Chapter 8 – Theoretical Biophysics  349

The persistence length values of typical biopolymers can vary from as low as ~0.2 nm

(equivalent to a chain segment length of ~0.4 nm, which is the same as the length of a single

amino acid residue) for flexible molecules such as denatured proteins and single-​stranded

nucleic acids, through tens of nanometers for stiffer molecules such as double-​stranded

DNA, up to several microns for very stiff protein filaments (e.g., the F-​actin filament in

muscle tissue has a persistence length of ~15 μm). The general trend for real biopolymers is

that the estimated RG is often much larger than the length scale of the cellular “container” of

that molecule. This implies the presence of significant confining forces beyond just the purely

entropic spring forces of the biomolecules themselves. These are examples of out of thermal

equilibrium behavior and so require an external free energy input from some source to be

maintained.

Deviations to models based on the assumption can also occur in practice due to

repulsive self-​avoidance forces (i.e., excluded volume effects) but also to attractive

forces between segments of the biopolymer chain in the case of a poor solvent (e.g.,

hydrophobic-​driven interactions). Flory’s mean field theory can be used to model these

effects. Here, the radius of gyration of a polymer relates to n through the Flory exponent

υ as RG ~ n^υ. In the case of a so-​called theta solvent, the polymer behaves as an ideal

chain and υ =​ ½. In reality, the solvent is somewhere between extremes of being a good

solvent (results in additional repulsive effects between segments such that the polymer

conformation is an excluded volume coil) and υ =​ 3/​5 and a bad solvent (the polymer is

compacted to a sphere) and υ =​ 1/​3.

KEY POINT 8.7

The radius of gyration of a biopolymer consisting n of segments in an equivalent chain

varies as ~n^υ where 1/​3 ≤ υ ≤ 3/​5.

In practice, the realistic modeling of biopolymer mechanics may involve not just the

intrinsic force response of the biopolymer but also the effects of external forces derived

from complicated potential energy functions, more similar to the approach taken for

MD simulations (see earlier in this chapter). For example, modeling the translocation of

a polymer through a nanopore, a common enough biological process, is nontrivial. The

theoretical approaches that agree best with experimental data involve MD simulations

combined with biopolymer mechanics, and the manner in which these simulations

are constructed illustrates the general need for exceptionally fine precision not only in

experimental measurement but in theoretical analysis also at this molecular level of bio­

physical investigation.

Table 8.1  Biopolymer Model Parameters of Real Molecules

Molecule

Number of

Nucleotides or

Amino Acids

Contour

Length

b

Ip

RG

Confining Structure

dsDNA; λ virus

49,000

15,000

100 nm

50 nm

0.5 μm

60 nm; capsid diameter

dsDNA; T2 virus

150,000

50,000

100 nm

50 nm

900

100 nm; capsid diameter

dsDNA; E coli bacteria

4.6 × 10⁶

15 mm

100 nm

50 nm

5 μm

1–​2 μm; cell width–​length

dsDNA; human

4 × 10⁹

  2 m

100 nm

50 nm

0.2 mm

5–​6 μm; nucleus diameter

ssDNA; STMV virus

1,063

  0.7 μm

  0.6 nm

  2 nm

15 nm

7 nm; capsid diameter

Titin protein; Human muscle

30,000

  1.7 μm

  4 nm

  2 nm

30 nm

3 nm; titin spacing

Titin protein; “random coil”

1,100

  0.7 μm

  0.4 nm

  0.5 nm

5 nm

3 nm; titin spacing

Notes:  Selection of polymer elasticity parameters from different types of DNA (ss/​ds =​ single-​strand/​double-​stranded) and muscle protein titin used

in various biophysical elasticity experiments. RG has been calculated from the measured b values. The general trend is that the estimated RG is

much larger than the length scale of their respective cellular “containers.”