### 2.1 Physical properties of DNA

This lecture shows how physical properties of biological polymers molecules such as DNA and RNA can be modelled (Slide 1). The first model discussed is the freely jointed chain model for DNA. After developing the model and discussing properties of an entropic spring, a more refined model, the wormlike chain is introduced.

The pressure exerted by packing 50 μm of phage DNA into a capsid that is a few tens of nanometres across is several tens of atmospheres. Each human cell, of the order of 100 μm across, contains approximately 2 metres of DNA – compacted and organized by proteins. The mechanical properties of DNA play important roles in determining how it is stored and accessed.

Optical tweezers are discussed in the lecture on single-molecule techniques in the topic “Molecular Machines”. Polystyrene beads with dimensions on the order of 1 micron (10^{-6} m) are trapped in focussed laser beams and can be moved around by steering the beams. The forces and displacements of the beads can be measured with pN and nm sensitivity.

In the movie shown in Slide 2, a single double-helical strand of DNA is attached as a tether between the beads. Each bead, of the order of 1μm in diameter, is trapped in a harmonic potential where a laser is focused by a high-numerical-aperture objective. The left trap is steadily moved away from the stationary right trap. Movement of the right bead within its trap can be used to deduce the force exerted on it by the DNA and the extension of the DNA molecule is equal to the distance between the beads. The stiffness of the trap can be calibrated by measuring the thermal fluctuations of the position of the bead when no other force is applied.

The movie is included courtesy of the Steve Block Lab, Stanford University California USA:

### 2.1 The freely jointed chain (FJC) model

**2.1.1 Introducing the model**

A polymer can be modelled as a chain of links. In the simplest model, a **freely-jointed chain** (FJC) the chain is allowed to cross itself, and the direction of each link is random and independent of all the others, including its neighbours. Slide 3 shows how a freely joined chain can be modelled as a **random walk** with *N* links each of length *b*.

Equation 1

The **rms end-to-end distance** *R* gives a measure of the mean size of a random coil:

Equation 2

For large *N*, *R* is much less than the contour length *L* where:

Equation 3

See also “Elements of Statistical Mechanics and Soft Condensed Matter, Lecture 2: Brownian Motion”.

**2.1.2 Entropic spring**

An FJC molecule is an **entropic spring**: there are fewer ways to arrange the chain the more it is extended, leading to a decrease in entropy and thus an increase in free energy. In other words, the restoring force increases as the length increases.

A well-known known entropic spring is rubber. The polymer molecules in rubber are all entangled, but force is generated because each is an entropic spring. Stretching the rubber aligns and extends the chains, increasing the free energy. This elasticity is a purely entropic effect – in the absence of an applied force all configurations of the chain have equal energy.

To explore this ‘spring-like’ behaviour of a polymer molecule, it is helpful to use the **partition function**, *Z*, for the system (Slide 4). Each segment of the chain contributes independently to the energy, so the partition function for the chain is the product of identical partition functions for the individual segments.

Equation 4

where

Equation 5

If a tension *f* is applied to the ends of the chain, this gives the energy of a particular configuration as

Equation 6a

Equation 6b

where *z* is the end-to-end distance in the direction of the applied force, so

Equation 7

Using Equation 4:

Equation 8a

Equation 8b

Equation 8c

where

Equation 9

### 2.2 Statistical mechanics – a reminder

Before proceeding further with the FJC model, it will be helpful to review some key results from statistical thermodynamics.

Slide 5 is a reminder of an important general result from statistical mechanics which uses the partition function, *Z*:

Equation 10

and the related expression for the average value of any quantity x:

Equation 11

If the energy of a particular microstate *E*_{k} depends upon *x* only via a single term *fx*, ie:

Equation 12

then

Equation 13

In our FJC example, the quantity *x* will be the end-to-end distance *z* and thus *f* will be the end-to-end force, but note that Equations 11 and 13 are true for any pair of quantities *x* and *f* that satisfy Equation 12.

See also “Elements of Statistical Mechanics and Soft Condensed Matter, Lecture 1: Statistical mechanics”.

### 2.3 Behaviour of an entropic spring

**2.3.1 Stretching an entropic spring**

We can use the above results from statistical mechanics to develop the FJC model further.

Note that the Helmholtz free energy *F* is

Equation 14

so using Equation 13 for the mean length

Equation 15

Slide 6 gives the expected end-to-end distance

Equation 16

For an entropic spring, natural units for the force are

. With *T* = 300 K (room temperature) and *b* = 1 nm, *f* = 4.1 pN. Note that force→0 as T→0.

The graph on Slide 6 shows the relationship between force and length for a FJC. Since **spring constant**

Equation 17

this shows that the spring constant of an entropic spring depends on temperature – a characteristic of an entropic force, where the entropy but not the internal energy depends on the length.

Slide 7 shows how we can explore the behaviour of the spring constant in the low force (high temperature) regime.

If

(low force, high temperature) then a power series expansion of Equation 8 yields:

Equation 18

where *L* is the contour length, and here

Hence

Equation 19

Provided terms of order *z*^{3} are small and can be ignored, Equation 19 describes a linear spring that obeys Hooke’s Law (Equation 17) and we can identify the spring constant, λ:

Equation 20

(Note that

On the other hand, Slide 8 shows the result for a high force, low temperature regime i.e.:

Then, with coth(α) ≈ 1, we have

Equation 21

Equation 22

Here we have a non-linear spring. The graph (shown in Slides 6, 7 and 8) diverges from a straight line as the end-to-end spring length approaches the contour length *N*b.

**2.3.2 Work done**

To explore the work done in stretching a FJC we could just integrate the analytical expression for force vs. displacement, but it is also instructive to look at limiting cases (Slide 9). The work done in stretching the chain could be recovered by allowing it to relax slowly against some load. These processes are occurring at constant temperature, so Δ*W* represents the free energy of a stretched polymer relative to a random coil.

For small forces and small extensions, (*f* < i>kBT/b, z < i>L/3) the spring is linear (Equation 19) and we have

Equation 23a

Equation 23b

where *R* is the mean square end-to-end distance:

The free energy cost reaches *k*_{B}*T* when the end-to-end length reaches *R*_{0}.

For high forces and large extensions the spring is non-linear (Equation 13) and then

Equation 24a

Equation 24b

Equation 24c

The various versions of Equation 24 show the functional form of divergence of the stretching free energy as the chain approaches the contour length *Nb*.

**2.3.3 Experimental data**

Slide 10 shows experimental results of stretching DNA molecules. The best-fit FJC curve has segment length *b* = 106 nm, corresponding to about 300 base-pairs or about 30 turns. The fit is less good for other segment lengths because the backbone is *not* freely jointed but held in the double helix.

The data and all calculated curves are consistent with Hooke’s Law for small extensions. The freely-jointed-chain model correctly predicts divergence in force as extension approaches contour length, but is not a good fit to the data for intermediate and high forces.

For a better fit to the data, we need a model that takes account of the energy required to bend DNA.

### 2.4 Worm-like chain model of DNA

The discrepancy between the experimental results in Slide 10 and the theoretical predictions can be resolved using a **worm-like chain** (WLC) model, also known as the **Kratky-Porod model** (Slide 11).

This model takes account of the energy involved in distorting the chain:

Equation 25

We can ignore the twist contribution if the DNA is nicked so as to allow free rotation around one phosphate linkage of the opposite strand. And DNA is stiff enough for the stretch to be ignored at small extensions, though this is important when the molecule is stretched close to its contour length. So the main contribution is *E*_{bend}.

Adding an energy cost of bending the chain gives the WLC model. We can express the bending energy in terms of the local curvature, *c*, and the bending modulus κ (Slide 12). ** u** is a unit vector indicating the direction of the chain,

*s*is the distance along the chain.

*c* = d*θ*/d*s* = ⌊d*u*/d*s*⌊

Equation 26

We then have

Equation 27

The conformational distribution of the polymer is given by the Boltzmann distribution:

Equation 28

where the **persistence length** *A* is

Equation 29

For a long polymer the persistence length plays the same role as the segment length b of the freely jointed chain. In this model (Slide 13), the mean square end-to-end distance, *R*, is

Equation 30

In the limiting case of a freely-jointed chain (*L* >> *A*), Equation 30 reduces to:

Equation 31a

Compare this with Equations 2 and 3 which combine to give:

Equation 31b

At the other extreme, when *L* < i>A (a rigid rod) we have *R*^{2} = *L*^{2}.

The limiting behaviour of a very long flexible polymer is that of a freely jointed chain with segment length *b* = 2*A* (known as the **Kuhn length**).

Without derivation, we will simply state expressions for force and extension of a WLC (Slide 14).

The force response is:

Equation 32

Compare this with the high-force limit of the freely joined chain:

The stronger divergence of the force exerted by the polymer at large extensions is a result of its bending elasticity.

For small extensions a freely jointed chain and worm-like chain both have a linear force-extension relationship.

### 2.5 Other biological polymers

In addition to DNA, another very common biological polymer is an amino-acid or polypeptide (Slide 15), also called a protein.

For an unstructured, single-stranded nucleic acid *b* ~ 0.6 nm

For a 300-residue *denatured* polypeptide:

*L*= *Nb* ~ 90 nm

~50nm

force required to straighten significantly

~ 14 pN

‘Denatured’ means not in native folded state. Here it means unfolded so that all amino acid residues are in contact with the surrounding water and the polypeptide backbone can be considered as random coil. Some regions of proteins behave like this under physiological conditions. Protein can be denatured by heating (typically to about. 80°C).

Proteins are the subject of the next lecture where it will be clear that simple physical models of polymers, reasonable for double-strand DNA, are completely inadequate to explain the important properties of proteins.

The passive tension developed by muscle when stretched is largely due to the rubberlike properties of the giant protein titin (Slide 16). The mechanically active region of titin is like a chain of beads, each bead a folded protein domain (see next lecture). In the experiment in Slide 16, an atomic force microscope was used to measure the mechanical properties of titin directly, by stretching it between tip and surface.

The characteristic sawtooth pattern of unfolding can be explained as stepwise increases in the contour length of a polymer whose elastic properties are described by the wormlike chain model. The figure at the lower right of Slide 16 shows a force extension curve obtained by stretching of a single Ig8 titin fragment (a genetically engineered protein consisting of 8 identical domains extracted from the much longer titin molecule).

The force extension curve shows a characteristic sawtooth pattern with seven peaks. The force extension curve leading up to each peak is well described by the WLC equation with a persistence length *A* = 0.4 nm and a contour length *L* that begins at 58 nm for the first peak and then increases by 28 to 29 nm to fit consecutive peaks, reaching a maximum of 227 nm for the last peak. The WLC model predicts that the contour length of the polypeptide chain increases by 28 to 29 nm each time an Ig domain unfolds. This value is close to the 30 nm predicted by fully extending a polypeptide chain comprising 89 amino acids (minus a folded length of 4 nm).

At a force of 150 to 300 pN, the polypeptide chain is not fully extended, hence the peaks are spaced by only ~25 nm. Unfolding of the first domain reduces the force close to zero, whereas unfolding of consecutive domains reduces the force to a lesser extent. This effect is also well explained by this simple model: Upon reaching a certain force (peaks), the abrupt unfolding of a domain lengthens the polypeptide by 28 to 29 nm and reduces the force (troughs) to that of the value predicted by the force extension curve of the enlarged polypeptide.

Note that Slides 15 and 16 show just two examples of biological polymers – there are many more.

Finally, Slides 17 and 18 show examples of two other biological structures than can be modelled as worm-like chains: an actin filament and a microtubule.

In Slide 17, two protofilaments of actin wind round each other to form a helix a few μm in length, with pitch 37 nm and diameter ~6 nm. The structure is stiff, with a persistence length ~15 μm.

In Slide 18, 13 parallel protofilaments of tubulin form a stiff, hollow tube of diameter 25 nm and length ~10 μm. The tube is very stiff, with a persistence length ~6 mm.