Stokes Radius and Stokes-Einstein Equation

Published Date:
October 27, 2023
Courtney Herms & Maja Wasilczyk
FIDA Basics

What is Stokes (hydrodynamic) Radius?

The hydrodynamic radius (Rh), also known as the Stokes radius, is the measurement of the absolute size of a molecule. It is named ‘’Stokes’’ after the Irish physicist Sir George Gabriel Stokes, who made significant contributions to fluid dynamics and the understanding of the behaviour of particles in a fluid.

In the context of the Stokes-Einstein equation, Stokes radius (Rh) represents the effective hydrodynamic size of a particle or molecule in a solution. The Stokes-Einstein equation is  a fundamental concept in the field of statistical physics. It  describes the relationship between the diffusion coefficient (D) of a particle in a fluid, the Boltzmann constant (k), the absolute temperature (T), the viscosity of the solvent (η), and the particle's hydrodynamic radius (Rh).

It has been used by scientists since the early 1900s to describe molecular geometry: that includes the shape, the mass, and the interactions of the molecule with the fluid it is suspended in. Albert Einstein was the first to define Rh as the radius of a hypothetical sphere that diffuses at the same rate as the particle or molecule in question under the same conditions

Thanks to advances in technology, Rh is no longer a measurement limited only to spheres. Today, it can be used to gather information about complex molecules of all shapes and sizes, and it is used as a crucial parameter in drug discovery, biotechnology, and environmental science, where the behavior of molecules in solution is essential to their functions and properties.

How to measure hydrodynamic radius?

One of the most precise, efficient, and clinically relevant ways to measure the hydrodynamic radius is flow induced dispersion analysis – FIDA. It is an in-solution, capillary-based technology, with no buffer constraints, which allows to measure the Rh in clinically relevant settings.

How is hydrodynamic radius measured and calculated?

As the sample passes through the capillary, differences in velocity at the center and at the walls of the capillary shape the sample into a parabola. As the molecules diffuse radially, away from the flow axis, this shape is recorded as a Gaussian signal by the detector. The radial diffusion is directly determined by the size of the molecules. Small molecules diffuse faster and create a more compact dispersion profile, while large molecules diffuse slower, which results in a more extended dispersion profile. Thus, peak dispersion becomes a direct measurement of molecular size: the hydrodynamic radius.

Hydrodynamic Radius of Molecules: Measurement Visualisation

Mathematically, the width of the peak, 2σ, is used to calculate the diffusivity, D, of the sample using Ficks Law of diffusion. Diffusivity is taken forward to calculate the hydrodynamic radius as a function of the temperature and size of the capillary, which is held constant during FIDA, using the Stokes-Einstein equation. Measurement of Rh using FIDA is a first principle technique; that is, it does not make assumptions or models and instead makes an absolute measurement of hydrodynamic radius.

As complexity increases, the need for absolute measurements increases.

Complex interactions – such as those between drugs and their targets – are a reality in natural biological systems. The assumptions made during relative measurements of hydrodynamic radius can hide or distort the true state of these interactions. FIDA’s application of first principle thinking to measure in absolute terms uses transparent and verifiable physics and fluid mechanics to provide the full picture. First principle based, absolute measurements not only ensure that the measurements delivered are correct, but also ensure precision and increase design flexibility.

What information can hydrodynamic radius provide?

With a precise measurement of Rh comes the ability to acquire essential information about complex molecules: affinity (Kd), quantity, stickiness, polydispersity, oligomerization, agglutination, and conformational changes. Additionally, FIDA can be integrated into processes for buffer screening, clone selection, and studies involving immunogenicity. All of this, from only a few nL of sample without the need for purification or fixation, which means all measurements are made on the native state of the molecule. If there is information to be learned about a complex construct or interaction between molecules, FIDA outputs them during the determination of hydrodynamic radius.

How is Rh and Stokes-Einstein used to study binding?

Measuring the hydrodynamic radius with FIDA opens the door to another crucial biophysical property: the binding between two or more biomolecules. As biomolecules come together and bind in solution, their diffusivity (D) decreases; that is, the bound molecules will diffuse slower through the capillary and generate a more extended dispersion profile and diffusion coefficient (D). Using the Stokes-Einstein equation, FIDA software reveals the increase in size (Rh) of the molecules as they bind together. Titrating your molecule of interest with its binding partners in a simple FIDA experiment can uncover a wide range of binding parameters, such as affinity (Kd), from just the hydrodynamic radius and the Stokes-Einstein equation.

Binding curve

If you are used to working with molecular weight, and are only starting to work with molecular size (Rh) you can try our molecular size to weight calculator

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