DNA Persistence Length and Contour Calculator

Find the contour length, root-mean-square end-to-end distance, radius of gyration and stiffness regime of a DNA molecule using worm-like chain theory.

🧬 DNA Persistence Length and Contour Calculator
DNA length (N)1000 bp
bp
10 bp20000 bp
Rise per base pair
nm/bp
Persistence length (Lp)
nm
Contour length (L)
—
End-to-end distance (Rrms)
—
Radius of gyration (Rg)
—
Kuhn segments (L / 2Lp)
—
Flexibility ratio (L / Lp)
—
Compaction (Rrms / L)
—
Regime
—
Step-by-step working

🧬 What is the DNA Persistence Length and Contour Calculator?

The DNA persistence length and contour calculator converts a base-pair count into the physical dimensions of a real DNA molecule in solution. It first multiplies base pairs by the helical rise to give the contour length, the full stretched-out length of the double helix, and then applies Kratky-Porod worm-like chain theory to find how large that molecule actually is once thermal motion has coiled it up. The outputs are the root-mean-square end-to-end distance, the Benoit-Doty radius of gyration, the number of Kuhn segments, the flexibility ratio L divided by Lp, and a plain-language regime label.

These numbers matter in a surprising number of settings. Single-molecule biophysicists fitting force-extension curves from optical tweezers or magnetic tweezers start from exactly this worm-like chain model. Gel electrophoresis and microfluidic device designers need the coil size to predict how a fragment threads through pores or channels. Structural biologists interpreting small-angle X-ray scattering compare a measured radius of gyration against the theoretical Rg to test whether their construct is compact or extended. Chromatin researchers use the fact that a nucleosome wraps roughly one persistence length of DNA to explain why nucleosome formation is energetically expensive, and nanotechnologists designing DNA origami need to know when a strut is short enough to count as rigid.

The most common misconception is that contour length describes how big a DNA molecule is. It does not. Contour length is what you would measure if you pulled the molecule perfectly straight, whereas a free molecule in solution is a fluctuating coil that occupies a far smaller volume. Lambda phage DNA has a contour length of about 16.5 micrometres but an Rrms of only about 1.28 micrometres. A second misconception is that persistence length is a fixed constant of DNA. It is buffer-dependent, rising in low salt and falling as monovalent or divalent counterions screen the backbone charge. A third is that single-stranded and double-stranded nucleic acids behave alike, when in fact single-stranded DNA and RNA are one to two orders of magnitude more flexible.

This calculator is useful because it puts all of those quantities side by side and shows the crossover explicitly. The reference table lists common DNA sizes from a 100 bp fragment up to a 50 kb genome with their contour lengths, end-to-end distances and regimes, and your own molecule is inserted into that table in the right position. The two log-log charts plot Rrms against length, where the slope changes from 1 in the rod regime to 0.5 in the coil regime, and the compaction ratio Rrms divided by L, which falls steadily as the molecule gets longer.

📐 Formula

L  =  N × rise   and   ⟨R²⟩  =  2 Lp L [1 − (Lp÷L)(1 − e−L/Lp)]
N = number of base pairs
rise = axial rise per base pair, 0.34 nm for B-form DNA
L = contour length, the fully extended length of the duplex (nm)
Lp = persistence length, about 50 nm for double-stranded DNA in 100 mM monovalent salt
⟨R²⟩ = mean square end-to-end distance from the Kratky-Porod worm-like chain (nm²)
Rrms = √⟨R²⟩, the root-mean-square end-to-end distance (nm)
Radius of gyration (Benoit-Doty): Rg² = L Lp÷3 − Lp² + (2Lp³÷L)[1 − (Lp÷L)(1 − e−L/Lp)]
Kuhn segment length is 2Lp, so the segment count is L ÷ 2Lp
Limits: when L « Lp, Rrms → L (rigid rod). When L » Lp, ⟨R²⟩ → 2LpL (ideal coil, Rrms grows as √L)
Example: N = 1000 bp, rise = 0.34 nm, Lp = 50 nm gives L = 340 nm, Rrms = 170.31 nm and Rg = 61.60 nm.
Worm-like chain averages follow Kratky and Porod (1949); the radius of gyration expression follows Benoit and Doty (1953).

📖 How to Use This Calculator

Steps

1
Enter the DNA length in base pairs. Type or drag the number of base pairs in your DNA fragment, anywhere from a short oligonucleotide to a whole viral genome.
2
Set the helical rise. Enter the rise per base pair, which is 0.34 nm for standard B-form DNA and slightly different for A-form or Z-form.
3
Set the persistence length. Enter the persistence length for your buffer conditions, about 50 nm for double-stranded DNA in physiological salt and 1 to 3 nm for single-stranded nucleic acid.
4
Read size, shape and regime. See the contour length, root-mean-square end-to-end distance, radius of gyration, Kuhn segment count and whether the molecule behaves as a rod, a semi-flexible chain or a coil.

💡 Example Calculations

Example 1 — A 1 kb PCR Product in Physiological Salt

N = 1000 bp, rise = 0.34 nm/bp, Lp = 50 nm

1
L = 1000 × 0.34 = 340.00 nm, so L / Lp = 340 ÷ 50 = 6.80, the semi-flexible regime
2
⟨R²⟩ = 2 × 50 × 340 × [1 − (50÷340)(1 − e−6.8)] = 34000 × 0.85310 = 29005.55 nm²
3
Rrms = √29005.55 = 170.31 nm, which is 50.09% of the contour length
4
Rg = 61.60 nm and the chain contains 340 ÷ 100 = 3.40 Kuhn segments
L = 340.00 nm, Rrms = 170.31 nm, Rg = 61.60 nm
Try this example →

Example 2 — Nucleosomal DNA at Exactly One Persistence Length

N = 150 bp (one nucleosome wrap), rise = 0.34 nm/bp, Lp = 50 nm

1
L = 150 × 0.34 = 51.00 nm, so L / Lp = 1.02, sitting right at the rod-to-coil crossover
2
Rrms = 43.62 nm, still 85.54% of the contour length because the fragment is barely longer than one persistence length
3
Rg = 13.38 nm and the chain contains only 0.51 Kuhn segments, far too few for random-coil statistics
4
Bending this nearly straight fragment into the tight nucleosome superhelix is exactly why histones must supply binding energy
L = 51.00 nm, Rrms = 43.62 nm (85.54% of L)
Try this example →

Example 3 — Lambda Phage Genome as a Flexible Coil

N = 48502 bp (bacteriophage lambda), rise = 0.34 nm/bp, Lp = 50 nm

1
L = 48502 × 0.34 = 16490.68 nm, about 16.5 micrometres if pulled straight
2
L / Lp = 329.81, deep in the flexible-coil regime where ⟨R²⟩ approaches 2 Lp L
3
Rrms = 1282.21 nm, only 7.78% of the contour length, and Rg = 521.88 nm
4
The molecule contains 164.91 Kuhn segments, enough for ideal-chain square-root scaling to hold well
L = 16490.68 nm, Rrms = 1282.21 nm, Rg = 521.88 nm
Try this example →

Example 4 — A 100 bp Fragment Behaves Like a Rigid Rod

N = 100 bp, rise = 0.34 nm/bp, Lp = 50 nm

1
L = 100 × 0.34 = 34.00 nm, so L / Lp = 0.68, below 1 and therefore rod-like
2
Rrms = 30.55 nm, which is 89.84% of the contour length, so the fragment is close to straight
3
Rg = 9.20 nm, near the rigid-rod limit of L ÷ √12 = 9.81 nm
4
Only 0.34 Kuhn segments are present, so coil formulas would badly overestimate how compact this fragment is
L = 34.00 nm, Rrms = 30.55 nm, regime = Rod-like (stiff)
Try this example →

❓ Frequently Asked Questions

What is the persistence length of DNA?+
About 50 nm, roughly 147 base pairs, for double-stranded B-form DNA in physiological monovalent salt near 100 mM. Persistence length is the contour distance over which the direction of the polymer backbone stays correlated, so it is a direct measure of bending stiffness.
How do you calculate the contour length of DNA?+
Multiply the number of base pairs by the helical rise: L = N x 0.34 nm for B-form DNA. A 1000 bp fragment therefore has a contour length of 340 nm, and the 48502 bp lambda phage genome stretches to 16490.68 nm, about 16.5 micrometres.
What is the worm-like chain formula for end-to-end distance?+
The Kratky-Porod result is mean square end-to-end distance = 2 Lp L [1 - (Lp/L)(1 - exp(-L/Lp))]. Taking the square root gives Rrms. For a 1000 bp fragment with Lp = 50 nm and L = 340 nm this gives Rrms = 170.31 nm, almost exactly half the contour length.
What is the radius of gyration of a DNA molecule?+
The Benoit-Doty expression gives Rg squared = L Lp/3 - Lp squared + (2 Lp cubed / L)[1 - (Lp/L)(1 - exp(-L/Lp))]. For 1000 bp of B-form DNA with Lp = 50 nm the radius of gyration is 61.60 nm. Rg is what light scattering and small-angle scattering experiments measure directly.
What is a Kuhn segment and how many are in my DNA?+
A Kuhn segment is twice the persistence length, 100 nm or about 294 base pairs for double-stranded DNA. The number of Kuhn segments is L divided by 2Lp, so a 1000 bp fragment contains 3.4 Kuhn segments, too few for ideal-chain statistics to apply cleanly.
When does DNA behave like a rigid rod instead of a coil?+
When the contour length is shorter than the persistence length, so L/Lp is less than 1. A 100 bp fragment has L = 34 nm and L/Lp = 0.68, and its Rrms of 30.55 nm is 89.84 percent of its contour length, meaning it is nearly straight.
How does salt concentration change DNA persistence length?+
Higher monovalent salt screens phosphate-phosphate repulsion along the backbone and lowers the electrostatic contribution to stiffness, so Lp drops. Typical values run from above 60 nm below 10 mM sodium to about 45 to 50 nm at 100 mM and slightly lower still in the presence of magnesium.
What is the persistence length of single-stranded DNA or RNA?+
Roughly 1 to 3 nm, one to two orders of magnitude shorter than double-stranded DNA, because a single backbone has no base-pair stacking to resist bending. Enter that value in the persistence length field and the same worm-like chain equations apply.
Why is the end-to-end distance so much smaller than the contour length?+
Because thermal bending randomises backbone direction over each persistence length. In the flexible coil limit Rrms grows only as the square root of L, so lambda phage DNA with a 16490.68 nm contour has an Rrms of just 1282.21 nm, about 7.78 percent of its stretched-out length.
What is the difference between Rrms and Rg?+
Rrms is the root-mean-square distance between the two chain ends, while Rg is the root-mean-square distance of all chain segments from the centre of mass. For an ideal flexible coil Rrms is about the square root of 6 times Rg, so Rg is always the smaller of the two.
Does the worm-like chain model include excluded volume?+
No. The Kratky-Porod model is an ideal chain that ignores self-avoidance, so for very long DNA in good solvent the true Rrms is somewhat larger than the value computed here, scaling closer to the Flory exponent of 0.588 rather than 0.5. The ideal result remains an excellent approximation up to tens of kilobases.
How many base pairs correspond to one persistence length?+
About 147 base pairs, since 50 nm divided by 0.34 nm per base pair gives 147. That is very close to the 147 bp wrapped around a histone octamer in a nucleosome core particle, a coincidence that biophysicists use to explain the elastic cost of chromatin packaging.
Tags: Dna Persistence Length Calculator Contour Length Calculator Worm-Like Chain Radius of Gyration Dna Dna Flexibility Calculator Formula Free Online

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What is the persistence length of DNA?

About 50 nm, roughly 147 base pairs, for double-stranded B-form DNA in physiological monovalent salt near 100 mM. Persistence length is the contour distance over which the direction of the polymer backbone stays correlated, so it is a direct measure of bending stiffness.

How do you calculate the contour length of DNA?

Multiply the number of base pairs by the helical rise: L = N x 0.34 nm for B-form DNA. A 1000 bp fragment therefore has a contour length of 340 nm, and the 48502 bp lambda phage genome stretches to 16490.68 nm, about 16.5 micrometres.

What is the worm-like chain formula for end-to-end distance?

The Kratky-Porod result is mean square end-to-end distance = 2 Lp L [1 - (Lp/L)(1 - exp(-L/Lp))]. Taking the square root gives Rrms. For a 1000 bp fragment with Lp = 50 nm and L = 340 nm this gives Rrms = 170.31 nm, almost exactly half the contour length.

What is the radius of gyration of a DNA molecule?

The Benoit-Doty expression gives Rg squared = L Lp/3 - Lp squared + (2 Lp cubed / L)[1 - (Lp/L)(1 - exp(-L/Lp))]. For 1000 bp of B-form DNA with Lp = 50 nm the radius of gyration is 61.60 nm. Rg is what light scattering and small-angle scattering experiments measure directly.

What is a Kuhn segment and how many are in my DNA?

A Kuhn segment is twice the persistence length, 100 nm or about 294 base pairs for double-stranded DNA. The number of Kuhn segments is L divided by 2Lp, so a 1000 bp fragment contains 3.4 Kuhn segments, too few for ideal-chain statistics to apply cleanly.

When does DNA behave like a rigid rod instead of a coil?

When the contour length is shorter than the persistence length, so L/Lp is less than 1. A 100 bp fragment has L = 34 nm and L/Lp = 0.68, and its Rrms of 30.55 nm is 89.84 percent of its contour length, meaning it is nearly straight.

How does salt concentration change DNA persistence length?

Higher monovalent salt screens phosphate-phosphate repulsion along the backbone and lowers the electrostatic contribution to stiffness, so Lp drops. Typical values run from above 60 nm below 10 mM sodium to about 45 to 50 nm at 100 mM and slightly lower still in the presence of magnesium.

What is the persistence length of single-stranded DNA or RNA?

Roughly 1 to 3 nm, one to two orders of magnitude shorter than double-stranded DNA, because a single backbone has no base-pair stacking to resist bending. Enter that value in the persistence length field and the same worm-like chain equations apply.

Why is the end-to-end distance so much smaller than the contour length?

Because thermal bending randomises backbone direction over each persistence length. In the flexible coil limit Rrms grows only as the square root of L, so lambda phage DNA with a 16490.68 nm contour has an Rrms of just 1282.21 nm, about 7.78 percent of its stretched-out length.

What is the difference between Rrms and Rg?

Rrms is the root-mean-square distance between the two chain ends, while Rg is the root-mean-square distance of all chain segments from the centre of mass. For an ideal flexible coil Rrms is about the square root of 6 times Rg, so Rg is always the smaller of the two.

📌 Quick Tips

💡Double-stranded B-form DNA has a persistence length near 50 nm, about 147 base pairs, in physiological monovalent salt. Single-stranded DNA and RNA are far more flexible, with persistence lengths around 1 to 3 nm.
💡Persistence length falls as salt concentration rises, because screening reduces the electrostatic repulsion between neighbouring phosphates along the backbone. Below about 10 mM monovalent salt Lp can climb above 60 nm.
💡The ratio L/Lp is the single number that tells you which regime you are in. Below 1 the molecule behaves like a stiff rod, between 1 and 10 it is semi-flexible, and above 10 it behaves as a flexible random coil.
💡A nucleosome wraps about 147 base pairs, almost exactly one persistence length, which is why bending DNA into a nucleosome costs a large amount of elastic energy and requires the histone octamer to pay for it.
💡Rrms and Rg describe the average size of a freely fluctuating chain in solution, not a fixed shape. Any single molecule at any instant may be far more extended or far more compact than these ensemble averages.