Calculation of hydrodynamic properties of globular proteins from their atomic-level structure

The solution properties, including hydrodynamic quantities and the radius of gyration, of globular proteins are calculated from their detailed, atomic-level structure, using bead-modeling methodologies described in our previous article (, Biophys. J. 76:3044-3057). We review how this goal has been pursued by other authors in the past. Our procedure starts from a list of atomic coordinates, from which we build a primary hydrodynamic model by replacing nonhydrogen atoms with spherical elements of some fixed radius. The resulting particle, consisting of overlapping spheres, is in turn represented by a shell model treated as described in our previous work. We have applied this procedure to a set of 13 proteins. For each protein, the atomic element radius is adjusted, to fit all of the hydrodynamic properties, taking values close to 3 A, with deviations that fall within the error of experimental data. Some differences are found in the atomic element radius found for each protein, which can be explained in terms of protein hydration. A computational shortcut makes the procedure feasible, even in personal computers. All of the model-building and calculations are carried out with a HYDROPRO public-domain computer program.     

MULTIHYDRO and MONTEHYDRO: conformational search and Monte Carlo calculation of solution properties of rigid or flexible bead models

A computer program, MULTIHYDRO, has been constructed for the calculation of hydrodynamic coefficients and other solution properties of multiple possible conformations of a bead model. With minimal additional programming to describe the model under study, this program interfaces efficiently with HYDRO for the calculation of solution properties, including hydrodynamic coefficients, radius of gyration, covolume, etc. A useful application is the conformation search of rigid macromolecules, because many possible conformations can be evaluated in a single run of the program. In this paper we also pay attention to the properties of flexible macromolecules, in the so-called Monte Carlo rigid-body approximation, which is virtually exact for the simpler solution properties. The theoretical aspects of the procedure are described, and we show how MULTIHYDRO can be employed for this calculation. However, for flexible molecules, a more general simulation scheme is importance-sampling Monte Carlo generation. We describe how this procedure is implemented in another computer program, MONTEHYDRO. Examples of the usage of these tools are provided.     

HYDROMIC: prediction of hydrodynamic properties of rigid macromolecular structures obtained from electron microscopy images

We have developed a procedure for the prediction of hydrodynamic coefficients and other solution properties of macromolecules and macromolecular complexes whose volumes have been generated from electron microscopy images. Starting from the structural files generated in the three-dimensional reconstructions of such molecules, it is possible to construct a hydrodynamic model for which the solution properties can be calculated. We have written a computer program, HYDROMIC, that implements all the stages of the calculation. The use of this procedure is illustrated with a calculation of the solution properties of the volume of the cytosolic chaperonin CCT, obtained from cryoelectron microscopy images.     

Hydrodynamic properties of rigid particles: comparison of different modeling and computational procedures.

The hydrodynamic properties of rigid particles are calculated from models composed of spherical elements (beads) using theories developed by Kirkwood, Bloomfield, and their coworkers. Bead models have usually been built in such a way that the beads fill the volume occupied by the particles. Sometimes the beads are few and of varying sizes (bead models in the strict sense), and other times there are many small beads (filling models). Because hydrodynamic friction takes place at the molecular surface, another possibility is to use shell models, as originally proposed by Bloomfield. In this work, we have developed procedures to build models of the various kinds, and we describe the theory and methods for calculating their hydrodynamic properties, including approximate methods that may be needed to treat models with a very large number of elements. By combining the various possibilities of model building and hydrodynamic calculation, several strategies can be designed. We have made a quantitative comparison of the performance of the various strategies by applying them to some test cases, for which the properties are known a priori. We provide guidelines and computational tools for bead modeling.     

Building hydrodynamic bead-shell models for rigid bioparticles of arbitrary shape.

The calculation of hydrodynamic and other solution properties of rigid macromolecules, using bead-shell model methodologies, requires the specification of the macromolecular shape in a format that can be interfaced with existing programs for hydrodynamic computations. Here, a procedure is presented for such a structural specification that is applicable to arbitrarily shaped particles. A computer program (MAKEPIXB), in which the user inserts the code needed to determine the structure, produces an structural file that is interpreted by another program (HYDROPIX) which is in charge of the computation of properties. As simple and yet illustrative examples we consider two cases: (1) dimeric structures composed of ellipsoidal subunits; and (2) toroidal structures, presenting simple equations that predict the properties of toroids with varying radial ratios.     

Translational diffusion and intrinsic viscosity of globular proteins. Theoretical predictions using hydrated hydrodynamic models. Application to BPTI.

In this paper, we present a way to make hydrodynamic models of globular proteins, including the hydration shell associated with them in aqueous solutions. Theoretical calculations using these models are made in order to determine the hydrodynamic properties of these proteins, employing rigorous and approximate methods of calculation. These will be applied to the bovine pancreatic trypsin inhibitor, BPTI. Several hydrodynamic models are constructed: the A-model for the unhydrated protein BPTI and a set of H-models for hydrated protein with different hydration degrees. Theoretical results for the translational diffusion coefficient Dt and the intrinsic viscosity [eta] are obtained from different models. From the analysis of the A-model and hydrodynamic properties, there is not a clear assignation of an ellipsoidal shape to this protein molecule. An amount of approximately 0.5 g H2O/g protein could be assigned to the BPTI.     

Calculation of translational friction and intrinsic viscosity. II. Application to globular proteins.

The translational friction coefficients and intrinsic viscosities of four proteins (ribonuclease A, lysozyme, myoglobin, and chymotrypsinogen A) are calculated using atomic-level structural details. Inclusion of a 0.9-A-thick hydration shell allows calculated results for both hydrodynamic properties of each protein to reproduce experimental data. The use of detailed protein structures is made possible by relating translational friction and intrinsic viscosity to capacitance and polarizability, which can be calculated easily. The 0.9-A hydration shell corresponds to a hydration level of 0.3-0.4 g water/g protein. Hydration levels within this narrow range are also found by a number of other techniques such as nuclear magnetic resonance spectroscopy, infrared spectroscopy, calorimetry, and computer simulation. The use of detailed protein structures in predicting hydrodynamic properties thus allows hydrodynamic measurement to join the other techniques in leading to a unified picture of protein hydration. In contrast, earlier interpretations of hydrodynamic data based on modeling proteins as ellipsoids gave hydration levels that varied widely from protein to protein and thus challenged the existence of a unified picture of protein hydration.     

Fast ignition upon the implosion of a thin shell onto a precompressed deuterium-tritium ball

Fast ignition of a precompressed inertial confinement fusion (ICF) target by a hydrodynamic material flux is investigated. A model system of hydrodynamic objects consisting of a central deuterium-tritium (DT) ball and a concentric two-layer shell separated by a vacuum gap is analyzed. The outer layer of the shell is an ablator, while the inner layer consists of DT ice. The igniting hydrodynamic flux forms as a result of laser-driven acceleration and compression of the shell toward the system center. A series of one-dimensional numerical simulations of the shell implosion, the collision of the shell with the DT ball, and the generation and propagation of thermonuclear burn waves in both parts of the system are performed. Analytic models are developed that describe the implosion of a thin shell onto a central homogeneous ball of arbitrary radius and density and the initiation and propagation of a thermonuclear burn wave induced by such an implosion. Application of the solution of a model problem to analyzing the implosion of a segment of a spherical shell in a conical channel indicates the possibility of fast ignition of a spherical ICF target from a conical target driven by a laser pulse with an energy of 500?C700 kJ.     

Refractive-Index Sensing with Ultrathin Plasmonic Nanotubes

We study the refractive-index sensing properties of plasmonic nanotubes with a dielectric core and ultrathin metal shell. The few nanometer thin metal shell is described by both the usual Drude model and the nonlocal hydrodynamic model to investigate the effects of nonlocality. We derive an analytical expression for the extinction cross section and show how sensing of the refractive index of the surrounding medium and the figure of merit are affected by the shape and size of the nanotubes. Comparison with other localized surface plasmon resonance sensors reveals that the nanotube exhibits superior sensitivity and comparable figure of merit.     

Calculation of NMR relaxation, covolume, and scattering-related properties of bead models using the SOLPRO computer program

The hydrodynamic properties of macromolecules and bioparticles, represented by bead models, can be calculated using methods implemented in the computer routine HYDRO. Recently, a new computer routine, SOLPRO, has been presented for the calculation of various SOLution PROperties. These include (1) time-dependent electro-optic and spectroscopic properties related to rotational diffusion, (2) non-dynamic properties like scattering curves, and (3) dimensionless quantities that combine two or more solution properties in a form which depends on the shape of the macromolecule but not on its size. In the present work we describe the inclusion of more of those types of properties in a new version of SOLPRO. Particularly, we describe the calculation of relaxation rates in nuclear magnetic resonance (NMR). For dipolar coupling, given the direction of the dipole the program calculates values of the spectral density, from which the NMR relaxation times can be obtained. We also consider scattering-related properties, namely the distribution of distances for the bead model, which is directly related to the angular dependence of scattered intensity, and the particle's longest distance. We have devised and programmed a procedure to calculate the covolume of the bead model, related to the second virial coefficient and, in general, to the concentration dependence of solution properties. Various shape-dependent dimensionless quantities involving the covolume are calculated. In this paper we also discuss some aspects, namely bead overlapping and hydration, that are not explicitely included in SOLPRO, but should be considered by the user. Received: 25 May 1998 / Revised version: 30 July 1998 / Accepted: 30 July 1998     

Calculation of the solution properties of flexible macromolecules: methods and applications

While the prediction of hydrodynamic properties of rigid particles is nowadays feasible using simple and efficient computer programs, the calculation of such properties and, in general, the dynamic behavior of flexible macromolecules has not reached a similar situation. Although the theories are available, usually the computational work is done using solutions specific for each problem. We intend to develop computer programs that would greatly facilitate the task of predicting solution behavior of flexible macromolecules. In this paper, we first present an overview of the two approaches that are most practical: the Monte Carlo rigid-body treatment, and the Brownian dynamics simulation technique. The Monte Carlo procedure is based on the calculation of properties for instantaneous conformations of the macromolecule that are regarded as if they were instantaneously rigid. We describe how a Monte Carlo program can be interfaced to the programs in the HYDRO suite for rigid particles, and provide an example of such calculation, for a hypothetical particle: a protein with two domains connected by a flexible linker. We also describe briefly the essentials of Brownian dynamics, and propose a general mechanical model that includes several kinds of intramolecular interactions, such as bending, internal rotation, excluded volume effects, etc. We provide an example of the application of this methodology to the dynamics of a semiflexible, wormlike DNA.     

A biomechanical hypothesis explaining upstream movements by the freshwater snail Elimia

1. Many taxa of freshwater invertebrates show active upstream movements, particularly the snails. Hypotheses explaining this behaviour invoke the search for food or space, compensation for drift, avoidance of predation and hydrodynamic effects. The pervasiveness of upstream movements among remote lineages of snails (two subclasses, three orders, 10 families), however, suggests that snails may move upstream for mechanical rather than adaptive reasons.
2. It is proposed that upstream movements by snails are a function of torque on the snail's foot generated by hydrodynamic drag on the shell. When subject to high broadside drag-forces on their shells, snails are able to reduce torque and stabilize orientation only by directing their anterior aspect upstream.
3. Movements of the freshwater pleurocerid snail Elimia were studied by following marked free-ranging individuals in six streams in Alabama, USA (four species, eight populations).
4. Populations showed either no net movement (two streams) or significant upstream movements ranging to a mean of ≈40 m over a 3-month period (four streams). Movement patterns were stream specific rather than species or population specific. Within populations showing significant upstream movements, snails with shell lengths ≤10 mm showed little net movement. Larger snails showed movements from 0 to >200 m upstream.
5. A torque-constrained random walk model was used to perform a post hoc test of the hypothesis that upstream movements were a function of torque on the snail's foot generated by hydrodynamic drag on the shell. The model predicted upstream and size-dependent movement patterns that approximated those observed for snails in the field.     

Motion of nanobeads proximate to plasma membranes during single particle tracking

Drag and torque on nanobeads translating within the pericellular layer while attached to glycolipids of the plasma membrane are calculated by a novel hydrodynamic model. The model considers a bead that translates proximate to a rigid planar interface that separates two distinct Brinkman media. The hydrodynamic resistance is calculated numerically by a modified boundary integral equation formulation, where the pertinent boundary conditions result in a hybrid system of Fredholm integrals of the first and second kinds. The hydrodynamic resistance on the translating bead is calculated for different combinations of the Brinkman screening lengths in the two layers, and for different viscosity ratios. Depending on the bead-membrane separation and on the hydrodynamic properties of both the plasma membrane and the pericellular layer, the drag on the bead may be affected by the properties of the plasma membrane. The Stokes-Einstein relation is applied for calculating the diffusivity of probes (colloidal gold nanobeads attached to glycolipids) in the plasma membrane. This approach provides an alternative way for the interpretation of in vitro observations during single particle tracking procedure, and predicts new properties of the plasma membrane structure.     

Solution structure of biopolymers: a new method of constructing a bead model

We propose a new, automated method of converting crystallographic data into a bead model used for the calculations of hydrodynamic properties of rigid macromolecules. Two types of molecules are considered: nucleic acids and small proteins. A bead model of short DNA fragments has been constructed in which each nucleotide is represented by two identical, partially overlapping spheres: one for the base and one for the sugar and phosphate group. The optimum radius sigma = 5.0 A was chosen on the basis of a comparison of the calculated translational diffusion coefficients (D(T)) and the rotational relaxation times (tau(R)) with the corresponding experimental data for B-DNA fragments of 8, 12, and 20 basepairs. This value was assumed for the calculation D(T) and tau(R) of tRNA(Phe). Better agreement with the experimental data was achieved for slightly larger sigma = 5.7 A. A similar procedure was applied to small proteins. Bead models were constructed such that each amino acid was represented by a single sphere or a pair of identical, partially overlapping spheres, depending on the amino acid's size. Experimental data of D(T) of small proteins were used to establish the optimum value of sigma = 4.5 A for amino acids. The lack of experimental data on tau(R) for proteins restricted the tests to the translational diffusion properties.     

Finite element modeling of shell shape in the freshwater turtle Pseudemys concinna reveals a trade‐off between mechanical strength and hydrodynamic efficiency

Aquatic species can experience different selective pressures on morphology in different flow regimes. Species inhabiting lotic regimes often adapt to these conditions by evolving low‐drag (i.e., streamlined) morphologies that reduce the likelihood of dislodgment or displacement. However, hydrodynamic factors are not the only selective pressures influencing organismal morphology and shapes well suited to flow conditions may compromise performance in other roles. We investigated the possibility of morphological trade‐offs in the turtle Pseudemys concinna. Individuals living in lotic environments have flatter, more streamlined shells than those living in lentic environments; however, this flatter shape may also make the shells less capable of resisting predator‐induced loads. We tested the idea that “lotic” shell shapes are weaker than “lentic” shell shapes, concomitantly examining effects of sex. Geometric morphometric data were used to transform an existing finite element shell model into a series of models corresponding to the shapes of individual turtles. Models were assigned identical material properties and loaded under identical conditions, and the stresses produced by a series of eight loads were extracted to describe the strength of the shells. “Lotic” shell shapes produced significantly higher stresses than “lentic” shell shapes, indicating that the former is weaker than the latter. Females had significantly stronger shell shapes than males, although these differences were less consistent than differences between flow regimes. We conclude that, despite the potential for many‐to‐one mapping of shell shape onto strength, P. concinna experiences a trade‐off in shell shape between hydrodynamic and mechanical performance. This trade‐off may be evident in many other turtle species or any other aquatic species that also depend on a shell for defense. However, evolution of body size may provide an avenue of escape from this trade‐off in some cases, as changes in size can drastically affect mechanical performance while having little effect on hydrodynamic performance. J. Morphol. 2011. © 2011 Wiley‐Liss, Inc.     

Computation of the dipole moments of proteins.

A simple and computationally feasible procedure for the calculation of net charges and dipole moments of proteins at arbitrary pH and salt conditions is described. The method is intended to provide data that may be compared to the results of transient electric dichroism experiments on protein solutions. The procedure consists of three major steps: (i) calculation of self energies and interaction energies for ionizable groups in the protein by using the finite-difference Poisson-Boltzmann method, (ii) determination of the position of the center of diffusion (to which the calculated dipole moment refers) and the extinction coefficient tensor for the protein, and (iii) generation of the equilibrium distribution of protonation states of the protein by a Monte Carlo procedure, from which mean and root-mean-square dipole moments and optical anisotropies are calculated. The procedure is applied to 12 proteins. It is shown that it gives hydrodynamic and electrical parameters for proteins in good agreement with experimental data.     

Calculation of hydrodynamic properties of macromolecular bead models with overlapping spheres

For the calculation of hydrodynamic properties of rigid macromolecules using bead modelling, models with overlapping beads of different sizes are used in some applications. The hydrodynamic interaction tensor between unequal overlapping beads is unknown, and an oversimplified treatment with the Oseen tensor may introduce important errors. Here we discuss some aspects of the overlapping problem, and explore an ad hoc form of the interaction tensor, proposed by Zipper and Durchschlag. We carry out a systematic numerical study of the hydrodynamic properties of a two-spheres model, showing how the Zipper-Durchschlag correction removes efficiently the numerical instabilities, and predicts the correct limits. Received: 15 February 1999 / Revised version: 29 April 1999 / Accepted: 11 May 1999     

Calculation of hydrodynamic properties of small nucleic acids from their atomic structure

Hydrodynamic properties (translational diffusion, sedimentation coefficients and correlation times) of short B-DNA oligonucleotides are calculated from the atomic-level structure using a bead modeling procedure in which each non-hydrogen atom is represented by a bead. Using available experimental data of hydrodynamic properties for several oligonucleotides, the best fit for the hydrodynamic radius of the atoms is found to be ~2.8 Å. Using this value, the predictions for the properties corresponding to translational motion and end-over-end rotation are accurate to within a few percent error. Analysis of NMR correlation times requires accounting for the internal flexibility of the double helix, and allows an estimation of ~0.85 for the Lipari–Szabo generalized order parameter. Also, the degree of hydration can be determined from hydrodynamics, with a result of ~0.3 g (water)/g (DNA). These numerical results are quite similar to those found for globular proteins. If the hydrodynamic model for the short DNA is simply a cylindrical rod, the predictions for overall translation and rotation are slightly worse, but the NMR correlation times and the degree of hydration, which depend more on the cross-sectional structure, are more severely affected.     

Structural organization of the multienzyme complex of mammalian aminoacyl-tRNA synthetases

The multienzyme complexes of mammalian aminoacyl-tRNA synthetases were purified from rat liver, rabbit liver, and rabbit reticulocytes according to the procedure slightly modified from Kellermann et al. [Kellermann, O., Brevet, A., Tonetti, H., & Waller, J.-P. (1979) Eur. J. Biochem. 99, 541-550]. Three forms of the synthetase complex with slightly different protein compositions were identified, suggesting a microheterogeneity of the synthetase complex. The hydrodynamic properties and the protein composition of the purified complexes were determined. The electron micrographs of the complex showed mostly amorphous particles and some hollow rings with an outer diameter of 164 A and an inner diameter of 42 A. The predicted hydrodynamic properties of several models of the complex were calculated. The properties of a ring model appear to best fit with those of the synthetase complex.