Coarse-graining a polymer

A chromosome has far too many degrees of freedom to simulate monomer by monomer, so monomers get grouped into single effective beads. The question the paper asks is what happens to the physics when you do that, because the coarse beads need their own bond lengths, their own spring constants and their own bending stiffness, and none of those are free parameters. They follow from the chain you started with, and they can be measured.

This page performs that measurement on the reference system: an ideal chain. That is deliberately the least interesting polymer available, and it is exactly why it is the right control. Whatever structure appears here cannot have come from chromatin biology, because there is none in the model.

What is being computed

The fine chain has bond lengths drawn from \(P(r) \propto r^2 e^{-\beta k_s (r-r_0)^2/2}\) with \(k_s = 100\), \(r_0 = 1\), \(k_BT = 1\), and directions uniform on the sphere, which is a genuinely three-dimensional walk and is what the rotating view shows. Because the bonds of an ideal chain are independent, this can be sampled exactly: no Monte Carlo, no equilibration, every conformation an independent and correctly weighted sample. That is why the histograms fill in as fast as they do.

Consecutive blocks of \(n_b\) monomers are then replaced by their centre of mass, blocks disjoint. Taking one monomer as a nucleosome and its linker, roughly 200 bp, the levels plotted correspond to roughly 1 kb (\(n_b = 5\)), 2 kb (\(n_b = 10\)) and 5 kb (\(n_b = 25\)). On that coarse chain four quantities are measured:

• the bond length \(b\) between neighbouring coarse beads
• the bond angle \(\theta_{cg} = \pi - \alpha\), where \(\alpha = \cos^{-1}(\hat{l}_i \cdot \hat{l}_{i+1})\), so a straight chain gives \(\theta = 180^\circ\)
• the effective spring constant \(k = k_BT/\mathrm{Var}(b)\)
• the bending stiffness \(\kappa = k_BT/\mathrm{Var}(\theta)\)

Why the angle distribution is plotted twice

The raw \(P(\theta_{cg})\) is misleading on its own, and for a purely geometric reason. There are many more ways to arrange a bond at \(90^\circ\) than at \(0^\circ\) or \(180^\circ\), because the azimuthal direction is free at right angles and pinched to a point at the extremes. An ideal chain with no coarse-graining therefore gives

\[P(\theta_{cg}) = \tfrac{1}{2}\sin\theta_{cg}\]

not a flat line, and that peak at \(90^\circ\) says nothing whatsoever about the polymer. Dividing the measure out gives the scaled density \(\tilde P(\theta_{cg}) = P(\theta_{cg})/\sin\theta_{cg}\), which for the same chain is exactly flat. The bottom panel is that curve, and the \(n_b = 1\) trace in it is the honest statement that an ideal chain has no angular preference at all.

Read the two panels together. The middle one shows what the raw statistics look like; the bottom one shows what is actually left once geometry is accounted for.

The result worth pausing on

Now raise the coarse-graining. The \(n_b = 1\) curve stays flat, but \(n_b = 2\), \(5\), \(10\) and \(25\) tilt progressively towards \(\theta > 90^\circ\). The chain has acquired a preference for extended configurations, which is to say a bending stiffness, that was demonstrably not present in the polymer we started from. As the paper puts it, this is “a lesser-discussed fact about polymers (even for the ideal chain)”: coarse-graining itself introduces the bias.

The mechanism is that neighbouring coarse bonds are not independent. Each is built partly from the monomers of the block lying between them, so they share information. Working that through gives an exact prediction for the correlation between consecutive coarse bond vectors,

\[\frac{\langle \mathbf{B}_i \cdot \mathbf{B}_{i+1}\rangle}{\langle |\mathbf{B}|^2\rangle} = \frac{n_b^2-1}{2(2n_b^2+1)}\]

which is \(0\) at \(n_b = 1\), \(0.235\) at \(n_b = 5\), and tends to \(1/4\). The bond lengths obey a companion law, \(\langle b^2 \rangle = \langle u^2 \rangle (2n_b^2+1)/3n_b\) visible as the steady march of \(P(b)\) to the right, since \(b\) grows as \(\sqrt{n_b}\). Both predictions sit on the readout beside the measured values, and both agree to better than a per cent.

What this is the baseline for

Everything above is the null result. In the paper, the same analysis applied to real chromatin, coarse-grained from nucleosome-resolution Micro-C contact maps, departs from these curves substantially, most strikingly with a second population of angles near \(60^\circ\) that no ideal chain or self-avoiding walk produces. That peak is the signal. It only reads as a signal because the baseline on this page is known, and known to be featureless.

The practical consequence follows directly. A coarse-grained chromatin model given zero bending stiffness, on the reasoning that the underlying fibre has no intrinsic stiffness, is already wrong before any biology is added, and wrong by an amount you can calculate rather than guess.