Hierarchy of a polymer coil

Polymer chains in dilute and semi-dilute solutions display a statistical structural hierarchy that differs in essence from the explicit structural hierarchy displayed by proteins.  The hierarchy begins with the persistence unit that builds upon short-range interactions at low chain index difference between interacting species.  Chain persistence can be measured using viscometry, dyanmic light scattering or static scattering measurements.  For sizes on the order of the persistence unit the coil size follows the scaling law,

bPersistence ~ n1 c

where c is the bond length and n is the number of bonds in a persistence unit.

The coil is composed of persistence steps and the coil size is described by a root-mean-square (RMS) radius called the coil end-to-end distance.  For a self-avoiding walk the coil end-to-end distance, R,  scales with,

RSAW ~ N3/5 b

where N is the number of persistence units of length b.  If self-avoidance is removed by screening of excluded volume the coil can take a Gaussian configuration where the coil size scales with,

RGaussian ~ N1/2 b

The hierarchy of the polymer coil can include scaling transitions other than the persistence transition.  Such scaling transitions can occur due to temperature, application of external force, concentration and a variety of other factors.  The details of such scaling transitions differ but the general feature is that the coil is decomposed into sub-structural features in a process termed renormalization.  Renormalization allows for thermodynamic accommodation of variable conditions by the coil and represents a new approach to thermodynamics where structural scaling transitions are a result of thermodynamic considerations.  For the concentration screening length, concentration blob, we consider the overlap concentration as a normalization factor for solution concentration,

c* = k N/R3 = k N-4/5

The concentration screening length must be proportional to RSAW, and to a power of c/c*,

x = RF (c/c*)P = k N(3+4P)/5

x is not dependent on N since it is a sub-structural unit of the coil that can not depend on N.  Then, P = -3/4.  The coil accommodates changes in concentration via a shift in the screening length.

We have investigated two drastically different kinds of hierarchy in the explicit structure of proteins and the statistical hierarchy of polymer coils.  Both hierarchies are built on thermodynamic conditions but the manifestation of thermodynamics on the detailed folding mechanism of the native state in a protein is much less subtle compared with the thermodynamic accommodation of changes in concentration by a concentration blob.  Statistical hierarchies are important even for proteins in the unfolded state and during the folding process. 

We will also consider dynamic hierarchies for polymer coils that are loosely related to statistical hierarchy of the coil.