free-energy perturbation (FEP) theory. Further testing was carried out for two cases of strong and one case of weak cation-π interactions between druglike molecules and their protein hosts, namely, the JH2 domain of JAK2 kinase and macrophage migration inhibitory factor. FEP results reveal greater binding by 1.5-4.4 kcal/mol from the addition of the explicit cation-π contributions. Thus, in the absence of such treatment of cation-π interactions, errors for computed binding or inhibition constants of 10¹-10³ are expected.

SARS-CoV-2 Virion Stabilization by Zn Binding

Zinc plays a crucial role in the process of virion maturation inside the host cell. The accessory Cys-rich proteins expressed in SARS-CoV-2 by genes ORF7a and ORF8 are likely involved in zinc binding and in interactions with cellular antigens activated by extensive disulfide bonds. In this report we provide a proof of concept for the feasibility of a structural study of orf7a and orf8 proteins. A conceivable hypothesis is that lack of cellular zinc, or substitution thereof, might lead to a significant slowing down of viral maturation.

Salacia mulbarica leaf extract mediated synthesis of silver nanoparticles for antibacterial and ct-DNA damage via releasing of reactive oxygen species

In this examination, we researched the advantages of DNA fragmentation and metallic nanoparticles well-appointed with biomolecules. A novel interpretation of DNA damage by Silver Nano-Clusters (AgNCs) which were developed by the utilization of green synthesis method was demonstrated. The green synthesis of AgNCs was accomplished by utilizing the leaf extract of Salacia mulbarica (SM). The preparation of SM-AgNCs was developed by estimating surface plasmon resonance peak around 449 nm by using a UV-Visible spectrophotometer. The effect of phytochemicals in SM leaf extract on the development of stable SM-AgNCs was confirmed by FTIR spectroscopy. The size of the fabricated SM-AgNCs was estimated by dynamic light scattering and zeta-sizer analysis and the morphology of the SM-AgNCs was examined by transmission electron microscopy. The presence of clusters of Ag particles in the prepared SM-AgNCs was recognized by energy dispersion X-ray analysis. The results show that saponins, phytosterols, and phenolic compounds present in plant extract may play a great part in developing the SM-AgNCs in their specialized particles. The succeeded SM-AgNCs shows incredible anti-bacterial action towards Escherichia coli and Bacillus subtilis. In-light of the antibacterial study, these SM-AgNCs were analyzed with calf thymus-DNA and found significant damage to the strand of thymus-DNA.

AMBER force field parameters for the Zn (II) ions of the tunneling-fold enzymes GTP cyclohydrolase I and 6-pyruvoyl tetrahydropterin synthase

The folate biosynthesis pathway is an essential pathway for cell growth and survival. Folate derivatives serve as a source of the one-carbon units in several intracellular metabolic reactions. Rapidly dividing cells rely heavily on the availability of folate derivatives for their proliferation. As a result, drugs targeting this pathway have shown to be effective against tumor cells and pathogens, but drug resistance against the available antifolate drugs emerged quickly. Therefore, there is a need to develop new treatment strategies and identify alternative metabolic targets. The two de novo folate biosynthesis pathway enzymes, GTP cyclohydrolase I (GCH1) and 6-pyruvoyl tetrahydropterin synthase (PTPS), can provide an alternative strategy to overcome the drug resistance that emerged in the two primary targeted enzymes dihydrofolate reductase and dihydropteroate synthase. Both GCH1 and PTPS enzymes contain Zn²⁺ ions in their active sites, and to accurately study their dynamic behaviors using all-atom molecular dynamics (MD) simulations, appropriate parameters that can describe their metal sites should be developed and validated. In this study, force field parameters of the GCH1 and PTPS metal centers were generated using quantum mechanics (QM) calculations and then validated through MD simulations to ensure their accuracy in describing and maintaining the Zn²⁺ ion coordination environment. The derived force field parameters will provide accurate and reliable MD simulations involving these two enzymes for future in-silico identification of drug candidates against the GCH1 and PTPS enzymes. Communicated by Ramaswamy H. Sarma.