Efficient identification and analysis of substructures in proteins using the kappa-tau framework: left turns and helix c-cap motifs.

We have recently (1) introduced an exact mathematical description of chain molecules based on discrete analogues of the curvature and torsion concepts in differential geometry called the kappa-tau framework, and applied it to the analysis of all known protein structures. We here use it to extract and analyze defined structural motifs from the Protein Data Bank. We find a class of left handed 4 residues long, open turns, tentatively called left alpha-tums, which do not exhibit internal backbone H-bonding and join both alpha and beta1 strands at the surface of many different proteins. Further, we discuss results on helix capping substructures such as the Schellman and alpha(L) motifs. The kappa-tau methodology proves to be the most efficient methodology to handle such problems available to date, and will greatly expand our capability to decipher the information hidden in the known structures and understand the fundamental relation between sequence and structure.

Potential of mean force treatment of salt-mediated protein crystallization.

In the initial stages of crystallization of proteins, monomers aggregate rapidly and form nuclei and large fractal clusters, as previously shown by dynamic light scattering experiments (Georgalis, Y., J. Schüler, J. Frank, D. M. Soumpasis, and W. Saenger. 1995. Protein crystallization screening through scattering techniques. Adv. Colloid Interface Sci. 58:57-86). In this communication we initiate an effort to understand the effective interactions controlling charged protein aggregation and crystallization using the potential of mean force (PMF) theory. We compute the PMFs of the system lysozyme-water-NaCl within the framework of the hypernetted chain approximation for a wide range of protein and salt concentrations. We show that the computed effective interactions can rationalize the experimentally observed aggregation behavior of lysozyme under crystallization conditions.

Hydration of an alpha-helical peptide: comparison of theory and molecular dynamics simulation.

We present a statistical mechanical description of biomolecular hydration that accurately describes the hydrophobic and hydrophilic hydration of a model alpha-helical peptide. The local density of water molecules around a biomolecule is obtained by means of a potential-of-mean-force (PMF) expansion in terms of pair- and triplet-correlation functions of bulk water and dilute solutions of nonpolar atoms. The accuracy of the method is verified by comparing PMF results with the local density and site-site correlation functions obtained by molecular dynamics simulations of a model alpha-helix in solution. The PMF approach quantitatively reproduces all features of the peptide hydration determined from the molecular dynamics simulation. Regions of hydrophobic hydration near the C alpha and C beta atoms along the helix are well reproduced. The hydration of exposed polar groups at the N- and C-termini of the helix are also well described by the theory. A detailed comparison of the local hydration by means of site-site radial distribution functions evaluated with the PMF theory shows agreement with the molecular dynamics simulations. The formulation of this theory is general and can be applied to any biomolecular system. The accuracy, speed of computation, and local character of this theory make it especially suitable for studying large biomolecular systems.

Electrostatics of a simple membrane model using Green's functions formalism.

The electrostatics of a simple membrane model picturing a lipid bilayer as a low dielectric constant slab immersed in a homogeneous medium of high dielectric constant (water) can be accurately computed using the exact Green's functions obtainable for this geometry. We present an extensive discussion of the analysis and numerical aspects of the problem and apply the formalism and algorithms developed to the computation of the energy profiles of a test charge (e.g., ion) across the bilayer and a molecular model of the acetylcholine receptor channel embedded in it. The Green's function approach is a very convenient tool for the computer simulation of ionic transport across membrane channels and other membrane problems where a good and computationally efficient first-order treatment of dielectric polarization effects is crucial.

Structural prediction of A- and B-DNA duplexes based on coordinates of the phosphorus atoms.

The sequence-dependent structure of DNA double helices was studied extensively during the past 10 years. How the backbone structure correlates with the base structure in a duplex conformation is still an important yet open question. Using a set of reduced coordinates and a least-squares fitting procedure, we have developed a method to predict structures for B-DNA duplexes based on coordinates of the phosphorus atoms. This method can be used to predict all-atom structures for both bent and straight molecules. We estimated the accuracies of the predictions by studying a set of 10 oligonucleotides with their structures available from the Protein Data Bank. We used this method to construct a modeled structure for the bacteriophage lambda cro operator for which the phosphorus coordinates were known from 3.5-angstrum resolution crystal data (4CRO).

Theoretical description of biomolecular hydration. Application to A-DNA.

The local density of water molecules around a biomolecule is constructed from calculated two- and three-points correlation functions of polar solvents in water using a Potential-of-Mean-Force (PMF) expansion. As a simple approximation, the hydration of all polar (including charged) groups in a biomolecule is represented by the hydration of water oxygen in bulk water, and the effect of non-polar groups on hydration are neglected, except for excluded volume effects. Pair and triplet correlation functions are calculated by molecular dynamics simulations. We present calculations of the structural hydration for ideal A-DNA molecules with sequences [d(CG)5]2 and [d(C5G5)]2. We find that this method can accurately reproduce the hydration patterns of A-DNA observed in neutron diffraction experiments on oriented DNA fibers (P. Langan et al. J. Biomol. Struct. Dyn., 10, 489 (1992)).

A statistical mechanical description of biomolecular hydration.

An efficient and accurate theoretical description of the structural hydration of biological macromolecules is presented. The hydration of molecules of almost arbitrary size (tRNA, antibody-antigen complexes, photosynthetic reaction centre) can be studied in solution and in the crystalline environment. The biomolecular structure obtained from X-ray crystallography, NMR or modelling is required as input information. The structural arrangement of water molecules near a biomolecular surface is represented by the local water density, analogous to the corresponding electron density in an X-ray diffraction experiment. The water-density distribution is approximated in terms of two- and three-particle correlation functions of solute atoms with water using a potentials-of-mean-force expansion.

The construction of DNA helical duplexes along prescribed 3-D curves.

Curved DNA structures can be either intrinsic or induced through interaction with proteins. Large DNA molecules can adopt complex curved shapes in space. Many curved DNA structures (e.g., Cro-operator, Cap-operator, nucleosomal DNA, etc.) are believed to have important biological functions. To model curved DNA molecules is a challenging task. In this work, we introduce a method for the computer generation of DNA structures having any prescribed 3-D shape in space. The approach is purely geometrical and highly efficient. This method is used successfully to construct an atomic level nucleosomal DNA that is consistent with the experimental data. The smallest closed DNA circle we are able to construct with the method is a 51 bp DNA duplex.

Hydration of nucleic acid fragments: comparison of theory and experiment for high-resolution crystal structures of RNA, DNA, and DNA-drug complexes.

A computationally efficient method to describe the organization of water around solvated biomolecules is presented. It is based on a statistical mechanical expression for the water-density distribution in terms of particle correlation functions. The method is applied to analyze the hydration of small nucleic acid molecules in the crystal environment, for which high-resolution x-ray crystal structures have been reported. Results for RNA [r(ApU).r(ApU)] and DNA [d(CpG).d(CpG) in Z form and with parallel strand orientation] and for DNA-drug complexes [d(CpG).d(CpG) with the drug proflavine intercalated] are described. A detailed comparison of theoretical and experimental data shows positional agreement for the experimentally observed water sites. The presented method can be used for refinement of the water structure in x-ray crystallography, hydration analysis of nuclear magnetic resonance structures, and theoretical modeling of biological macromolecules such as molecular docking studies. The speed of the computations allows hydration analyses of molecules of almost arbitrary size (tRNA, protein-nucleic acid complexes, etc.) in the crystal environment and in aqueous solution.

Dynamics and relative stabilities of parallel- and antiparallel-stranded DNA duplexes.

The dynamics and stability of four DNA duplexes are studied by means of molecular dynamics simulations. The four molecules studied are combinations of 4, 15 bases long, single-stranded oligomers, F1, F2, F3, and F4. The sequence of these single strand oligomers are chosen such that F1-F2 and F3-F4 form parallel (ps) DNA double helices, whereas F1-F4 and F2-F3 form antiparallel-stranded (aps) DNA double helices. Simulations were done at low (100 K) and room (300 K) temperatures. At low temperatures the dynamics are quasi-harmonic and the analysis of the trajectories gives good estimates of the low frequency vibrational modes and density of states. These are used to estimate the linear (harmonic) contribution of local fluctuations to the configurational entropy of the systems. Estimates of the differences in enthalpy between ps and aps duplexes show that aps double helices are more stable than the corresponding ps duplexes, in agreement with experiments. At higher temperatures, the distribution of the fluctuations around the average structures are multimodal and estimates of the configurational entropy cannot be obtained. The multi-basin, nonlinear character of the dynamics at 300 K is established using a novel method which extracts large amplitude nonlinear motions from the molecular dynamics trajectories. Our analysis shows that both ps DNA exhibit much larger fluctuations than the two aps DNA. The large fluctuations of ps DNA are explained in terms of correlated transitions in the beta, epsilon, and zeta backbone dihedral angles.

An extension of the rigorous base-unit oriented description of nucleic acid structures.

Our proposed description for DNA base/base-pair structures (1), though rigorous, does not satisfy some of the requirements as established at the Cambridge Workshop (2). Here, we propose a revised description for base/base-unit structures of nucleic acids. This new description is as rigorous and satisfies all the requirements (2). Following the original approach, the moment-of-inertia frame is still the choice of the internal coordinate system for a base/base-unit. The revised description has the minimum number of parameters (i.e., six parameters per rigid body) in the set. Besides regular Watson-Crick type of helices (e.g., A-DNA, A-RNA, B-DNA, Z-DNA, etc.), the revised description also works for non-Watson-Crick, multiple stranded molecules (e.g., triplex, quadruplex, etc.) as well as parallel stranded molecules.

Computation of ionic distributions around charged biomolecular structures: results for right-handed and left-handed DNA.

We introduce an efficient computational methodology employing the potentials of mean force approach for estimating the detailed three-dimensional ionic distributions around arbitrarily complex charged biomolecular structures for all monovalent salt concentrations of practical interest (e.g., 0.1-5.0 M NaCl). Such distributions are required for specifying thermodynamic and structure-specific features of ion-mediated interactions of charged proteins, DNA and RNA, membranes, and macromolecular assemblies. As a first application, we present results for distributions around the B and ZI conformers of the DNA oligomer d(C-G)18.d(C-G)18. The ionic microenvironment depends strongly on the DNA conformation, sequence, and bulk salt concentrations.

Rigorous description of DNA structures. II. On the computation of best axes, planes, and helices from atomic coordinates.

We show that diagonalization of the moments of inertia tensor of an arbitrary molecular structure automatically yields the best (least squares) points, line, and plane through the structure. Furthermore, we show how to compute best helices using atomic coordinates. These results supplement the general methodology for description of DNA structures proposed recently (D.M. Soumpasis and C.-S. Tung, J. Biomol. Struct. Dyn. 6, 397-420, 1988).

Energetics of the hairpin to mismatched duplex transition of d(GCCGCAGC) on NaCl solution.

We report Potential of Mean Force studies to describe the relative thermodynamic stabilities of d(GCCGCAGC) in a mismatched duplex and a hairpin monomer conformation in NaCl solution. The PMF calculations are combined with previous molecular mechanics and normal mode analysis in order to estimate the role of different components of the free energy in determining the relative stability of the duplex and hairpin structures. The high entropy associated with the loop region and the lack of minor groove phosphate-phosphate interactions in the hairpin compete against the gain in enthalpic contribution to the free energy due to base pairing in the mismatched duplex. The combined free energy calculations show that the hairpin is the most stable conformation at low salt and that a hairpin to duplex transition takes place at approximately 0.47 M NaCl. In addition, we studied the hairpin to partially stacked single helical conformation equilibrium at low salt. We found a small variation in transition temperature in salt concentration, delta Tm/delta log10(cs) approximately 2-3 degrees K/decade, in contrast to the duplex to hairpin or duplex to partially stacked single helix transition where the transition temperature exhibited marked dependence on salt concentration. This is in qualitative agreement with experimental data. Based on the Potential of Mean Force free energy calculation, the order of relative stability of the three-conformations studied varies with salt concentration. We observed the following orders of stability: stacked single helix greater than hairpin greater than duplex for cs less than 0.77 M NaCl; single helix greater than duplex greater than hairpin for 0.77 less than Cs less than 2.1 M; and duplex greater than hairpin greater than single strand for cs greater than 2.1 M. From the calculated PMF free energy curves in the NaCl concentration range, 0.012 less than cs less than 5.0 M, we can assign upper and lower bounds for the non-ionic differences in free energy between the duplex, hairpin, and stacked single helical states (at standard conditions: cs = 1.0 M, T = 25 degrees C, and 1 M oligomer concentration). We found that for delta G duplex single helix = G duplex - 2 x G single helix less than -7.38 Kcal/mol, the single helix is the least stable state. For the duplex-to-hairpin free energy difference in the range, -1.87 less than delta G duplex-hairpin less than 0.03 Kcal/mol, there will always be a salt-induced hairpin-to-duplex transition for 0.01 less than cs less than 1.6 M NaCl. If delta G duplex-hairpin less than -1.87, the duplex is always more stable than the hairpin; and for delta G duplex-hairpin greater than Kcal/mol, the hairpin state is always more stable than the duplex, for all salt concentrations.

Inclusion of ionic interactions in force field calculations of charged biomolecules--DNA structural transitions.

The potential of mean force (PMF) approach for treating polyion-diffuse ionic cloud interactions [D. M. Soumpasis (1984) Proceedings of the National Academy of Sciences USA 81, 5116-5120] has been combined with the AMBER force field describing intramolecular interactions. The resultant generalized AMBER-PMF force field enables one to treat the conformational stabilities and structural transitions of charged biomolecules in aqueous electrolytes more realistically. For example, we have used it to calculate the relative stabilities of the B and Z conformations of d(C-G)6, and the B and heteronomous (H) conformations of dA12.dT12, as a function of salt concentration. In the case of d(C-G)6, the predicted B-ZI transition occurs at 2.4M and is essentially driven by the phosphate-diffuse ionic cloud interactions alone as suggested by the results of earlier PMF calculations. The ZII conformer is less stable than the B form under all conditions. It is found that the helical parameters of the refined B and Z structures change with salt concentration. For example, the helical rise of B-DNA increases about 10% and the twist angle decreases by the same amount above 1M NaCl. In the range of 0.01-0.3M NaCl, the H form of dA12.dT12 is found to be more stable than the B form and its stability increases with increasing salt concentration. The computed greater relative stability of the H conformation is likely due to noninclusion of the free energy contribution from the spine of hydration, a feature presumed to stabilize the B form of this sequence.

Harmonic vibrations and thermodynamic stability of a DNA oligomer in monovalent salt solution.

We compute the full harmonic vibrational spectrum and eigenmodes of a DNA oligomer, d(C-G)3, in optimized B and Z conformations in various ionic environments (0.01-5.0 M NaCl). The statistical interactions of DNA with the diffuse ionic cloud surrounding it in solution are approximately represented within the potential of mean force framework. The lowest eigenfrequency of the B conformation is found to drastically decrease with increased NaCl concentration. This suggests that a soft mode mechanism may be a precursor for the B-to-Z conversion. The free energy balance governing the B-Z isomerization of d(C-G)3 is dominated by the solvent-averaged effective phosphate-phosphate interactions due to substantial cancelations between the much larger intramolecular energy contributions.

A rigorous basepair oriented description of DNA structures.

We propose new, rigorous definitions for (i) basepair fixed coordinate systems and (ii) the twist, tilt, and roll angles (called tau, t, rho) describing the relative orientation of adjacent basepairs and bases in a pair, in arbitrary DNA structures obtained from x-ray diffraction, 2D NMR, or energy calculations. In contrast to the corresponding angular parameters (tg, theta T, theta R) and coordinate systems introduced by Dickerson and co-workers and currently in use, our angular parameters and coordinate systems, together with a set of three displacement parameters, dx, dy, dz, provide a mathematically correct and general description of DNA conformations at the basepairs and/or base level. For instance, our description is applicable when the DNA structure considered is inherently curved, irregular, and/or does not possess dyad (or pseudodyad) axes. We develop a computationally convenient algorithm for rigorous DNA conformational analysis and apply it to some of the known crystal structures. We establish the connection to the currently used parameters and test the consistency and efficiency of our methodology by reconstructing the Dickerson B dodecamer using only the sequence and the set of parameters obtained from the atomic coordinates. The six parameter (tau, t, rho, dx, dy, dz) basepair level reconstruction is good but not perfect. Perfect reconstruction is obtained when one also considers each base in a basepair (consideration of propeller twist alone is not sufficient). The variation of the rigorous parameters proposed along the sequence is much larger, but their average values agree with fiber and solution data much better than in the case of the currently used set. The results of our analysis do not support Trifonov's AA.TT wedge model for DNA curvature but provide some evidence in favor of the Crothers junction-bend model. We point out some of the limitations of basepair level approaches when applied to DNA structure prediction and quantitative understanding of sequence-dependent variations in structure.

Salt dependence of DNA structural stabilities in solution. Theoretical predictions versus experiments.

The predictions of currently available theories for treating DNA-diffuse ionic cloud free energy contributions to conformational stability have been tested against experimental data for salt induced B-Z and B-A transitions. The theories considered are (i) Manning's counterion condensation approach (CC), (ii) the idealized Poisson-Boltzmann approximation (PB), and (iii) the potentials of mean force (PMF) approach proposed by Soumpasis. As far as we can judge from comparison with the set of experimental data currently available, it is found that only the latter theory yields satisfactory quantitative results for the dependence of the B-Z and B-A relative stabilities on monovalent salt concentration. The correct application of the PB and CC theories does not yield very low salt Z-B transitions, in contradiction to earlier assertions. At low salt concentrations the PB theory is qualitatively correct in predicting that the B form is electrostatically more favorable than both the A and Z forms, whereas the CC theory is qualitatively wrong predicting that Z-DNA is more stable than both B and A DNA.

B-Z DNA conformational transition in 1:1 electrolytes: dependence upon counterion size.

We have studied the B-Z transition of poly[d(G-C)] in the presence of alkali metal, tetramethylammonium and tetraethylammonium chlorides at room temperature. The measured critical salt concentrations increase in the order Na, K, Rb, TMA, Cs and are in good agreement with the theoretical values predicted from a statistical-mechanical treatment of the transition.

Relative stabilities and transitions of DNA conformations in 1:1 electrolytes: a theoretical study.

We use a recently developed formalism (1) to calculate the salt dependent part of the free energy determining DNA conformational stability in 1:1 electrolytes. The conformations studied are the A, B, C and alternating-B right-handed forms and the ZI, ZII left-handed forms of DNA. In the case of the B-ZI transition of d(G-C).d(G-C) helices in NaCl solution, the free energy contribution considered suffices to describe the transition in a quantitative manner. The theory also predicts the occurrence of salt-induced B-A transitions which have been recently observed with poly[d(n2 A-T)] and poly[d(G-C)]. In other cases, additional terms in the free energy balance, particularly due to hydration effects, must be at least as important as salt effects in determining conformational stability and structural transitions in solution. If diffuse ionic cloud electrostatic effects alone would dominate in all cases, the relative helical stabilities at 0.2 M monovalent salt would decrease in the order C greater than B greater than A greater than ZII greater than ZI greater than alternating-B. At high salt concentrations (2.0 M-5.0 M), the order would be alternating-B greater than ZI greater than A greater than ZII greater than B greater than C.