alexander v. vologodskii- distributions of knots and links in circular dna

8/3/2019 Alexander V. Vologodskii- Distributions of Knots and Links in Circular DNA

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Distributions of Knots and Links in Circular DNA

Alexander V. VOLOGODSKII

ABSTRACT

The distinctive feature of closed circular molecules is that theyoccupy particular topological states that cannot be altered by anyconformational rearrangement short of breaking DNA strands. Thistopological constraint opens unique possibilities for experimentalstudies of the distributions of topological states created by different

ways. Primarily, the equilibrium distributions of topologicalproperties are considered in the review. We describe how suchdistributions can be obtained and measured experimentally and howthey can be computed. Comparison of the calculated and measuredequilibrium distributions of knots and links formed by circular

molecules gave a lot of valuable information about properties of thedouble helix. Study of the steady state fraction of knots and linkscreated by type II DNA topoisomerases exposed surprising propertyof the enzymes, their ability to reduce these fractions essentially

below the equilibrium level.

1. Introduction

A typical DNA molecule adopts many different conformations insolution, and therefore its properties have to be analyzed inprobability terms. The probability distributions of DNA properties

can be calculated with good accuracy now and can be measuredexperimentally in many cases. From the experimental point of viewthe distributions of topological states are especially attractive. Thetopology of a circular DNA is not altered as long as the strands of thedouble helix stay intact, and one can manipulate samples in different ways to determine the distributions. In particular, it is possible toresolve different topological forms by gel electrophoreses and obtainthe distributions by measuring the intensity of different bands. Onthe other hand, there are methods which allow reliable computingtopological properties of circular DNA. Due to these experimentaland theoretical possibilities, distributions of topological states arenot only very interesting objects, but also powerful instruments inbiophysical studies. In this review we will consider distributions ofknots and links between pairs of circular DNA molecules and showhow they can be used to study DNA conformational properties. In thelast Section of the review we will show how these concepts andmethods can be used to study enzymes which change DNA topology,DNA topoisomerases.


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There are three levels of topological variables which are needed todescribe a state of double-stranded circular DNA. For the first andsecond levels we need to consider DNA molecule as a simple curvewhich coincides with DNA axis. The fist level describes topology of

isolated closed curve which topology can correspond to unknottedcircle or to a knot of a particular type. If we have many DNAmolecules, part of them can form topological links with others, andthe second level of description specifies types of these links. In thisreview we will consider only links between two chains, also giganticnetwork of topologically linked DNA circles are known in biology [1].The third level of the topological variables specify the links formed bycomplementary strands of the double helix. The links exist only in so-called closed circular DNA where both complementary DNA strandsare closed. The linking of the DNA complementary strands, which is very important for DNA conformational and biological properties(see reviews [2,3], for example), will not be considered here.

Our main emphasis will be the equilibrium distributions oftopological states. There is one distinctive feature of suchdistributions. Normally, initial distribution of conformationalproperties relaxes, as a result of thermal motion, to its equilibriumform. This is true for all properties of linear polymer chains. Aftersome time the distribution of the end-to-end distance of a polymerchain, for example, or the distribution of loops randomly formed bythe molecules will correspond to the thermodynamic equilibrium.This is not the case for the equilibrium distributions of topologicalstates, because exchange between different topological states is

impossible for circular molecules. The system of circular polymerchains does not relax to the equilibrium distribution of topologicalstates. The concept of the equilibrium distribution, however, is veryuseful in this case.

The definition of the equilibrium distribution of topological states ofcircular molecules does not differ from the corresponding definitionsfor other properties. This is the most probable distribution which hasto be reached by random exchange between different states insolution. Thus, the distribution minimizes the free energy of thepolymer molecules. We would get the equilibrium distribution oftopological states if our circular molecules had phantom backbones,

so that their segments could pass one through another during thethermal motion. Thus, to calculate the distribution we should forgetthat the exchange between different topological states is forbiddenand use usual rules of statistical mechanics. It means that, regardless

of the topology, the probability of conformation i , iP, is specified by

the Boltzmann distribution:


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)/exp( kTEP ii , (1)

where iE is the energy of conformation iand kTis the Boltzmann

temperature factor. To calculate the probability of a particulartopological state we have to take a sum of iP over all conformations

which correspond to this state. In the Section 3 we considercomputer simulation of such equilibrium conformationaldistributions as well as the analysis of the simulated conformations.

Experimentally we can obtain thermodynamic equilibrium overtopological states by slow cyclization of linear molecules. In the caseof double-stranded DNA, the cyclization can be provided by joiningcomplementary single-stranded ends (cohesive ends) [4]. If thecohesive ends are long enough, about dozen of nucleotides or longer,a circular form is stable at room temperature. Such circularmolecules have single-stranded nicks, so only topology specified bythe double helix axis is defined there. Comparison of theexperimentally measured distribution with computer simulationsprovides a lot of valuable information about DNA properties. Suchstudies will be described in Sections 4-6. Distributions of topologicalstates created by type II DNA topoisomerases will be described in thelast section of the review.

2. Description of topological states of circular DNA

2.1. Knots

A knot is understood to be any closed curve, and in particularthe topological equivalent of a circle (in this case it is called a trivialknot). There is infinite number of topologically distinguished knots.For the classification, knots have to be deformed to obtain thestandard form of their projection on a plane. A standard form of aknot projection is such an image thereof when the minimum number

Figure 1. Table of simple knots with less than seven intersections in thestandard form.

of intersections on the projection is achieved. One must be alwayscareful, though, that there should be no self-intersection of chains in


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the course of such reduction. Deformations of this kind are known intopology as isotopic deformations. Two knots which can betransformed into each other by way of isotopic deformation belong tothe same isotopic type.

Knots can be simple or composite. A knot is composite if there is anunlimited surface crossed by the knot at two points, which thusdivides it into two nontrivial knots. The simplest knot has threeintersections in the standard form and is called a trefoil. Fig. 1 showsthe initial part of the table of simplest knots - namely, all knots withless than seven intersections in the standard form. A knot and itsmirror image are considered to belong to the same type of knot,although they may belong to the same or different isotopic types. Inparticular, the trefoil and its mirror image cannot be transformedinto each other by way of continuous deformation without self-intersection, and therefore belong to different isotopic types. The

figure eight knot (41) and its mirror image belong to the sameisotopic type. The table of knots is, in fact, a table of knot types, for itfeatures only one representative of mirror pairs. With the growth ofthe number of intersections the number of types of simple knotsgrows very fast. As it turns out, there are 49 types of simple knots with nine intersections, 165 with ten and about 552 types of knots with eleven intersections [5,6]. The table of simple knots with lessthan 17 crossings in the standard form was completed recently [7],there are 1,701,936 of such knots.

2.2. Links

In chemistry and biology links of two or more circles are usuallycalled catenanes. The tables of two-contour links are based on thesame principle as the tables of knots [5]. The initial part of the tableof link types is shown in Fig. 2. Although three links in this table, 21,

41 and 61, belong to the torus class [8] which corresponds to

Figure 2. Table of links with less than seven intersections in the standardform.


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the links of the complementary strands of the double helix in closedcircular unknotted DNA, the fraction of torus links among morecomplex links is very small.

2.3. Identification of topological statesSuppose we want to determine the type of knot formed by highlycoiled closed cord. The fact that persistent attempts to untangle thecord have produced no result cannot be taken as a proof that we aredealing with a nontrivial knot. What is needed for resolving thisproblem is an algorithm of verification of the topological identity ofthe configuration in question. Topologists developed many of suchalgorithms based on invariants of the topological states,characteristics thereof which remain unchanged with any isotopicdeformations of the chains, which are possible without the disruptionof their integrity [8]. The simplest invariant of topological state is theGauss integral which defines the linking number of two chains (seebelow). Of course, for classifying the state of chains with a topologicalinvariant, the latter must assume different values for differenttopological states. Not a single topological invariant meets thisrequirement in full measure, but there are very powerful ones amongthem which help identify many elementary types of knots (links) anddistinguish them from more complex ones. The Gauss integral is afairly weak topological invariant and is of no use for distinguishingmany linked states of chains from unlinked states. Still the integral is very useful for analysis of DNA supercoiling, since it identifies alllinks of the torus class. Another important invariant is the Alexanderpolynomial which is very convenient in computer simulation

(reviewed in [9]). It is a polynomial of one variable in the case ofknots and of two variables in the case of links of two closed contours.The invariant was used in all studies dealing with the computation ofthe equilibrium distribution of topological states. The great majorityof the simplest knots have different Alexander polynomials, (t) , and

thus the invariant allows to distinguish all simplest knots one fromanother and from unknotted chains [9]. For unknotted chain

1)( = t . )(t for all knots with less then 11 crossings in standard

form can be found in Ref. [5].

An invariant in the form of an Alexander polynomial also exists for

links, but it is a function of two variables rather then one. Thealgorithm of calculating the Alexander polynomial for two chains,),( ts , is described in ref. [10].


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3. Computer simulations of equilibrium conformationaldistributions of circular DNA

3.1. DNA model and its relation to the actual molecule

If we want to compare computed results with measured properties ofan actual polymer we have to establish the correspondence between amodel chain and actual polymer molecule. This is done over theconcept of Kuhn statistical segment, b. The idea of thiscorrespondence can be described as following. It can be shown thatfor any model of sufficiently long polymer chain the average square

of the end-to-end distance,2R , is proportional to the chain

contour length,L:

bLR =2 . (2)

Eq. (2) corresponds to random walk with step length b. It does notaccount for excluded volume effect, related with the chain thickness, but the effect does not change Eq. (2) for DNA molecules a few

thousand base pairs in length. The value of2R can be determined

experimentally and, if we knowL, Eq. (2) allows us to determine b.The value ofb is directly related to the chain persistence length, a =b/2, and characterizes the bending rigidity of a polymer chain [11].For DNA molecule b is close to 100 nm or 300 base pairs. Now wecan express the molecule length in the number of Kuhn segments,

L/b. The concept of Kuhn statistical segment is equally applied to anytheoretical model of the polymer chain. Modeling conformationalproperties of a polymer chain we should use a model chain of thesame number of Kuhn segments.

DNA double helix ofn Kuhn statistical segments can be modeled, forour purposes, as a closed chain ofkn rigid cylinders of equal lengthand diameter (d). The energy of the chain, E, is specified by angulardisplacements i of each segment i relative to segment 1+i :

=

=1

1

2

2

kn

i i

gE

. (3)

Eq. (3) can be naturally transformed for the case of circular chain. Allconformational properties considered here do not change, within agood accuracy, if k 10 , and for majority of these properties we canobtain accurate results even for k = 1 ( 0= ). In the polymer


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statistical physics the chain with k = 1 is called the freely-jointedchain, since all its conformations have the same energy. The freely-jointed chain is the simplest model of a polymer chain, often used ingeneral analysis of polymer properties. If we set a value ofk, we can

calculateg[12].DNA effective diameter, d, specifies the electrostatic interaction

between DNA segments: dis the diameter of an uncharged polymerchain which mimics the conformational properties of actualelectrically charged DNA. Its value depends strongly on ionic

conditions. The dependence of d on [Na+] and ][Mg

+2is known

with accuracy [13-16]. The value ofd can be several times greaterthan the geometric diameter of the double helix. For the purposes ofcomputer simulation it is convenient to express d in the units ofKuhn length, so the topological properties of two chains should be

similar if they have the same number Kuhn segments and the samediameter. For physiological ionic conditions DNA effective diameteris close to 4 nm, or 0.04 in Kuhn length units. Thus, the DNA modelhas two parameters, the chain length measured in the number ofKuhn statistical lengths, n, and the effective diameter, d.

3.2. Computation of the conformational properties

Despite the simplicity of the model described above, few of itsstatistical properties can be evaluated analytically. There are,however, well developed computational methods which allowestimating many conformational properties, including topologicalproperties of circular chains. The methods are based on the random

sampling of the equilibrium conformational distribution followed bydirect analysis of the constructed conformations. Metropolis MonteCarlo procedure is the most general way to perform the samplingwhich satisfies condition (1) [17].

At each step of the procedure, a trial conformation of the chain isgenerated by displacement from the previous conformation. Thestarting conformation is chosen arbitrarily. The probability ofaccepting the trial conformation is obtained by applying the energy

test. If the energy of the trial conformation, newE , is lower than that

of the previous conformation, oldE , then the trial conformation is

accepted. If the energy of the trial conformation is greater than theenergy of the previous conformation, then the probability ofacceptance of the trial conformation is equal to

]/)exp[( kTEE newold . It is assumed that a conformation has

infinite energy if the distance between two non-adjacent segments issmaller than the dvalue. Therefore, the minimum distance between


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all pairs of non-adjacent segments of a trial conformation iscalculated. If any distance is less than d, the trial conformation isrejected. This procedure is repeated numerous times to obtain anensemble of conformations that represents the equilibrium

distribution. If for any reason a trial conformation is rejected, thecurrent conformation is taken as the next one in the constructed setof conformations.

Displacement of the chain is usually performed by a crankshaftrotation of a randomly chosen subchain [18]. A subchain containingan arbitrary number of adjacent segments is rotated by a randomlychosen angle, , around the straight line connecting the vertices

bounding the subchain. The value of is uniformly distributed over

an interval ),( 00 chosen so that about half of the moves are

accepted.

Segments of the chain are allowed to cross each other during thedisplacements. With such a phantom, or incorporeal, chain, topologyof the chain is changing and we obtain a sampling of the equilibriumdistribution over topological states. To analyze the distribution weneed to determine knot type of each constructed conformation. It can

be done by calculating )(t for t = -1 and t = -2 for all

conformations of the set [9,19,20].

Figure 3. Typical simulated conformations of circular DNA 10,000 base

pairs in length. The illustration was obtained by computer simulation of theequilibrium conformational set.

Sometimes we need to evaluate equilibrium properties of the chains with a particular topology. In this case we have to start simulationfrom a conformation with the desired topology and keep the topology


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during the displacements by rejecting all trial conformations whichtopology is different from the topology of the starting conformation.

A typical DNA conformation from the equilibrium set is shown inFig. 3. The size of this DNA corresponds to DNA of bacteriophage P4.

Similar approaches are used to compute equilibrium properties oftwo closed DNA molecules which can be unlinked or form links ofdifferent types [9,10,21,22].

5. Equilibrium distribution of knots

Let us consider a diluted solution of nicked circular DNAs atthermodynamic equilibrium, so the probability of links between twomolecules is negligible. In this case the equilibrium distribution oftopological states is reduced to the probabilities of unknotted chainsand various knots. Let us first consider theoretical aspect of this

problem.Starting from 1974 [19,20], about a dozen of studies have beendevoted to computations of the equilibrium probability of knots(reviewed in [9,23]; see also [24] and refs. therein). The results showthat, for DNA molecules thousands base pairs in length, theequilibrium fractions of nontrivial knots are rather small althoughincreases with DNA length (Fig. 4).

0 10 20 300

1

2

3

4

DNA length, thousands of base pairs

51

52

Fractionofknots,

%

31

41

Figure 4. Equilibrium probability of the simplest knots in circular DNA.Results of computer simulations [15] are shown by solid lines, symbolscorrespond to the experimental data for trefoils [15,42]. The datacorrespond to the physiological ionic conditions (0.2 M of NaCl).


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The computations also showed, that the probabilities of knots reducedramatically when the thickness of a polymer chain increases[25,26]. The probability that closed chain of n Kuhn segments

( 50


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Figure 5. Electrophoreticseparation of knotted (right lane)and linked (left lane) DNA

molecules 4363 bp length. Each band corresponds to a knot orlink with a specific number ofintersections in the standardform. These numbers are shownnext to each band. All links

belong to the torus family. OCdimer is open circular DNAmolecule of double length.(Illustration provided by E. M.Shekhtman and D. E. Adams.)

taking into account that there were no adjustable parameters in theDNA model used in the computations. One of two model parameters,DNA effective diameter, increases greatly when the ion concentrationreduces, and this causes the decline of the trefoil fraction shown inthe figure. Comparison between measured and calculated fractions oftrefoils allowed determination of DNA effective diameter for other

ionic conditions [16].

6. Equilibrium probabilities of links

If the concentration of DNA molecules during their randomcyclization in solution is not too small, a fraction of the moleculesforms catenanes at thermodynamic equilibrium. Let us suggest thatthe concentration is still small enough so that only small fraction ofthe molecules is linked with others at the equilibrium; in this caseformation of links from 3 or more molecules can be neglected. Thus we can consider formation of catenanes as bimolecular reaction which is characterized by its equilibrium constant, B . Equilibrium

concentration of linked DNA molecules, catc , can be found from the

equation


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Equilibriumf

ractionoftrefoils,

%

0.01 0.1 10

1

2

3

4

5

[Na+], M

Figure 6. Comparison of the measured and simulated equilibrium

fractions of trefoils for different concentrations of sodium ions. P4 DNA, 10kb in length, was cyclized in solution of different NaCl concentrations via

joining the cohesive ends [15]. Each point on the graph (filled circles) is themean of 6 to 20 determinations. Computational results (open circles)account for the dependence of DNA effective diameter on NaClconcentration [13,14].

22)2( BcccBc catcat = , (5)

where c is the total concentration of circular molecules. The value ofB depends on DNA length and ionic conditions in solutions.Dependence ofB on DNA length for physiological ionic conditions isshown in Fig. 7.

We can also consider topological equilibrium for a system whichincludes two types of circular DNAs (A and B). Such equilibriumcatenation was studied first by Wang and Schwarz more than 30 years ago [32]. We recently used the approach to probeconformations of supercoiled molecules [22]. The system consisted

of supercoiled molecules of one length (molecules A) and cyclizingmolecules of another length (molecules B). The system was prepared by such a way that there was only partial topological equilibrium:there were no AA catenanes although the concentration of moleculesA was rather high but there was the equilibrium regarding AB and BBlinks. Since the goal was to measure the equilibrium constant for ABlinks, such partial equilibrium simplified the analysis of various


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products in this system. Comparison between simulation andmeasured data obtained in this study was used to test simulatedconformational properties of supercoiled DNA [22].

Figure 7. Equilibrium constant of catenation of two identical nickedcircular DNAs. The data obtained by computer simulation for thephysiological ionic conditions (see ref. [22] for details of the computation).

7. Unknotting and unlinking circular DNA by type IItopoisomerases

Type II topoisomerases are important enzymes that pass one DNAthrough another and thereby change DNA topology. They make atransient double-stranded break in one DNA segment that allowspassage of another segment of the same or another molecule andthen reseal the break (reviewed in [33,34]). Thus, these enzymesmight convert real DNA molecules into phantom chains that freelypass through themselves. There was a widely held belief that, withthe obvious exception of DNA gyrase, which introduce supercoilingin circular DNA [35], topological equilibrium is also achieved in thereactions catalyzed by type II DNA topoisomerases. They can unlinkknotted [36,37] or catenated DNA [38], and catalyze catenation if theDNA concentration is high enough [39]. In all these reactions the

systems approach equilibrium. However, the question of whether theenzymes create an equilibrium distribution of catenated or knottedmolecules had not been carefully examined till the study byRybenkov et al. [40].

It was unexpectedly found that the fractions of knotted and catenatedDNAs produced by type II topoisomerases are up to two orders of


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magnitude lower than at equilibrium (Fig. 8). Thermodynamically,there is no contradiction in this finding because the enzymes utilizethe energy of ATP hydrolysis. This property of topoisomerases has animportant biological consequence. It helps to explain how they can

remove all DNA links under the crowded cellular conditions whichfavor the opposite outcome.

Whereas ATP hydrolysis allows type II topoisomerases to removetopological links from DNA below their equilibrium values, it was notclear from the beginning how this topology simplification is achieved.Free strand passage during the thermal motion of circular DNAmolecules should establish an equilibrium distribution of topologicalforms. This distribution would not change even if only a fraction ofthe segment collisions results in strand passage, as long as theprobability of passage is independent of DNA topology. However, asingle topoisomerase, given its small size compared to DNA, cannot

recognize the topological state of DNA by binding the wholemolecule. Topology is a global property of circular DNA molecules,

Figure 8. Type II topoisomerase, topo IV fromE. coli, removes topologicallinks from DNA to level below equilibrium. The reaction reached its steadystate at substoichiometric values of enzyme/DNA ratio. Equilibrium valueof the knot fraction for the given conditions and DNA length, 7 kb, is shownas a reference level by dashed line. Also shown, as a control, the fraction ofknots found for topo III, type I topoisomerase from E. coli, which does notconsume the energy during the catalysis and thus must shift the fraction oftrefoils to the equilibrium level [40].

and it is impossible to design a machine which could determinetopology by using only a local interaction with these long molecules.Enzymes are very small compared with DNA molecules, so theydefinitely belong to the category of such locally acting machines,


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although they can interact simultaneously with two DNA segmentsseparated along the molecular contour. We found, however, thatthese enzymes do so by using internal statistical properties of DNAmolecules rather than by determining their overall topology [41].

Type II topoisomerases create a sharp bend upon binding with aDNA segment, forming a hairpin (Fig. 9).

Figure 9. Model of type II topoisomerase action. The enzyme (black

pacman) bends the bound DNA segment into a hairpin. The entrance gatefor another segment of DNA is inside the hairpin. Thus, the second segmentcan pass through the first segment only from inside to outside the hairpin.

It turns that the probability to find another segment of the circularDNA inside the hairpin is a few times higher for knotted DNA thanfor unknotted one. Therefore, if the strand passing goes only frominside the hairpin, which is provided by specific orientation of theentrance gate of the enzyme relative to the hairpin, the strandpassing should reduce the steady state fraction of knots. It is

important to note that the hairpin itself does not notably affect theequilibrium fraction of knots. The steady-state fraction of knots isreduced dramatically in this model only because the enzymecatalyzes the strand passage reaction in one direction, from inside tooutside the hairpin. The quantitative analysis of the model predictionas well as supporting it experimental data can be found in ref. [41].

This work was supported by NIH grant GM54215 to the author.

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New York University, Department of Chemistry, 31 Washington

Place, New York 10003, USA

alexander v. vologodskii- distributions of knots and links in circular dna

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