Learning Lab · Polymer theory
Comparing polymer models against chromatin data
Four candidate polymers, four observables, and the measured chromatin ranges shaded behind them. No model fits everything.
This is the summary page for the theory sequence. Each of the four models has its own page; this one puts them on the same axes so they can be compared where it matters, which is against data.
Four observables, four panels. \(R(s)\) and \(P(s)\) describe structure; \(\tau(s)\) and the mean-squared displacement describe dynamics. The shaded bands are the ranges reported for real chromatin. Toggle models on and off, and watch which panels each one passes.
The numbers
| \(\nu\) | \(\eta\) | \(\gamma\) | \(\alpha\) | |
|---|---|---|---|---|
| Ideal chain | 0.50 | 1.50 | 2.00 | 0.50 |
| Self-avoiding walk | 0.59 | 2.20 | 2.20 | 0.54 |
| Equilibrium globule | 0.33, then flat | 1.50, then flat | 3.00 | 0.25 |
| Fractal globule | 0.33 | 1.00 | 1.67 | 0.40 |
| Measured in chromatin | 0.22 – 0.37 | ≈ 1 | not reported | 0.4 – 0.5 |
Everything in the first four rows comes from the earlier pages. The size exponents are derived on the ideal chain, self-avoiding walk and two globule pages; the dynamic exponents follow from one line of algebra in every case except the equilibrium globule, which reptates instead, and the Zimm value for the fractal globule, which needs screened hydrodynamics.
Reading the panels
The ideal chain is ruled out by structure. It sits far above the measured band on both \(R(s)\) and \(P(s)\): real chromatin is much more compact than a random walk. It remains useful as a reference and as the local behaviour inside a globule, but not as a description of a chromosome.
The self-avoiding walk is worse, not better. Adding excluded volume swells the chain further, moving it away from the data in the same direction. This is worth noticing, because “real chains are self-avoiding, so use a self-avoiding walk” is a natural instinct and it points the wrong way. Confinement matters more than self-avoidance at these scales.
The equilibrium globule fails on dynamics. Its structure is defensible, but reptation gives \(\alpha = 0.25\), well below the measured 0.4 to 0.5, and \(\gamma = 3\) makes large domains effectively frozen. Chromatin visibly moves on timescales relevant to transcription, so this model is hard to sustain.
The fractal globule comes closest, matching \(\eta \approx 1\) and landing at \(\alpha = 0.4\), at the bottom edge of the measured range. A live collapse under spherical confinement reproduces both of its structural exponents to within about twenty per cent, which you can watch happen on its own page. It is the only one of the four that is in the right neighbourhood on both structure and dynamics at once, which is why it became the standard picture after the first Hi-C papers.
What “closest” is worth
Not as much as it looks. Three points are worth keeping in view.
The measured \(\nu\) for chromatin, 0.22 to 0.37, is a range rather than a number, and it varies systematically with activity state: active regions are more open, repressed regions more compact. A single exponent cannot describe a polymer whose exponent depends on where you look, and no model in the table has that property.
The models are all homogeneous. Chromatin is not. Its folding is driven by loop extrusion and by attraction between similarly modified regions, both of which act at specific positions determined by the sequence and its marks. Those mechanisms produce domains, stripes and compartments, none of which is a scaling law at all. That is the subject of the next section of this lab.
And matching an exponent over a limited range is weak evidence. The window in which \(P(s) \approx 1/s\) holds in Hi-C data is roughly one decade wide. Several quite different mechanisms can produce something near \(1/s\) over one decade.
The honest summary is that these models establish what to expect from geometry alone, so that departures from them can be attributed to mechanism. That is what they are for. They are the null hypotheses, not the answer.
A note on the dynamics column
The \(\tau(s)\) panel has no measured band because relaxation time is not usually reported in the same form: experiments measure displacement over time, not a relaxation exponent, and converting between them requires assuming the model you are trying to test. The MSD panel carries the dynamic comparison instead.
Why it matters · Every one of these models can be made to look right if you pick the right panel. Judging them on all four at once is what turns a scaling argument into a test, and on that basis none of the simple models is sufficient.