hydrodynamic modeling - auc solutionsultrascan.aucsolutions.com/data/brookes140615somo.pdf · rigid...

Brookes, E. US Workshop 15 June 2014

Emre Brookes

UltraScan Workshop

15 June 2014

Hydrodynamic Modeling


Rigid Body Hydrodynamics : Overview

Stokes flow methods

Hydrodynamic tensor

Boundary Element

Zeno method

Implementation within US-SOMO

Hands on with US-SOMO


Rigid Body Hydrodynamics : Stokes flow

Translation and rotational dynamics of a rigid body of arbitrary shape can be described by a 6x6 resistance matrix which, under low Reynolds number conditions, directly relates the three forces and torques acting upon the the particle to its linear and angular velocities:

Spotorno, B., Piccinini, L., Tassara, G., Ruggiero, C., Nardini, M., Molina, F., Rocco, M. (1996) BEAMS (BEAds Modelling System): a set of computer programs for the generation, the visualization and the computation of the hydrodynamic and conformational properties of bead models of bead models of proteins. Eur Biophys J 25:373-84

Happel, J., Brenner, H (1973) Low Reynolds Number Hydrodynamics, ch 5. Nordhoff, Leyden.

6 dimensionalvelocity vector

6 dimensional force and torque

vector

6x6 resistance

matrix

creeping flow or creeping motionviscous forces dominate


Reynolds number


Rigid Body Hydrodynamics

This equation can be reformulated as:

Spotorno, B., Piccinini, L., Tassara, G., Ruggiero, C., Nardini, M., Molina, F., Rocco, M. (1996) BEAMS (BEAds Modelling System): a set of computer programs for the generation, the visualization and the computation of the hydrodynamic and conformational properties of bead models of bead models of proteins. Eur Biophys J 25:373-84

force andtorquevectors

translational,rotational and

roto-translational (coupling)frictional tensors

linear andangular

velocities

origin dependent

transpose

only symmetric at a particular point:“The center of reaction R”



An analogous equation can be written for the diffusion and a center of diffusion exists where the roto-translational (coupling) diffusion tensor is symmetric. The relationship between the diffusion matrix and the resistance matrix is given by the generalized Stokes-Einstein equation:

Brenner, H (1967) Coupling between the translational and rotational Brownian motions of rigid particles of arbitrary shape. J Colloid Interface Sci 23:407-436

diffusionmatrix

Boltzmann'sconstant absolute

temperature

resistancematrix



For an ensemble of N beads, it is possible to calculate from Stokes' law the frictional force exerted on the solvent by each bead. However, the motion of each bead creates an internal velocity field in the solvent that must be added to the external one. This “hydrodynamic interaction” tensor can be described by:

Rotne, J. Prager, S. (1969) Variational treatment of hydrodynamic interaction on polymers. J Chem Phys 50:4831-48

Yamakawa, H. (1970) Transport properties of polymer chains in dilute solutions. Hydrodynamic interaction. J Chem Phys 53:436-43

García de la Torre, J., Bloomfield, V.A. (1977) Hydrodynamic properties of macromolecular complexes. I. Translation. Biopolymers 16:1747-63.

solventviscosity

frictional force exerted upon the solvent by the i-th bead



This N bead hydrodynamic tensor equation can be rewritten as:

García de la Torre, J. (1989) Hydrodynamic properties of macromolecular assemblies. In: Harding S.E., Rowe, A.J. (eds) Dynamic properties of biomolecular assemblies. The Royal Society of Chemistry Special Publication No 74, Cambridge, UK, pp 3-31.

Dirac delta



Assemble the 3Nx3N “supermatrix” and invert:

Computationally expensive

Ω(n2 log n)


3Nx3N supermatrixcomposed of NxN blocks Bij



And now we can compute:

Harvey, S.C., Mellado, P., Garcia de la Torre, J. (1983) Hydrodynamic resistance and diffusion coefficients of segmentally flexible macromolecules with two subunits. J Chem Phys 78:2081-90

García de la Torre, J., Rodes, V. (1983) Effects from bead size and hydrodynamic interactions on the translational and rotational coefficients of macromolecular bead models. J Chem Phys 79:2454-60


matrix whose elements are the componentsof the vectors joining the center of

bead i with the origin of the reference system

“volume” correction total volume of all beads



And now we can compute:

translationalfrictional

coefficient

translationaldiffusion

coefficient

translationalStokes' radius



And the rotational properties must be evaluated from the center of reaction:

García Bernal, J.M., García de la Torre, J. (1980) Transport properties and hydrodynamic centers of rigid macromolecules with arbitrary shapes. Biopolymers 10:751-66

dyadic productdistance vector relating the bead coordinate

origin with with the center of reaction



Relaxation times and intrinsic viscosity:

Wegener, W.A., Dowben, R.M., Koester, V.J. (1979) Time-dependent birefrigence, linear dichroism, and optical rotation resulting from rigid-body rotational diffusion. J Chem Phys 70:622-32

eigenvalues of therotational diffusion tensor

intrinsic viscosity


Rigid Body Hydrodynamics : BEST

A boundary element method for solving the Stokes flow equations

Does not depend on hydrodynamic interaction tensors

Requires tessellation of the arbitrary shape

Computationally intensive

Must be repeated multiple times to generate a trajectory

Sergio Aragon and Dina Flamik, Precise computation of transport properties of cylinders by the boundary element method, Macromolecules, 2009, 42 (16), 6290:6299



Rigid Body Hydrodynamics : Zeno

The Zeno method can be used to compute the hydrodynamic radius and intrinsic viscosity of arbitrarily shaped objects. From the hydrodynamic radius, the translational diffusion can be directly computed utilizing the Stokes-Einstein relation. The method proceeds by solving for the electrostatic capacity and electrostatic polarizability of a perfect conductor having the same size and shape. From the electrostatic capacity, the hydrodynamic radius can be computed which has shown to be accurate with 1%, and from the electrostatic polarizability the intrinsic viscosity can be computed to within 2-3%. The procedure utilizes a path integral method derived from probabilistic potential theory incorporating random walks. The method can be used on bead models or atomic structures defined as bead models (e.g. utilizing Van der Walls radii). Importantly, the bead models may contain overlaps and the individual beads can be arbitrarily sized, allowing high resolution structures to be processed. Additionally, the required computation time scales linearly in the number of beads as opposed to cubically as in methods solving the hydrodynamic interaction tensor [HYDROPRO,SOMO] or utilizing the Stokes flow equations [BEST], making the Zeno computation of the hydrodynamic radius of high-resolution bead models feasible.

Mansfield M.L., Douglas J.F., and Garboczi, E.J. Intrinsic viscosity and electric polarizability of arbitrarily-shaped objects, Phys. Rev. E 2001, 64, 061601



US-SOMOHydrodynamic calculations from structure

Batch & cluster modes to process large number of structures

Model classifier compares against experimental data

http://somo.uthscsa.edu


Hands on

Hydrodynamic Modeling Principles & Applications

Loading single PDB files

dealing with incomplete or non-coded residues, editing a PDB file

Generating, visualizing and retrieving bead models

SOMO bead model method

DAMMIN bead models

AtoB Grid Models

Computing rigid-body hydrodynamics

Optional analysis of attendees-supplied structures

Batch-mode operation

saving parameters to a file

model classifier

hydrodynamic modeling - auc solutionsultrascan.aucsolutions.com/data/brookes140615somo.pdf · rigid...

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