Calculate coil distance
Use residue count, backbone length, and a selected polymer model.
Example data
| Residues | Residue length | Kuhn length | Model | Estimated RMS distance |
|---|---|---|---|---|
| 100 | 0.38 nm | 1.50 nm | Freely jointed chain | 7.55 nm |
| 250 | 0.38 nm | 1.50 nm | Worm-like chain | 11.74 nm |
| 500 | 0.38 nm | 1.50 nm | Empirical scaling chain | 14.68 nm |
The values are illustrative. Experimental conditions can change coil dimensions.
Formula used
For a freely jointed chain, the calculator uses the root-mean-square end-to-end distance:
Here, n is the residue count, l is contour length per residue, L is contour length, and b is Kuhn length. The worm-like chain option uses persistence length, where Lp = b ÷ 2. The empirical option uses Rrms = a × Nν.
The radius of gyration is estimated as Rg = Rrms ÷ √6. The percentile distance is obtained from a three-dimensional Gaussian-coil distribution.
How to use this calculator
- Enter the total amino acid residues in your peptide or protein region.
- Set the contour length per residue. Use 0.38 nm when appropriate.
- Enter a Kuhn length that reflects the chain flexibility.
- Choose freely jointed, worm-like, or empirical scaling behavior.
- Add scaling values when you choose the empirical model.
- Choose a percentile and preferred display unit.
- Press Calculate distance. Review the RMS result and supporting metrics.
Understanding random coil dimensions
Random coils occupy statistical space
A flexible peptide does not hold one permanent shape. Its backbone samples many conformations. Every conformation has a different end-to-end distance. A random coil model summarizes that changing ensemble. The root-mean-square distance is especially useful. It gives larger separations suitable statistical weight. It does not describe a fully stretched chain. It describes the expected scale of molecular separation across many conformations.
Residue count increases the coil size
More residues create a longer contour length. The coil grows more slowly than the contour. This happens because segments point in many directions. Their directional effects partly cancel. Doubling residues therefore does not normally double random coil distance. Ideal chain behavior often follows a square-root relationship. Real chains can follow a scaling exponent instead. Solvent quality, sequence composition, charge, and temperature can influence that exponent.
Flexibility controls effective segment length
Kuhn length translates local stiffness into an equivalent segment length. A larger Kuhn length means the chain remains directionally correlated over a longer distance. This usually raises the predicted coil size for the same contour length. A smaller Kuhn length represents a more flexible chain. The freely jointed model assumes independent Kuhn segments. It works best when many segments are present. Short chains may require a more detailed stiffness-aware model.
Sequence effects may also matter. Proline can restrict local angles. Glycine can increase flexibility. Charged residues may expand the chain under low salt. Hydrophobic residues may encourage compaction. Treat a single Kuhn length as a useful average. Refine it when sequence-specific measurements become available.
Choose a model that matches the question
The freely jointed chain model gives a fast ideal estimate. The worm-like chain model includes smooth local bending. It is useful when persistence length matters. The empirical scaling model uses measured or literature-derived parameters. It can better reflect particular solvents or sequences. No simple model captures every interaction. Use measured values when precision matters. Compare model outputs when conditions are uncertain. Large differences can reveal sensitivity to flexibility assumptions.
Read the outputs together
The RMS distance gives the central size estimate. Radius of gyration describes mass spread around the molecular center. Contour length shows the maximum path length after full extension. The percentile distance estimates a larger radial separation within a Gaussian distribution. These metrics answer different questions. A coil can have a small RMS distance while retaining a much larger contour length. That distinction is important for linkers, disordered regions, and polymer design.
Use sensible input assumptions
Start with a residue contour length near 0.38 nanometres when suitable. Then select a Kuhn length supported by your system. Ionic strength can alter charged chain behavior. Denaturants can expand some peptide chains. Binding partners can compact them. The calculated value should guide reasoning, not replace experiment. Small-angle scattering, fluorescence transfer, and simulations can provide useful checks. Record your selected parameters for reproducible comparisons across sequences.
Frequently asked questions
What distance does the calculator report?
The main result is the root-mean-square end-to-end distance. It represents the typical separation scale between chain ends across many random conformations.
Is a random coil completely unfolded?
Not necessarily. Random coil means the chain lacks one stable, repeating conformation. It can still show local preferences, transient contacts, or partial structure.
What residue length should I use?
A common polypeptide backbone estimate is 0.38 nm per residue. Use a different value only when your structural model or experimental convention supports it.
What is Kuhn length?
Kuhn length is an effective segment length. It converts a flexible chain into idealized independently oriented segments while preserving its large-scale dimensions.
When should I choose the worm-like chain model?
Choose it when local stiffness is important, especially for shorter chains or systems with known persistence length. It treats directional memory more realistically.
What does the scaling exponent mean?
The exponent describes how coil size changes with residue count. An exponent near 0.5 reflects ideal behavior. Larger values often indicate expanded-chain behavior.
Why is coil distance smaller than contour length?
The contour follows every bond along the backbone. Random orientations fold the chain through space, producing a much shorter straight-line end-to-end separation.
What is radius of gyration?
Radius of gyration measures how mass is distributed around a molecular center. For an ideal Gaussian chain, it is the RMS end-to-end distance divided by √6.
Does solvent affect the result?
Yes. Solvent quality, temperature, salt concentration, and charge can change chain expansion. Use empirical scaling parameters when experimental conditions are known.
Can I use this for folded proteins?
Use it mainly for flexible linkers, disordered regions, or denatured chains. A compact folded protein requires structural dimensions from a different model.
Are the percentile distances experimental limits?
No. They are statistical estimates from a Gaussian-coil distribution. They describe modeled conformations, not hard physical boundaries or guaranteed experimental measurements.