DNA topology

THEO CHEM ELSEVIER

Journal of Molecular Structure (Theochem) 336 (1995) 235-243

DNA topology Maxim D. Frank-Kamenetskii Centerjor

Advanced Biotechnology, Department of Biomedical Engineering, Boston University. Boston, MA 02215. USA

Received 22 September 1994; accepted 18 October 1994

Abstract Two topological different

levels are inherent in closed circular (cc) DNA. First, the double helix as a whole can form knots of types, or form the trivial knot (to be unknotted). Secondly, the two complementary strands are interlinked in

ccDNA, which results in a biologically very important phenomenon of DNA supercoiling. During the past 20 years we have elaborated a theoretical model, which makes it possible to treat the DNA topological properties at equilibrium in a rigorous way by the Monte Carlo method. Within the framework of the model, the double helix is considered as a homogeneous, isotropic elastic rod with given values of bending and torsional rigidity parameters and the effective diameter. We have predicted the equilibrium fraction of knotted DNA molecules for different DNA lengths and for different values of DNA effective diameter. These theoretical predictions have recently been fully confirmed by experiment. Comparison of theory with experiment also gave the value of DNA effective diameter under a variety of ambient conditions. Calculations of equilibrium distribution of ccDNA over topoisomers yielded a reliable estimation of the DNA torsional rigidity. As a result, all three parameters of the DNA model have been reliably estimated. This makes possible unambiguous predictions of various properties of DNA under topological constraints. In particular, extensive computer simulations of supercoiled DNA have been carried out.

1. Introduction

A stimulus for the development of the theory to be surveyed in this paper was the discovery of circular DNAs. The DNA molecule, which contains all the information on the structure of living organisms, consists of two polymer chains attached to one another by weak, noncovalent interactions. These chains form the double helix in which “/. = 10.5 monomer residues (base pairs) occur per turn. Actual DNAs contain from several thousand to hundreds of millions of monomer links. Initially the main attention was focused on studying the properties of linear DNA molecules, since this was precisely the form of DNA that could be extracted from cells and virus particles. 0166-1280/95/$09.50

It was unexpectedly found in 1963 that DNA existed in certain viruses in a closed circular (cc) form. In this state, the two single strands of which the DNA consists are each closed on themselves. Fig. 1 schematically illustrates ccDNA. One can see that the two complementary strands in ccDNA are linked. They form a high-order linkage (of the order of N/ye, where N is the number of pairs in the DNA). Initially, the discovery of circular DNA was not seen to be very significant, since this form of DNA was regarded as exotic. However, in the course of time, the cc form of DNA was discovered in an ever greater number of organisms. Currently it is generally acknowledged that precisely this form of DNA is typical of the simplest DNAs, and also of the cytoplasm

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DNAs of animals. Also most virus DNAs pass through a stage of the cc form in the course of infection of cells. Such a widespread occurrence of this form of DNA in nature has elicited an interest in its structure and properties that has been manifested in recent years. The discovery of ccDNA has led to the formulation of fundamentally new problems, since many of the physical properties of the cc form differ radically from those of the linear form. The difference between the properties of these two forms of DNA is not at all due to the existence of end effects in the one case but not in the other. Apparently the most striking difference between circular and linear DNA is the fact that circular DNA can be knotted.

2. Knots The first problem that arises in the theoretical analysis of ring polymer chains, including ccDNA, is formulated in the following way. Let a ring molecule be formed by fortuitous closure of a linear molecule consisting of IZsegments. What is the probability of forming a knotted chain, i.e. a nontrivial knot? This problem has been clearly formulated by Delbruck [l] and solved by our group [2,3]. To solve the problem one needs, first of all, a knot invariant. Indeed, a closed chain can

Fig. 1. Schematic

representation

of closed circular

(cc) DNA.

Fig. 2. Various

knots.

be unknotted or can be a knot of different type. The very beginning of the table of knots is shown in Fig. 2. However, an analytical expression for the knot invariant is unknown. Note that Edwards [4] proposed such an invariant but we showed that Edwards’ expression is actually not invariant toward deformations of the closed chain [2]. Therefore, we had to use a computer and an algebraic invariant elaborated in the topological theory of knots. We found that the most convenient invariant is the Alexander polynomial (see Refs. [5,6]). The next problem consists of generating closed polymer chains. In our first calculations, we simulated DNA as a freely-jointed polymer chain. Several methods exist to generate exclusively closed chains for this model [5,7]. Using these methods and teaching the computer to calculate the Alexander polynomials and therefore to distinguish the knots of different types, we could calculate the knotting probability.

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IO0

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n

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Fig. 3. Probability of knot formation, P, as a function of the number n of Kuhn statistical lengths for an infinitely thin polymer chain. Different symbols correspond to results obtained by different authors (data from Ref. [5]).

Analogous calculations have been. performed later by other researchers (see Ref. [5] for references). The data on the relationship between the probability of knot formation and the number of Kuhn lengths in the chain are collected together in Fig. 3. We see that the results obtained by various authors agree very well with each other. This is not surprising, since, in spite of a certain difference in the polymer models employed, to which certain differences in the results are due, the presented data in all cases fit the model of an infinitely thin polymer chain. One can see from Fig. 3 that the probability of knot formation has an evident tendency to approach unity as n increases, though it has been possible to perform the calculations only up to n values such that P barely exceeds 0.5. Of course, there is a limitation in principle, which involves the fact that certain nontrivial knots have the same Alexander polynomial as the unknotted chain (the trivial knot). However, such knots constitute such an infinitesimal fraction of knots formed in the course of computations, that this does not affect the results on the probability of knot formation. The above calculations were performed under the assumption that the polymer chain under consideration has zero diameter. Among all known polymers, duplex DNA seemed the most appropriate to

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conform this assumption. We realized very early on [2] that excluded volume effects should significantly decrease the knotting probability. However, the knotting probability proved to be even more sensitive to the excluded volume effects than we originally anticipated so these effects could not be neglected even in the case of DNA. We arrived at this conclusion after elaborating a Metropolis-Monte Carlo approach to calculate DNA topological characteristics [8]. In this approach, we model the polymer chain as a series of straight segment so that each Kuhn length contains k such segments. The total elastic energy is the sum of terms, each of which corresponds to a pair of adjacent straight segments and quadratically depends on the angle between them (see Ref. [6] for details). The final results are obtained, within the framework of the model, as asymptotics for the large k values. Fortunately, all characteristics we studied leveled off very quickly with increasing k so that k = 10 proved to be quite sufficient to get very reliable quantitative asymptotic results (see Fig. 4). The asymptotics of this model corresponds to the well-known wormlike model of the polymer chain. This approach made it possible to simulate the behavior of DNA molecules allowing for excluded

8 t

/

’ ’ ’ 1 .-

0-0

i

t 5

15

10

20

k Fig. 4. Typical results of Metropolis-Monte Carlo calculations on the dependence on the number of straight segments per Kuhn length, k, of a mean quantity (the mean writhing number in this particular case) for a closed polymer chain containing 6 Kuhn lengths. The data are from Refs. [6,28].

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3. DNA supercoiling

0.03

0.02

0.01

0 d

Fig. 5. Dependence of the equilibrium fraction of knotted molecules on DNA effective diameter, d, for closed DNA containing 14 Kuhn lengths (lower curve), 20 Kuhn lengths (middle curve) and 30 Kuhn lengths (upper curve). The data are from Ref. [7]. The diameter is measured in Kuhn lengths, so to obtain the d value in nanometers one has to multiply the figures on the abscissa by the factor of 100.

volume effects [7]. So we arrived at quantitative predictions about the dependence of knotting probability on the DNA effective diameter, d. Fig. 5 shows the results. One can see a dramatic dependence of the P value on d. Even in the case of the DNA geometric diameter, which corresponds to d = 0.02 in Fig. 5, the knotting probability is already significantly lower than for d = 0. However, in reality the effective diameter of DNA noticeably exceeds its geometric value due the excluded volume effects, which are determined by the screened electrostatic interactions between highly charged DNA segments. Therefore the d value can be varied by changing the ionic strength of the solution. Our theoretical predictions have been recently checked experimentally (see below).

From the schematics in Fig. 1 it is clear that the two complementary strands of DNA form a link, in the topological sense. One can present a Table of links similar to the Table of knots in Fig. 2 (see Ref. [5]). However, because the two complementary strands of DNA are attached to each other forming the double helix, the links which DNA can form, belong to a subclass of all possible links. Namely, they form the so-called torus links because the two strands could be put into a torus. For torus links the well-known Gauss integral, which defines the linking number value, Lk, is a strict topological invariant (see Ref. [5]). There is another viewpoint on the torus links. The two strands in this case could be treated as the edges of a ribbon. Therefore the topological theory of torus links is actually the theory of ribbons. The application of the topological ideas to studying the properties of ccDNA was started by Fuller [9], when he applied the results of the ribbon theory to analyzing the properties of these molecules. According to this theory [lO,l l] (a simple derivation can be found in Ref. [5]), besides the topological characteristic of a ribbon, the Lk value, two differential-geometric characteristics play an important role, the twist, Tw, of the ribbon, and its writhing, Wr. All three characteristics are interrelated by the condition: Lk=Tw+Wr

(1)

The ccDNA is generally not characterized by the total quantity Lk, but by the number of excess turns (the number of supercoils 7): r = Lk - N/y0

(4

The “/. value is rigorously fixed under given ambient conditions. However, upon changing the ambient conditions (temperature, composition of solvent, etc.), it can vary. Therefore, the number of supercoils, 7, in contrast to the Lk value, is a topological invariant of DNA only under fixed ambient conditions. Very valuable information on the energy and conformation characteristics of ccDNA has arisen from experiments in which the value of Lk could vary, and the equilibrium distribution of the cc

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molecules over the Lk value was studied. The most convenient way to vary Lk is to employ special enzymes, so called topoisomerases. The studies under discussion employed type I topoisomerases, which alter the topological state of ccDNA by breaking and rejoining only one of the strands of the double helix. The mechanism of action of these enzymes has recently been elaborated in great detail [12]. These enzymes relax the distribution of the molecules over the Lk value to its equilibrium state. The very sensitive gel-electrophoresis method was used to analyze the distribution of the ccDNA molecules over the Lk value. This method can easily separate two molecules of ccDNA that differ in Lk by just 1. Naturally, the maximum of the equilibrium distribution always corresponds to 7 = 0 because this minimizes the elastic energy. Note that, although the quantity T can only adopt discrete values that differ by no less that unity, it is not required to be an integer. Therefore, as a rule, molecules having 7 = 0 do not appear in a preparation. A distribution, in which the mo!ecules having positive and negative values of 7 are separated, is obtained when the electrophoresis is performed under conditions differing from those under which the reaction with the topoisomerase is conducted. The change in the conditions means that we must substitute some other value 7; instead of y. in Eq. (2) without changing the Lk value. This means that the entire distribution is shifted by the amount of ST = N[l/yo) - (l/r;)]. Then the molecules that had the value T in the original distribution will have the values 7’ = T + ST in the new distribution. If the 67 value is large enough, all of the topoisomers are well separated. Experiments have shown that the obtained distribution is always normal [13-151. The variance, {r-2), of this normal distribution was measured for different DNAs. These experiments have played a very important role in studying the physical properties of ccDNA. First of all, they made it possible to determine the free energy of supercoiling, which is directly connected to the variance: F = kBT7*/2(T2)

= llOOk,J-N-‘T*

(3) where kB is the Boltzmann constant and T is the absolute temperature.

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Secondly, these data made it possible for the first time to obtain a reliable estimation of the torsional rigidity of the double helix and to find what fraction of the supercoiling is realized in the form of a change in the twist value, ATw, and what fraction in the form of writhing, Wr. Under conditions allowing breaks and rejoining of one of the DNA strands, i.e. in the presence of topoisomerases, ATw and Wr are independent random quantities, while the resultant quantity 7 equals their sum: T=ATw+Wr

(4)

Evidently the mean values are (ATw) = (Wr) = (7) = 0. However, the quantities (( ATw)~), ((Wr)*) and (T*) differ from zero. Then, because of the independence of the random quantities ATw and Wr, one obtains [16] (T*) = ((ATw)~) + ((Wr)2)

(5)

The (T*) quantity is known from experiment, and the ((Wr)2) quantity could be calculated by computer simulation because the ribbon theory offers a simple analytical formula for the value provided that the shape of the chain is known [5,6,16-181. (Note that in so doing we extensively used our method of discrimination of knotted and unknotted chains.) To compare values of (?) and ((Wr)‘) for the same DNA we need to know the quantitative value of the DNA Kuhn length, which is known with a good accuracy to be 100 nm [ 191. Therefore we could find ((ATw)*) as a function of the DNA length by subtracting the calculated ((Wr)2) value from the experimental (TV) value. However, the ((ATw)~) quantity is directly related to the value of the torsional rigidity of the double helix, C: ((ATw)~) = N((A@)*) = hkBTN/4n2C

(6)

where h and 4 are the distance along the axis and the rotation angle between two adjacent base pairs in the double helix, respectively. In full agreement with this equation, the ((ATw)~) value proved to be strictly proportional to N. The slope of the straight line made it possible to determine the C value, which proved to be 3.0 x lo-t9 erg cm [17,18]. This value of the torsional rigidity of DNA corresponds to a root-mean-square amplitude of

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thermal fluctuations in the value of the angle between adjacent base pairs of 4-5”. The obtained results indicated that in sufficiently long supercoiled DNA one-third of the superhelical energy is stored in twisting and two-thirds are stored in writhing. Thus, the analysis of the experimental data on circular DNAs employing the topological approach made it possible to estimate one of the fundamental characteristics of the double helix. These estimations agree with the results obtained by other methods (see Ref. [20] and references therein).

4. Knotted DNAs As mathematical objects, knots and links have been studied already for more than 100 years. The question of possible existence of such topological states in molecules has been raised since at least the late 1940s (see Ref. [21]). It has acquired special interest since the discovery of closed circular molecules of DNA. The calculations of the probability of knot formation upon closing a polymer chain, the results of which are discussed above, have posed the problem of the possible existence of knotted DNAs. The results indicated that the equilibrium fraction of knotted DNAs must be appreciable for circular DNAs containing more than about lo4 base pairs (30 Kuhn lengths). In most cases, DNA molecules have an even greater length, and the hypothesis has been put forward of the existence in the cell of special mechanisms that prevent the formation of knotted DNAs [3]. In fact, in the course of replication of a knotted chain (at least for some types of knots) the daughter strands cannot separate. That is, the replication of knotted DNAs involves serious problems. Knotted molecules were first detected in preparations of single-stranded circular DNAs after they had been treated under special conditions with a type I topoisomerase [22]. This was the first case when a knotted molecule was observed. However, the problem of knotting of normal, double-stranded DNAs continued to be very intriguing. It turned out that there is a special subclass of topoisomerases called type II topoisomerases, which are capable of untying and tying knots in ccDNAs. Moreover, these enzymes

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catalyze the formation of catenanes from pairs or from a larger number of molecules of ccDNA. Here entire networks are formed, similarly to those observed in vivo in kinetoplasts. In contrast to type I topoisomerases, type II topoisomerases break, and then rejoin both strands of DNA molecules. It has been shown that the enzyme “draws” a segment of the same or of another molecule lying nearby through the “gap” that is formed in the intermediate state between the ends that arise through breakage. This operation with an individual ccDNA corresponds to a change in the writhing number by 2. However, it evidently does not alter Tw. Consequently, we have IA,% = 2. In fact, experiment shows that type II topoisomerases, in contrast to the type I enzyme, always alter Lk only by an even number. Thus, the type II topoisomerases catalyze the process of mutual penetration of segments of the double helix through one another. Consequently, these topoisomerases must lead to establishment of complete topological equilibrium, i.e. to a distribution of the molecules over the topological states that would correspond to freely permeable strands. As we have noted above, DNA molecules need not be very long for a reliable proof of the detection of knotted molecules, but then the fraction of knots, as our calculation showed, must be small. Liu et al. [22] were able to overcome this contradiction by using topoisomerase II in very large concentrations in which it substantially changed the macromolecular properties of the DNA itself. Moreover, they did not add ATP to the enzyme, which is necessary for its normal operation. Precisely under these extreme conditions, they found even in short DNAs having N = 4.5 x lo3 a considerable fraction of knotted molecules. They were able to detect them initially from the appearance of new bands in the gel electrophoregram that corresponded to a greater mobility. Study of the properties of these fractions by various methods including electron microscopy has made it possible to show that they correspond to knots of various types. If then topoisomerase II in the normal amount and ATP were added to a purified preparation of knotted molecules, rapid untying of the knots took place [22]. That is, the system rapidly relaxes to the equilibrium state for pure DNA

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molecules, in which, as our calculations predicted, there should be practically no knots for the given length. As to the reasons why the enzyme in high concentration sharply shifts the equilibrium toward knot formation, the most likely explanation is that the protein in high concentration decreases the dimensions of the polymer coil of DNA by changing the character of the interaction of segments remote along the chain. As our calculations showed [5], even a small change in the dimensions of the polymer coil sharply increases the equilibrium fraction of knots. Knotted molecules of DNA (and also catenanes) were obtained also by sophisticated methods employing various enzymes of DNA site-specific recombination [23]. Although the above experimental observations did not contradict our theoretical expectations, the question about quantitative validity of the theory remained open. Almost 20 years after we first published theoretical estimations of the probability of DNA knotting [2,3], quantitative experimental data have been reported [24,25], which fully agree with the theory. In these experiments, the equilibrium fraction of knotted DNA molecules at various ionic conditions was quantitatively measured while molecules carrying “cohesive” ends randomly closed, in the absence of any proteins. Comparing the fraction with our theoretical predictions [7] the value of the DNA effective diameter was determined as a function of salt concentration. The obtained dependence proved to be in complete quantitative agreement with theoretical predictions of Stigter [26], which were based on the polyelectrolyte theory.

5. Computer simulations of DNA supercoiling Quantitative explanation and prediction of a variety of DNA topological characteristics, most notably the data on the equilibrium knotting probability and on the equilibrium distribution of ccDNA over topoisomers (see above), have demonstrated a remarkable success of our basic model. Within the framework of this model, we treat DNA as an isotropic homogeneous elastic road, which is characterized by three parameters: the bending rigidity measured in terms of Kuhn

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length, torsional rigidity C, and DNA effective diameter, d. It should be emphasized that under a variety of ambient conditions quantitative values of all three parameters are known with a good accuracy. Therefore, under these conditions our model has no uncertainty and provides us with unambiguous quantitative predictions. Several years ago we initiated the application of our model to predict the properties of supercoiled molecules [27,28] (see also review articles in Refs. [6,29]). Note that in parallel another group conducted ideologically similar theoretical studies [30,3 11,although with quite different objectives. In its traditional form, the Monte Carlo approach does not permit simulating highly or even moderately supercoiled molecules because the probability of their occurrence due to thermal motion is negligible. We have extended our previous Metropolis-Monte Carlo calculations to make it possible to generate supercoiled DNA molecules with arbitrary supercoiling. In brief. our computational procedure is as follows (see Ref. [6] for a detailed description of the method). We consider a phantom closed chain, in which senf-intersections are allowed. Elementary steps to change the conformations are introduced. After each elementary step, the energy is calculated [6]: E,({r]) = E(P))

+2~‘(ClhN)[~-Wr({r})l’

(7)

where E({r}) is the elastic energy of the DNA chain. Then the regular Metropolis-Monte Carlo rules are applied: if the energy difference between the step under consideration and the previous energy LIE, < 0, then the new conformation is accepted; if AE, > 0, the new conformation is accepted with the probability of exp (-AE,/kaT). H owever, this is only a conditional acceptance. The new conformation needs to meet two additional criteria. First, none of all possible pairs of the straight segments could approach each other closer than by the value d. Secondly, the chain should remain unknotted as the result of the conformational change. The knot checking procedure is carried out as in the case of knotting probability calculations described above. An ensemble of chains thus generated is used to calculate different averaged characteristics of

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supercoiled molecules and enables one to obtain theoretical images of supercoiled molecules. Fig. 6 gives examples of such images. Our theoretical predictions about the shape of supercoiled DNA molecules agree with most of the available experimental data [27,28,32].

References

G= -0.07

Fig. 6. Results of computer simulations of supercoiled DNA molecules for different values of superhelical density o = 7-yO/N. The data are from Ref. [27].

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