Smoluchowski dynamics of the vnd/NK‐2 homeodomain from Drosophila melanogaster: First‐order mode‐coupling approximation

This work is the first in a series devoted to applying mode coupling diffusion theory to the derivation of local dynamics properties of proteins in solution. The first‐order mode‐coupling approximation, or optimized Rouse–Zimm local dynamics (ORZLD), is applied here to derive the rotational dynamics of the bonds and compare the calculated with the experimental nmr ¹⁵N spin–lattice relaxation time behavior of the vnd/NK‐2 homeodomain from Drosophila melanogaster. The starting point for the calculations is the experimental three‐dimensional structure of the homeodomain determined by multidimensional nmr spectroscopy. The results of the computations are compared with experimentally measured ¹⁵N spin–lattice relaxation times T₁, at 34.5 and 60.8 MHz, to check the first‐order approximation. To estimate the relative importance of internal and overall rotation, both rigid and fluctuating dynamic models are examined, with fluctuations evaluated using molecular dynamics (MD) simulations. The correlation times for the fundamental bond vector time correlation function and for the second‐order bond orientational TCF are obtained as a function of the residue number for vnd/NK‐2. The stability of the corresponding local dynamics pattern for the fluctuating structure as a function of the length of the MD trajectory is presented. Diffusive dynamics, which is essentially free of model parameters even at first order in the mode‐coupling diffusion approach, confirm that local dynamics of proteins can be described in terms of rotational diffusion of a fluctuating quasi‐rigid structure. The comparison with the nmr data shows that the first‐order mode coupling diffusion approximation accounts for the correct order of magnitude of the results and of important qualitative aspects of the data sensitive to conformational changes. Indications are obtained from this study to efficiently extend the theory to higher order in the mode‐coupling expansion. These results demonstrate the promise of the mode‐coupling approach, where the local dynamics of proteins is described in terms of rotational diffusion of a fluctuating quasi‐rigid structure, to analyze nmr spin–lattice relaxation behavior. © 1999 John Wiley & Sons, Inc. Biopoly 49: 235–254, 1999     

Mode‐coupling Smoluchowski dynamics of a double‐stranded DNA oligomer

The local dynamics of a double‐stranded DNA d(TpCpGpCpG)₂ is obtained to second order in the mode‐coupling expansion of the Smoluchowski diffusion theory. The time correlation functions of bond variables are derived and the ¹³C‐nmr spin–lattice relaxation times T₁ of different ¹³C along the chains are calculated and compared to experimental data from the literature at three frequencies. The DNA is considered as a fluctuating three‐dimensional structure undergoing rotational diffusion. The fluctuations are evaluated using molecular dynamics simulations, with the ensemble averages approximated by time averages along a trajectory of length 1 ns. Any technique for sampling the configurational space can be used as an alternative. For a fluctuating three‐dimensional (3D) structure using the three first‐order vector modes of lower rates, higher order basis sets of second‐rank tensor are built to give the required mode coupling dynamics. Second‐ and even first‐order theories are found to be in close agreement with the experimental results, especially at high frequency, where the differences in T₁ for ¹³C in the base pairs, sugar, and backbone are well described. These atomistic calculations are of general application for studying, on a molecular basis, the local dynamics of fluctuating 3D structures such as double‐helix DNA fragments, proteins, and protein–DNA complexes. © 1999 John Wiley & Sons, Inc. Biopoly 50: 613–629, 1999     

Diffusive Dynamics in a Detailed Potential: Application to Biological Macromolecules

Abstract

The local dynamics of macromolecules is obtained to second-order in the mode-coupling expansion of the Smoluchowski diffusion theory. The NMR spin-lattice relaxation times of different ¹³C or ¹⁵N nuclei along the chains are calculated and compared to experimental data from the literature. The macromolecules are considered as fluctuating 3D structures undergoing rotational diffusion. The fluctuations can be evaluated with any technique for sampling the configurational space. In the presented test cases Molecular Dynamics simulations have been applied to a DNA fragment and to the NK-2 homeodomain. In the case of the double-stranded DNA fragment d(TpCpGpCpG)₂, second and even first order theories are found to be in close agreement with experimental results. The major advantage of the diffusion technique is that only a good statistics is important as input while the solvent dynamic effects enter through hydrodynamic theory. Application based on Hybrid Monte Carlo schemes coupled with J-walking, are now in progress.     

Solution structure and backbone dynamics of the defunct domain of calcium vector protein.

CaVP (calcium vector protein) is a Ca(2+) sensor of the EF-hand protein family which is highly abundant in the muscle of Amphioxus. Its three-dimensional structure is not known, but according to the sequence analysis, the protein is composed of two domains, each containing a pair of EF-hand motifs. We determined recently the solution structure of the C-terminal domain (Trp81-Ser161) and characterized the large conformational and dynamic changes induced by Ca(2+) binding. In contrast, the N-terminal domain (Ala1-Asp86) has lost the capacity to bind the metal ion due to critical mutations and insertions in the two calcium loops. In this paper, we report the solution structure of the N-terminal domain and its backbone dynamics based on NMR spectroscopy, nuclear relaxation, and molecular modeling. The well-resolved three-dimensional structure is typical of a pair of EF-hand motifs, joined together by a short antiparallel beta-sheet. The tertiary arrangement of the two EF-hands results in a closed-type conformation, with near-antiparallel alpha-helices, similar to other EF-hand pairs in the absence of calcium ions. To characterize the internal dynamics of the protein, we measured the (15)N nuclear relaxation rates and the heteronuclear NOE effect in (15)N-labeled N-CaVP at a magnetic field of 11.74 T and 298 K. The domain is mainly monomeric in solution and undergoes an isotropic Brownian rotational diffusion with a correlation time of 7.1 ns, in good agreement with the fluorescence anisotropy decay measurements. Data analysis using a model-free procedure showed that the amide backbone groups in the alpha-helices and beta-strands undergo highly restricted movements on a picosecond to nanosecond time scale. The amide groups in Ca(2+) binding loops and in the linker fragment also display rapid fluctuations with slightly increased amplitudes.     

Backbone motions in a crystalline protein from field-dependent 2H-NMR relaxation and line-shape analysis

We have used 2H-nmr to study backbone dynamics of the 2H-labeled, slowly exchanging amide sites of fully hydrated, crystalline hen egg white lysozyme. Order parameters are determined from the residual quadrupole coupling and values increase from S2 = 0.85 at 290 K to S2 = 0.94 at 200 K. Dynamical rates are determined from spin-lattice relaxation at three nmr frequencies (38.8, 61.5, and 76.7 MHz). The approach used here is thus distinct from solution nmr studies where dynamical amplitudes and rates are both determined from relaxation measurements. At temperatures below 250 K, relaxation is independent of the nmr frequency indicating that backbone motions are fast compared to the nmr frequencies. However, as the temperature is increased above 250 K, relaxation is significantly more efficient at the lowest frequency, which shows, in addition, the presence of motions that are slow compared to the nmr frequencies. Using the values of S2 determined from the residual quadrupole coupling and a model-free relaxation formalism that allows for fast and slow internal motions, we conclude that these slow motions have correlation times in the range of 0.1 to 1.0 microsecond and are effectively frozen out at 250 K where fast motions of the amide planes with approximately 15 ps effective correlation times and 9 degrees rms amplitudes dominate relaxation. The fast internal motions increase slightly in amplitude as the temperature rises toward 290 K, but the correlation time, as is also observed in solution nmr studies of RNase H, is approximately constant. These findings are consistent with hypotheses of dynamic glass transitions in hydrated proteins arising from temperature-dependent damping of harmonic modes of motion above the transition point.     

Configurational and dynamic properties of different length superhelical DNAs measured by dynamic light scattering

The solution conformation and internal motions of five superhelical DNAs between 2100 and 10200 base-pairs in length have been characterized by dynamic light scattering (DLS). Variations in the diffusion coefficients and rotational relaxation times with molecular weight are both indicative of an anisotropic extended structure of these DNAs; we therefore conclude that under our conditions the interwound superhelical structure prevails. The internal dynamics can be described by a superposition of rotational diffusion and internal relaxation. The latter process is characterized by the internal diffusion of persistence length size segments within the DNA chain and faster bending motions within these segments.     

Diffusion coefficients of segmentally flexible macromolecules with two subunits: a study of broken rods

A general treatment for the solution dynamics of segmentally flexible macromolecules having two subunits is presented. Bead modeling allows for a complete inclusion of hydrodynamic interactions in this treatment. The finite size of the beads is also considered, so that it is therefore possible to account properly for torsional motions of the subunits. Expressions for the components of the resistance matrix are derived. From them, the translational and rotational diffusion coefficients can be calculated. Distinction is made between hinged macromolecules, whose only internal motion is bending, and swivel-jointed macromolecules, for which torsions of the subunits are also allowed. Numerical results are presented for broken rods with the two types of flexibility. The effects of hydrodynamic interaction between arms of broken rods are about 25% for translation and under 10% for rotation. These findings give support to the treatments of Harvey, Wegener, and co-workers in which interactions were neglected. The rotational dynamics of hinged and swivel-jointed rods are compared. Although there are differences in the short-time behavior, the longest relaxation time is the same for the two cases. Finally, the validity of Wegener's rotational diffusion constants is discussed.     

Internal dynamics of tRNA(Phe) studied by depolarized dynamic light scattering

The collective internal dynamics of transfer RNA(Phe) from brewer's yeast in solution was studied by depolarized dynamic light scattering (DDLS). Within the melting region of tRNA the depolarized spectra consist of two Lorentzian, where the narrow (slow) component describes the overall rotation of the macromolecule. The broad component is attributed to the collective reorientation of the bases within the biopolymer. At high temperature only this relaxation process is observed in the spectrum. The viscosity dependence of the collective internal relaxation process is described by the Stokes-Einstein-Debye equation for rotational diffusion. Estimates of the internal orientational pair correlation factor from the integral depolarized intensities of tRNA(Phe) solutions indicates that the observed dynamics correspond to the collective reorientation of approximately 5 bases. A comparison of the results presented with DDLS studies on the aggregation of the mononucleotide guanosine-5'-monophosphate confirms this result. For a further characterization of the relaxation process we studied the effect of hydrostatic pressure (1-1000 bar) on the depolarized spectra of tRNA. While other spectroscopic methods like nmr, fluorescence polarization anisotropy decay, or ESR give information about the very local motion of a single base within the DNA or RNA, this study shows that by DDLS one can characterize collective internal motions of macromolecules.     

Dynamics of vibrational frequency fluctuations in deuterated liquid ammonia: roles of fluctuating hydrogen bonds and free ND modes

We present an ab initio molecular dynamics study of the roles of fluctuating hydrogen bonds and free ND modes in the dynamics of ND stretch frequency fluctuations in deuterated liquid ammonia. We have also looked at some of the other dynamical quantities such as diffusion and orientational relaxation and also structural quantities such as pair correlations and hydrogen bonding properties which are relevant in the current context. The time correlation function of ND stretch frequencies is found to decay with primarily two time scales: A short-time decay with a time scale of less than 100 fs arising from intermolecular motion of intact hydrogen bonds and also from fast hydrogen bond breaking and a longer time scale of about 500 fs which can be assigned to the lifetime of free ND modes. Unlike water, in liquid ammonia an ND mode is found to remain free for a longer period than it stays hydrogen bonded and this longer lifetime of free ND modes determines the long-time behaviour of frequency fluctuations. Our hole dynamics calculations produced results of vibrational spectral diffusion that are similar to the decay of frequency time correlation. Inclusion of dispersion corrections is found to make the dynamics slightly faster.     

Weak self-association of human growth hormone investigated by nitrogen-15 NMR relaxation

The self-association of human growth hormone(hGH) was investigated using 15N NMR relaxation.The investigation relies on the 15N R1 and R2 relaxation rates and the heteronuclear{1H}-15N NOEs of the backbone amide groups at multiple protein concentrations. It is shown that the rotational correlation time of hGH in solution depends strongly on its concentration, indicating a significant degree of self-association.The self-association is reversible and the monomers in the aggregates are noncovalently linked. Extrapolation of the relaxation data to zero concentration predicts a correlation time of 13.4 ns and a rotational diffusion anisotropy of 1.26 for monomeric hGH, in agreement with the rotational diffusion properties estimated by hydrodynamic calculations. Moreover, the extrapolation allows characterization of the backbone dynamics of monomeric hGH without interference from self-association phenomena, and it is found that hGH is considerably more flexible than originally thought. A concerted least-squares analysis of the 15N relaxations and their concentration dependence reveals that the self-association goes beyond a simple monomer-dimer equilibrium, and that tetramers or other multimeric states co-exist in fast exchange with the monomeric and dimeric hGH at sub-millimolar concentrations. Small changes in the 1H and 15N amide chemical shifts suggest that a region around the C-terminus is involved in the oligomer formation.     

Sampling of protein dynamics in nanosecond time scale by 15N NMR relaxation and self-diffusion measurements.

This paper presents a procedure for detection of intermediate nanosecond internal dynamics in globular proteins. The procedure uses 1H-15N relaxation measurements at several spectrometer frequencies and hydrodynamic calculations based on experimental self-diffusion coefficients. New heteronuclear experiments, using pulse field gradients, are introduced for the measurement of translation diffusion coefficients of 15N labeled proteins. An advanced interpretation of recently published (Luginbühl et al., Biochemistry, 36, 7305-7312 (1997)) backbone amide 15N relaxation data, measured at two spectrometers (400 and 750 MHz for 1H) for N-terminal DNA-binding domain (1-63) of 434 repressor, is presented. Non-applicability of commonly used fast (picosecond) dynamics model (FD) was justified by (i) poor fit of relaxation data by the FD model-free spectral density function both for isotropic and anisotropic models of the overall molecular tumbling; (ii) specific dependence of the overall rotation correlation times calculated from T1/T2 ratio on the spectrometer frequency; (iii) mismatch of the ratio of longitudinal 15N relaxation times T1, measured at different spectrometer frequencies, in comparison with that anticipated for the FD model; (iv) significantly underestimated overall rotation correlation time provided by the FD model (5.50+/-0.15 and 5.80+/-0.15 ns for 750 and 400 MHz spectrometer frequency respectively) in comparison with correlation time obtained from hydrodynamics. On the other hand, all relaxation and hydrodynamics data are in good correspondence with the model of intermediate (nanoseconds) dynamics. Overall rotation correlation time of 7.5+/-0.7 ns was calculated from experimental translation self-diffusion rate using hydrodynamics formalism (Garcia de la Torre, J. and Bloomfield, V.A. Quart. Rev. Biophys., 14, 81-139 (1981)). The statistical analysis of 15N relaxation data along with the hydrodynamic consideration clearly revealed that most of the residues in 434(1-63) repressor are involved in the nanosecond internal dynamics characterized by the the mean order parameters of 0.59+/-0.06 and the correlation times of ca. 5 ns.     

The use of the AMBER force field in conformational analysis of carbohydrate molecules: Determination of the solution conformation of methyl α-lactoside by NMR spectroscopy,assisted by molecular mechanics and dynamics calculations

The solution conformation of methyl α-lactoside has been studied through nmr spectroscopy and molecular mechanics calculations using the assisted model building with energy refinement (AMBER) force field. The nmr data have included nuclear Overhauser effect (NOE) measurements hot It in the laboratory and rotating frames, longitudinal relaxation times, and homonuclear and heteronuclear coupling constants. The steady-state and transient NOEs have been interpreted in terms of an ensemble average distribution of conformers, making use of the complete relaxation matrix approach. The molecular mechanics calculations have been performed at two dielectric constants [ε = 1 * r and 80 Debyes (D)] in an exhaustive way, and have been complemented with specific calculations at intermediate ε values. Relaxed energy maps and adiabatic surfaces have been generated for the different dielectric constants. The probability distribution of conformers has been estimated from these steric energy maps. Molecular dynamics simulations in vacuo have also been performed. The experimental results indicate that the β(1 → 4)-glycosidic linkage shows some fluctuations among three low energy regions, although spends ca. 85% of its lime in the region close to the global minimum. It is shown that the over estimation of the electrostatic contributions in AMBER is responsible for the failure of this force field to explain the experimental results when Used at low dielectric constant (ε < 20 D). The matching between the expected and observed facts increases for ε > 40 D. Different conditions have been tested to perform temperature constant molecular dynamics simulations in vacuo, which have indicated that, when used without explicit solvent, this force field should only be employed in a qualitatively way when analyzing dynamical properties of oligosaccharides. © 1995 John Wiley & Sons, Inc.     

Hydrodynamics of segmentally flexible macromolecules

Segmentally flexible macromolecules are composed of a few rigid subunits linked by joints which are more or less flexible. The dynamics in solution of this type of macromolecule present special aspects that are reviewed here. Three alternative approaches are described. One is the rigid-body treatment, which is shown to be valid for overall dynamic properties such as translational diffusion and intrinsic viscosity. Another approach is the Harvey-Wegener treatment, which is particularly suited for rotational diffusion. The simplest version of this treatment, which ignores hydrodynamic interaction (HI) effects, is found to be quite accurate when compared to a more rigorous version including HI. A third approach is the Brownian dynamics simulation that, albeit at some computational cost, might describe rigorously cases of arbitrary complexity. This technique has been used to test the approximations in the rigid-body and Harvey-Wegener treatments, thus allowing a better understanding of their validity. Brownian trajectories of simplified models such as the trumbbell and the broken rod have been simulated. The comparison of the decay rates of some correlation functions with the predictions of the two treatments leads to a general conclusion: the Harvey-Wegener treatment determines the initial rate, while the long-time behavior is dominated by the rigid-body relaxation time. As an example of application to a specific biological macromolecule, we present a simulation of an immunoglobulin molecule, showing how Brownian Dynamics can be used to predict rotational and internal dynamics. Another typical example is myosin. Literature data of hydrodynamic properties of whole myosin and the myosin rod are compared with predictions from the Harvey-Wegener and rigid-body treatments. The present situation of the problem on myosin flexibility is analyzed, and some indications are given for future experimental and simulation work.     

Rotational diffusion of short DNA fragments in polyacrylamide gels: an electric birefringence study

Using electric birefringence we have examined the rotational diffusion of five short DNA fragments (55 to 256 base pairs) both in polyacrylamide gels as a function of gel concentration and in solution. The length dependence of the measured rotational relaxation times in the gels is in good agreement with the prediction from the Odijk theory for the dynamics of slightly flexible rods in a network. The rotational relaxation times were found to depend on the gel concentration, contrary to the prediction from the Odijk theory. Possible reasons for this observation are discussed. The birefringence decay curves for DNA fragments in the gel were single exponential only at small electric field strength.     

The measurement of the rotational diffusion coefficient of bovine plasma fibronectin by electric birefringence technique

Although the conformation of fibronectin has been widely investigated by various techniques, there has not yet been any determination of its rotational diffusion coefficient. We report here this determination by the transient electric birefringence study of solutions of bovine plasma fibronectin at physiological ionic strength. The solutions showed a positive birefringence. A linear relationship was observed between the intensity of the birefringence at equilibrium and the square of the electric field within the range of fields applied (up to 12.5 kV.cm-1). The field-independent decay of the induced birefringence was described by a single exponential with a relaxation time of 0.76 (+/- 0.08) microsecond at 23 degrees C. This establishes fibronectin in solution as a globally rigid structure with a rotational diffusion coefficient, at 20 degrees C, of 202,000 s-1. This result allows the first rigorous determination of the low-resolution structure of fibronectin. It is important to notice that the analysis combines only results obtained in physiological conditions on native molecules and follows a strict hydrodynamic interpretation. The conclusion of this work is that a hollow sphere of about 20 nm external diameter can be proposed as a model for the three-dimensional structure of the fibronectin molecule in solution. This new model suggests the fibronectin could have the structure of a carrier protein.     

Sequence-dependent elastic properties of DNA

Harmonic elastic constants of 3-11 bp duplex DNA fragments were evaluated using four 5 ns unrestrained molecular dynamics simulation trajectories of 17 bp duplexes with explicit inclusion of solvent and counterions. All simulations were carried out with the Cornell et al. force-field and particle mesh Ewald method for long-range electrostatic interactions. The elastic constants including anisotropic bending and all coupling terms were derived by analyzing the correlations of fluctuations of structural properties along the trajectories. The following sequences have been considered: homopolymer d(ApA)(n) and d(GpG)(n), and alternating d(GPC)(n) and d(APT)(n). The calculated values of elastic constants are in very good overall agreement with experimental values for random sequences. The atomic-resolution molecular dynamics approach, however, reveals a pronounced sequence-dependence of the stretching and torsional rigidity of DNA, while sequence-dependence of the bending rigidity is smaller for the sequences considered. The earlier predicted twist-bend coupling emerged as the most important cross-term for fragments shorter than one helical turn. The calculated hydrodynamic relaxation times suggest that damping of bending motions may play a role in molecular dynamics simulations of long DNA fragments. A comparison of elasticity calculations using global and local helicoidal analyses is reported. The calculations reveal the importance of the fragment length definition. The present work shows that large-scale molecular dynamics simulations represent a unique source of data to study various aspects of DNA elasticity including its sequence-dependence.     

Transport properties and hydrodynamic centers of rigid macromolecules with arbitrary shapes

A general formalism, which includes translation–rotation coupling, is proposed for calculating translational and rotational transport properties, as well as intrinsic viscosities, of rigid macromolecules with an arbitrary shape. This formalism is based on Brenner's theory of translational–rotational dynamics and on methods for the calculation of hydrodynamic properties that have been already presented, and can be regarded as a generalization of the one proposed by Nakajima and Wada. The calculated transport properties depend on the origin as predicted by Brenner's theory, but in a disagreement with him, the center of resistance and the center of diffusion do not coincide. As one can define several hydrodynamic centers, which in practice turn out to be located at different points, the influence of the choice of the center on the calculated transport properties is discussed. An analysis of the translation–rotation coupling effects in translational diffusion reveals that they arise exclusively from hydrodynamic interactions and are rather small in some cases of interest. Finally, we present a study of the rotational diffusion of rigid bent rods with a fixed length-to-diameter ratio. The diffusion coefficients obtained can be useful to estimate changes with respect to a straight rod.     

Molecular dynamics analysis of the engrailed homeodomain-DNA recognition

Molecular dynamics (MD) simulations were performed for investigating the role of Gln50 in the engrailed homeodomain-DNA recognition. Employing the crystal structure of free engrailed homeodomain and homeodomain-DNA complex as a starting structure, we carried out MD simulations of: (i) the complex between engrailed homeodomain and a 20 base-pair DNA containing TAATTA core sequence; (ii) the free engrailed homeodomain. The simulations show that homeodomain flexibility does not depend on its ligation state. The engrailed homeodomain shows similar flexibility, and the recognition helix-3 shows very similar characteristic of high rigidity and limited conformational space in two complexation states. At the same time, DNA structure has also no obvious conformational fluctuations. These results preclude the possibility of the side chain of Gln50 forming direct hydrogen bonds to the core DNA bases. MD simulations confirm a few well-conserved sites for water-mediated hydrogen bonds from protein to DNA are occupied by water molecules, and Gln50 interacts with corresponding core DNA bases through water-mediated hydrogen bonds. So Gln50 plays a relatively modest role in determining the affinity and specificity of the engrailed homeodomain. In addition, the electrostatic interaction between homeodomain and phosphate backbone of the DNA is a main factor for N- and C-terminal arm becoming ordered upon DNA binding.