Coil Globule Transition

1. Look at the 5th of these polymer lecture notes Links to an external site. for a discussion of the coil globule transition.

hw7/coil_globule.py is an implementation using "reptation dynamics" for a lattice model describing a polymer chain with attractive interactions with its neighbors, but is also self avoiding (i.e. it can't occupy a lattice site twice.)

Try running the code with different parameters. Try two chain lengths, length=32 and length=64. Try two values of the attractive coupling, J=0 (no attraction between nearest neighbors) and J=.5. J=0 should be a self avoiding walk, and J=0.5 should be in the collapsed (globule) phase.

When you run the simulation, it starts off in an overlapping configuration and the code moves the chain until it's no longer overlapping. Then it starts taking data and computes the radius of gyration squared, averaged over many measurements.

In the coil phase (J=0), the polymer should be self avoiding. The average radius of gyration RR as a function of chain length NN should obey R \propto N^\nuRNν where \nuν in three dimensions is about 0.6.

In the collapsed (globule) phase, the chain should be of constant density, that is N/R^3N/R3 should be constant so that \nu = 1/3ν=1/3. In order to test these predictions, run obtain the average radii of gyration by running the code for the matrix of values J=0,0.5, N=32,64. Then you can get an estimate of \nuν for the self avoiding and collapsed phases by considering the ratio of the radius of gyration for J=0 and J=.5. From this you can estimate the two different values of \nuν. But you've got to do it by taking ratios as you don't know the proportionality factor between R and N. It's not much algebra but it very nicely illustrates the power of scaling in fractal systems.

2. Applications to biology:

  • Describe an experimental paper that studies the coil globule transition for long DNA. What method was used to induced a transition?
  • Compare the coil globule transition with the transition of a protein from denatured to folded.

 

3. Suppose you had a long single stranded DNA molecule with the sequence ATAT...AT. Describe how you would expect it to condense starting from a high temperature denatured phase. Would this look like a coil-globule transition? Think about high entropy states that also have fairly low energies. Sketch a picture of a typical configuration you might expect to see. Note that we only expect one long hairpin at extremely low temperatures that aren't experimentally accessible (water freezes). So you'd expect hairpins but also a lot of randomness, leading to many configurations and hence a pretty high entropy.