| # Manual

## Zeno - Hydrodynamic Calculations Options Module

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Last updated: August 2025

In this module, the options and settings controlling the ZENO method for computing the hydrodynamics and other properties of a model are set. The following description of the ideas behind ZENO reproduced below are taken from the Zeno website http://www.stevens.edu/zeno/ (see Douglas, Some Applications of Fractional Calculus to Polymer Science, Adv. Chem. Phys. 102:121?191, 1997; Douglas et al., Hydrodynamic friction and the capacitance of arbitrarily shaped objects, Phys. Rev. E 49:5319-5331, 1994; Mansfield et al., Intrinsic Viscosity and the Electric Polarizability of Arbitrarily Shaped Objects, Phys. Rev. E, 64:61401-61416, 2001).

Purpose of Zeno: Algorithm (Zeno) for calculating the Stokes friction coefficient, electrostatic capacity, intrinsic viscosity, intrinsic conductivity and electrical polarizability of essentially arbitrarily-shaped 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. 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.

Electrostatic-Hydrodynamic 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.

Computational Method: The Zeno computational method involves enclosing an arbitrary-shaped probed object within a sphere and launching random walks from this sphere.The probing trajectories either hit or return to the launch surface ('loss'), whereupon the trajectory is either terminated or reinitiated.

Compute Zeno checkbox. This will launch 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, Dt
- Translational friction coefficient, f
- Intrinsic viscosity, [η]
- Hydrodynamic volume, Vh

Enter the number of steps in the Zeno Steps (thousands): field (Default: 1000).

The Skin Thickness (current units): The skin thickness is utilized in the Zeno calculation. It is a multiplier of the launch radius. The launch radius is the radius of a sphere centered at zero which contains the model. Leaving the value at 0 uses the internal default value of 1E-6. The authors of Zeno recommend that the value should be on the order of 1E-5 to 1E-6. (Default: 0.000)

More recently, we have discovered a discrepancy between the translational diffusion coefficient as calculated by ZENO and GRPY. We found that the ZENO calculation of Dt(20,w) constantly produced relatively larger (~+1-2%) values than those computed by GRPY for our test set of 25 protein structures. A symmetrical behavior was found when comparing the intrincic viscosity [η] calculated values, with ZENO underestimating the values by roughly the same percentages. We also used more elongated structures (derived from the fibrinogen main body crystal structure 3GHG.pdb), confirming these trends. As described in our recent publication (Brookes, Zuk and Rocco, Bead model hydrodynamics: an in-depth comparison between GRPY and ZENO. Eur Biophys J (2025). https://doi.org/10.1007/s00249-025-01758-8), we found heuristic dependencies as a function of the number of beads in the model, described by power laws in the form y = a + b * xc), whose coefficients are listed in this panel. Applying these corrections reduced the discrepancy between ZENO and GRPY calculated Dt(20,w) values to an average of -0.11±0.22% (range -0.56 +0.27%) for globular proteins, and to +0.25±0.16% (range +0.06 +0.52%) for elongated structures. The second power-law fit corrects ZENO-calculated [η] values reducing the average percent difference for globular proteins to -0.03±0.44% (range -0.79 +0.92%), and to +0.48±0.42% (range +0.21 +0.85%) for elongated structures. These heuristic ZENO corrections are now offered as a default option from the August 2025 release of US-SOMO.

The last entry in this option panel is the Zeno repetitions. ZENO could be relaunched sequentially on the same model, providing additional statistics on the calculated parameters. Default: 1 (meaning ZENO will run just once for each model).

www contact: Emre Brookes

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Last modified on June 13, 2024