Stokes-Einstein equation: symbols and SI units
This page is the reference entry for the Stokes-Einstein equation: the formula written out, every symbol with its SI unit, where the name comes from, and what the equation assumes. For the fuller account of Stokes radius and hydrodynamic radius themselves, see Stokes radius and the Stokes-Einstein equation.
The Stokes-Einstein equation describes how quickly a particle spreads through a fluid by diffusion, and it says that rate depends on only three things: how big the particle is, how viscous the fluid is, and how warm it is. Nothing else enters. Particles diffuse faster in less viscous fluids, faster at higher temperatures, and faster when they are smaller.
What makes the relationship useful is that it runs in both directions. Diffusion is something you can measure. The size of a molecule in solution is not something you can see. The equation connects the two, so a measured diffusion coefficient becomes a radius. That is why it sits underneath every sizing method that works by watching molecules move, Flow Induced Dispersion Analysis included, and why such a method can report an absolute size in nanometres without a calibration curve or a reference protein.
The radius it returns is the hydrodynamic radius: the effective size a molecule appears to have as it moves through a liquid, including the layer of water that travels with it. It therefore describes the molecule as it behaves in solution, rather than as it appears in a crystal structure.
What is the formula?
D = kBT / (6πηr)
Written in plain characters, which is how it is usually typed into a search box: D = kB T / (6 pi eta r).
Rearranged for the radius, which is the direction used when size is the quantity of interest:
r = kBT / (6πηD)
What does each symbol mean?
D is the diffusion coefficient, sometimes called diffusivity: how quickly the particle spreads through the fluid. Unit: square metres per second.
kB is the Boltzmann coefficient, a fundamental constant in physics that connects the microscopic world of individual particles to the macroscopic properties of materials such as temperature and energy. Its value is fixed by definition at 1.380649 x 10-23 joules per kelvin. It acts as a temperature translator, telling you how much energy each tiny particle carries at a given temperature.
T is the absolute temperature, in kelvin.
η, the Greek letter eta, is the viscosity of the fluid: its resistance to flow, or more precisely the internal friction between layers of the fluid as they move relative to each other. Unit: pascal seconds.
r is the hydrodynamic radius of the particle, in metres. This is the effective radius of the molecule based on its diffusion rate in solution, which includes the layer of water that moves with it.
Why is it also called the Stokes-Einstein-Sutherland equation?
Three names attach to the same relationship. Sir George Gabriel Stokes supplied the drag on a sphere moving through a viscous fluid, in 1851. Albert Einstein and William Sutherland then derived the link between that drag and diffusion independently of one another, both publishing in 1905. Sutherland's paper used the result to estimate the molecular mass of albumin, which is why the relationship arrives in protein science with his name on it. Physics and biophysics literature uses Stokes-Einstein, Einstein-Stokes and Stokes-Einstein-Sutherland interchangeably for the same equation.
What does the equation actually tell you?
Read it as three proportionalities, and it stops being algebra.
Diffusion is faster in less viscous fluids. Viscosity sits in the denominator, so doubling the viscosity halves the diffusion coefficient.
Diffusion is faster at higher temperatures. Temperature sits in the numerator: more thermal energy, more motion.
Diffusion is faster for smaller particles. Radius sits in the denominator, so a molecule twice the radius diffuses at half the rate.
The practical consequence for measurement is that if you can measure how fast something diffuses, and you know the temperature and the viscosity, the equation gives you its size. That is the whole basis of size measurement by diffusion.
What does the equation assume?
Four conditions, and each one is part of the formal definition rather than a convention that grew up around it. The dissolved molecule is isolated, which in practice means a dilute solution. The fluid is treated as a continuum rather than as individual solvent molecules. The particle is treated as a sphere. And the motion is slow enough to be free of turbulence, which is the condition Stokes placed on the drag term the equation contains.
Only one of those is obviously untrue of a protein, and it is the one that matters least. Proteins are not spheres.
Does it work for proteins that are not spheres?
Yes, and the departure from a sphere is useful information rather than an error term.
What the equation returns is the radius of the sphere that would diffuse at the observed rate. That is what hydrodynamic radius means, and it is why it is described as an effective radius rather than a geometric one. A protein in water is also surrounded by a hydration shell, so it behaves as if it were bigger than its dry crystal size, and the hydrodynamic radius reflects that effective size, which is the size that governs how proteins move and interact in the body and in an in-vitro experiment.
The measurement does not assume a shape, because diffusion is measured directly. A shape assumption only enters if you take the second, optional step of converting the radius into a molecular weight. A linear correlation between hydrodynamic radius and molecular weight holds for globular proteins, and the molecular weight to size calculator uses it. Hydrodynamic radius, however, depends strongly on the conformation and compactness of a molecule, so for intrinsically disordered proteins, partially disordered or degraded proteins the correlation moves off that line.
The direction it moves in is the information. Plot measured hydrodynamic radius against molecular weight and the space around the globular line divides into three zones. Above the line, beyond even the radius of a fully unfolded monomer, sits oligomerisation. Between the globular line and the unfolded line sits a protein larger than a globular protein of the same mass, which means an elongated structure, disordered regions, or oligomerisation. Below the globular line sits a radius smaller than the most compact form the polypeptide could take, which means degradation or the loss of a subunit.
Those zones were measured rather than assumed. Standard globular proteins were run in triplicate at 25 degrees C to establish the globular line. BSA and beta-lactoglobulin were then run in a high concentration of guanidinium chloride to unfold them, giving a significantly larger radius and a second trendline for unfolded protein. BSA boiled in a low ionic strength tris buffer oligomerised and landed above the unfolded line. Partially degraded aldolase and ferritin both fell below the globular line. The full experiment is in the technical note linked at the end of this article.
So a result that sits off the globular line is a readout, not a failure. That is the basis of several standard uses.
Folded proteins are compact and give a smaller hydrodynamic radius. Unfolded or denatured proteins expand and give a larger one. That makes the measurement useful for studying stability, misfolding diseases such as Alzheimer's, and thermal unfolding.
When proteins form dimers, oligomers or aggregates, the hydrodynamic radius increases, so monitoring it detects self-association, protein-ligand binding, and aggregation problems during formulation or storage.
If a crystal structure or an AlphaFold prediction exists, the expected hydrodynamic radius can be calculated from it and compared with the measured one, which gives an indication of whether the protein is folded as expected or is forming higher-order structures. Proteins are dynamic, and their structure in solution can differ from the crystal structure: in a crystal, packing forces can distort flexible regions, stabilise particular conformations or restrict motion, while in solution a protein is free to adopt a range of conformations influenced by solvent interactions, pH, ionic strength and temperature. Small differences between predicted and measured radius are expected for that reason.
How is the equation used to measure molecular size?
Using in-capillary diffusion of proteins in a laminar flow, the Fida instrument records the fluorescence signal from diffusing species as a function of time, the time taken by the sample to flow through the capillary. The signal appears as a Gaussian curve. Smaller, faster diffusing molecules give a narrow, sharp peak; larger, slower diffusing molecules give a wider, broader peak.
Knowing the capillary radius and the peak residence time, the peak width is put into Taylor's dispersion law to obtain the diffusion coefficient, D. The diffusion coefficient is then converted into hydrodynamic radius using the Stokes-Einstein equation.
Why does the viscosity have to be measured, not assumed?
Look at the equation again. Of the four quantities on the right, kB is a constant of nature and T is held constant during the measurement. Viscosity is the only term that varies from sample to sample, and it sits in the denominator, so it scales the answer directly. An incorrect viscosity yields a wrong size.
This is an uncomfortable fact for diffusion-based sizing in general. Most techniques that rely on diffusion require you to supply the sample viscosity, which means measuring it separately on a viscometer, or assuming the viscosity of the buffer and hoping the protein has not changed it. That is error prone, and a wrong viscosity input changes the entire size calculation.
The Fida instrument measures viscosity on every single run instead, and compensates for it automatically. The mechanism falls out of the same physics. The sample passes through the capillary in the laminar flow range at constant temperature and constant pressure, so the only parameter that can change the residence time is the viscosity of the sample. Run a sample of known viscosity, such as PBS buffer, and the shift in retention time from that reference gives the change in viscosity directly.
Two things are worth keeping in mind. Viscosity is temperature dependent, and it is concentration dependent, and the changes follow exponential functions. Both are reasons why a viscosity measured once, elsewhere, on a different day, is not a safe substitute for one measured on the sample in front of you. There is more on this in What is sample viscosity?
Why is this called a first principle measurement?
In physics, work is said to be based on first principles if it starts directly at the level of established science and does not rely on assumptions such as empirical modelling and parameter fitting.
Flow Induced Dispersion Analysis (FIDA) is an efficient, in-solution method for protein characterisation, measuring molecular size (hydrodynamic radius), aggregation, and complex molecular interactions including binding affinity and kinetics. It is a first principle technology in that sense: it relies on basic laws of nature, it is absolute, and it does not require calibration. On the theoretical side it rests on the established descriptions and models of Geoffrey Taylor and Albert Einstein.
That is what the Stokes-Einstein equation buys. Size comes out of measured diffusion, a known temperature and a measured viscosity, with no standard curve, no reference protein and no molecular weight assumption anywhere in the chain.
Frequently asked questions
What is the difference between the Stokes-Einstein equation and the Stokes radius?
The equation is the relationship; the Stokes radius is one of its terms. Stokes radius and hydrodynamic radius are the same quantity, the effective radius a molecule appears to have as it moves through a liquid. The full account of the radius is in Stokes radius and the Stokes-Einstein equation.
What are the units in the Stokes-Einstein equation?
In SI units: the diffusion coefficient D in square metres per second, the Boltzmann coefficient kB in joules per kelvin, the temperature T in kelvin, the viscosity eta in pascal seconds, and the radius r in metres. The units are consistent: joules per kelvin times kelvin, divided by pascal seconds times metres, gives metres squared per second.
Does the Stokes-Einstein equation work for intrinsically disordered proteins?
It does. The equation returns the radius of a sphere that would diffuse at the observed rate, and a disordered or elongated protein diffuses more slowly than a compact one of the same mass, so it reports a larger hydrodynamic radius. The measurement itself is unaffected; it is the conversion from radius to molecular weight that assumes a globular shape, and the deviation from that correlation is what identifies the protein as disordered, oligomeric or degraded.
Why does viscosity have to be measured rather than assumed?
Because viscosity is a term in the equation, and it is the only term that varies between samples. An incorrect viscosity yields a wrong size, and viscosity is both temperature dependent and concentration dependent. The Fida instrument measures it on every run rather than asking you to supply it. See What is sample viscosity?
Can hydrodynamic radius be converted to molecular weight?
For globular proteins there is a linear correlation between hydrodynamic radius and molecular weight, and the molecular weight to size calculator uses it. The correlation depends strongly on conformation and compactness, so it holds poorly for intrinsically disordered proteins, partially disordered or degraded proteins, and oligomers.
Who derived the Stokes-Einstein equation?
Stokes derived the drag on a sphere in a viscous fluid in 1851. Einstein and Sutherland independently linked that drag to diffusion in 1905, Sutherland using the result to estimate the molecular mass of albumin. The combined relationship carries all three names.
References
Bureau International des Poids et Mesures. (2019). The International System of Units (SI) (9th ed.). BIPM. https://www.bipm.org/en/publications/si-brochure
Einstein, A. (1905). Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik, 322(8), 549-560. https://doi.org/10.1002/andp.19053220806
International Union of Pure and Applied Chemistry. Stokes law. In Compendium of Chemical Terminology (the Gold Book), entry S06028. https://goldbook.iupac.org/terms/view/S06028
International Union of Pure and Applied Chemistry. Stokes-Einstein equation. In Compendium of Chemical Terminology (the Gold Book), entry 12260. https://goldbook.iupac.org/terms/view/12260
Stepto, R., Chang, T., Kratochvíl, P., Hess, M., Horie, K., Sato, T., & Vohlídal, J. (2015). Definitions of terms relating to individual macromolecules, macromolecular assemblies, polymer solutions, and amorphous bulk polymers (IUPAC Recommendations 2014). Pure and Applied Chemistry, 87(1), 71-120. https://doi.org/10.1515/pac-2013-0201
Stokes, G. G. (1851). On the effect of the internal friction of fluids on the motion of pendulums. Transactions of the Cambridge Philosophical Society, 9, 8-106.
Sutherland, W. (1905). A dynamical theory of diffusion for non-electrolytes and the molecular mass of albumin. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 9(54), 781-785. https://doi.org/10.1080/14786440509463331
Related resources
The technical note The relationship between hydrodynamic radius and molecular weight contains the calibration experiment behind the three zones described above, including the unfolded protein trendline, and is the practical companion to the section on non-spherical proteins. For the equation inside a working measurement, the application note Assessment of Sample Quality with Every Measurement shows how absolute viscosity is accounted for in every hydrodynamic radius calculation, alongside the other quality control parameters read from the same run.
On this site, Stokes radius and the Stokes-Einstein equation covers the radius itself in depth, What is hydrodynamic radius? defines the quantity the equation returns, What is sample viscosity? covers the one term that varies between samples, and the Molecular Size readout shows what the measurement looks like in practice.
