What is hydrodynamic radius?
Hydrodynamic radius (Rh), is a parameter used to describe the size of a particle or molecule in a fluid, taking into account its shape, mass, and interactions with the surrounding fluid molecules. The hydrodynamic radius is defined as the radius of a hypothetical sphere that diffuses at the same rate as the particle or molecule in question, under the same conditions (Einstein, 1905).
The hydrodynamic radius is used to describe the behaviour of biomolecules, such as proteins, in solution. Techniques such as flow-induced dispersion analysis (FIDA) are used to determine the hydrodynamic radius experimentally. The hydrodynamic radius is a crucial parameter in drug discovery, biotechnology, and environmental science, where the behaviour of molecules in solution is essential to their function and properties.
A useful way to picture it is to follow a protein as it moves through water. As it moves it drags water along with it, so it occupies more space than the molecule alone. That total space it takes up while moving through the liquid is the hydrodynamic radius, and it includes both the particle itself and the water molecules travelling with it.
How is hydrodynamic radius calculated?
In practice, the hydrodynamic radius is calculated from the diffusion coefficient of the molecule using the Stokes-Einstein equation (Einstein, 1905):
D = kBT / (6πηRh)
which rearranges to
Rh = kBT / (6πηD)
where D is the diffusion coefficient in m2 s-1, kB is the Boltzmann constant (1.380649 × 10-23 J K-1), T is the absolute temperature in K, η is the solvent viscosity in Pa s, and Rh is the hydrodynamic radius in m. This relation links molecular motion in a liquid to its effective size.
Two of those terms have to be controlled rather than assumed for the result to mean anything. In a FIDA measurement the temperature is held constant and the viscosity of the sample is determined on every single run, which is what allows the radius to be reported as an absolute size in nanometres rather than a value relative to a standard. The symbols, SI units and primary sources are set out in full on Stokes-Einstein equation: symbols and SI units.
Is hydrodynamic radius the same as the geometric radius?
Importantly, the hydrodynamic radius is not always the same as the geometric radius. It reflects how a molecule behaves in solution, which can be influenced by hydration layers, conformational flexibility, oligomerisation, or interactions with other molecules. As a result, Rh is often considered a more physiologically relevant parameter than molecular weight alone.
Proteins in water are surrounded by a hydration shell, which makes them behave as if they were larger than their dry, crystal size. The radius that matters for how a protein moves and interacts, in the body or in an experiment, is this effective size rather than the dimensions read off a structure. It captures the real size and behaviour of the molecule in its natural environment, which is what makes it a more complete and more useful description than molecular weight.
Because of these features, hydrodynamic radius is widely applied in the study of protein folding, aggregation, stability, molecular interactions, and formulation science, where solution behaviour must be understood in detail.
How does hydrodynamic radius relate to molecular weight?
Molecular weight represents the total mass of a molecule, while hydrodynamic radius reflects its effective size in solution. For globular proteins the two follow a linear correlation on a double logarithmic plot, and that standard curve can be used to convert between them in either direction. The molecular weight to size protein radius calculator on fidabio.com does that conversion: enter a molecular weight to get the expected hydrodynamic radius of a globular protein, or enter a measured radius to get the mass it corresponds to.
The correlation holds only as far as the globular assumption holds, because the radius depends heavily on the conformation and compactness of the molecule. For intrinsically disordered, partially disordered or degraded proteins the relationship departs from the globular line, and it does so in directions that are themselves informative. Three regions fall away from it.
Protein oligomers appear above the line, because mass and radius both rise as higher-order structure forms.
Elongated or disordered proteins also appear above the line, since an extended chain sweeps out far more volume than a compact one of the same mass.
Degraded proteins, or proteins with missing domains, appear below the line.
A point that sits off the calibration curve is therefore not a failed measurement. It is the readout, and it is the basis of the three quality control zones. The technical note The relationship between hydrodynamic radius and molecular weight sets out the calibration experiment behind those zones, the standard protein set it was built on, and the measurements made under denaturing and degrading conditions.
What information can hydrodynamic radius provide?
Size and shape in solution. Hydrodynamic radius is an effective measure of the molecule's size. It is particularly useful for non-spherical or complex-shaped molecules that might not be well described by simple geometric dimensions.
Conformational changes. Changes in the Rh can indicate alterations in the molecule's conformation. To exemplify, when a protein undergoes structural change due to denaturation, ligand binding, or similar, its hydrodynamic radius can change accordingly. Folded proteins are compact and give a smaller radius, while unfolded or denatured proteins expand and give a larger one, which makes the measurement useful for studying stability, misfolding and thermal unfolding.
Aggregation. Changes in hydrodynamic radius can serve as a signal of the aggregation state of molecules. Aggregates tend to have larger hydrodynamic radii than individual molecules due to increased effective volume, and the same applies to dimers and oligomers, so the radius reports on self-association and on aggregation arising during formulation or storage. A sample containing more than one size population is also reported as a rise in polydispersity.
Binding, affinity and kinetics. The radius generally increases when two or more molecules associate, so titrating one partner against the other under equilibrium conditions and following the change in radius gives a binding curve, and from it the affinity. If the reaction time is limited so that the system has not reached equilibrium at the point of detection, the curve carries kinetic information as well, and because reaction time is controlled by in-capillary mixing and pressure, one set of experiments gives KD, kon and koff. This is the principle on which flow-induced dispersion analysis was originally described (Jensen and Østergaard, 2010). Where binding produces no size change, as is common for a small molecule binding a protein, Lambda Dynamics is used instead.
Solution conditions. Conditions such as temperature, pH, ionic strength, and viscosity can influence the Rh. Because viscosity enters the Stokes-Einstein equation directly, it is measured on every run and reported as a readout in its own right, covered on the sample viscosity page.
Comparative studies. Hydrodynamic radius is used for comparing the sizes of different molecules, such as proteins, polymers, or nanoparticles, under similar conditions.
Dynamics and transport. Knowing the hydrodynamic radius is necessary to predicting how a molecule will diffuse through a solution, which has implications for transport within biological systems and industrial processes.
Biophysical characterisation. Hydrodynamic radius is a critical parameter for the biophysical characterisation of proteins, nucleic acids, and other biomolecules. It complements other measurements such as structural techniques, providing information about a molecule's overall behaviour. It is often used as a quality control parameter before structural assays, alongside aggregation, polydispersity and sample loss.
How is hydrodynamic radius measured?
Hydrodynamic radius is the primary readout of a FIDA measurement, and it is obtained from the diffusion of the sample in a flowing liquid, using nanolitre amounts of material (Jensen and Østergaard, 2010). The technology page covers the instrument and the principle in full.
The sample is carried through a microfluidic capillary in laminar flow, where the liquid moves in smooth parallel streams and nothing mixes. As it travels, molecules spread sideways across those streams by radial diffusion: small molecules diffuse quickly and spread between layers, while larger ones diffuse slowly and stay closer to their original path. The signal from the diffusing species is recorded against time and appears as a Gaussian peak, narrow and sharp for a small fast diffusing molecule and wide and broad for a large slow one. With the capillary radius and the peak residence time known, the peak width gives the diffusion coefficient through Taylor's dispersion law (Taylor, 1953), and the Stokes-Einstein equation converts that into a hydrodynamic radius (Einstein, 1905).
.svg)
Why is hydrodynamic radius reported as an absolute size?
FIDA is a first principles technology, which in physics means that it starts at the level of established science rather than relying on empirical modelling and parameter fitting. It rests on the descriptions of Taylor and Einstein, it is absolute, and it requires no calibration.
In practice that means the number is a real size in nanometres rather than a position relative to something else. There is no reference protein, no standard curve and no column to calibrate against, so the result does not depend on the standards that happen to be available, and it can be compared directly against a radius predicted from a structure. The molecule is measured in solution, in its native state, with no immobilisation and no separation step, so what is measured is the molecule as it actually is in that buffer, in serum or in a crude matrix.
Can aggregation and polydispersity be measured at the same time as size?
The radius is the main measurement, and it does not arrive alone. The same run also reports polydispersity index, aggregation, stickiness, sample loss, viscosity and labelling quality, which together form the quality control picture for that sample. Two further readouts run alongside it: Lambda Dynamics, a ratiometric fluorescence measurement well suited to protein and small molecule interactions, and BRIC, Binding Related Intensity Change, which picks up binding that does not change the size.
That matters for a size measurement in particular, because a radius on its own can be misread: an aggregate, a polydisperse sample and a genuinely larger species all push the number up. The application note Assessment of Sample Quality with Every Measurement shows the radius measured alongside aggregation, polydispersity and stickiness from a single sample, which is how you tell the three apart.
Is hydrodynamic radius the same as Stokes radius?
Yes. The two terms describe the same quantity, the effective radius a molecule appears to have as it moves through a liquid. Stokes radius takes its name from Sir George Gabriel Stokes, whose work on the drag experienced by a sphere in a viscous fluid underlies the relationship. The fuller account is in Stokes radius and the Stokes-Einstein equation.
How does hydrodynamic radius differ from radius of gyration?
Hydrodynamic radius and radius of gyration describe different things. Rh is the effective radius a molecule has as it diffuses, including the water it carries, and comes from a transport measurement. The radius of gyration, Rg, is the mass-weighted root mean square distance of the molecule's mass elements from its centre of mass, and comes from a scattering measurement or from a structure.
The two are not interchangeable, and the ratio between them reflects shape rather than size. The PDB correlator in the Fida software takes a PDB file, from a crystal structure or an AlphaFold prediction, and calculates Rh, Rg and the ratio between them, along with the name, shape, atom count and molecular weight of the structure. A measured radius can then be compared directly against the one expected from the structure, and a large difference usually means the protein is not in the form it was assumed to be in, for instance because it has oligomerised at working concentration.
Frequently asked questions
What is the hydrodynamic radius in simple terms?
The hydrodynamic radius is the size a molecule appears to have as it moves through a liquid. It is the radius of a hypothetical sphere that would diffuse at the same rate, under the same conditions (Einstein, 1905), and it includes both the molecule itself and the water travelling with it.
Does the hydrodynamic radius change when a molecule binds?
It generally increases when two or more molecules bind, because the complex diffuses more slowly than either partner alone. This is the basis for measuring affinity from a size readout (Jensen and Østergaard, 2010).
What techniques measure hydrodynamic radius?
Flow Induced Dispersion Analysis measures it from in-capillary diffusion in a laminar flow, in solution and in nanolitre volumes, and returns it as an absolute size in nanometres with no calibration and no immobilisation (Jensen and Østergaard, 2010). Because the measurement is made on the sample as it is, it works in buffer, in serum and in crude matrices such as unpurified lysate, and the same run returns the aggregation, polydispersity and viscosity of that sample as well.
Can hydrodynamic radius be measured without labelling or immobilisation?
Nothing is immobilised in a FIDA measurement: both partners stay free in solution, so there is no surface, no tethering and no geometry constraint on binding. Detection is fluorescence based, which in the native format uses the intrinsic fluorescence of the protein and otherwise requires one partner to carry a fluorescent tag.
References
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. doi.org/10.1002/andp.19053220806
Jensen, H., & Østergaard, J. (2010). Flow induced dispersion analysis quantifies noncovalent interactions in nanoliter samples. Journal of the American Chemical Society, 132(12), 4070-4071. doi.org/10.1021/ja100484d
Taylor, G. I. (1953). Dispersion of soluble matter in solvent flowing slowly through a tube. Proceedings of the Royal Society of London A, 219(1137), 186-203. doi.org/10.1098/rspa.1953.0139
Related resources
The technical note The relationship between hydrodynamic radius and molecular weight contains the calibration experiment and the quality control zones. The application note Assessment of Sample Quality with Every Measurement shows the radius measured alongside aggregation, polydispersity and stickiness from a single sample. Immobilisation Free In-solution Kinetics Using Flow Induced Dispersion Analysis covers affinity and kinetics from the same size readout, and Screening de novo designed protein binders in unpurified lysate shows it working in a crude matrix.
On this site, the molecular weight to size protein radius calculator converts between mass and radius, the molecular size readout covers the measurement, and the PDB correlator predicts a radius from a structure.
