2.5 Chromatin Structure

Step 1: Start with naked B-DNA contour length

The human diploid genome has $N = 6.4 \times 10^9$ bp. The contour length of B-DNA is:

$$L_{DNA} = N \times 0.34 \text{ nm} = 6.4 \times 10^9 \times 3.4 \times 10^{-10} \text{ m} \approx 2.18 \text{ m}$$

Diameter: 2 nm. This must fit into a nucleus of ~10 $\mu$m = $10^{-5}$ m diameter.

Step 2: Level 1 -- Nucleosome (beads on a string, 10 nm fiber)

Each nucleosome wraps 147 bp (contour length = $147 \times 0.34 = 50.0$ nm) into a disk of diameter ~11 nm and height ~5.5 nm. With NRL = 200 bp:

$$\text{Compaction}_1 = \frac{200 \times 0.34 \text{ nm}}{11 \text{ nm}} \approx 6.2\text{-fold}$$

Effective length after Level 1: $2.18 \text{ m} / 6.2 \approx 0.35$ m = 35 cm.

Step 3: Level 2 -- Chromatin fiber folding (~40-fold total)

Whether through a 30 nm fiber or irregular folding, the 10 nm fiber is further compacted by an additional ~6-7 fold:

$$\text{Compaction}_2 \approx 6\text{--}7\text{-fold additional}$$

$$\text{Cumulative} = 6.2 \times 7 \approx 40\text{-fold}$$

Effective length: ~5.5 cm. Fiber diameter ~30 nm.

Step 4: Level 3 -- Chromatin loops (~1,000-fold total)

CTCF/cohesin-mediated loops of 50-200 kb bring distant regions together. Each loop compresses ~100 kb of 30 nm fiber into a ~300 nm diameter domain:

$$\text{Compaction}_3 \approx 25\text{-fold additional}$$

$$\text{Cumulative} = 40 \times 25 \approx 1{,}000\text{-fold}$$

Effective length: ~2 mm. This is the interphase chromatin state.

Step 5: Level 4 -- Metaphase condensation (~10,000-fold total)

Condensin complexes further compact the loop domains into the characteristic metaphase chromosome shape. An additional ~10-fold condensation:

$$\text{Total compaction} \approx 1{,}000 \times 10 = 10{,}000\text{-fold}$$

$$L_{metaphase} = \frac{2.18 \text{ m}}{10{,}000} = 218 \, \mu\text{m}$$

Distributed across 46 chromosomes: ~$218/46 \approx 4.7$ $\mu$m average per chromosome (consistent with observed metaphase chromosome lengths of 2-10 $\mu$m).

Step 6: Volume verification

As a check, the volume of DNA itself is approximately:

$$V_{DNA} = \pi r^2 L = \pi (1 \text{ nm})^2 (2.18 \times 10^9 \text{ nm}) \approx 6.8 \times 10^9 \text{ nm}^3 = 6.8 \, \mu\text{m}^3$$

The nuclear volume is ~$\frac{4}{3}\pi(5)^3 \approx 524$ $\mu$m$^3$. So DNA occupies ~1.3% of the nuclear volume by itself, but with histones, the total chromatin volume is ~10-20% of the nucleus. This leaves space for the nucleoplasm, nuclear bodies, and RNA processing machinery.

Step 1: Define linking number for two closed curves

Consider two closed curves C1 and C2 in 3D space (representing the two strands of a closed circular DNA). The Gauss linking integral defines their linking number:

$$Lk = \frac{1}{4\pi} \oint_{C_1} \oint_{C_2} \frac{(\vec{r}_1 - \vec{r}_2) \cdot (d\vec{r}_1 \times d\vec{r}_2)}{|\vec{r}_1 - \vec{r}_2|^3}$$

This integral counts the algebraic number of times one curve passes through the surface bounded by the other. It is always an integer for closed curves.

Step 2: Decompose into twist and writhe

White (1969) and Fuller (1971) showed that Lk can be decomposed into two geometric contributions. Twist (Tw) measures the winding of one strand around the helical axis, and writhe (Wr) measures the coiling of the helical axis itself in space.

Step 3: Define twist mathematically

Let $\vec{t}(s)$ be the tangent to the helical axis and $\vec{u}(s)$ be a unit vector pointing from the axis to strand 1. Twist is:

$$Tw = \frac{1}{2\pi} \int_0^L \left(\vec{u} \times \frac{d\vec{u}}{ds}\right) \cdot \vec{t} \, ds$$

For relaxed B-DNA: $Tw = N/10.5$ where N is the number of base pairs.

Step 4: Define writhe as a self-linking integral

Writhe is the Gauss integral of the helical axis curve with itself:

$$Wr = \frac{1}{4\pi} \oint \oint \frac{(\vec{r}(s_1) - \vec{r}(s_2)) \cdot \left(\frac{d\vec{r}}{ds_1} \times \frac{d\vec{r}}{ds_2}\right)}{|\vec{r}(s_1) - \vec{r}(s_2)|^3} \, ds_1 \, ds_2$$

Wr = 0 for a planar curve. Wr is positive for right-handed coiling of the axis and negative for left-handed.

Step 5: White's theorem -- the topological invariance

White's theorem (1969) proves that for any smooth ribbon (two nearby closed curves), the Gauss linking number decomposes exactly as:

$$\boxed{Lk = Tw + Wr}$$

The key insight: Lk is a topological invariant (cannot change without cutting a strand), while Tw and Wr are geometric quantities that can interconvert freely. This means you can change the shape of DNA (converting twist to writhe) without changing Lk.

Step 6: Application to nucleosomal DNA

Each nucleosome wraps DNA in ~1.67 left-handed toroidal turns. The linking number change per nucleosome is:

$$\Delta Lk_{nuc} = \Delta Tw + \Delta Wr \approx (-0.2) + (-0.8) = -1.0$$

For the human genome (~30 million nucleosomes): total $\Delta Lk \approx -30 \times 10^6$. When nucleosomes are removed (e.g., during replication or transcription), this stored negative supercoiling is released, facilitating strand separation. This is a fundamental mechanism for regulating DNA accessibility in chromatin.