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.     

Transport properties of rigid bent-rod macromolecules and of semiflexible broken rods in the rigid-body treatment. Analysis of the flexibility of myosin rod.

The translational diffusion coefficients, rotational relaxation times and intrinsic viscosities of rigid bent rods, composed by two rodlike arms joined rigidly at an angle alpha, have been evaluated for varying conformation using the latest advances in hydrodynamic theory. We have considered semiflexible rods in which the joint is an elastic hinge or swivel, with a potential V(alpha) = 1/2Q alpha 2 with constant Q. Accepting the rigid-body treatment, we calculate properties of broken rods by averaging alpha-dependent values for rigid rods. The results are finally used to interpret literature values of the properties of myosin rod. Q is regarded as an adjustable parameter, and the value fitted is such that the average bending angle of myosin rod is approximately 60 degrees.     

Internal motions in myosin.

High-resolution proton nuclear magnetic resonance (1H NMR) measurements were made on myosin, heavy meromyosin (HMM), myosin subfragment 1 (S1), light meromyosin (LMM), and actin. A strong signal from amino acid side chains undergoing motions too fast to be accounted for by simple rotations of groups on a rigid backbone was obtained from myosin. Comparison of myosin, HMM, S1, and LMM showed that the mobile region is located almost entirely in S1 and accounts for approximately 22% of its structure. Adenosine triphosphate (ATP) and ATP analogues had no measurable effect on the S1 spectrum. Actin, on the other hand, quenched the internal motions of S1. When S1 was titrated with actin, an association was obtained which was in agreement with other measured values. The actin effect was reversed by adding magnesium pyrophosphate (MgPPi) or adenyl-5'-yl imidophosphate (MgAMPPNP). Quantitative treatment of the broad signals from myosin and its subfragments substantiated the existence of two flexible regions in myosin. The highly mobile portion of myosin may be located in the "swivel" between S1 and the rest of myosin or in the actin binding site or in both. These possibilites are discussed, and a new possible mechanism for muscle cross bridge elasticity is proposed.     

Theory of light scattering from hollow spheres

Light-scattering from a dilute, fluid disperson of rigid, hollow spheres is treated in the Rayleigh—Debye approximation. The hollow spheres are composed of cylindrically symmetric, radially oriented scattering elements. Analytic expressions for all form factors are derived. The time correlation function of the scattered intensity depends only on the translational diffusion coefficient of the sphere.     

Large-scale rotational motions of proteins detected by electron paramagnetic resonance and fluorescence.

Direct spectroscopic measurements of rotational motions of proteins and large protein segments are crucial to understanding the molecular dynamics of protein function. Fluorescent probes and spin labels attached to proteins have proved to be powerful tools in the study of large-scale protein motions. Fluorescence depolarization and conventional electron paramagnetic resonance (EPR) are applicable to the study of rotational motions in the nanosecond-to-microsecond time range, and have been used to demonstrate segmental flexibility in an antibody and in myosin. Very slow rotational motions, occurring in the microsecond-to-millisecond time range, are particularly important in supramolecular assemblies, where protein motions are restricted by association with other molecules. Saturation transfer spectroscopy (ST-EPR), a recently developed electron paramagnetic resonance (EPR) technique that permits the detection of rotational correlation times as long as 1 ms, has been used to detect large-scale rotational motions of spin-labeled proteins in muscle filaments and in membranes, providing valuable insights into energy transduction mechanisms in these assemblies.     

The conformation of serum albumin in solution: a combined phosphorescence depolarization-hydrodynamic modeling study

There is a striking disparity between the heart-shaped structure of human serum albumin (HSA) observed in single crystals and the elongated ellipsoid model used for decades to interpret the protein solution hydrodynamics at neutral pH. These two contrasting views could be reconciled if the protein were flexible enough to change its conformation in solution from that found in the crystal. To investigate this possibility we recorded the rotational motions in real time of an erythrosin-bovine serum albumin complex (Er-BSA) over an extended time range, using phosphorescence depolarization techniques. These measurements are consistent with the absence of independent motions of large protein segments in solution, in the time range from nanoseconds to fractions of milliseconds, and give a single rotational correlation time phi(BSA, 1 cP, 20 degrees C) = 40 +/- 2 ns. In addition, we report a detailed analysis of the protein hydrodynamics based on two bead-modeling methods. In the first, BSA was modeled as a triangular prismatic shell with optimized dimensions of 84 x 84 x 84 x 31.5 A, whereas in the second, the atomic-level structure of HSA obtained from crystallographic data was used to build a much more refined rough-shell model. In both cases, the predicted and experimental rotational diffusion rate and other hydrodynamic parameters were in good agreement. Therefore, the overall conformation in neutral solution of BSA, as of HSA, should be rigid, in the sense indicated above, and very similar to the heart-shaped structure observed in HSA crystals.     

Prediction of protein motions from amino acid sequence and its application to protein-protein interaction

Background  

Structural flexibility is an important characteristic of proteins because it is often associated with their function. The movement of a polypeptide segment in a protein can be broken down into two types of motions: internal and external ones. The former is deformation of the segment itself, but the latter involves only rotational and translational motions as a rigid body. Normal Model Analysis (NMA) can derive these two motions, but its application remains limited because it necessitates the gathering of complete structural information.     

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.     

Characterization of the overall and internal dynamics of short oligonucleotides by depolarized dynamic light scattering and NMR relaxation measurements

The dynamics of three synthetic oligonucleotides d(CG)4, d(CG)6, and d(CGCGTTGTTCGCG) of different length and shape were studied in solution by depolarized dynamic light scattering (DDLS) and time-resolved nuclear Overhauser effect cross-relaxation measurements. For cylindrically symmetric molecules the DDLS spectrum is dominated by the rotation of the main symmetry axis of the cylinder. The experimental correlation times describe the rotation of the oligonucleotides under hydrodynamic stick boundary conditions. It is shown that the hydrodynamic theory of Tirado and Garcia de la Torre gives good predictions of the rotational diffusion coefficients of cylindrically symmetric molecules of the small axial ratios studied here. These relations are used to calculate the solution dimensions of the DNA fragments from measured correlation times. The hydrodynamic diameter of the octamer and dodecamer is 20.5 +/- 1.0 A, assuming a rise per base of 3.4 A. The tridecamer, d(CGCGTTGTTCGCG), adopts a hairpin structure with nearly spherical dimensions and a diameter of 23.0 +/- 2.0 A. The DDLS relaxation measurements provide a powerful method for distinguishing between different conformations of the oligonucleotides (e.g., DNA double-helix versus hairpin structure). Furthermore, the rotational correlation times are a very sensitive probe of the length of different fragments. The NMR results reflect the anisotropic motion of the molecules as well as the amount of local internal motion present. The experimental correlation time from NMR is determined by the rotation of both the short and long axes of the oligonucleotide.(ABSTRACT TRUNCATED AT 250 WORDS)     

Diffusion coefficients for rigid macromolecules with irregular shapes that allow rotational-translational coupling

We consider six-dimensional diffusion and frictional tensors for a rigid macromolecule immersed in a viscous fluid at low Reynolds number. Our treatment allows for screwlike properties which couple rotational and translational movements. We show that the center of diffusion of a screwlike body can be distinct from its hydrodynamic center of reaction. Symmetry conditions which ensure coincidence are examined. The center of diffusion is found to be the point of a body with the slowest diffusive movements, while rotations about the center of reaction encounter the least average resistance. The macroscopic translational diffusion coefficient is evaluated from a perturbation analysis of the six-dimensional diffusion equation. We show that methodologies which ignore translational–rotational coupling will necessarily underestimate the diffusion rate of screwlike particles. A procedural framework is presented to calculate diffusion coefficients of complicated bodies. As an example we treat a long bent rod.     

Expression of Dictyostelium myosin tail segments in Escherichia coli: domains required for assembly and phosphorylation

The assembly of myosins into filaments is a property common to all conventional myosins. The ability of myosins to form filaments is conferred by the tail of the large asymmetric molecule. We are studying cloned portions of the Dictyostelium myosin gene expressed in Escherichia coli to investigate functional properties of defined segments of the myosin tail. We have focused on five segments derived from the 68-kD carboxyl-terminus of the myosin tail. These have been expressed and purified to homogeneity from E. coli, and thus the boundaries of each segment within the myosin gene and protein sequence are known. We identified an internal 34-kD segment of the tail, N-LMM-34, which is required and sufficient for assembly. This 287-amino acid domain represents the smallest tail segment purified from any myosin that is capable of forming highly ordered paracrystals characteristic of myosin. Because the assembly of Dictyostelium myosin can be regulated by phosphorylation of the heavy chain, we have studied the in vitro phosphorylation of the expressed tail segments. We have determined which segments are phosphorylated to a high level by a Dictyostelium myosin heavy chain kinase purified from developed cells. While LMM-68, the 68-kD carboxyl terminus of Dictyostelium myosin, or LMM-58, which lacks the 10-kD carboxyl terminus of LMM-68, are phosphorylated to the same extent as purified myosin, subdomains of these segments do not serve as efficient substrates for the kinase. Thus LMM-58 is one minimal substrate for efficient phosphorylation by the myosin heavy chain kinase purified from developed cells. Taken together these results identify two functional domains in Dictyostelium myosin: a 34-kD assembly domain bounded by amino acids 1533-1819 within the myosin sequence and a larger 58-kD phosphorylation domain bounded by amino acids 1533-2034 within the myosin sequence.     

Segmental flexibility of the C1q subcomponent of human complement and its possible role in the immune response

Fluorescence polarization techniques were used to study the rotational dynamics of the C1q subcomponent of human complement. C1q was covalently labeled with dansyl (DNS) chloride. Digestion of either C1q-DNS4.0 or C1q-DNS1.8 conjugates with pepsin showed that about 75% of the DNS probes were attached to the C1q globular heads and that the remainder were on the collagen-like stalk (peptic fragment). C1q-DNS conjugates readily agglutinated IgG-coated latex beads and combined with C1r2C1s2 to form hemolytically active 16 S C1-DNS. Both C1q-DNS and C1-DNS samples displayed steady-state rotational correlation time and fluorescence lifetime transitions near 48 degrees C. Hydrodynamic studies showed that C1q formed soluble aggregates near the transition temperature. In contrast, stalk samples with a DNS probe apparently attached to the large central fibril showed no thermal transitions or aggregation even when heated above 50 degrees C. Nanosecond fluorescence depolarization measurements detected restricted flexible motions of the C1q heads with an associated rotational correlation time, phi s, of about 25 ns. The C1q anisotropy decay was dominated, however, by a long component, phi L, of perhaps 1000 ns. Except for probe wiggle, the stalk-DNS anisotropy profile was essentially flat. The rapid rotations associated with phi s could represent restricted twisting motions of the arm-head segments or wobbling motions of the heads themselves. Such motions may facilitate binding of the C1q heads to immune complexes. Straightforward diffusion calculations indicated that phi L could represent either global tumbling of the entire C1q molecule or wagging motions of the individual arm-head segments, as suggested by electron micrographs. Upon binding of the C1q heads to an activator, some of the C1q segments may be held in a slightly more open or more closed conformation, which in turn may trigger activation of the C1 proenzymes. In conclusion, we suggest a plausible triggering mechanism for C1 activation that is compatible with the flexible properties of its subcomponents.     

Light chain dependent effects of actin binding on the S-1/S-2 swivel in myosin

The S-1/S-2 swivel in myosin provides a flexible link between the head and tail portions of the molecule. We have investigated the properties of the swivel by employing limited proteolysis methods. Our results indicate that the binding of actin to heavy meromyosin inhibits both the chymotryptic and papain cleavage of the S-1/S-2 swivel, and that this effect is dependent on the presence of intact LC-2 light chains. Actin did not slow digestions carried out using heavy meromyosin previously treated with proteases to nick the LC-2 chains to 17,000 or 14,000 Mr fragments. Although the integrity of the LC-2 light chain appears to be required to transmit the effects of actin binding from the myosin head to the S-1/S-2 swivel, the binding of Ca2+ to the 17,000 Mr LC-2 fragment can still affect the chemical reactivity of SH1 thiol groups. Both chymotryptic and papain digestions of heavy meromyosin containing intact or fragmented LC-2 light chain show substantial temperature sensitivity between 5 degrees C and 35 degrees C. Calculated apparent activation energies for this process indicate that the S-1/S-2 swivel in myosin can undergo temperature-dependent structural changes independently of the state of the LC-2 light chain. Thus, both actin binding and temperature variations can induce structural transitions in the S-1/S-2 swivel.     

Immunoglobulin G antibody bound to the C1q subcomponent of human complement exhibits segmental flexibility

The rotational dynamics of rabbit immunoglobulin G (IgG) anti-dansyl antibodies bound to the C1q subcomponent of human complement were studied by nanosecond fluorescence spectroscopy. Deconvoluted anisotropy decays of IgG-C1q mixtures were fitted to a two-exponential expression and were corrected for the effects of unbound IgG, which was determined with an analytical ultracentrifuge. Compared with the anisotropy parameters for free IgG, the pre-exponential weighting factors and the short correlation time of the C1q-bound antibody were nearly unchanged, and the long correlation time increased by only about 45 nanoseconds. These results, together with rotational diffusion calculations, indicate that the Fab arms of the C1q-bound antibody exhibited considerable flexibility. This finding may have biological relevance because it suggests that C1q can bind to the Fc segments of IgG molecules anchored in an immune complex, even though the angles between the two Fab arms of the different antibodies may vary. The results of this study also support our earlier interpretation that both the short and long correlation times of IgG principally represent flexible motions of the Fab segments.     

Elastic properties of ribosomal RNA building blocks: molecular dynamics of the GTPase-associated center rRNA

Explicit solvent molecular dynamics (MD) was used to describe the intrinsic flexibility of the helix 42–44 portion of the 23S rRNA (abbreviated as Kt-42+rGAC; kink-turn 42 and GTPase-associated center rRNA). The bottom part of this molecule consists of alternating rigid and flexible segments. The first flexible segment (Hinge1) is the highly anharmonic kink of Kt-42. The second one (Hinge2) is localized at the junction between helix 42 and helices 43/44. The rigid segments are the two arms of helix 42 flanking the kink. The whole molecule ends up with compact helices 43/44 (Head) which appear to be modestly compressed towards the subunit in the Haloarcula marismortui X-ray structure. Overall, the helix 42–44rRNA is constructed as a sophisticated intrinsically flexible anisotropic molecular limb. The leading flexibility modes include bending at the hinges and twisting. The Head shows visible internal conformational plasticity, stemming from an intricate set of base pairing patterns including dynamical triads and tetrads. In summary, we demonstrate how rRNA building blocks with contrasting intrinsic flexibilities can form larger architectures with highly specific patterns of preferred low-energy motions and geometries.     

Hydrodynamic modes of arbitrarily shaped flexible macromolecules: Influence on dynamic light scattering

The diffusional motions of flexible macromolecules are analyzed with an increasingly realistic Rouse–Zimm model, i.e., by modeling the molecule as an arbitrary set of spheres connected by nearly harmonic springs. New features include (1) nearly arbitrary arrangements of spheres, (2) arbitrary arrangements of translational and torsional springs, (3) significant anharmonic corrections to the elastic potential surface, and (4) inclusion of torsional damping and various hydrodynamic cross-coupling effects (including two types of translational-rotational coupling) with no additional fitted parameters. The hydrodynamic interactions [R. F. Goldstein (1985) Journal of Chemical Physics, Vol. 83, pp. 2390–2397] contain no adjustable parameters other than temperature, viscosity, and the radii and positions of the spheres. These hydrodynamic interactions allow accurate calculations of rigid body diffusion as well as flexible motions. Given the positions, radii, and spring constant matrix, one can calculate a full set of three-dimensional diffusional modes. Because one uses an off-diagonal hydrodynamic resistance matrix instead of a diagonal mass matrix, the diffusional modes are different in structure from vacuum normal modes, and give rise to different rms motions in the laboratory frame. These hydrodynamic modes include the effects of vibrational-translational cross-coupling (i.e., motion along a vibrational coordinate may give rise to a translational force, and vice versa). The diffusional modes are used to simulate dynamic light scattering (DLS). I examine various molecules with different shapes, flexibilities, and with different scattering vectors. Radial and angular motions influence DLS decays differently. These effects are dependent upon the molecular shape (straight, bent, or curved) and type of flexibility (stretching or bending). Furthermore, small cubic corrections to the potential surface can be significant for DLS of certain geometries such as straight rods and semicircles. © 1993 John Wiley & Sons, Inc.     

Structures of smooth muscle myosin and heavy meromyosin in the folded, shutdown state

Remodelling the contractile apparatus within smooth muscle cells allows effective contractile activity over a wide range of cell lengths. Thick filaments may be redistributed via depolymerisation into inactive myosin monomers that have been detected in vitro, in which the long tail has a folded conformation. Using negative stain electron microscopy of individual folded myosin molecules from turkey gizzard smooth muscle, we show that they are more compact than previously described, with heads and the three segments of the folded tail closely packed. Heavy meromyosin (HMM), which lacks two-thirds of the tail, closely resembles the equivalent parts of whole myosin. Image processing reveals a characteristic head region morphology for both HMM and myosin, with features identifiable by comparison with less compact molecules. The two heads associate asymmetrically: the tip of one motor domain touches the base of the other, resembling the blocked and free heads of this HMM when it forms 2D crystals on lipid monolayers. The tail of HMM lies between the heads, contacting the blocked motor domain, unlike in the 2D crystal. The tail of whole myosin is bent sharply and consistently close to residues 1175 and 1535. The first bend position correlates with a skip in the coiled coil sequence, the second does not. Tail segments 2 and 3 associate only with the blocked head, such that the second bend is near the C-lobe of the blocked head regulatory light chain. Quantitative analysis of tail flexibility shows that the single coiled coil of HMM has an apparent Young's modulus of about 0.5 GPa. The folded tail of the whole myosin is less flexible, indicating interactions between the segments. The folded tail does not modify the compact head arrangement but stabilises it, indicating a structural mechanism for the very low ATPase activity of the folded molecule.     

Numerical Study of Propulsion Mechanism for Oscillating Rigid and Flexible Tuna-Tails

Numerical study on the unsteady hydrodynamic characteristics of oscillating rigid and flexible tuna-tails in viscous flow-field is performed.Investigations are conducted using Reynolds-Averaged Navier-Stokes (RANS) equations with a moving adaptive mesh.The effect of swimming speed,flapping amplitude,frequency and flexure amplitude on the propulsion performance of the rigid and flexible tuna-tails are investigated.Computational results reveal that a pair of leading edge vortices develop along the tail surface as it undergoes an oscillating motion.The propulsive efficiency has a strong correlation with various locomotive parameters.Peak propulsive efficiency can be obtained by adjusting these parameters.Particularly,when input power coefficient is less than 2.8,the rigid tail generates larger thrust force and higher propulsive efficiency than flexible tail.However,when input power coefficient is larger than 2.8,flexible tail is superior to rigid tail.     

Myosin VI walks "wiggly" on actin with large and variable tilting

Myosin VI is an unconventional motor protein with unusual motility properties such as its direction of motion and path on actin and a large stride relative to its short lever arms. To understand these features, the rotational dynamics of the lever arm were studied by single-molecule polarized total internal reflection fluorescence (polTIRF) microscopy during processive motility of myosin VI along actin. The axial angle is distributed in two peaks, consistent with the hand-over-hand model. The changes in lever arm angles during discrete steps suggest that it exhibits large and variable tilting in the plane of actin and to the sides. These motions imply that, in addition to the previously suggested flexible tail domain, there is a compliant region between the motor domain and lever arm that allows myosin VI to accommodate the helical position of binding sites while taking variable step sizes along the actin filament.