ZENO Efficient Method for Characterizing Object Shape and for Calculating Transport Properties of Nanoparticles and Synthetic and Biological Macromolecule

Movie (will be available soon)  Computational Elements:  
ZENO computation: 

Summary of Important Properties (Stevens Institute of Technology)

A Monte Carlo numerical path integration that generates a large number of random walks in the space outside the body. Because the Laplacian operator governs the statistics of these walks, sums taken over these random walks yield: 

electrostatic capacity, C polarizability tensor, a intrinsic conductivity, [s] hydrodynamic radius, Rh translational diffusion coefficient, D translational friction coefficient, f intrinsic viscosity, [h] hydrodynamic volume, Vh 

INTERIOR computation: 

Protein Properties/Zeno:
PDB ID code : (will be available soon) contact: ekang1@stevens.edu 
A Monte Carlo integration that generates a large number of points distributed randomly throughout the interior of the body. Sums taken over these points yield: 

particle volumes, V radius of gyration (evaluated for the interior of the particle) Rgi particle scattering function, P(q)


SURFACE computation: 

A Monte Carlo integration that generates a large number of points distributed randomly over the surface of the body. Sums takes over these points yield: 

surface area, S radius of gyration (evaluated for the surface of the particles) Rgs translational diffusion coefficient, D, by the Kirkwood doublesum formula 

Acknowledgements: We thank CTCMS at NIST for funding support of this development Figure 1:
Collection of spheres (green) represents a model soot particles (clustercluster
aggregate) and the path
(yellow) represents a probing random walk trajectory. C is determined by fraction trajectories that hit the
sphere. f (translational) ยป 6p h C
Links: 
Purpose of Zeno:
Algorithm (Zeno) for calculating the Stokes friction coefficient, electrostatic capacity, intrinsic viscosity, intrinsic conductivity and electrical polarizability of essentially arbitrarilyshaped objects to unprecedented accuracy.
Idea Behind Calculation:
There is a fundamental relation between the Laplacian operator ^{ }and random paths whose step size has a finite variance. This correspondence allows for the solution of the equations of mathematical physics to be formally expressed as averages over random walk trajectories. There are many famous mathematicians that have contributed to this formal computational scheme.
Kakutani, Wiener, Kac, Ito and MacKean and many others
The advantage of this method is that it allows for the calculation of transport properties for objects having essentially arbitrary shape. This method becomes a practical and highly accurate method for performing transport property calculations when random walks are generated by computer.
Computational Method:
The Zeno computational method involves enclosing an arbitraryshaped probed object within a sphere and launching random walks from this sphere. The probing trajectories either hit or return to the launch surface ('loss') as shown in the fig.1 for a model soot particles aggregate, whereupon the trajectory is either terminated or reinitiated. ElectrostaticHydrodynamic Analogy:
Hydrodynamic and electrostatic properties are determined, respectively, by the Navier Stokes and Laplacian equation. However, a specific orientational averaging of the Navier Stokes equations brings them into the form of Laplace's equation. This means that an approximate analogy exists between certain hydrodynamic and electrostatic properties. Particularly, the hydrodynamic radius and the intrinsic viscosity of a macromolecule are proportional, respectively, to the capacitance and polarizability of a perfect conductor having the same shape as the macromolecule. These proportionalities have been extensively tested on diverse shapes, and are always found to be accurate to 1% for the hydrodynamic radius and 5% for the intrinsic viscosity. Zeno determines the electrostatic properties directly by Monte Carlo path integration, and then infers hydrodynamic properties from these proportionalities. See References [1],[3], and [4]. 

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Last updated: 8/04/06
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