In the previous section we showed three important results for the random flight model (sometimes called the freely jointed chain, or random walk polymer):

1.  If no forces are applied to the chain, the average of its end-to-end vector is zero

which  is just an expression of the symmetry of the unperturbed polymer under rotations.

2. If no forces are applied to the chain, it has a mean squared end-to-end vector of

or in other words, the overall `random coil size' is about  b N^1/2 which is much less than its contour length L = N b if N >> 1.

3. If a tension (force) f is applied so as to pull the ends apart, then the average of the end-to-end vector is nonzero.  The end-to-end vector points in the direction of the applied force, and has magnitude

where the dimensionless quantity  q = b f / (k_B T)  is just the force in units of   k_B T/ b .

These three results will be very useful.  To start with, we can use the coil size to estimate the overall size of a random-coil polymer:

Example: What is the approximate size of a denatured (unfolded random coil) protein made of 300 amino acids?

The only hard part of this question is determining the segment length  b  for the polypeptide backbone.  Since the peptide unit involves the sequence of singly-bonded atoms   N-C-C, and since the C-C and C-N bonds are around 2 A each, we can estimate that each peptide unit will contribute a segment length of around 3 A (not quite 4 - note the bond angles).   Plugging into R = b N^1/2 tells us R is about 50 A = 5 nm.
 

If we need to estimate the size of a denatured nucleic acid, looking at the backbone tells us that each nucleotide unit contributes a contour length of about b = 6 A.

Just a note about the term `denatured' - in order to talk about flexible-polymer behavior with some concrete examples, we have to suppose that the protein in the above example is unfolded, so that its amino-acid residues are in contact with the surrounding water, so that it can be considered to be a flexible, random-coil polymer.  Some regions of proteins behave this way in the cell, i.e. regions with hydrophilic (usually charged) residues.  But you can easily unfold a protein so that it is entirely random coil by e.g. raising the temperature so that the weak interactions which hold it in a folded state are overwhelmed by thermal agitation (usually 80 C is sufficient to do this).   A protein which has been thus thermally (or chemically) unfolded is called denatured, since it is no longer in its `native' or `natural' folded state.

Nucleic acids can also be `melted' or 'denatured' by exposing them to elevated temperatures, or to chemical conditions which cause them to expose their bases to the surrounding solution.

Another thing we can now do:

Example: Roughly estimate the force that needs to be applied to a denatured protein chain to stretch it out.

We will calculate this in more detail below, but it is already clear that e.g. a protein will be perturbed from its random-coil conformation when the tension on it exceeds roughly

You should remember that  1 k_B T / nm  = 4.1 x 10^-12 J/m = 4.1 pN  at room temperature.  Thus we need to apply about 15 pN to extend a protein chain at room temperature.

This force is associated with the work that must be done solely to reduce the entropy of the random coil (recall that the energy of all states of the freely-jointed polymer are the same, so no work is going into `internal energy' as the chain is being extended).  If you like, you can think that to overwhelm the thermal forces which tend to fold the polymer up into a random coil, you must apply comparable forces to each segment, which end up doing work of k_B T when you force the segment to align in the force direction, so as to have an average projected length b.

Forces of the form  k_B T / b  (ratios of the thermal energy to some characteristic length) are often called entropic forces.   In this case we might call the elastic response of the freely-jointed case its entropic elasticity.  The elastic response of polymer materials (e.g. rubber bands made of crosslinked polymers) is basically due to the entropic elasticity of the flexible polymers.