Unbound Structures Research Articles

Cofactor binding triggers rapid conformational remodelling of the active site of a methyltransferase ribozyme

The methyltransferase ribozyme SMRZ-1 utilizes S-adenosyl-methionine (SAM) and Cu (II) ions to methylate RNA. Comparison of the SAM bound and unbound RNA structures has shown a conformational change in the RNA. However, the contribution of specific interactions and the role of a pseudo-triplex motif in the catalytic centre on the methylation reaction is not completely understood. In this study, we have used atomic substitutions and mutational analysis to investigate the reaction specificity and the key interactions required for catalysis. Substitution of the fluorescent nucleotide 2-aminopurine within the active ribozyme enabled the conformational dynamics of the RNA upon co-factor binding to be explored using fluorescence spectroscopy. We show that fast co-factor binding (t1/2 ∼ 0.7 seconds) drives a conformational change in the RNA to facilitate methyl group transfer. The importance of stacking interactions at the pseudo-triplex motif and chelation of the Cu (II) ion were shown to be essential for SAM binding.

In Vitro Activation of Paraoxonase 1 by Steroids: An Experimental, Molecular Docking, and Molecular Modelling Study

AbstractParaoxonase 1 (PON1) is an antioxidant enzyme that prevents lipid oxidation by hydrolysing lipid peroxides in the oxidised low‐density lipoprotein (LDL) structure, bound to high‐density lipoprotein (HDL) in serum, and exhibits esterase and lactonase activity. In this work, the hPON1 enzyme was purified from human serum using a hydrophobic gel with Sepharose 4B‐L‐tyrosine‐naphthylamine, and the affinity of some steroid derivatives previously isolated from fungal steroid biotransformations was examined on the pure hPON1 enzyme. The results indicate that all these derivaties activate the hPON1 enzyme to a different extent. Additionally, the binding potential of the most active five steroid derivatives to the PON1 enzyme to form a stable complex was explored through molecular docking and molecular dynamics (MD) simulation. The compounds with the highest potency in the enzymatic assay, S‐8 and S‐20, had the highest binding potential to the enzyme. The stability of the complexes formed by the two compounds was assessed and compared to the stability of the unbound enzyme structure. The enzyme‐compound complexes generally had similar stability to the unbound enzyme structure. Together with this, the MD simulation revealed that compound S‐8 would remain inside the enzyme's binding site during the simulation period, unlike compound S‐20. This situation varies according to the respective derivatives’ functional groups and hydrophobic characteristics.

Computational Methods to Predict Conformational B-Cell Epitopes.

Accurate computational prediction of B-cell epitopes can greatly enhance biomedical research and rapidly advance efforts to develop therapeutics, monoclonal antibodies, vaccines, and immunodiagnostic reagents. Previous research efforts have primarily focused on the development of computational methods to predict linear epitopes rather than conformational epitopes; however, the latter is much more biologically predominant. Several conformational B-cell epitope prediction methods have recently been published, but their predictive performances are weak. Here, we present a review of the latest computational methods and assess their performances on a diverse test set of 29 non-redundant unbound antigen structures. Our results demonstrate that ISPIPab performs better than most methods and compares favorably with other recent antigen-specific methods. Finally, we suggest new strategies and opportunities to improve computational predictions of conformational B-cell epitopes.

Collision of rare-gas atoms on helium nanodroplets: Theoretical evidence for an efficient coagulation of heavy rare-gas atoms.

The coagulation of rare-gas atoms (RG = Ne, Ar, Kr, Xe, and Rn) in helium nanodroplets (HNDs) composed of 1000 atoms is investigated by zero-point averaged dynamics where a He-He pseudopotential is used to make the droplet liquid with proper energies. This method reproduces the qualitative abundances of embedded Arn+1 structures obtained by Time-Dependent Density Functional Theory and Ring Polymer Molecular Dynamics for Ar + ArnHe1000 collisions at realistic projectile speeds and impact parameters. More generally, coagulation is found to be much more efficient for heavy rare-gases (Xe and Rn) than for light ones (Ne and Ar), a behavior mainly attributed to a slower energy dissipation of the projectile in the HND. When coagulation does not occur, the projectile maintains a speed of 10-30m s-1 within the HND, but its velocity vector is rarely oriented toward the dopant, and the projectile roams in a limited region of the droplet. The structure of embedded RGn+1 clusters does not systematically match their gas-phase global minimum structure, and more than 30% of RGn-RG unbound structures are due to one He atom located in between the projectile and a dopant atom.

Revealing the Role of the Arg and Lys in Shifting Paradigm from BTK Selective Inhibition to the BTK/HCK Dual Inhibition - Delving into the Inhibitory Activity of KIN-8194 against BTK, and HCK in the Treatment of Mutated BTKCys481 Waldenström Macroglobulinemia: A Computational Approach.

Despite the early success of Bruton's tyrosine kinase (BTK) inhibitors in the treatment of Waldenström macroglobulinemia (WM), these single-target drug therapies have limitations in their clinical applications, such as drug resistance. Several alternative strategies have been developed, including the use of dual inhibitors, to maximize the therapeutic potential of these drugs. Recently, the pharmacological activity of KIN-8194 was repurposed to serve as a 'dual-target' inhibitor of BTK and Hematopoietic Cell Kinase (HCK). However, the structural dual inhibitory mechanism remains unexplored, hence the aim of this study. Conducting predictive pharmacokinetic profiling of KIN-8194, as well as demonstrating a comparative structural mechanism of inhibition against the above-mentioned enzymes. Our results revealed favourable binding affinities of -20.17 kcal/mol, and -35.82 kcal/mol for KIN-8194 towards HCK and BTK, respectively. Catalytic residues Arg137/174 and Lys42/170 in BTK and Arg303 and Lys75/173/244/247 in HCK were identified as crucial mediators of the dual binding mechanism of KIN-8194, corroborated by high per-residue energy contributions and consistent high-affinity interactions of these residues. Prediction of the pharmacokinetics and physicochemical properties of KIN-8194 further established its inhibitory potential, evidenced by the favourable absorption, metabolism, excretion, and minimal toxicity properties. Structurally, KIN-8194 impacted the stability, flexibility, solvent-accessible surface area, and rigidity of BTK and HCK, characterized by various alterations observed in the bound and unbound structures, which proved enough to disrupt their biological function. These structural insights provided a baseline for the understanding of the dual inhibitory activity of KIN- 8194. Establishing the cruciality of the interactions between the KIN-8194 and Arg and Lys residues could guide the structure-based design of novel dual BTK/HCK inhibitors with improved therapeutic activities.

Quantum Chemistry-Based Protein-Protein Docking without Empirical Parameters.

This study developed a novel protein-protein docking approach based on quantum chemistry. To judge the appropriateness of complex structures, we introduced two criterion values, EV1 and EV2, computed using the fragment molecular orbital method without any empirical parameters. These criterion values enable us to search complex structures in which patterns of the electrostatic potential of the two proteins are optimally aligned at their interface. The performance of our method was validated using 53 complexes in a benchmark set provided for protein-protein docking. When employing bound state structures, docking success rates reached 64% for EV1 and 76% for EV2. On the other hand, when employing unbound state structures, docking success rates reached 13% for EV1 and 17% for EV2.

Deep learning of antibody epitopes using positional permutation vectors

BackgroundThe accurate computational prediction of B cell epitopes can vastly reduce the cost and time required for identifying potential epitope candidates for the design of vaccines and immunodiagnostics. However, current computational tools for B cell epitope prediction perform poorly and are not fit-for-purpose, and there remains enormous room for improvement and the need for superior prediction strategies. ResultsHere we propose a novel approach that improves B cell epitope prediction by encoding epitopes as binary positional permutation vectors that represent the position and structural properties of the amino acids within a protein antigen sequence that interact with an antibody. This approach supersedes the traditional method of defining epitopes as scores per amino acid on a protein sequence, where each score reflects each amino acids predicted probability of partaking in a B cell epitope antibody interaction. In addition to defining epitopes as binary positional permutation vectors, the approach also uses the 3D macrostructure features of the unbound protein structures, and in turn uses these features to train another deep learning model on the corresponding antibody-bound protein 3D structures. This enables the algorithm to learn the key structural and physiochemical features of the unbound protein and embedded epitope that initiate the antibody binding process helping to eliminate “induced fit” biases in the training data. We demonstrate that the strategy predicts B cell epitopes with improved accuracy compared to the existing tools. Additionally, we show that this approach reliably identifies the majority of experimentally verified epitopes on the spike protein of SARS-CoV-2 not seen by the model during training and generalizes in a very robust manner on dissimilar data not seen by the model during training. ConclusionsWith the approach described herein, a primary protein sequence and a query positional permutation vector encoding a putative epitope is sufficient to predict B cell epitopes in a reliable manner, potentially advancing the use of computational prediction of B cell epitopes in biomedical research applications.

Predicting binding events in very flexible, allosteric, multi-domain proteins.

Knowledge of the structures formed by proteins and small ligands is of fundamental importance for understanding molecular principles of chemotherapy and for designing new and more effective drugs. Due to the still high costs and to the several limitations of experimental techniques, it is most often desirable to predict these ligand-protein complexes in silico, particularly when screening for new putative drugs from databases of millions of compounds. While virtual screening based on molecular docking is widely used for this purpose, it generally fails in mimicking binding events associated with large conformational changes in the protein, particularly when the latter involve multiple domains. In this work, we describe a new methodology aimed at generating bound-like conformations of very flexible and allosteric proteins bearing multiple binding sites. Validation was performed on the enzyme adenylate kinase (ADK), a paradigmatic example of proteins that undergo very large conformational changes upon ligand binding. By only exploiting the unbound structure and the putative binding sites of the protein, we generated a significant fraction of bound-like structures, which employed in ensemble-docking calculations allowed to find native-like poses of substrates, inhibitors, and catalytically incompetent binders. Our protocol provides a general framework for the generation of bound-like conformations of flexible proteins that are suitable to host different ligands, demonstrating high sensitivity to the fine chemical details that regulate protein's activity. We foresee applications in virtual screening for difficult targets, prediction of the impact of amino acid mutations on structure and dynamics, and protein engineering.

A comprehensive investigation of the interactions of human serum albumin with polymeric and hybrid nanoparticles.

Nanoparticles (NPs) engineered as drug delivery systems continue to make breakthroughs as they offer numerous advantages over free therapeutics. However, the poor understanding of the interplay between the NPs and biomolecules, especially blood proteins, obstructs NP translation to clinics. Nano-bio interactions determine the NPs' in vivo fate, efficacy and immunotoxicity, potentially altering protein function. To fulfill the growing need to investigate nano-bio interactions, this study provides a systematic understanding of two key aspects: (i) protein corona (PC) formation and (ii) NP-induced modifications on protein's structure and stability. A methodology was developed by combining orthogonal techniques to analyze both quantitative and qualitative aspects of nano-bio interactions, using human serum albumin (HSA) as a model protein. Protein quantification via liquid chromatography-mass spectrometry, and capillary zone electrophoresis (CZE) clarified adsorbed protein quantity and stability. CZE further unveiled qualitative insights into HSA forms (native, glycated HSA and cysteinylated), while synchrotron radiation circular dichroism enabled analyzing HSA's secondary structure and thermal stability. Comparative investigations of NP cores (organic vs. hybrid), and shells (with or without polyethylene glycol (PEG)) revealed pivotal factors influencing nano-bio interactions. Polymeric NPs based on poly(lactic-co-glycolic acid) (PLGA) and hybrid NPs based on metal-organic frameworks (nanoMOFs) presented distinct HSA adsorption profiles. PLGA NPs had protein-repelling properties while inducing structural modifications on HSA. In contrast, HSA exhibited a high affinity for nanoMOFs forming a PC altering thereby the protein structure. A shielding effect was gained through PEGylation for both types of NPs, avoiding the PC formation as well as the alteration of unbound HSA structure.

DynamicBind: predicting ligand-specific protein-ligand complex structure with a deep equivariant generative model

While significant advances have been made in predicting static protein structures, the inherent dynamics of proteins, modulated by ligands, are crucial for understanding protein function and facilitating drug discovery. Traditional docking methods, frequently used in studying protein-ligand interactions, typically treat proteins as rigid. While molecular dynamics simulations can propose appropriate protein conformations, they’re computationally demanding due to rare transitions between biologically relevant equilibrium states. In this study, we present DynamicBind, a deep learning method that employs equivariant geometric diffusion networks to construct a smooth energy landscape, promoting efficient transitions between different equilibrium states. DynamicBind accurately recovers ligand-specific conformations from unbound protein structures without the need for holo-structures or extensive sampling. Remarkably, it demonstrates state-of-the-art performance in docking and virtual screening benchmarks. Our experiments reveal that DynamicBind can accommodate a wide range of large protein conformational changes and identify cryptic pockets in unseen protein targets. As a result, DynamicBind shows potential in accelerating the development of small molecules for previously undruggable targets and expanding the horizons of computational drug discovery.

Flexible protein-protein docking with a multitrack iterative transformer.

Conventional protein-protein docking algorithms usually rely on heavy candidate sampling and reranking, but these steps are time-consuming and hinder applications that require high-throughput complex structure prediction, for example, structure-based virtual screening. Existing deep learning methods for protein-protein docking, despite being much faster, suffer from low docking success rates. In addition, they simplify the problem to assume no conformational changes within any protein upon binding (rigid docking). This assumption precludes applications when binding-induced conformational changes play a role, such as allosteric inhibition or docking from uncertain unbound model structures. To address these limitations, we present GeoDock, a multitrack iterative transformer network to predict a docked structure from separate docking partners. Unlike deep learning models for protein structure prediction that input multiple sequence alignments, GeoDock inputs just the sequences and structures of the docking partners, which suits the tasks when the individual structures are given. GeoDock is flexible at the protein residue level, allowing the prediction of conformational changes upon binding. On the Database of Interacting Protein Structures (DIPS) test set, GeoDock achieves a 43% top-1 success rate, outperforming all other tested methods. However, in the standard DIPS train/test splits, we discovered contamination of close homologs in the training set. After decontaminating the training set, the success rate is 31%. On the DB5.5 test set and a benchmark dataset of antibody-antigen complexes, GeoDock outperforms the deep learning models trained using the same dataset but falls behind most of the conventional methods and AlphaFold-Multimer. GeoDock attains an average inference speed of under 1 s on a single GPU, enabling its application in large-scale structure screening. Although binding-induced conformational changes are still a challenge owing to limited training and evaluation data, our architecture sets up the foundation to capture this backbone flexibility. Code and a demonstration Jupyter notebook are available at https://github.com/Graylab/GeoDock.

Prospects for Connections to the Outer Halo of the Milky Way with LSST

Over its ten-year mission, Rubin Observatory's Legacy Survey of Space and Time will gather enough time series data to identify variable stars much fainter than previously observed. This makes ideal for finding RR Lyrae stars and mapping previously unexplored regions of our Galaxy's stellar halo. This project creates several mock data sets from both simulations and known satellite properties to model what might be found. We use these to examine the prospects for making connections between stellar structures already known around the Galaxy (satellites, streams and shells) and newly discovered stars. Such connections could be used to place constraints on the mass distribution in the outer regions of the Galaxy’s dark matter halo. Our results are encouraging, suggesting: that tens of percent of the known structures should have counterparts in the outer halo; and that several entirely new unbound structures could be discovered.

Influence of roll crushing of plant material on the consumer characteristics of «Ukrainske» biscuits

The study revealed that adding crushed flax seeds to the dough system prolonged the duration of its formation by 5 minutes. It was also noted that the amount of gluten in the dough was 17% less than in the control sample, which had a loose, unbound structure and lower extensibility. The impact of the rollers ensures a high-quality grinding process of flax seeds, which allows mucilage-forming polysaccharides to pass into the liquid phase of the dough and prevent the formation of a continuous gluten structure. It was also found that the necessary rheological properties are provided by adding 4–5% of crushed flax seeds to the dough recipe, while increasing the amount to 6% and above leads to a deterioration in organoleptic characteristics.

Transport and inhibition mechanism of the human SGLT2-MAP17 glucose transporter.

Sodium-glucose cotransporter 2 (SGLT2) is imporant in glucose reabsorption. SGLT2 inhibitors suppress renal glucose reabsorption, therefore reducing blood glucose levels in patients with type 2 diabetes. We and others have developed several SGLT2 inhibitors starting from phlorizin, a natural product. Using cryo-electron microscopy, we present the structures of human (h)SGLT2-MAP17 complexed with five natural or synthetic inhibitors. The four synthetic inhibitors (including canagliflozin) bind the transporter in the outward conformations, while phlorizin binds it in the inward conformation. The phlorizin-hSGLT2 interaction exhibits biphasic kinetics, suggesting that phlorizin alternately binds to the extracellular and intracellular sides. The Na+-bound outward-facing and unbound inward-open structures of hSGLT2-MAP17 suggest that the MAP17-associated bundle domain functions as a scaffold, with the hash domain rotating around the Na+-binding site. Thus, Na+ binding stabilizes the outward-facing conformation, and its release promotes state transition to inward-open conformation, exhibiting a role of Na+ in symport mechanism. These results provide structural evidence for the Na+-coupled alternating-access mechanism proposed for the transporter family.

DeepProSite: structure-aware protein binding site prediction using ESMFold and pretrained language model.

Identifying the functional sites of a protein, such as the binding sites of proteins, peptides, or other biological components, is crucial for understanding related biological processes and drug design. However, existing sequence-based methods have limited predictive accuracy, as they only consider sequence-adjacent contextual features and lack structural information. In this study, DeepProSite is presented as a new framework for identifying protein binding site that utilizes protein structure and sequence information. DeepProSite first generates protein structures from ESMFold and sequence representations from pretrained language models. It then uses Graph Transformer and formulates binding site predictions as graph node classifications. In predicting protein-protein/peptide binding sites, DeepProSite outperforms state-of-the-art sequence- and structure-based methods on most metrics. Moreover, DeepProSite maintains its performance when predicting unbound structures, in contrast to competing structure-based prediction methods. DeepProSite is also extended to the prediction of binding sites for nucleic acids and other ligands, verifying its generalization capability. Finally, an online server for predicting multiple types of residue is established as the implementation of the proposed DeepProSite. The datasets and source codes can be accessed at https://github.com/WeiLab-Biology/DeepProSite. The proposed DeepProSite can be accessed at https://inner.wei-group.net/DeepProSite/.

The alteration of structural network upon transient association between proteins studied using graph theory.

Proteins such as enzymes perform their function by predominant non-covalent bond interactions between transiently interacting units. There is an impact on the overall structural topology of the protein, albeit transient nature of such interactions, that enable proteins to deactivate or activate. This aspect of the alteration of the structural topology is studied by employing protein structural networks, which are node-edge representative models of protein structure, reported as a robust tool for capturing interactions between residues. Several methods have been optimized to collect meaningful, functionally relevant information by studying alteration of structural networks. In this article, different methods of comparing protein structural networks are employed, along with spectral decomposition of graphs to study the subtle impact of protein-protein interactions. A detailed analysis of the structural network of interacting partners is performed across a dataset of around 900 pairs of bound complexes and corresponding unbound protein structures. The variation in network parameters at, around, and far away from the interface are analyzed. Finally, we present interesting case studies, where an allosteric mechanism of structural impact is understood from communication-path detection methods. The results of this analysis are beneficial in understanding protein stability, for future engineering, and docking studies.

Rational Prediction of PROTAC-Compatible Protein-Protein Interfaces by Molecular Docking.

Proteolysis targeting chimeras (PROTACs) are heterobifunctional ligands that mediate the interaction between a protein target and an E3 ligase, resulting in a ternary complex, whose interaction with the ubiquitination machinery leads to target degradation. This technology is emerging as an exciting new avenue for therapeutic development, with several PROTACs currently undergoing clinical trials targeting cancer. Here, we describe a general and computationally efficient methodology combining restraint-based docking, energy-based rescoring, and a filter based on the minimal solvent-accessible surface distance to produce PROTAC-compatible PPIs suitable for when there is no a priori known PROTAC ligand. In a benchmark employing a manually curated data set of 13 ternary complex crystals, we achieved an accuracy of 92% when starting from bound structures and 77% when starting from unbound structures, respectively. Our method only requires that the ligand-bound structures of the monomeric forms of the E3 ligase and target proteins be given to run, making it general, accurate, and highly efficient, with the ability to impact early-stage PROTAC-based drug design campaigns where no structural information about the ternary complex structure is available.

Putative Role of Cholesterol in Shaping the Structural and Functional Dynamics of Smoothened (SMO).

The smoothened (SMO) receptor belongs to the superfamily of class F G protein-coupled receptors (GPCRs) and is a potential drug target in several types of cancer. It has two ligand binding sites, respectively, in the cysteine-rich domain (CRD) and the transmembrane domain (TMD). It has been shown that cholesterol is important for its activation and function. However, the molecular-level understanding of SMO dynamics in the presence of cholesterol has not been explored in sufficient detail. In this work, we have carried out atomistic molecular dynamics simulations totaling 3.6 μs to analyze the effect of cholesterol binding to TMD and/or CRD on the structure and dynamics of the SMO receptor. Our results show that the presence of cholesterol in the CRD and TMD, respectively, alters the conformational dynamics of SMO differently. We reported that the reorganization of the D-R-E network at the extracellular end of the TMD is important for the high activity of SMO. In general, the transmembrane helices 5, 6, and 7 and helix 8 are most affected, which, in turn, leads to changes in the CRD and intracellular cytoplasmic domain (ICD) dynamics patterns depending on the presence or absence of cholesterol in the CRD and/or the TMD. We have also reported that the interaction of membrane lipids with SMO is different in different SMO states. These results agree with the experimental structural observations and data of cholesterol-bound and unbound structures of SMO and add to our molecular understanding of the SMO-cholesterol interaction.

Role of Conformational Changes in DNA Recognition in Transcription Factors having Helix-Turn-Helix Motif

DNA-binding proteins play a crucial role in various cellular processes, some of these have a unique structural arrangement, the helix-turn-helix motif, which enables them to bind to DNA molecules in a specific manner. Upon binding to DNA, conformational changes occur in both the protein structure and the DNA molecule, making it essential to quantify and characterise these changes. This work aims to quantify and characterise the conformational changes in the protein structures. This is achieved by calculating the root-mean-square deviation of the protein structure in the free state and DNA bound state. A database is curated which has a significant number of DNA-bound and unbound pairs available. For the unbound structure of the proteins, pre-processing is required to remove chains other than those found in its corresponding DNA-bound structure followed by superimposing it to the DNA-bound structures obtained from the Protein Data Bank (PDB) to calculate the RMSD. After quantifying the conformational changes using RMSD values, the proteins are categorised into six types based on their observed conformational changes. In this study, the primary objective is to conduct a comprehensive analysis of conformational changes in DNA-binding proteins and investigate their correlation with various protein properties. By exploring these correlations, the study aims to unravel potential patterns or dependencies between the extent of conformational changes and the diverse properties of DNA-binding proteins. Such insights can shed light on the functional implications of structural variations in DNA-protein interactions, providing valuable information about how protein dynamics contribute to DNA recognition and binding. In addition, this work endeavours to broaden its scope by attempting to replicate DNA-bound complexes through the utilisation of freely available structures. By doing so, it aims to assess the reliability and accuracy of docking methods in predicting and reconstructing these complexes. The primary objective is to evaluate the feasibility and effectiveness of such techniques in the context of complex formation between DNA and protein molecules.

Computational exploration of natural compounds targeting Staphylococcus aureus: inhibiting AgrA promoter binding for antimicrobial intervention

Staphylococcus aureus is a highly virulent nosocomial pathogen that poses a significant threat to individuals exposed to healthcare settings. Due to its sophisticated machinery for producing virulence factors, S. aureus can cause severe and potentially fatal infections in humans. This study focuses on the response regulator AgrA, which plays a crucial role in regulating the production of virulence factors in S. aureus. The objective is to identify natural compounds that can inhibit the binding of AgrA to its promoter site, thus inhibiting the expression of virulence genes. To achieve this, a pharmacophore model was generated using known drugs and applied to screen the ZINC natural product database. The resulting compounds were subjected to molecular docking-based virtual screening against the C-terminal DNA binding domain of AgrA. Three compounds, namely ZINC000077269178, ZINC000051012304, and ZINC000004266026, were shortlisted based on their strong affinity for key residues involved in DNA binding and transcription initiation. Subsequently, the unbound and ligand-bound complexes were subjected to a 200 ns molecular dynamics simulation to assess their conformational stability. Various analyses, including RMSD, RMSF, Rg, SASA, Principal Component Analysis, and Gibbs free energy landscape, were conducted on the simulation trajectory. The RMSD profile indicated similar fluctuations in both bound and unbound structures, while the Rg profile demonstrated the compactness of the protein without any unfolding during the simulation. Furthermore, Principal component analysis revealed that ligand binding reduced the overall atomic motion of the protein whereas free energy landscape suggested the energy variations obtained in complexes. Communicated by Ramaswamy H. Sarma