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* Institut Romand de Recherche Numérique en Physique des Matériaux (IRRMA), Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland; Laboratoire d'Analyse Ultrastructurale (LAU), Université de Lausanne, CH-1015 Lausanne, Switzerland; Institut de Physique Théorique, EPFL, CH-1015 Lausanne, Switzerland; and Laboratoire de Physique de la Matière Vivante, EPFL, CH-1015 Lausanne, Switzerland
Correspondence: Address reprint requests to Giovanni Dietler, E-mail, giovanni.dietler{at}epfl.ch.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTS AND DISCUSSIONCONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTS AND DISCUSSIONCONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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–8).
Most experimental and theoretical studies of the electrophoresis process deal with linear or circular DNA (9–12). But DNA comes also in knotted forms. Various classes of enzymes (topoisomerases and site-specific recombination enzymes) produce different types of knots or catenanes by acting on circular DNA molecules (13,14). The analysis of these knots provides important information about the mechanisms of action of these enzymes that are involved in the proper functioning of chromosomes (see, for example, Duplantier et al. (15) and Schvartzman and Stasiak (16)). Knots also arise as a result of DNA packing in phage heads; analysis of these knots sheds light on the arrangement of DNA in the tightly packed state (17). Being able to study knots produced by a given enzyme in prescribed conditions implies being able to perform some sort of "knot spectroscopy" that can be done, for example, by electron microscopy, where knots are observed one by one. Yet, if large numbers of knots need to be classified, then some high throughput technique is needed. Such a technique is gel electrophoresis. Indeed, experimental work has shown a linear relationship between the distance of electrophoretic migration on agarose gel of different types of DNA knots (all with the same number of basepairs) and the average crossing number of the ideal geometrical representations of the corresponding knots (closely related to the complexity of the knot) (18). As a consequence, the type of a knot can be simply identified by measuring its position on the gel, without the need of electron microscopy experiments as required until recently. An exception to this rule concerns right- and left-handed forms of the same knot that cannot, generally, be distinguished by gel electrophoresis.
At low electric field, the usual observation is that the more complex the knot is, the higher is its mobility. A simple intuitive explanation for this behavior is that the compactness of a knot increases with its complexity (for a constant string length) and the friction coefficient 
= 6
0RH (with RH the hydrodynamic radius of the knot and 0 the viscosity of the solvent) is smaller, resulting in higher mobilities. A more refined calculation of the friction coefficient relies on the Kirkwood-Riseman formula (19)
 | (1) |
i and friction coefficient 
i = 6oi, and Rij is the distance between beads i and j. The term 
due to hydrodynamic interactions between beads, in the second factor of Eq. 1 explains the observed behavior: more compact molecules have smaller distances Rij, and thus a smaller friction coefficient. The calculations of an average friction coefficient 
for an equilibrium set of thermally agitated DNA molecules forming different types of knots agree with the experimental results (20).
Recently, it was observed by two-dimensional agarose gel electrophoresis that when the strength of the electric field is increased, the electrophoretic mobility of DNA knots changes behavior (Fig. 1) (21,22). Two-dimensional gels are run in two steps: a low strength electric field of 0.6–1 Vcm–1 is first applied along one direction in the gel. At this step, more complex knots show a higher mobility, in agreement with the Kirkwood-Riseman formula. During a second step, a stronger electric field (
5 Vcm–1) is applied perpendicularly to the first one. In this case, the opposite behavior is observed: more complex knots cover smaller distances than simple ones.
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drifts over a distance equal to its size along the direction of the applied electric field 
in a time 
where 
is the drift velocity in the stationary regime and q the total electrical charge carried by the DNA molecule. During the same time, the drifting knot diffuses laterally over a distance 
where T is the absolute temperature and kB is the Boltzmann's constant. If the transverse diffusion explores distances d much larger than the typical size of the knot, , then, on average, the knot will be deflected by the gel fiber and the knots will drift with a slight renormalization of the friction coefficient. If instead 
then whenever a knot is on a collision course with a gel strand, it can be trapped by the fiber. As a consequence, after being trapped at the obstacle, the knot needs to crawl around it to free itself. Crawling around an obstacle is much more difficult for more complex knots than for simple ones, due to the self-avoidance constraint. Following this argument, the two electric field regimes are separated by a critical field 
that can be obtained by setting 
giving 
To estimate 
we use the typical values for an 11,000 basepairs DNA knot: the size of a closed DNA ring is about = 300 nm; this value is estimated from atomic force microscope images of complex knots (23). The knots were kindly provided by Dr. J. Roca (IBMB, CID-CSIC, Barcelona, Spain) and the atomic force microscope images were taken by Dr. F. Valle (LPMV, EPFL, Lausanne, Switzerland). The total charge q depends on the gel conditions, since every nucleotide carries a P– group, hence one electron negative charge, which can be strongly screened by charges in the solvent. As a consequence, we use q 10–15–10–16 C. We then obtain a critical field 
in reasonable agreement with experiments (21,22). The expression for the critical electric field holds also in the case when the gel is concentrated, as it is the case in many experiments. Under the condition of high gel density, the DNA is filling the pores, and the expression for the collision condition is that the DNA lying between two gel strands will collide with one gel strand before the DNA can drift transversally to the direction of the electric field. Instead of , one has to insert the gel pore size 
. The lateral diffusion constant D* has to be rescaled to include the effect of the gel. The condition for the critical electric field reads again 
One has also to note that the two conditions for the critical electric field actually mean that the energy gained by the DNA when moving one diameter or one pore size along the electric field is twice the thermal energy: 
(or ) 
Although the above model gives a first hint of the origin of the observed behavior, here we want to address the issue more thoroughly using lattice Monte Carlo simulations.
Lattice Monte Carlo simulations are widely used in biophysical studies aimed to grasp basic principles of such complex phenomena as protein folding (24) or polymer knotting (25). Most frequently, a simple cubic lattice is applied where any point of the lattice cannot be visited more than once. Although standard simulations on such a cubic lattice are not precise enough to model fine structural aspects like, for example, the effect of varying excluded volume on the structure of modeled polymers, this simulation procedure reproduces perfectly scaling properties of self-avoiding polymers and gives the scaling exponent v = 0.588 in practically the same way as in off-lattice models where the effective diameter can be easily varied (26,27). Also, investigations of random knotted polygons of a given type revealed that their writhe (a measure of chirality) is the same for random polygons on a cubic lattice and random polygons off lattice, although the latter ones were modeled as very thin chains (28,29). This demonstrates that lattice models are very valuable for grasping essential phenomena such as scaling of self-avoiding polymers or their topological properties.
Our study is not intended to model the exact conformation of DNA molecules in the gel nor the exact shape of gel fibers. Our aim is to grasp the basic principles of the inversion of gel mobility of DNA knots upon the increase of the field intensity.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTS AND DISCUSSIONCONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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compared to the knot lattice, so that no points of the knot lie on the gel. Knots are not allowed to cross the gel network. The coordinates of the N monomers in the configuration at time t are written as
 | (2) |
The dynamics is followed using the Berg-Foerster-Aragao de Carvalho-Caracciolo-Froehlich (BFACF) algorithm (30–32). Two types of moves are allowed: a), the creation/destruction of a handle, and b), the flip of a corner into the mirror position (see Fig. 2). The first move clearly does not preserve the knot length, which can vary by ±2 at every step, but introduces the knot elasticity. The BFACF algorithm preserves knot classes, within which it is ergodic (33). Self-avoidance is imposed by disallowing monomers to visit any site that is already occupied by other monomers. Furthermore, knots are not allowed to cross gel rods, so that corner flips and handle creation/destruction are forbidden when a rod has to be crossed.
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the electrostatic energy at time t is given by
 | (3) |
N(t) is the length of the knot at time t, and it is associated with an elastic energy
 | (4) |
At each time step, we choose a point at random on the chain and propose alternatively one of the two moves. If it satisfies the self-avoiding and gel-avoiding constraints, it is accepted with a probability given by the Metropolis algorithm: if the energy of the new trial configuration, Etrial, is lower than that of the previous configuration, Eold = E(t), the move is accepted and 
; otherwise, the probability of acceptance of the trial configuration is equal to exp{–[Etrial – E(t)]}/kBT. If the move is rejected, then 
We used a set of 100 trial initial knot configurations, for each different knot type consisting of 150 segments drawn manually on the cubic lattice (without distinction of chirality). Simulations were done for each of these starting trial configurations, and then the data were averaged over all starting knot configurations. The Alexander polynomial (34) was calculated to check that the knot type did not change during the simulation. Starting from the trial knot configuration, we let the system freely relax to thermodynamic equilibrium in the absence of an external field ( 
) until correlations from the initial configuration have disappeared. Then the electric field is switched on, and we let the knots migrate in the gel. The computed quantities are the position of the center-of-mass and the average crossing number (ACN) of the knot along a trajectory.
Time is measured in Monte Carlo iterations, length in lattice spacing. The initial length N0 of our polymers was set to 150, and the mean length of the knot depends generally on the electric field and on the gel parameter. However, the mean length is 145 (146) for C = 0.1 (C = 0.4) and b = 20, and we checked that during the simulations it fluctuates around that value. The average length is slightly shorter that N0, since the probability of shortening the polymer is a slightly larger than the probability of lengthening it due to the self-avoiding condition. The gel parameter was set to b = 20 (in units of a), corresponding to a relatively sparse gel with big pores. For each initial knot, 20 x 106 iterations were performed. The center-of-mass position has been measured every 1000 Monte Carlo steps, and it was then averaged over the trajectories obtained by the migration of 100 different initial knots.
One feature of the Monte Carlo algorithm is that more complex knots have a smaller drift velocity than less complex ones even in the absence of the gel, when time is measured in Monte Carlo steps. This is due to the fact that already in the absence of the gel, the number of rejected moves is proportional to the complexity of the knot, due to the self-avoidance condition. Therefore,
 | (5) |
(ACN) is an internal friction coefficient related to the Monte Carlo algorithm that depends on the average crossing number of the knots. To find the relation between the real time (tR) and the number of Monte Carlo steps (tMC), we used the Kirkwood-Riseman formula (1) to compute the relation between the friction coefficient KR and the ACN of the different knots.
Imposing (ACN) = KR(ACN) gives a relation between the real time and the number of Monte Carlo steps, tMC = F(ACN) x tR. Interestingly enough, we find that F is a linear function of the ACN such that tMC/tR = F(ACN) = 
x ACN – ß, finding = 0.067(1) and ß = 0.993(1). However, for a given knot, the collision of the knot with the gel will make it more elongated along the electric field direction, and therefore its geometry will be anisotropic. Since the mean cross section, which is responsible for the Kirkwood drag, will be smaller due to this deformation, we correct slightly the time rescaling by multiplying it with the following geometry factor: 
where Rfree (Rgel) is the mean radius of gyration in the free solution (gel), and ACNfree (ACNgel) is the mean ACN of the knot moving along a trajectory in the free solution (gel).
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTS AND DISCUSSIONCONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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) in the case of linear DNA molecules and also for simple and complex DNA knots. The simulations performed for linear DNA molecules reveal that in the tested range of the electric field (0 < C < 0.5), the speed of electrophoretic migration v is well fitted with a quadratic dependence upon C: v = aC2 + b, where b is a small offset that permits a better fit. These results are in agreement with experimental data (35).
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Fig. 5 presents the effect of strong and weak electric fields (
and 
respectively) on the migration speed of 11 types of knots (31, 41, 51, 52, 61, 62, 63, 71, 72, 73, and 81) all having a rest length of 150 monomers. For each knot type, we extract the average velocity from the distance of migration versus time curve. The velocity of migration is then plotted as a function of the measured ACN that is related to the knot type. This relation is presented in Fig. 5, a and b, for high and low electric fields, respectively. We observe that there is a fairly linear relationship between the average velocity of knots and the ACN (measure of complexity). More complex knots migrate slower than simpler ones at strong electric fields (although much noisier than for weak fields). These results are in agreement with experiments. The data for low electric field (
) are shown in Fig. 5 b. Table 1 summarizes the obtained results. We have also compared the average velocity of migration of different knots at low electric field in a free solution and in a gel (see Fig. 6). We conclude that at low electric field, the migration speed is only weakly affected by the gel, and the migration order of different knot types is the same in the two cases.
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We must emphasize that our dynamic suffers from the fact that we consider only thermal activated processes for describing the knot going around an obstacle, which are known to have an exponentially long relaxation time (6,36). Yet, adding suitable nonlocal moves would be enough to eliminate the artificial slowing down of the dynamics due to our local dynamic. However, the closed knot topology of DNA in our numerical simulations does not allow us to introduce the long range moves.
However, we argue that exponentially long relaxation times present in our simulations affect equally, in first approximation, all the knots, irrespective of their topology. In the ideal case of purely mechanical and frictionless unbinding, one can easily check that the knot complexity introduces at most a small logarithmic correction according to which more complex knots would anyway unpin faster, in contradiction with both experiments and our simulation. Therefore, the simulated absolute drift velocity vabs is affected by a timescale that is artificially stretched in essentially the same way independent of the knot class. So, by looking at the ratios of the absolute velocities, this timescale should in first approximation cancel out.
In addition, we have tested our algorithm in the simple case of linear DNA molecules, to check that the plateaus induced in the knotted polymer dynamics aren't an artifact of the method. Indeed, no plateaus induced by any temporary trapped configurations of the linear polymers were found even at very high electric field C = 0.7 (see Fig. 7). Also, the quadratic behavior of the speed versus the electric field of the linear DNA match well the experiments (35). When a knot hits a gel rod, it is slowed down, because the probability of a backward step is very small, and it is a growing function of the temperature T. Crawling around a gel fiber introduces plateaus in the migration distance as a function of time for individual knots, hence reducing the average migration velocity.
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for comparison. The question mark index in the experimental data (in 6?, for example) indicates that only the minimal crossing number of the analyzed knots was known. The agreement between experimental data and simulations is remarkable. We show a graphic interpretation of Table 1 in Fig. 8.
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CONCLUSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTS AND DISCUSSIONCONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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,22), namely that complex knots migrate quicker than simple knots at low voltages, whereas at high voltages the behavior is inverted. |
ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTS AND DISCUSSIONCONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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This work was partially supported by the Swiss National Science Foundation (grants Nos. 21-50805.97, 200021-101851, 3100A0-103962 and 3152-68151).
Submitted on July 21, 2005; accepted for publication January 12, 2006.
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REFERENCES |
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) knots as function of 
Also plotted is the drift velocity of the linear DNA (
). The lines are only guides for the eyes. (Inset) Quadratic fit of the drift velocity of the linear DNA.
).
and 
and