Topological patterns in two-dimensional gel electrophoresis of DNA knots

Abstract

Gel electrophoresis is a powerful experimental method to probe the topology of DNA and other biopolymers. Although there is a large body of experimental work that allows us to accurately separate different topoisomers of a molecule, a full theoretical understanding of these experiments has not yet been achieved. Here we show that the mobility of DNA knots depends crucially and subtly on the physical properties of the gel and, in particular, on the presence of dangling ends. The topological interactions between these and DNA molecules can be described in terms of an "entanglement number" and yield a nonmonotonic mobility at moderate fields. Consequently, in 2D electrophoresis, gel bands display a characteristic arc pattern; this turns into a straight line when the density of dangling ends vanishes. We also provide a novel framework to accurately predict the shape of such arcs as a function of molecule length and topological complexity, which may be used to inform future experiments.DNA knots | topology | gel electrophoresis T opology plays a key role in the biophysics of DNA and is intimately related to its functioning. For instance, transcription of a gene redistributes twist locally to create what is known as supercoiling, whereas catenanes or knots can prevent cell division; hence they need to be quickly and accurately removed by specialized enzymes known as topoisomerases. How can one establish experimentally the topological state of a given DNA molecule? By far the most successful and widely used technique to do so is gel electrophoresis (1, 2). This method exploits the empirical observation that the mobility of a charged DNA molecule under an electric field depends on its size, shape, and topology (2). Gel electrophoresis is so reliable that it can be used, for instance, to map replication origins and stalled replication forks (3), to separate plasmids with different amount of supercoiling (3, 4), and to identify DNA knots (5, 6). The most widely used variant of this technique nowadays is 2D gel electrophoresis, where a DNA molecule is subjected to a sequence of two fields, applied along orthogonal directions (2). The two runs are characterized by different field strengths and sometimes also gel concentrations (4); with suitable choices, the joint responses lead to increased sensitivity.Although gel electrophoresis is used very often, and is extremely well characterized empirically, there is still no comprehensive theory to quantitatively understand, or predict, what results will be observed in a particular experiment. Some aspects are reasonably well established. For instance, it is now widely accepted that the physics of the size-dependent migration of linear polymers can be explained by the theory of biased polymer reptation (7-12). Likewise, the behavior of, for example, nicked, torsionally relaxed, DNA knots in a sparse gel and under a weak field is analogous to that of molecules sedimenting under gravity (13-15). The terminal velocity can be estimated via a balance between the applied force and the fr...

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Knot theory in modern chemistry

“…27 Simulations by Michieletto et al suggest another possible explanation for the electrophoresis results. 28 Modelling the gel as an imperfect cubic mesh, where some of the edges were cut, these authors examined the effect of knots becoming impaled on the dangling ends of the gel. Although more complex knots were less likely to become impaled (due to their compactness), their entanglement was more likely to be severe, resulting in a longer delay before the molecule could resume its drift and causing an overall decrease in mobility.…”

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