Ch 24 pages Lecture 9 – Flexible macromolecules

 Viscosity measures the rate of energy dissipation in a flowing fluid. The addition of a macromolecule to a solution greatly increases the viscosity and thus increases the rate of energy dissipation  The rate of energy dissipation in a macromolecular solution is greater than the rate of dissipation in pure solvent: Summary of lecture 8  represents the fraction of the solution volume occupied by macromolecules and is a numerical factor related to the shape of the macromolecule.

 relative viscosity Summary of lecture 8  specific viscosity Where C is the concentration of the solute in gm/mL and is the partial specific volume of the hydrated macromolecule  intrinsic viscosity both hydration and shape (asymmetry parameter) contribute to it

Chain Polymers (e.g. DNA) Many biological polymers (e.g. high molecular weight DNA) may be idealized as a repeating structural unit connected by a series of bonds (see diagram). The shape of the polymer is determined by rotations around the “bonds”. The rotational state of the bond between the i-1 and the i unit is designated by the torsional angle  i. The shape of the polymer chain is determined by the set of torsion angles {  i }.

Chain Polymers (e.g. DNA) Two parameters that depend on conformation and are associated with a chain polymer are the end-to-end distance and the radius of gyration Suppose a polymer chain is composed of n+1 repeating units. Imagine these units to be connected by n bonds, and associated with each bond is a vector l i, which is parallel to the bond axis. The end-to-end distance is: r is the sum of the n bond vectors

Chain Polymers (e.g. DNA) The magnitude of r is: A polymer can have many possible shapes, and therefore we can introduce a mean-squared end-to-end distance representing the average property of a population of polymeric molecules:

Chain Polymers (e.g. DNA) The radius of gyration R G can be measured by an experiment called dynamic light scattering and is defined as: Where r ij is the distance between structural unit i and structural unit j. Both r and R G are measures of chain structure

Random Coil Polymers A particularly simple model of a chain polymer is called a random coil polymer. Basically at each repeating unit the direction of the bond is random, and the shape of the polymer can be visualized as following a random walk model, making the structural description of such a polymer an exact analog of the diffusion of a particle through a continuous medium or the hops of fleas A consequence of the random nature of bond directions is that in the equation:

Random Coil Polymers Note this is the same result that we obtained for the mean squared displacement of a randomly jumping particle that executes n jumps each of length l In accordance with the random walk model, we can identify the number of conformations that have a given length. In one dimension, this was shown to be in Lecture 5:

Random Coil Polymers In accordance with the random walk model, we can identify the number of conformations that have a given length. In one dimension, this was shown to be in Lecture 5: However, we must correct this expression by taking into account the three-dimensional nature of the random walk problem associated with the polymer (treating each dimension as independent:

Random Coil Polymers We can use this expression to calculate the mean-squared end-to-end distance:

Random Coil Polymers We can introduce the concept of contour length L: The contour length is of course the length measured along the links of the DNA (or polymer in general)

Random Coil Polymers For very flexible polymers (e.g. polythene), l is simply the length of each CC bond. However, DNA is a very rigid polymer, so the relationship between mean end-to-end square distance reported above for a random coil polymer does not apply to DNA, if we use the base-to-base distance, for example, as l. We can introduce nonetheless the concept of effective segment, as the length l over which a polymer like DNA behaves as a random coil object. If we measure the diffusion coefficient of a number of DNA molecules of a different size, we will find that l is independent of size (if the DNA is large enough) and equal to approximately 1,000A (or about 300 base pairs)

Viscosity of Flexible Polymers For hydrated spheres: (the asymmetry factor >5/2 for asymmetrical objects) From this expression, we can obtain an expression for the hydrodynamic, or effective radius R e of a flexible polymer in solution: where M is the molecular weight and N A is Avogadro number

Viscosity of Flexible Polymers In the limit that C approaches zero, i.e. very dilute solutions is the root-mean-squared end-to-end distance of the flexible polymer; from the previous paragraph:

Viscosity of Flexible Polymers for a random coil polymer. Assuming In reality the intrinsic viscosity of a flexible polymer deviates from this behavior because the mean-squared end-to-end distance is not proportional to N. In fact, the expression assumes the ends of the random coil polymer do not interact. In reality, such interactions do exist and are called excluded volume effects, i.e. the ends of the coil cannot approach too closely. In such cases:

Viscosity of Flexible Polymers In reality, such interactions do exist and are called excluded volume effects, i.e. the ends of the coil cannot approach too closely. In such cases: For short DNA oligonucleotides which behave as rigid rods in solution For longer DNAs, which behave as flexible coils