386 8.6  Rigid-Body and Semirigid-Body Biomechanics

Worked Case Example 8.4: Biopolymer Compaction

The E. coli genome is a circle of, largely, B-​DNA containing ~4.6 million base pairs.

a

If the DNA acts as a “floppy” molecule whose persistence length lp is roughly 50 nm,

estimate the volume that would be occupied by randomly coiled DNA assuming it is a

sphere whose radius was equal to DNA radius of gyration.

b

If the DNA instead acts as a rigid rod, what is its volume?

c

SIM imaging of fluorescently labeled nucleoids of E. coli (the region inside the bac­

terium in which the DNA is found) suggests it is roughly ellipsoidal with long and

short axes of ~1.6 µm and ~0.5 µm length, respectively. Explain if you think the DNA

inside a nucleoid is a randomly coiled floppy molecule or a rigid rod.

Answers

a The radius of gyration is given by Equation 8.37 as √<R²

G^{> ≈ √<R2}

FJC^{>/​√6 or (√n,b)/​}

√6 where b is the Kuhn length or ~2lp for a “floppy” molecule and n is the number

of equivalent segments in the freely jointed chain. Here, n is the total length L of

DNA divided by b (or 2lp). Using the known dimensions of B-​DNA (see Chapter 2)

indicates:

L =​ 4.6 × 10⁶ × 0.34 nm =​ 4.6 × 10⁶ × 0.34 × 10^–​9 =​ 1.5 × 10^–​3 m or 1.5 mm

n =​ L/​b ≈ (1.5 × 10^-​3)/​(2 × 50 × 10^–​9) =​ 15,000 links in the chain. Thus:

√<R²

G^{> ≈ √(15,000) × (2 × 50 × 10–​9)/​√6 =​ 5 × 10–​6 m or 5 µm. So, the random coil}

volume is:

Vrc =​ (4π/​3) × (5 × 10^–​6)³ =​ 5.2 × 10^–​16 m³ or ~520 µm³.

b For a rigid rod, the volume is the total DNA length L multiplied by the circular

cross-​sectional area of B-​DNA (which has a radius of ~1 nm, see Chapter 2), so the

volume is:

Vrr =​ (1.5 × 10^–​3) × π × (1 × 10^–​9)² =​ 4.7 × 10^–​21 m³ or ~0.005 µm³.

c Without worrying about the exact formula for an ellipsoid, it is clear that its volume

is going to be of the order of ~1 µm³. This is therefore 500-​fold less than Vrc but 200-​

fold greater than Vrr, so the nucleoid is far more compacted than a simple random

coil (indeed, it needs to be since the volume of a whole E. coil cell itself is only a few

µm³), but it is certainly not as rigid as to be in the rodlike regime. So, the answer

to this question is it is neither. In reality, there are so-​called nucleoid associated

proteins (NAPs), very similar to histones in eukaryotic cells, which serve a role to

compact the bacterial nucleoid in such a way as it doesn’t form a filamentous rod

but rather cross-​links across different segments of DNA, which can potentially be

separately by hundreds and thousands of base pairs.

Worked Case Example 8.5: Bayes Theorem and COVID Testing

A lateral flow test (LFT) for COVID testing was found to have a probability also known

as the sensitivity) of 85% of giving a positive result if someone is infected with one of

the variants of coronavirus, but a 99% probability of correctly giving a negative result if

someone is not infected (also known as the sensitivity). At the height of the pandemic in