Bruno H. Zimm

Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity, Flow Birefringence and Dielectric Loss

The problem of the motions of a chain molecule diffusing in a viscous fluid under the influence of external forces or currents is considered for a particular model. This model is a chain of beads connected by ideal springs. Hydrodynamic interaction between the beads is introduced in the approximate form due to Kirkwood and Riseman. It is possible to solve this problem exactly with the use of a transformation to a set of normal coordinates. The viscosity, birefringence of flow, and dielectric and tensile relaxation behavior are calculated explicitly. The intrinsic viscosity in steady flow is somewhat different from the Kirkwood‐Riseman result, and there is no change of viscosity with shear rate. The spectrum of relaxation times is similar to that found by Rouse and by F. Bueche, but has its maximum at a lower frequency than those obtained by Kuhn and Kuhn and by Kirkwood and Fuoss in other ways.

Theory of the Phase Transition between Helix and Random Coil in Polypeptide Chains

The transition between the helical and randomly coiled forms of a polypeptide chain is discussed by reference to a simple model that allows bonding only between each group and the third preceding. Two principal parameters are introduced, a statistical parameter that is essentially an equilibrium constant for the bonding of segments to a portion of the chain that is already in helical form, and a special correction factor for the initiation of a helix. A third parameter which specifies the minimum number of segments in a random section between two helical portions has only a minor effect on the results. The partition function for this model is handled in two alternative ways, either as a summation suitable for short chains, or in terms of the eigenvalues and eigenvectors of a characteristic matrix; the latter is more suitable for long chains. A transition from the random to the helical form is encountered as either the bonding parameter or the chain length is increased. The critical value of the bonding pa...

The Dimensions of Chain Molecules Containing Branches and Rings

Formulas for the mean square radii of various branched and ringed polymer molecules are developed under the usual assumptions regarding the statistics of chain configuration. For branched molecules, the mean square radii vary less rapidly with molecular weight than for strictly linear molecules, while for systems containing only rings and unbranched chains the variation is more rapid than for the linear case. These results show that in principle the quantity of branches or of rings can be determined from light‐scattering measurements.

The Scattering of Light and the Radial Distribution Function of High Polymer Solutions

A radial distribution function of polymer segments in a solution of a high polymer may be defined as a quantity proportional to the density of segments at a given distance from some given segment. An approximate expression is derived for this function for dilute solutions of chain molecules of moderate degrees of polymerization. By Fourier inversion a simple expression for the intensity of light scattering, as a function of angle and concentration, may be obtained.

Apparatus and Methods for Measurement and Interpretation of the Angular Variation of Light Scattering; Preliminary Results on Polystyrene Solutions

A photoelectric apparatus for the measurement of the angular dependence of light scattering from solutions is described in detail and its performance is discussed. Methods of calculation for the determination of the average extension of the scattering molecules from the data are described. Data are presented for two fractions of polystyrene in various solvents, showing the effect of changing solvent power and temperature, and also confirming a theoretically derived formula for the concentration dependence of the scattering.

Viscosity and sedimentation of the DNA from bacteriophages T2 and T7 and the relation to molecular weight

The viscosity and sedimentation behavior of the DNA from T2 and T7 bacteriophages has been studied as a function of concentration and of shear stress or of rotor speed. It was found that it is possible to obtain meaningful values of the intrinsic viscosity [ η ] and sedimentation coefficient S w.20 o of these DNA polymers of high molecular weight if care is taken to work under conditions where the properties of the molecule are not seriously affected by changing its rate of movement through the solvent. In addition, an extrapolation to zero polymer concentration is essential. The hydrodynamic properties of T2 and T7 DNAs indicate that these polymers are members of the homologous series which includes DNA samples of lower molecular weight. Establishing a general relation between the molecular weight and the intrinsic viscosity or sedimentation coefficient of the members of this series has been difficult, largely because of the curvature of the conventional plot of log [ η ] or log S versus log M . In this paper we combine existing theoretical ideas concerning the hydrodynamics of chain molecules of limited flexibility with a plotting method which permits empirical correction for excluded volume effects. The result is a linear representation of the results on a log-log plot, covering the molecular weight range from 2 × 10 5 to 1·3 × 10 8 . The equations of these lines can be used to calculate molecular weights of polymers within this range. Furthermore, it appears reasonable to suppose that the relations can be safely extrapolated to higher molecular weights.

Theory of twisting and bending of chain macromolecules; analysis of the fluorescence depolarization of DNA

An elastic model of semiflexible chain macromolecules is developed in order to treat internal rotatory Brownian motion in the DNA helix. Dynamical equations for torsion and bending of the chain are generated, using results from classical elasticity and hydrodynamic theories. The rotational diffusion equation in normal coordinates is derived, and the initial‐boundary value problem solved for the conditions of a nanosecond fluorescence depolarization experiment. The resulting time distribution function of the angular orientation of a fluorescent probe, embedded in a chain at thermal equilibrium, is used to compute the emission anisotropy. The predicted decay law is unusual, with exponentials in ∼t due to twisting and in ∼t1/4 due to bending. Comparison with published data for ethidium–DNA complex reveals that the decay of the anisotropy arises primarily from twisting of the DNA helix, with a small contribution from bending. By fitting theory and experiment, the torsional rigidity C of DNA may be obtained.

Application of the Methods of Molecular Distribution to Solutions of Large Molecules

The equations for the thermodynamic potentials of the solvent in solutions of ordinary organic molecules are extended to solutions of large molecules by methods using continuous molecular distribution functions. Particular attention is given to the coefficient, A2, of the second term in the expansion of the osmotic pressure in terms of the concentration, since this coefficient has a simple molecular meaning and is sufficient to describe the deviation of the system from ideality at low concentrations. A2 is calculated by direct integration for two rigid shapes, the sphere and the long thin rod. A general expression is then developed for A2 for flexible chain molecules in terms of the interactions of the segments of the chains. In favorable cases it is found possible to relate A2 for a chain molecule to the solution properties of its small molecule homologues by an equation very similar to those developed by Flory, Huggins, and Miller. In general, however, interactions that depend both on the local structur...

Theory of the melting transition of synthetic polynucleotides: Evaluation of the stacking free energy*

Expressions are derived for the expected form of the helix-coil transitions of certain types of synthetic polynucleotides. The steepness of the transition is related to the “stacking free energy” in the helix; this can be thought of as the free energy gained when base pairs are stacked on each other in the helical configuration. Applying the theory to the results of Inman & Baldwin (1964) on dAT† and mixtures of dC and dI, we estimate the stacking free energy to be about 7 kcal, per mole of base pairs. The stacking free energy must therefore be the source of a major part of the free energy stabilizing the helix.

Theory of ``Melting'' of the Helical Form in Double Chains of the DNA Type

Previous theoretical treatments of the transition between the helical and random forms of the desoxyribose nucleic acid (DNA) molecule are extended to include formally the explicit consideration of the dissociation into two separate chains and the consideration of the effects of the ends of the chains. An approximate form for the fraction of base pairs that are bonded is obtained in terms of two parameters, a stability constant for base pairing and a constant representing the interaction of adjacent base pairs. The matrix method of statistical mechanics proves to be adaptable to this problem. Some numerical examples are worked out for very long molecules, for which case it is found that the effect of concentration is small.