Icosahedral Lattice Counts
Calculate the Caspar–Klug triangulation number, capsomere count, and protein-subunit count for a conventional icosahedral lattice.
Description
Calculate Caspar–Klug triangulation, conventional capsomere count, and protein-subunit count from nonnegative lattice coordinates.
Many regular viral capsids organize protein subunits with icosahedral symmetry. The Caspar–Klug construction describes a conventional quasi-equivalent lattice using two nonnegative integer coordinates, h and k. Its triangulation number T distinguishes allowed lattice sizes.1
In this construction a capsid contains 60T protein subunits organized into 12 pentamers and 10(T−1) hexamers. Counting the pentamer and hexamer groups gives 10T+2 capsomeres. A capsomere is a group of subunits, so its count differs from the total protein count.
Icosahedral Lattice Counts applies these conventional formulas to supplied h and k. It does not infer coordinates or architecture from an image, sequence, or named virus.
When to calculate a conventional capsid lattice
Use the tool for a specified Caspar–Klug geometry or a teaching comparison of its allowed counts. Confirm that this construction is appropriate before applying the counts to a biological structure.
- Calculate T from a pair of lattice coordinates.
- Distinguish capsomere groups from protein subunits.
- Compare coordinate pairs such as (1,0) and (1,1).
How to enter lattice coordinates and calculate capsid counts
- Enter H and K as nonnegative integers, with at least one greater than zero.
- Run the calculation and read Triangulation number, Capsomere count, and Protein subunits.
- Interpret the counts as the conventional geometry defined by the inputs.
T = h² + h×k + k²; capsomereCount = 10T + 2; proteinSubunits = 60T The implementation calculates T first, requires a positive safe integer, and then checks both output counts remain exact safe integers.
{
"input": {
"h": 1,
"k": 1
},
"output": {
"triangulationNumber": 3,
"capsomereCount": 32,
"proteinSubunits": 180
}
}Swapping h and k preserves these counts; this tool does not report handedness. Zero/zero coordinates, negative or fractional coordinates, and excessively large combinations exceeding exact integer representation produce errors.
Limitations and interpretation
- The construction is a structural model, not a universal description of every icosahedral-looking capsid.
- The output does not identify protein species, dimers, accessory components, maturation state, or pseudo-triangulation architecture.
- Counts alone do not specify physical radius, shell thickness, packing density, or genome capacity.
- No sequence analysis, structural fitting, or assembly prediction is performed.
Alternative or Complementary methods
Use a helical-symmetry calculation for a rise-and-rotation description of a different architecture. A spherical-volume calculation needs a physical radius and answers a size question that h and k alone cannot resolve.
References
-
Physical principles in the construction of regular viruses — Cold Spring Harbor Symposia on Quantitative Biology; D. L. D. Caspar and A. Klug (1962)
Similar or alternative tools
- Helical Capsid Pitch
Calculate subunits per turn and axial pitch from supplied helical rise and rotation.
- Sphere Volume Calculator
Compute an ideal spherical volume from a measured or supplied radius.