45824-AC6
Theoretical Study of the Electro-Optic Properties of Branched Polymers
During this reporting period, we have carried out Monte Carlo simulations to investigate the effect of local chemistry on the properties of hyper-branched polymer chains. The main focus of effort during this reporting period was the development of computer programs to study this problem. These programs will be used to elucidate the electro-optic properties of hyper-branched polymers.
As a first step we studied freely-jointed tangent sphere hyper-branched polymers. There have been previous simulations of this model1, and we tested our simulation algorithms against these previous simulations. We studied degrees of polymerization ranging from N = 10 to 2000 and calculated conformational properties of the polymers. In most cases the simulation results were in excellent agreement with previous simulations. The exception was for short chains where there were small discrepancies. We were able to show that this discrepancy arose because the simulation algorithm used in reference 1 did not satisfy detailed balance!
We then developed algorithms to study united-atom models with an additional bond-bending potential, which effectively constrained the bond angles to be near the experimental value, but there were no rotational barriers. Finally, we included a torsional potential, thus included all the important intra-molecular interactions. A feature of the latter model peculiar to hyper-branched polymers is that the energy difference between an all trans 4 monomer segment and the corresponding star configuration must be specified. We obtained this information from quantum calculations of 4-mers. We are not aware of this issue being tackled in the polymer literature.
The basic steps to simulate the hyperbranched polymer chains are as follows. First, an initial self-avoiding linear chain is produced. After this polymer chain is equilibrated over certain runs, one bond is chosen at random and the polymer chain is cut to two pieces. A branching point and an attaching point are chosen at random from the larger portion and the smaller portion of the chain separately. Then the smaller portion is attached to the larger portion and rotated as a rigid body to the branching point by a random angle. We call this cutting and reconnecting process as "the cluster move" in our simulation. The new configuration is accepted not only when less than four bonds are attached to the branching and attaching points respectively after the cluster move, but also when the monomers of the smaller portion do not overlap with those of the larger portion. Otherwise, this new configuration is rejected, a new bond is randomly chosen and the cluster move starts again. After many successful moves, a fully equilibrated configuration is obtained and used as the initial configuration for the calculation run.
We investigated the monomer number dependence of the mean-square radius of gyration and the mean number of tri-functional branching points. Our results are consistent with previous simulation results of Cui and Chen1, and show a power-law dependence of chain size on N, as expected. The universal behavior is similar when bond-bending and torsional rotations are incorporated.
We have also studied the behavior of freely-jointed hyperbranched polymer chains confined within a cylinder, which is a model for the behavior of the nucleoid in E. coli. cells2. In E. coli, the nucleoid is the (ring-like) DNA that is twisted into a plectonemic conformation. The topology is very similar to a hyper-branched polymer, and we are in a position to study the conformations of the nucleoid and the diffusion of proteins inside the cell. We are collaborating with the group of Professor Weisshaar who are doing experiments for the diffusion of proteins in E. coli.
In our model, the monomers are hard spheres of diameter 0.2( and the bond length equals to (. The number of monomers in one chain is N = 4000. The cell is modeled as a cylinder: The length of the cylinder ranges from L = 10( to 50( and the diameter of the cylinder is D = 5.0(.
We find that the un-perturbed (in free space) dimensions of the nucleoid are much larger than that of the cell. When the nucleoid is confined within the cell, it essentially fills space. The number of tri-functional nodes is relatively insensitive to the length of the cell, but the chain expands in the axial direction as the cell becomes longer.
The monomer density in the middle region of the cylinder is larger than that in region close to the surface of the cylinder. However, except for the immediate vicinity of the surface, the density of monomers is essentially constant across the cell. We will extend this work to study the diffusion of proteins inside the nucleoid.
1 S. M. Cui and Z. Y. Chen, Physical Review E 53 (6), 6238 (1996).
2 S. Cunha, C. L. Woldringh, and T. Odijk, Journal of Structural Biology 136 (1), 53 (2001).
