J. Mol. Biol. (1998) 278, 1±3 COMMUNICATION Sedimentation and Electrophoretic Migration of DNA Knots and Catenanes Alexander V. Vologodskii1, Nancy J. Crisona2, Ben Laurie3 Piotr Pieranski4, Vsevolod Katritch5, Jacques Dubochet6 and Andrzej Stasiak6* 1 Department of Chemistry New York University New York, NY 10003, USA 2 Department of Molecular and Cell Biology, University of California, Berkeley, CA 94720 USA 3 A.L. Digital, London, England 4 Institute of Physics, Poznan University of Technology 60-965, Poznan and Institute of Molecular Physics 60-159, Poznan, Poland 5 Department of Chemistry Rutgers the State University of New Jersey, New Brunswick NJ 08903, USA Various site-speci®c recombination enzymes produce different types of knots or catenanes while acting on circular DNA in vitro and in vivo. By analysing the types of knots or links produced, it is possible to reconstruct the order of events during the reaction and to deduce the molecular ``architecture'' of the complexes that different enzymes form with DNA. Until recently it was necessary to use laborious electron microscopy methods to identify the types of knots or catenanes that migrate in different bands on the agarose gels used to analyse the products of the reaction. We reported recently that electrophoretic migration of different knots and catenanes formed on the same size DNA molecules is simply related to the average crossing number of the ideal representations of the corresponding knots and catenanes. Here we explain this relation by demonstrating that the expected sedimentation coef®cient of randomly ¯uctuating knotted or catenated DNA molecules in solution shows approximately linear correlation with the average crossing number of ideal con®gurations of the corresponding knots or catenanes. # 1998 Academic Press Limited 6 Laboratoire d'Analyse Ultrastructurale, BaÃtiment de Biologie, Universite de Lausanne, CH-1015 Lausanne-Dorigny, Switzerland *Corresponding author Keywords: DNA knots; DNA catenanes; DNA sedimentation; DNA topology; DNA gel electrophoresis We investigated recently the relation between ideal geometric representations of knots or catenanes and the physical behaviour of knotted or catenated DNA molecules (Katritch et al., 1996, 1997; Stasiak et al., 1996; Laurie et al., 1998). Ideal geometric con®gurations of knots or catenanes are the trajectories that allow maximal radial expansion of a virtual tube of uniform diameter centred around the axial trajectory of the knot (Grosberg et al., 1996; Katritch et al., 1996). In characterizing ideal geometric forms of knots and catenanes we have chosen scale-independent measures, such as the ratio between the length and the diameter of the tube used to generate ideal con®gurations (Grosberg et al., 1996), or the average crossing number, which describes how many crossings one 0022±2836/98/160001±03 $25.00/0/mb981696 perceives on average when the knot or catenane is viewed from an in®nite number of directions equisampling the sphere (Katritch et al., 1996). When different knots are made from the same length of tube, the radial expansion of the tube becomes more limited as the number of crossings, or complexity, of the knot increases. As a result, the ideal con®gurations of knots get more and more compact as the average crossing number increases (see Figure 1). The analogous situation applies to DNA knots made from molecules of the same length. The more complex the knot, the more compact its structure. Knowing that the electrophoretic mobility of the molecules with the same molecular mass and the same overall charge increases with compactness of the molecules, we expected that # 1998 Academic Press Limited 2 Figure 1. Linear relation between the electrophoretic migration of different DNA knots and the compactness of ideal con®gurations of the corresponding knots. Torus and twist knots were produced by site-speci®c recombination mediated by a mutant form of the Gin recombinase acting on a 7 kb plasmid. All of the knots have the same molecular mass since they arise from the same unknotted substrate molecules (Crisona et al., 1994). Before gel electrophoresis, the knotted DNA molecules were nicked to eliminate supercoiling of the molecules so that the difference in the knot type was the only reason for the electrophoretic separation of knots into different bands. The DNA from each band was analysed by electron microscopy using the RecA coating method (Krasnow et al., 1983) and identi®ed as the knot type drawn. Drawings of the knots present the ideal geometric con®guration of each knot type. To help visual tracing of the knots, the tubes forming ideal geometric representations are presented with a smaller diameter than in their maximally expanded con®gurations (Katritch et al., 1996). However, the axial trajectory remains unchanged. Note that with increasing complexity of the knots, the overall dimensions of the knots decrease, although the total length of the DNA is the same. Notice also that knots migrate with a speed directly proportional to the average crossing number of their ideal con®gurations. the speed of electrophoretic migration of different knots made from DNA molecules of the same length would increase with the average crossing number measured for the ideal con®guration of the corresponding knot. In fact this correlation turned out to be approximately linear (Figure 1) as preliminarily described by Stasiak et al. (1996). Trying to explain this linear correlation between the migration of different types of knots and the average crossing number of their ideal con®gurations, we observed that one standard measure of molecular compactness, the mean of inverse distances measured on ideal representations of different Physical Behaviour of DNA Knots and Catenanes knots having the same axial length, was directly proportional to the average crossing number of ideal con®gurations of the corresponding knots (Stasiak et al., 1996). However, knotted DNA molecules in solution do not adopt ideal geometric con®gurations. Therefore, in the present study, we decided to calculate the molecular compactness of knotted DNA molecules undergoing thermal motion and thus not adopting regular trajectories of ideal con®gurations. One of us (A.V.) developed recently a method to calculate the expected sedimentation coef®cient of DNA molecules with a given topology (Rybenkov et al., 1997). Since the sedimentation coef®cient is an accepted measure of molecular compactness of molecules in solution, the values obtained are more suited to be compared with electrophoretic migration than the average inverse distance calculated for the ideal con®gurations of the corresponding knots or catenanes. We decided therefore to simulate an equilibrium set of thermally agitated DNA molecules forming different types of knots and to calculate the time-averaged sedimentation coef®cient for the different knot types tied on a torsionally relaxed DNA chain 3600 bp long, the size of a typical bacterial plasmid. Figure 2(a) shows that there is a linear correlation between the average crossing number of ideal con®gurations of knots and the computed sedimentation coef®cient of knotted polymers forming a given type of knot. The slight deviation from linearity may result from inaccuracies of the modelling approach or may indicate that in fact linearity only approximates a more complex relation. We reported recently that the compactness of ideal con®gurations of catenanes correlates well with the electrophoretic migration of different types of DNA catenanes (Laurie et al., 1998). We therefore decided to calculate the sedimentation coef®cients of different types of DNA catenanes formed by two torsionally relaxed DNA molecules of 3600 bp each and to compare them with the compactness of ideal catenanes of a given type. As shown in Figure 2(b) there is an approximately linear relation between the calculated sedimentation coef®cients of different types of DNA catenanes in solution and the average crossing number of their ideal representations. Since the electrophoretic mobility of different DNA knots and catenanes is proportional to the average crossing number of their ideal geometric representations, there is a colinearity of the sedimentation coef®cient with the electrophoretic migration of DNA knots and catenanes analysed on agarose gels run at low voltage (Crisona et al., 1994). This last observation may advance our understanding of the gel electrophoretic separation of cyclic polymers. The results presented here strengthen the evidence that ideal con®gurations of knots and catenanes contain information about the actual physical behaviour of knotted and catenated polymers in solution (Katritch et al., 1996, 1997; Stasiak et al., 1996; Laurie et al., 1998). Earlier studies by 3 Physical Behaviour of DNA Knots and Catenanes and the gel migration of the corresponding DNA knots (Simon, 1996). Acknowledgements We thank Alicja Stasiak and Eric Larquet for help in preparation of the Figures. This work was supported by Swiss National Science Foundation grant 31-42158.94, Foundation Herbette University of Lausanne, US public Health Service grants GM34809, GM31657-15 and GM 54215, Polish Scienti®c Committee grant 8T11-F01008P04 and by A. L. Digital. References Figure 2. Linear relation between the compactness of ideal con®gurations of (a) knots and (b) catenanes, and the sedimentation coef®cients of DNA molecules forming knots and catenanes of a given type. To calculate the sedimentation coef®cients of knotted or catenated DNA molecules in solution, it is necessary to create a rich ensemble of con®gurations corresponding to momentary conformations of thermally agitated DNA molecules with a required size and topology. The approach of Metropolis Monte Carlo simulations was used to create a Boltzmann population (ca 106 con®gurations) of the desired type of DNA molecules (Katritch et al., 1996; Vologodskii et al., 1992). For each con®guration, the sedimentation coef®cient was then calculated using the Kirkwood-Riseman approximation (Bloom®eld et al., 1974). Finally, the average of the obtained sedimentation coef®cient values for individual conformations was taken as the time-averaged sedimentation coef®cient for molecules of a given size and topology. The sedimentation coef®cient of unknotted circles of 3600 bp was calculated to be 16.35 S. other authors also demonstrated the connection between the entropy of knotted polymers and the ideal con®gurations of corresponding knots (Grosberg et al., 1996) and investigated the relation between energy-minimized con®gurations of knots Bloom®eld, V. A., Crothers, D. M. & Tinoco, I. (1974). Physical Chemistry of Nucleic Acids, Harper & Row, New York. Crisona, N. J., Kanaar, R., Gonzalez, T. N., Zechiedrich, E. L., Klippel, A. & Cozzarelli, N. R. (1994). Processive recombination by wild-type Gin and an enhancer-independent mutant. Insight into the mechanisms of recombination selectivity and strand exchange. J. Mol. Biol. 243, 437 ± 457. Grosberg, A. Y., Feigel, A. & Rabin, Y. (1996). Florytype theory of a knotted ring polymer. Phys. Rev. ser. A, 54, 6618± 6622. Katritch, V., Bednar, J., Michoud, D., Scharein, R. G., Dubochet, J. & Stasiak, A. (1996). Geometry and physics of knots. 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