Learning Lab · Polymer theory
The ideal chain
Why a chain's size grows as the square root of its length: the scaling every coarse-grained chromatin model inherits.
The ideal chain is the starting point for essentially every polymer calculation, including the chromatin models on the other pages of this lab. It is a chain of segments each pointing in a direction chosen at random, with no forces and no chemistry. It is worth understanding precisely because so much survives that severity.
What it predicts immediately is a scaling law. Doubling the number of segments does not double the size of the molecule; it multiplies it by about \(\sqrt{2}\). Size grows as
\[R \sim N^{\nu}, \qquad \nu = \tfrac{1}{2}\]and the log–log plot beside the chain builds that relationship up as you drag the length, with the theoretical slope drawn over it for comparison.
Two measures of size
The end-to-end distance \(R_{ee}\) is the separation of the two ends. It is the natural thing to measure but a poor description of a coil, since it ignores everything between the endpoints and fluctuates enormously. A chain folded back on itself has a small \(R_{ee}\) while still occupying plenty of space.
The radius of gyration \(R_g\) is the root-mean-square distance of the segments from their own centre of mass, and it measures the space actually filled. It is the quantity that generalises: you can compute it for a whole chromosome or for a five-monomer block, and the coarse-graining lab does exactly the latter to assign a size to each effective bead.
For a long ideal chain the two are related by \(\langle R_{ee}^2\rangle / \langle R_g^2\rangle = 6\), which falls out of the definitions alone. The readout shows this ratio for the single conformation on screen, and it will scatter widely around 6, often falling between 2 and 12. That scatter is not measurement error. A polymer at equilibrium is an ensemble, and any one conformation is an unremarkable draw from a broad distribution. Keeping that in view is what stops a single simulation snapshot being over-interpreted.
Why this matters for chromatin
Real chromatin is not an ideal chain. It is self-avoiding, it is confined, it is acted on by motor proteins, and segments carrying the same histone modifications attract one another. Every one of those effects changes \(\nu\), or breaks the single-exponent description altogether.
That is exactly why the ideal chain is worth knowing precisely. It is the null model: the structure a stretch of chromatin would have if nothing in its epigenetic state mattered. Any measured departure from \(\nu = 1/2\), any excess of contacts over what a random walk predicts, is a quantity that some mechanism has to account for. The interesting number is never the exponent on its own. It is the difference between the observed value and this one.
The coarse-graining lab puts that logic to work: it establishes what an ideal chain does under coarse-graining so that the behaviour of real chromatin can be read against it.
Why it matters · Random-walk statistics are the reference against which real chromatin is measured, and the reason a coarse-grained bead's size can be calculated rather than chosen.