Calculate protein hydrodynamic radius and diameter in nm from molecular weight in kDa using a typical globular shape, or enter your own frictional ratio.
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Protein Hydrodynamic Radius Formula
The protein hydrodynamic radius is the radius of an equivalent hard sphere that would diffuse through solution at the same rate as the actual protein. This calculator estimates that radius from molecular weight and frictional ratio, using a typical protein partial specific volume of 0.73 mL/g.
When the default partial specific volume is assumed and molecular weight is entered in daltons, the equation can be simplified into a practical shortcut for nanometer output:
Variable Definitions
- Hydrodynamic radius is the estimated radius of the protein in solution, typically reported in nanometers, angstroms, or micrometers.
- Molecular weight is the mass of the protein or protein complex. Use the mass of the actual species present in solution, including tags or oligomeric assembly when relevant.
- Frictional ratio is f/fโ relative to a smooth unhydrated sphere of equal volume. It includes hydration and asymmetry; in this convention it is at least 1 and does not uniquely determine shape.
- Partial specific volume represents the volume occupied per unit mass of protein. This calculator uses a common protein approximation of 0.73 mL/g.
- Avogadro's constant converts molar mass into the volume of a single molecule.
Rearranged Forms
Because the calculator can solve for a missing variable when any two values are known, the same relationship can be rearranged as follows.
In this dimensional form, R_h must be in centimeters: multiply a radius in nanometers by 10โปโท before substitution. The calculator performs unit conversions automatically.
How to Calculate Protein Hydrodynamic Radius
- Enter the protein molecular weight in daltons or kilodaltons.
- Pick a protein shape (a typical globular protein is f/fโ โ 1.25), or choose Custom and enter a frictional ratio consistent with the unhydrated-sphere reference; hydration also contributes.
- Apply the hydrodynamic radius formula.
- Read the result in the desired unit. One nanometer corresponds to ten angstroms or one-thousandth of a micrometer.
The estimate is especially useful when you want a quick size approximation from mass and shape, rather than a direct measurement from solution diffusion data.
Example Calculation
For a protein with molecular weight of 50,000 Da and a frictional ratio of 1.5:
This means the protein behaves in solution like a hydrated sphere with a radius of about 3.66 nanometers.
How to Interpret the Result
- Larger radius means slower diffusion. At the same temperature and solvent viscosity, a larger hydrodynamic radius gives a lower diffusion coefficient.
- Frictional ratio matters linearly. At fixed mass and partial specific volume, a 10% increase in frictional ratio increases estimated radius by 10%.
- Mass matters by a cube-root relationship. Increasing molecular weight does increase radius, but not proportionally. Very large changes in mass produce more modest changes in radius.
- Compact proteins have smaller radii than extended ones of the same mass. Two proteins with similar molecular weights can have noticeably different hydrodynamic radii if one is folded compactly and the other is elongated or partially disordered.
Choosing a Reasonable Frictional Ratio
- 1.0: the ideal unhydrated reference sphere.
- Above 1.0: hydration and departures from the reference sphere increase friction; a value alone does not identify a unique shape.
- Below 1.0: inconsistent with this fixed reference and volume assumption; check the units, mass and model.
Use a measured or otherwise justified frictional ratio for the same species and reference convention. If it is unknown, compare explicitly stated assumptions rather than treating a guessed value as a measurement.
Connection to Diffusion Measurements
When a diffusion coefficient is available from experiment, hydrodynamic radius is more directly related to solution behavior through the Stokes-Einstein relation:
In that form, the radius depends on temperature, solvent viscosity, and the measured diffusion coefficient. The calculator on this page is a convenient approximation when those measurements are not available.
Practical Notes and Limitations
- Use the molecular weight of the full species in solution, not just the theoretical monomer sequence if the protein forms dimers, trimers, or larger complexes.
- Affinity tags, glycosylation, cofactors, and bound ligands can all affect the effective size.
- The fixed 0.73 mL/g partial specific volume is an approximation. Unusual composition can change the reference volume, while hydration also contributes through the frictional ratio.
- This estimate is most reliable for soluble proteins under near-native conditions and should be treated as an approximation rather than an exact experimental value.
