Theoretical insights into the binding interaction of Nirmatrelvir with SARS-CoV-2 Mpro mutants (C145A and C145S): MD simulations and binding free-energy calculation to understand drug resistance.

Mpro, the main protease and a crucial enzyme in SARS-CoV-2 is the most fascinating molecular target for pharmacological treatment and is also liable for viral protein maturation. For antiviral therapy, no drugs have been approved clinically to date. Targeting the Mpro with a compound having inhibitory properties against it can hinder viral replication. The therapeutic potential of the antiviral compound Nirmatrelvir (NMV) against SARS-CoV-2 Mpro was investigated using a systematic approach of molecular docking, MD simulations, and binding free energy calculation based on the MM-GBSA method. NMV, a covalent inhibitor with a recently revealed chemical structure, is a promising oral antiviral clinical candidate with significant in vitro anti-SARS-CoV-2 action in third-phase clinical trials. To explore the therapeutic ability and possible drug resistance, the Mpro system was studied for WT and two of its primary mutants (C145A & C145S). The protein-ligand (Mpro/NMV) complexes were further examined through long MD simulations to check the possible drug resistance in the mutants. To understand the binding affinity, the MM-GBSA method was applied to the Mpro/NMV complexes. Moreover, PCA analysis confirms the detachment of the linker region from the major domains in C145S and C145A mutants allowing for conformational alterations in the active-site region. Based on the predicted biological activities and binding affinities of NMV to WT and mutant (C145A & C145S) Mpro, it can be stipulated that NMV may have conventional potency to act as an anti-viral agent against WT Mpro, while the catalytic-dyad mutations may show substantial mutation-induced drug resistance.Communicated by Ramaswamy H. Sarma.

Correlation between protein conformations and water structure and thermodynamics at high pressure: A molecular dynamics study of the Bovine Pancreatic Trypsin Inhibitor (BPTI) protein.

Pressure-induced perturbation of a protein structure leading to its folding-unfolding mechanism is an important yet not fully understood phenomenon. The key point here is the role of water and its coupling with protein conformations as a function of pressure. In the current work, using extensive molecular dynamics simulation at 298 K, we systematically examine the coupling between protein conformations and water structures of pressures of 0.001, 5, 10, 15, 20 kbar, starting from (partially) unfolded structures of the protein Bovine Pancreatic Trypsin Inhibitor (BPTI). We also calculate localized thermodynamics at those pressures as a function of protein-water distance. Our findings show that both protein-specific and generic effects of pressure are operating. In particular, we found that (1) the amount of increase in water density near the protein depends on the protein structural heterogeneity; (2) the intra-protein hydrogen bond decreases with pressure, while the water-water hydrogen bond per water in the first solvation shell (FSS) increases; protein-water hydrogen bonds also found to increase with pressure, (3) with pressure hydrogen bonds of waters in the FSS getting twisted; and (4) water's tetrahedrality in the FSS decreases with pressure, but it is dependent on the local environment. Thermodynamically, at higher pressure, the structural perturbation of BPTI is due to the pressure-volume work, while the entropy decreases with the increase of pressure due to the higher translational and rotational rigidity of waters in the FSS. The local and subtle effects of pressure, found in this work, are likely to be typical of pressure-induced protein structure perturbation.

A strategy to optimize the peptide-based inhibitors against different mutants of the spike protein of SARS-CoV-2.

The SARS-CoV-2 virus has caused high-priority health concerns at a global level. Vaccines have stalled the proliferation of viruses to some extent. Yet, the emergence of newer, potentially more infectious, and dangerous mutants such as Delta and Omicron are among the major challenges in finding a more permanent solution for this pandemic. The effectiveness of antivirals Molnupiravir and Paxlovid, authorized for emergency use by the FDA, are yet to be assessed on a larger population. Patients with a high risk of disease progression have received treatment with antibody-cocktail. Most of the mutations leading to the new lineage of SARS-CoV-2 are found in the spike protein of this virus that plays a key role in facilitating host entry. The current study has investigated how to modify a promising peptide-based inhibitor of spike protein, LCB3, against common mutations, N501Y and K417N in the target protein so that it retains its efficacy against the spike protein. LCB3 being a prototype for protein-based inhibitors is an ideal testing system to learn about protein-based inhibitors. This study proposes the substitutions of amino acid residues of LCB3 inhibitor using a structure-based approach that considers free energy decomposition of residues, the distance between atoms, and the interaction among amino acids. The binding free energy calculations suggest a possible improvement in the binding affinity of existing inhibitor LCB3 to the mutant forms of the S-protein using simple substitutions at specific positions of the inhibitor. This approach, being general, can be used in different inhibitors and other mutations and help in fighting against SARS-CoV-2.Communicated by Ramaswamy H. Sarma.

Crustwater: Modeling Hydrophobic Solvation.

We describe Crustwater, a statistical mechanical model of nonpolar solvation in water. It treats bulk water using the Cage Water model and introduces a crust, i.e., a solvation shell of coordinated partially structured waters. Crustwater is analytical and fast to compute. We compute here solvation vs temperature over the liquid range, and vs pressure and solute size. Its thermal predictions are as accurate as much more costly explicit models such as TIP4P/2005. This modeling gives new insights into the hydrophobic effect: (1) that oil-water insolubility in cold water is due to solute-water (SW) translational entropy and not water-water (WW) orientations, even while hot water is dominated by WW cage breaking, and (2) that a size transition at the Angstrom scale, not the nanometer scale, takes place as previously predicted.

Calcium Ion Binding to the Mutants of Calmodulin: A Structure-Based Computational Predictive Model of Binding Affinity Using a Charge Scaling Approach in Molecular Dynamics Simulation.

The binding of calcium ions (Ca2+) to the calcium-binding proteins (CBPs) controls a plethora of regulatory processes. Among the roles played by CBPs in several diseases, the onset and progress of some cardiovascular diseases are caused by mutations in calmodulin (CaM), an important member of CBPs. Rationalization and prediction of the binding affinity of Ca2+ ions to the CaM can play important roles in understanding the origin of cardiovascular diseases. However, there is no robust structure-based computational method for predicting the binding affinity of Ca2+ ions to the different forms of CBPs in general and CaM in particular. In the current work, we have devised a fast yet accurate computational technique to accurately calculate the binding affinity of Ca2+ to the different forms of CaM. This method combines the well-known molecular mechanics Poisson-Boltzmann surface area (MM-PBSA) method and a charge scaling approach developed by previous authors that takes care of the polarization of CaM and Ca2+ ions. Our detailed analysis of the different components of binding free energy shows that subtle changes in electrostatics and van der Waals contribute to the difference in the binding affinity of mutants from that of the wild type (WT), and the charge scaling approach is superior in calculating these subtle changes in electrostatics as compared to the nonpolarizable force field used in this work. A statistically significant regression model made from our binding free energy calculations gives a correlation coefficient close to 0.8 to the experimental results. This structure-based predictive model can open up a new strategy to understand and predict the binding of Ca2+ to the mutants of CBPs, in general.

Synthetic Tunability and Biophysical Basis for Fabricating Highly Fluorescent and Stable DNA Copper Nanoclusters.

The real motivation in the present work is to tune the synthesis variables that can result in a highly fluorescent and stable DNA copper nanocluster (CuNC) and also to understand the intricate mechanism behind this process. Here, carefully optimized concentrations of various reactants enabled the creation of a DNA-encapsulated CuNC for AT-DNA, displaying a size of <1.0 nm as confirmed by transmission electron microscopy and dynamic light scattering. The extremely small size of the AT-DNACuNC supports the discrete electronic transitions, also characterized by an exceptionally strong negative circular dichroism (CD) band around 350 nm, whose intensity is well correlated with the observed strong fluorescence emission intensity. This remarkably strong CD can open new applications in the detection and quantification of a specific DNACuNC. Further, time-dependent fluorescence analysis suggested stronger photostabilization of these DNACuNCs. The simulation study, based on Cu ion distribution, explained how AT-DNA is a better candidate for NC formation than GC-DNA. In conclusion, the better-tuned synthesis procedure has resulted in a highly compact, well-defined three-dimensional conformation that promotes a more favorable microenvironment to sequester a DNA-based CuNC with high brightness and outstanding photostability.

Analytical 2-Dimensional Model of Nonpolar and Ionic Solvation in Water.

A goal in computational chemistry is computing hydration free energies of nonpolar and charged solutes accurately, but with much greater computational speeds than in today's explicit-water simulations. Here, we take one step in that direction: a simple model of solvating waters that is analytical and thus essentially instantaneous to compute. Each water molecule is a 2-dimensional dipolar hydrogen-bonding disk that interacts around small circular solutes with different nonpolar and charge interactions. The model gives good qualitative agreement with experiments. As a function of the solute radius, it gives the solvation free energy, enthalpy and entropy as a function of temperature for the inert gas series Ne, Ar, Kr, and Xe. For anions and cations, it captures relatively well the trends versus ion radius. This approach should be readily generalizable to three dimensions.

Calcium ion binding to calmodulin: binding free energy calculation using the molecular mechanics Poisson-Boltzmann surface area (MM-PBSA) method by incorporating implicit polarization.

Binding of calcium ion to calcium-binding proteins (CBP) triggers a large number of biological processes in a cell. CBP are known to play important roles in various diseases, such as cancer, alzheimer, and neuronal problems. However, the calculation of the binding affinity of calcium ion to CBP still possesses a significant challenge to the computational investigators. One of the main reasons for this difficulty is the polarization of CBP due to the binding of calcium. In the current work, we have used the implicit polarization method of Leontyev et al. (PCCP, 13.7 (2011): 2613-2626) to calculate the binding free energy of calcium ion binding to calmodulin, an important CBP. We have used the widely used MM-PBSA method to find a good protocol of calculation with implicit polarization. We have also optimized the best value of the calcium radius to match the experimental results. Our results show incorporation of polarization improves the agreement between the calculated and experimental results, although still, some discrepancy remains. On the whole, this work shows implicit polarization when combined with the MM-PBSA method can give results better than calculation without any polarization, and further improvement is necessary to get a quantitative match with experiments.Communicated by Ramaswamy H. Sarma.

A look inside the black box: Using graph-theoretical descriptors to interpret a Continuous-Filter Convolutional Neural Network (CF-CNN) trained on the global and local minimum energy structures of neutral water clusters.

We describe a method for the post-hoc interpretation of a neural network (NN) trained on the global and local minima of neutral water clusters. We use the structures recently reported in a newly published database containing over 5 × 106 unique water cluster networks (H2O)N of size N = 3-30. The structural properties were first characterized using chemical descriptors derived from graph theory, identifying important trends in topology, connectivity, and polygon structure of the networks associated with the various minima. The code to generate the molecular graphs and compute the descriptors is available at https://github.com/exalearn/molecular-graph-descriptors, and the graphs are available alongside the original database at https://sites.uw.edu/wdbase/. A Continuous-Filter Convolutional Neural Network (CF-CNN) was trained on a subset of 500 000 networks to predict the potential energy, yielding a mean absolute error of 0.002 ± 0.002 kcal/mol per water molecule. Clusters of sizes not included in the training set exhibited errors of the same magnitude, indicating that the CF-CNN protocol accurately predicts energies of networks for both smaller and larger sizes than those used during training. The graph-theoretical descriptors were further employed to interpret the predictive power of the CF-CNN. Topological measures, such as the Wiener index, the average shortest path length, and the similarity index, suggested that all networks from the test set were within the range of values as the ones from the training set. The graph analysis suggests that larger errors appear when the mean degree and the number of polygons in the cluster lie further from the mean of the training set. This indicates that the structural space, and not just the chemical space, is an important factor to consider when designing training sets, as predictive errors can result when the structural composition is sufficiently different from the bulk of those in the training set. To this end, the developed descriptors are quite effective in explaining the results of the CF-CNN (a.k.a. the "black box") model.

Calculation of salt-dependent free energy of binding of ß-lactoglobulin homodimer formation and mechanism of dimer formation using molecular dynamics simulation and three-dimensional reference interaction site model (3D-RISM): diffuse salt ions and non-polar interactions between the monomers favor the dimer formation.

There are several important phenomena in chemistry, biology, and physics where molecules (or parts of a molecule) having charges of the same sign come closer together and become stable. DNA condensation, RNA folding, colloid-colloid interactions are some of the examples of this kind. In the current work, we have investigated how ß-lactoglobulin, a protein found in milk, in spite of carrying +13 charge, favors the homodimer formation in the presence of salt. We have focussed on calculating the protein-protein binding free energy in the presence of salt and identifying the thermodynamic and microscopic mechanism of the process. Estimation of binding free energy of this salt-dependent process is done by combining molecular dynamics simulation with statistical mechanical theory of three-dimensional reference interaction site model (3D-RISM). Binding free energy is evaluated from the chemical potential of the solutes as opposed to potential of mean force calculation, which gives only a constrained free energy. Our calculated values semi-quantitatively match with the experimental results. By examining the different components of binding free energy, we have found that the role of salt ions (especially of Cl-) is to shift the equilibrium towards the dimer. Non-polar (Lennard-Jones) interactions between the monomers is also favorable to the binding free energy. However, water slightly disfavors the dimer formation. For the microscopic mechanism, heterogeneous of both Na+ and Cl- near the charged residues at the binding interface and change of this charge distribution on dimer formation contribute to the stability. A fine-tuning of enthalpic and entropic effects of salt ions is found to operate at different salt concentrations. Both thermodynamic and microscopic mechanism of dimer formation gives detailed insight into the complex electrostatics of charged protein-protein binding.

Atlas of putative minima and low-lying energy networks of water clusters n = 3-25.

We report a database consisting of the putative minima and ∼3.2 × 106 local minima lying within 5 kcal/mol from the putative minima for water clusters of sizes n = 3-25 using an improved version of the Monte Carlo temperature basin paving (MCTBP) global optimization procedure in conjunction with the ab initio based, flexible, polarizable Thole-Type Model (TTM2.1-F, version 2.1) interaction potential for water. Several of the low-lying structures, as well as low-lying penta-coordinated water networks obtained with the TTM2.1-F potential, were further refined at the Møller-Plesset second order perturbation (MP2)/aug-cc-pVTZ level of theory. In total, we have identified 3 138 303 networks corresponding to local minima of the clusters n = 3-25, whose Cartesian coordinates and relative energies can be obtained from the webpage https://sites.uw.edu/wdbase/. Networks containing penta-coordinated water molecules start to appear at n = 11 and, quite surprisingly, are energetically close (within 1-3 kcal/mol) to the putative minima, a fact that has been confirmed from the MP2 calculations. This large database of water cluster minima spanning quite dissimilar hydrogen bonding networks is expected to influence the development and assessment of the accuracy of interaction potentials for water as well as lower scaling electronic structure methods (such as different density functionals). Furthermore, it can also be used in conjunction with data science approaches (including but not limited to neural networks and machine and deep learning) to understand the properties of water, nature's most important substance.

Microscopic picture of water-ethylene glycol interaction near a model DNA by computer simulation: Concentration dependence, structure, and localized thermodynamics.

It is known that crowded molecular environment affects the structure, thermodynamics, and dynamics of macromolecules. Most of the previous works on molecular crowding have majorly focused on the behavior of the macromolecule with less emphasis on the behavior of the crowder and water molecules. In the current study, we have precisely focused on the behavior of the crowder, (ethylene glycol (EG)), salt ions, and water in the presence of a DNA with the increase of the EG concentration. We have probed the behavior of water and crowder using molecular dynamics (MD) simulation and by calculating localized thermodynamic properties. Our results show an interesting competition between EG and water molecules to make hydrogen bonds (H-bond) with DNA. Although the total number of H-bonds involving DNA with both EG and water remains essentially same irrespective of the increase in EG concentration, there is a proportional change in the H-bonding pattern between water-water, EG-EG, and EG-water near DNA and in bulk. At low concentrations of EG, the displacement of water molecules near DNA is relatively easy. However, the displacement of water becomes more difficult as the concentration of EG increases. The density of Na+ (Cl-) near DNA increases (decreases) as the concentration of EG is increased. The density of Cl- near Na+ increases with the increase in EG concentration. It was also found that the average free energy per water in the first solvation shell increases with the increase in EG concentration. Putting all these together, a microscopic picture of EG, water, salt interaction in the presence of DNA, as a function of EG concentration, has emerged.

Identification of inhibitors against α-Isopropylmalate Synthase of Mycobacterium tuberculosis using docking-MM/PBSA hybrid approach.

α-Isopropylmalate Synthase (α-IPMS) encoded by leuA in Mycobacterium tuberculosis (M.tb) is involved in the leucine biosynthesis pathway and is extremely critical for the synthesis of branched-chain amino acids (leucine, isoleucine and valine). α-IPMS activity is required not only for the proliferation of M.tb but is also indispensable for its survival during the latent phase of infection. It is absent in humans and is widely regarded as one of the validated drug targets against Tuberculosis (TB). Despite its essentiality, any study on designing of potential chemical inhibitors against α-IPMS has not been reported so far. In the present study, in silico identification of putative inhibitors against α-IPMS exploring three chemical databases i.e. NCI, DrugBank and ChEMBL is reported through structurebased drug design and filtering of ligands based on the pharmacophore feature of the actives. In the absence of experimental results of any inhibitor against α-IPMS, a stringent validation of docking results is done by comparing with molecular mechanics/Poisson- Boltzmann surface area (MM/PBSA) calculations by investigating two more proteins for which experimental results are known.

How Mg2+ ion and water network affect the stability and structure of non-Watson-Crick base pairs in E. coli loop E of 5S rRNA: a molecular dynamics and reference interaction site model (RISM) study.

The non-Watson-Crick (non-WC) base pairs of Escherichia coli loop E of 5S rRNA are stabilized by Mg2+ ions through water-mediated interaction. It is important to know the synergic role of Mg2+ and the water network surrounding Mg2+ in stabilizing the non-WC base pairs of RNA. For this purpose, free energy change of the system is calculated using molecular dynamics (MD) simulation as Mg2+ is pulled from RNA, which causes disturbance of the water network. It was found that Mg2+ remains hexahydrated unless it is close to or far from RNA. In the pentahydrated form, Mg2+ interacts directly with RNA. Water network has been identified by two complimentary methods; MD followed by a density-based clustering algorithm and three-dimensional-reference interaction site model. These two methods gave similar results. Identification of water network around Mg2+ and non-WC base pairs gives a clue to the strong effect of water network on the stability of this RNA. Based on sequence analysis of all Eubacteria 5s rRNA, we propose that hexahydrated Mg2+ is an integral part of this RNA and geometry of base pairs surrounding it adjust to accommodate the [Formula: see text]. Overall the findings from this work can help in understanding the basis of the complex structure and stability of RNA with non-WC base pairs.

Free Energy Contribution Analysis Using Response Kernel Approximation: Insights into the Acylation Reaction of a Beta-Lactamase.

A widely applicable free energy contribution analysis (FECA) method based on the quantum mechanical/molecular mechanical (QM/MM) approximation using response kernel approaches has been proposed to investigate the influences of environmental residues and/or atoms in the QM region on the free energy profile. This method can evaluate atomic contributions to the free energy along the reaction path including polarization effects on the QM region within a dramatically reduced computational time. The rate-limiting step in the deactivation of the ß-lactam antibiotic cefalotin (CLS) by ß-lactamase was studied using this method. The experimentally observed activation barrier was successfully reproduced by free energy perturbation calculations along the optimized reaction path that involved activation by the carboxylate moiety in CLS. It was found that the free energy profile in the QM region was slightly higher than the isolated energy and that two residues, Lys67 and Lys315, as well as water molecules deeply influenced the QM atoms associated with the bond alternation reaction in the acyl-enzyme intermediate. These facts suggested that the surrounding residues are favorable for the reactant complex and prevent the intermediate from being too stabilized to proceed to the following deacylation reaction. We have demonstrated that the free energy contribution analysis should be a useful method to investigate enzyme catalysis and to facilitate intelligent molecular design.

An analytical correlated random walk model and its application to understand subdiffusion in crowded environment.

Subdiffusion in crowded environment such as movement of macromolecule in a living cell has often been observed experimentally. The primary reason for subdiffusion is volume exclusion by the crowder molecules. However, other effects such as hydrodynamic interaction may also play an important role. Although there are a large number of computer simulation studies on understanding molecular crowding, there is a lack of theoretical models that can be connected to both experiment and simulation. In the current work, we have formulated a one-dimensional correlated random walk model by connecting this to the motion in a crowded environment. We have found the exact solution of the probability distribution function of the model by solving it analytically. The parameters of our model can be obtained either from simulation or experiment. It has been shown that this analytical model captures some of the general features of diffusion in crowded environment as given in the previous literature and its prediction for transient subdiffusion closely matches the observations of a previous study of computer simulation of Escherichia coli cytoplasm. It is likely that this model will open up more development of theoretical models in this area.

Molecular dynamics simulations indicate that tyrosineB10 limits motions of distal histidine to regulate CO binding in soybean leghemoglobin.

Myoglobin (Mb) uses strong electrostatic interaction in its distal heme pocket to regulate ligand binding. The mechanism of regulation of ligand binding in soybean leghemoglobin a (Lba) has been enigmatic and more so due to the absence of gaseous ligand bound atomic resolution three-dimensional structure of the plant globin. While the 20-fold higher oxygen affinity of Lba compared with Mb is required for its dual physiological function, the mechanism by which this high affinity is achieved is only emerging. Extensive mutational analysis combined with kinetic and CO-FT-IR spectroscopic investigation led to the hypothesis that Lba depended on weakened electrostatic interaction between distal HisE7 and bound ligand achieved by invoking B10Tyr, which itself hydrogen bonds with HisE7 thus restricting it in a single conformation detrimental to Mb-like strong electrostatic interaction. Such theory has been re-assessed here using CO-Lba in silico model and molecular dynamics simulation. The investigation supports the presence of at least two major conformations of HisE7 in Lba brought about by imidazole ring flip, one of which makes hydrogen bonds effectively with B10Tyr affecting the former's ability to stabilize bound ligand, while the other does not. However, HisE7 in Lba has limited conformational freedom unlike high frequency of imidazole ring flips observed in Mb and in TyrB10Leu mutant of Lba. Thus, it appears that TyrB10 limits the conformational freedom of distal His in Lba, tuning down ligand dissociation rate constant by reducing the strength of hydrogen bonding to bound ligand, which the freedom of distal His of Mb allows.

Low energy isomers of (H2O)25 from a hierarchical method based on Monte Carlo temperature basin paving and molecular tailoring approaches benchmarked by MP2 calculations.

We report new global minimum candidate structures for the (H2O)25 cluster that are lower in energy than the ones reported previously and correspond to hydrogen bonded networks with 42 hydrogen bonds and an interior, fully coordinated water molecule. These were obtained as a result of a hierarchical approach based on initial Monte Carlo Temperature Basin Paving sampling of the cluster's Potential Energy Surface with the Effective Fragment Potential, subsequent geometry optimization using the Molecular Tailoring Approach with the fragments treated at the second order Møller-Plesset (MP2) perturbation (MTA-MP2) and final refinement of the entire cluster at the MP2 level of theory. The MTA-MP2 optimized cluster geometries, constructed from the fragments, were found to be within <0.5 kcal/mol from the minimum geometries obtained from the MP2 optimization of the entire (H2O)25 cluster. In addition, the grafting of the MTA-MP2 energies yields electronic energies that are within <0.3 kcal/mol from the MP2 energies of the entire cluster while preserving their energy rank order. Finally, the MTA-MP2 approach was found to reproduce the MP2 harmonic vibrational frequencies, constructed from the fragments, quite accurately when compared to the MP2 ones of the entire cluster in both the HOH bending and the OH stretching regions of the spectra.

A new coarse-grained model for E. coli cytoplasm: accurate calculation of the diffusion coefficient of proteins and observation of anomalous diffusion.

A new coarse-grained model of the E. coli cytoplasm is developed by describing the proteins of the cytoplasm as flexible units consisting of one or more spheres that follow Brownian dynamics (BD), with hydrodynamic interactions (HI) accounted for by a mean-field approach. Extensive BD simulations were performed to calculate the diffusion coefficients of three different proteins in the cellular environment. The results are in close agreement with experimental or previously simulated values, where available. Control simulations without HI showed that use of HI is essential to obtain accurate diffusion coefficients. Anomalous diffusion inside the crowded cellular medium was investigated with Fractional Brownian motion analysis, and found to be present in this model. By running a series of control simulations in which various forces were removed systematically, it was found that repulsive interactions (volume exclusion) are the main cause for anomalous diffusion, with a secondary contribution from HI.

Wang-Landau density of states based study of the folding-unfolding transition in the mini-protein Trp-cage (TC5b).

We present the results of a high-statistics equilibrium study of the folding/unfolding transition for the 20-residue mini-protein Trp-cage (TC5b) in water. The ECEPP/3 force field is used and the interaction with water is treated by a solvent-accessible surface area method. A Wang-Landau type simulation is used to calculate the density of states and the conditional probabilities for the various values of the radius of gyration and the number of native contacts at fixed values of energy--along with a systematic check on their convergence. All thermodynamic quantities of interest are calculated from this information. The folding-unfolding transition corresponds to a peak in the temperature dependence of the computed specific heat. This is corroborated further by the structural signatures of folding in the distributions for radius of gyration and the number of native contacts as a function of temperature. The potentials of mean force are also calculated for these variables, both separately and jointly. A local free energy minimum, in addition to the global minimum, is found in a temperature range substantially below the folding temperature. The free energy at this second minimum is approximately 5 k(B)T higher than the value at the global minimum.