Single Molecule Force Spectroscopy (WLC) Calculator
Find the entropic restoring force of a stretched biopolymer from its persistence length, contour length, and extension using the worm-like chain model.
🔗 What is a Single Molecule Force Spectroscopy (WLC) Calculator?
A single molecule force spectroscopy (WLC) calculator converts the geometry of a stretched biopolymer into the entropic restoring force that molecule pulls back with. It applies the worm-like chain model, the standard polymer-physics description of a semi-flexible filament, using the Marko-Siggia interpolation formula. Feed it a persistence length, a contour length, an extension, and a temperature, and it returns the force in piconewtons along with the local spring stiffness of the chain at that extension.
The tool is used daily in three kinds of laboratory work. Optical-tweezers and magnetic-tweezers groups use it to predict how much force a DNA or RNA tether will exert at a chosen extension before they design the trap stiffness. AFM force-spectroscopy groups fit unfolding traces of titin, ubiquitin, or engineered polyprotein constructs, where each sawtooth peak is preceded by a worm-like chain rise whose persistence length and contour length are the fit parameters. Polymer chemists apply the same formula to synthetic chains such as polyethylene glycol to extract stiffness from stretching data.
Two misconceptions are worth clearing up. First, the restoring force does not depend on how long the molecule is in any absolute sense, it depends on the relative extension x/L0, so a 1 micron DNA and a 16 micron DNA held at the same fractional extension feel exactly the same force. Second, the force is entropic rather than enthalpic below about 10 pN for double-stranded DNA, meaning the chain resists stretching because it is losing accessible conformations, not because chemical bonds are being strained.
This calculator is useful because the Marko-Siggia expression is easy to mistype and the unit conversions (nanometers to meters, joules to piconewtons, Celsius to Kelvin) are the most common source of order-of-magnitude errors. It also plots the full force-extension curve and marks your working point on it, so you can see immediately whether you are on the flat entropic shoulder or on the steep wall near full extension.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Short DNA Tether on Magnetic Tweezers
Lp=50 nm, L0=1000 nm (about 2.9 kilobase pairs), stretched to 80% at 25°C
Example 2 — Lambda DNA Pulled Close to Full Extension
Lp=50 nm, L0=16500 nm (lambda phage DNA, about 48.5 kilobase pairs), stretched to 95% at 25°C
Example 3 — Unfolded Polypeptide in an AFM Pulling Experiment
Lp=0.4 nm, L0=100 nm (about 280 residues at 0.36 nm each), stretched to 60% at 25°C
❓ Frequently Asked Questions
🔗 Related Calculators
What is the worm-like chain (WLC) model?
The worm-like chain model treats a biopolymer as a continuously flexible rod with a bending stiffness set by its persistence length Lp. Stretching such a chain reduces the number of accessible thermal conformations, so the chain pulls back with an entropic restoring force that grows without bound as the extension approaches the contour length L0.
What is the Marko-Siggia formula for WLC force?
The Marko-Siggia interpolation formula is F = (kBT/Lp)[1/(4(1-x/L0)^2) - 1/4 + x/L0], where kB is the Boltzmann constant, T is absolute temperature, Lp is the persistence length, x is the end-to-end extension, and L0 is the contour length. It reproduces the exact WLC behaviour at both low and high force and stays within roughly 10% in between.
What is the persistence length of DNA?
Double-stranded B-form DNA has a persistence length of about 50 nm (roughly 150 base pairs) in standard physiological salt near 100 mM monovalent ions. Single-stranded DNA is far more flexible at about 1 to 2 nm, and an unfolded polypeptide chain is more flexible still at roughly 0.4 nm.
How much force does it take to stretch DNA to 80% of its contour length?
For DNA with a 50 nm persistence length at 25 degrees Celsius, stretching to 80% of the contour length requires about 0.56 pN. Because the persistence length sets the whole force scale, this value is independent of how long the DNA molecule is, only the relative extension x/L0 matters.
Why does the WLC force diverge near full extension?
The restoring force is entropic. As x approaches L0 the chain runs out of accessible thermal conformations, so the entropy loss per additional nanometre of extension grows without limit. Mathematically the (1 - x/L0)^-2 term blows up, which is why real experiments see force rise steeply above roughly 90% extension.
What is the difference between contour length and persistence length?
Contour length L0 is the total end-to-end length of the fully straightened chain, which scales with the number of monomers. Persistence length Lp is the length over which the chain's direction stays correlated, which measures stiffness and does not depend on how long the molecule is.
Does the WLC model apply to unfolded proteins?
Yes. Unfolded polypeptides are routinely fitted with the worm-like chain model in AFM and optical-tweezers experiments, typically using a persistence length between 0.35 and 0.8 nm and a contour length of about 0.36 nm per residue. The very short persistence length is why unfolded proteins require tens of piconewtons where DNA needs less than one.
How do I convert base pairs or residues into contour length?
Multiply the number of monomers by the rise per monomer: about 0.34 nm per base pair for B-form double-stranded DNA, about 0.59 nm per nucleotide for stretched single-stranded DNA, and about 0.36 nm per amino acid residue for an extended polypeptide.
What is the entropic spring stiffness of a stretched biopolymer?
The local stiffness is the derivative dF/dx = (kBT/(Lp L0))[0.5/(1-x/L0)^3 + 1], reported by this calculator in pN/nm. Stiffness increases sharply near full extension, which is why a molecule held at high extension acts as a much stiffer spring than the same molecule held slack.
Does temperature change the WLC force?
The force scales directly with absolute temperature through the kBT prefactor, so warming from 25 to 37 degrees Celsius raises the computed force by about 4%. The persistence length of real biopolymers also changes slightly with temperature, an effect the pure WLC formula does not include.
Which instruments measure single-molecule forces in this range?
Magnetic tweezers cover roughly 0.01 to 20 pN, optical tweezers cover roughly 0.1 to 100 pN, and atomic force microscopy covers roughly 10 pN up to several nanonewtons. Compare the calculator's output against these windows when planning a stretching experiment.
Why does the WLC model ignore enthalpic stretching?
The pure worm-like chain treats the backbone as inextensible, so all compliance is entropic. Above roughly 10 pN for double-stranded DNA the backbone itself begins to stretch, and researchers switch to the extensible worm-like chain, which adds a stretch modulus term of about 1000 pN.