The Mobility of Polyions in Gel Electrophoresis

ROGER WEST, Medical Research Council, Mammalian Genome Unit, Kings Buddwgs, West Mains Road, Edinburgh, Scotland

The reptation theory of polymer diffusion was introduced by de G e n n e ~ ? ~ and by Doi and Edwards? This work was the foundation of the reputation theory of gel electrophoresis developed by Lerman and Frisch4; Lumpkin and Zimm5; Lumpkin, DeJardin, and Zhn6; and Slater and N~odlandi.'.~ Lumpkin et al., and Slater and Noolandi, deduced equations relating the electrophoretic velocity of polyions in gel electrophoresis to the applied voltage gradient. These equations are as follows:

Eq. (13) of Ref. 6

Eq. (14) of Ref. 6 and Eq. (42) of Ref. 7

where v is the electrophoretic velocity; V the applied voltage gradient; Q is the total charge on the polyion; F is the frictional coefficient for the translational motion of the polyion; a is the length of a segment of the primitive tube through which the polyion is moving, p is the persistence length and N the number of persistence lengths in the polyion ( p N = the contour length of the polyion); E is the dimensionless reduced electric field, E = a . q . V/2kBT*, where q is the effective charge on a segment; k , is Boltzmann's constant; and Tabs is the absolute temperature. The quantities enclosed in angle brackets (e.g., (a)) are mean values. Equation (1) is a general equation that allows for the incorporation of a distribution function for a in evaluating the averages; Eq. (2) applies to the special (and physically implausible!) case in which all pore sizes are the same. A( E ) and A'( E ) are the Langevin functions of E and its first derivative, respectively. The success of the equations in simulating the relationship between electrophoretic velocity and voltage gradient may be seen by comparing Fig. 2 of Ref. 6 with Fig. 2 of Fangmang and Fig. 11 of McDonell et al."

The authors assume that for uniform polyions Q and F are both strictly propor- tional to the molecular lengths of the polyions. This is likely to be true only if the retarding force is entirely frictional (viscous) and is mediated through the solvent. In this case, F should be strictly proportional to the viscusity of the electrophoresis buffer. However, we must assume that in a gel there are collisions between the polyion and gel fibers, and that these collisions deform the polyion. We cannot assume a priori that these forces are negligible, or that their magnitude is strictly proportional to the molecular length of the polyion. The Q / F term in Eqs. (1) and (2) should at least be replaced by a term that allows for the influence of such forces. At constant voltage and gel concentration Eqs. (1) and (2) reduce to the form, u = (k,/'N) + k , , whereas Southed' found that the empirical relationship is better described by t) = ( k , / p N + k , ) + k , . The discrepancy may be due to these factors or to the explicit approxima- tions used by Lumpkin et al.? and are not necessarily a serious impediment to the acceptance of the theory.

Biopolymers, Vol. 26, 609-611 (1987) 8 1987 John Wiley & Sons, Inc. CCC oooS-3525/87/05osos-03$04.00

610 BIOPOLYMERS VOL. 26 (1987)

A phenomenon that any theory of gel electrophoresis must explain is the conver- gence of the mobilities of large polyions when electrophoresed in gels of increasing gel concentration at fairly small voltage gradients. This phenomenon is illustrated by Serwer" [Figs. 3(A) and 41 and by Fangmang (Table I). In the model of gel electro- phoresis proposed by Lumpkin et al.6 and by Slater and Noolandi7, convergence at high voltages is due to the increasing preponderance of the last term of Eqs. (1) or (2). This term is proportional to the second (or greater) power of a. In all models of gel architecture, the scaling factor that relates (a') to gel concentration G is a negative power. Therefore, the last terms of Eqs. (1) and (2) diminish as gel concentration increases, leading to the prediction that resolution is always improved by increasing gel concentration. Models of gel architecture combined with models of gel fiber structure predict values for the scaling factor. Og~ton's'~ model of gel architecture, taken with the assumption that gel fibers have constant thickness (fiber mass/length scaling as Go), leads to the prediction that ( u 2 ) scales as G-'.'. The scaling factor may also be determined experimentally. Waki et a111.14 found that the number of fibers per unit area in electron micrographs of agarose gels was proportional to implying that ( u 2 ) scales as G-0.75. The scaling factor can also be inferred directly from gel chromatogra- phy experiments. Published results (reviewed by Serwer15) do not give a coherent picture, but suggest that the scaling factor for agarose gels is about - 0.7- - 1.0. Hurley" was obliged to propose that ( u 2 ) scales as G-0.4 to G-0.3 in order to reconcile the theory of Lumpkin et al.6 with his experimental results. This proposition lies so far outside experimental evidence, and implies such an implausible gel structure, that, in my opinion, it casts grave doubt upon the theory.

Lumpkin et al.6 emphasize that the distribution of a about its mean value, as well as the mean itself, should strongly influence the resolving power of the gel. Waki et al.14 set agarose gels from aqueous solutions containing various concentrations of salt. They counted by electron microscopy, the number of gel fibers seen in fields of freeze-frac- tured surfaces of these gels, and estimated that in gels of 0.5 gm agarose/100 mL set from "high-salt" solutions, the total length of gel fiber per unit volume was about 20% less than in a gel of the same concentration set from water (presumably cornpensatcd by a slight thickening of the fibers). However, the distribution of fibers within the gel was greatly affected by the salt concentration of the solution from which the gel was set. They equilibriated the gels against a common electrophoresis buffer, then used them for the electrophoresis of double-stranded DNA in the size range of 1.9-8.4 kb at 2.0 v/cm. In the gels set from water the mobility of all DNA species was about 14% greater than in the gel set from the solution of greatest salt concentration. This may be explained satisfactorily by the difference in gel fiber length/unit volume between the two gels. However, the ratio of the mobilities of DNA species electrophoresed together was almost identical in all gels (data from Table I11 of Ref. 14), a result that contradicts the theory of Lumpkin et al.6

Conclusion. An acceptable theory of gel electrophoresis must agree with experiment in three principal respects: the variation of mobility as a function of polyion length at constant voltage and gel concentration, the variation of mobility as a function of voltage gradient at constant gel concentration, and the variation in mobility as a function of gel concentration at constant voltage gradient. The other principal vari- ables are buffer composition, whose influence on the electrical properties of dissolved polyelectrolytes is well understood (see Bloomfield et al.17), and temperature, whose effects, as I have argued elsewhere:' are unlikely to be useful in discriminating between possible alternative theories. Lumpkin et al.6 and Slater and Noolandi7 have made a bold attempt to solve this difficult problem, and have produced a theory that meets the first two criteria, although unfortunately, published experimental results are too limited in extent to establish agreement beyond doubt. The discrepancies may be due to the approximations that they used, and might be reduced by a more detailed

RESEARCH COMMUNICATIONS 61 1

analytical treatment based on the same principles. The predictions implied by the theory regarding the third criterion have not yet been properly tested, and there are, in my opinion, serious grounds to doubt that the theory agrees with experimental evidence in this respect. If it can be shown that the predictions of the theory are in reasonable agreement with experimental evidence, then it will have passed the most important test by which it might be falsified. Unless and until that has been shown, the acceptability of the theory must be in question.

References

1. deGennes, P.-G. (1971) J. C h m . Phys. 55, 572-578. 2. de Gennes, P.-G. (1979) Scaling Concepts in Polymer Physics, Cornell U. P., Ithaca, N.Y.,

3. Doi, M. & Edwards, S. (1978) J. Chem. SOC. F u r h y Trans. 74, 1789-1801. 4. Lerman, L. S. & Frisch, H. L. (1982) Biopolymers 2l, 995-997. 5. Lumpkin, 0. J. & Zimm, B. H. (1982) Bwplymrs 21,2315-2316. 6. Lumpkin, 0. J., DeJardin, P. & Zimm, B. H. (1985) Bwpolymrs 24, 1575-1593. 7. Slater, G. W. & Noolandi, J. (1986) Biopolymers 25, 431-454. 8. Slater, G. W. & Noolandi, J. (1985) Biopolymers 24,2181-2184. 9. Fangman, W. L. (1979) Nucleic Acids Res. 5, 653-665.

10. McDonell, M. W., Simon, M. W. & Studier, F. W. (1977) J. Mol. Bwl. 110, 119-146. 11. Southern, E. M. (1979) Anal. Bwchem. 100, 319-323. 12. Serwer, P. (1980) Biochemistry 19,3001-3004. 13. Ogston, A. G. (1958) Trans. F u r h y SOC. 54, 1754-1759. 14. Waki, S., Harvey, J. D. & Bellarny, A. R. (1982) Biopolymers 21, 1909-1926. 15. Serwer, P. (1983) Electrophoresis 4, 373-382. 16. Hurley, I. (1986 Biopolymers 25, 539-554. 17. Bloomtield, V. A., Crothers, D. M. & Tinoco, I. (1974) Physical Chemistry of Nucleic Acids,

18. West, R. M. (1987) Bwplymers 26,607-608.

pp. 34-35, 230.

Harper & Row, New York.

Received June 9, 1986 Accepted November 24, 1986