A Generalized Bead and Spring Model for the Dynamics of DNA in Solution. Application to the Intrinsic Viscosity

A generalized version of the Rouse-Zimm model is applied to the calculation of the intrinsic viscosity of DNA. Some approximations of the model are examined, and some corrections are introduced in the evaluation of the eigenvalues from which the intrinsic viscosity is calculated. The theoretical results are compared with experimental data to obtain the molecular parameters of the model. The statistical segment length so obtained is analyzed together with that extracted from the wormlike chain model and also from experimental data of other macroscopic properties. A first estimation of the excluded volume effects is also discussed. In the study of chain macromolecules in solution, two different models can be used: the continuous KratkyPorodl wormlike chain and the discrete Rouse-Zimm bead-and-spring The former has the great advantage of representing chain stiffness between the limiting cases of flexible coil and rigid rod, while the latter, in its original f~ rmula t ion ,~ is only valid for flexible chains. However, the Rouse-Zimm model is, by far, superior to the wormlike chain in the simplicity of the physical concepts and mathematical details embodied in the evaluation of dynamical properties, and, consequently, it is currently considered the most adequate description of the low-frequency dynamics of flexible polymers in solution. In the model, intramolecular motions are represented by a set of relaxation times or equivalent eigenvalues, from which many dynamical properties (intrinsic v i sco~i ty ,~ quasielastic light ~ca t t e r ing ,~ flow linear di~ h r o i s m , ~ etc.) are easily derived. The above considerations illustrate the potential usefulness of extending the applicability of the Rouse-Zimm model to stiff chains such as DNA, specially if the model parameters can have some molecular interpretations as has been recently suggested by Lin and Schurr.'j A first attempt in this direction was made by Simon,7 who proposed a generalized version of the model in which chain stiffness is taken into account as a first perturbation by introducing springs connecting second-neighboring beads. Comparison of this model with experimental data of total-intensity light scattering and sedimentation coefficients of high-molecular-weight DNA has been made8 and shows that the statistical length predicted by the model is similar to that 0024-9297/79/2212-0971$01.00/0 obtained for a wormlike chain. For these properties, however, explicit introduction of the relaxation times corresponding to the realistic nondraining limit of hydrodynamic interactions is not needed, and, therefore, the theory has also to be tested with other properties more directly related to the relaxation times. In the present work, we performed a comparison of the Simon model with existingg experimental data of the intrinsic viscosity of high-molecular-weight DNA. These data (along with others for DNA fragments of lower molecular weightlOJ1) have been previously i~~ te rp re t ed '~J~ by means of the Yamakawa-Fujii theory12J4 that describes the hydrodynamic behavior of a wormlike chain, and, consequently, they seem very adequate to check the theoretical expressions for the eigenvalues derived from the Simon model. In our revision of the calculations for these eigenvalues, we have arrived at an approximate expression significantly different from that of Simon. We have verified the validity of our new expression by comparing numerical eigenvalues calculated from it with those obtained by exact diagonalization. The comparison between the revised theory and the experimental data yields values for the two parameters of the model that do not completely agree with those obtained with other properties. This fact is discussed in terms of excluded volume effects, which are larger for the intrinsic viscosity. We have made an estimation of these effects that give a statistical length of DNA close to the one corresponding to an excluded volume wormlike m ~ d e l . ' ~ ~ ' Q 1979 American Chemical Society 972 Freire, de la Torre Macromolecules Our main conclusion is that the revised Simon theory essentially reproduces the results of the rigorous Yamakawa and Fujii treatment for the intrinsic viscosity and, therefore, the expressions for the eigenvalues presented here can be applied to theoretical calculations corresponding to other dynamic properties for which the mathematically complicated wormlike model would be impractical. Revision of Theoretical Expressions (A) Reciprocal Distance Averages. Considering a linear chain composed of N + 1 beads joined by N first-neighbor springs and N 1 second-neighbor springs, it is shown7 that the eigenvalues for the model matricial equation in the absence of hydrodynamic interactions (free-draining limit of the Rouse model') are given by the exact expression