PHYSICAL REVIEW E, VOLUME 63, 031909

Kinetic proofreading can explain the supression of supercoiling of circular DNA moleculesby type-II topoisomerases

Jie Yan,1 Marcelo O. Magnasco,2 and John F. Marko11Department of Physics, The University of Illinois at Chicago, 845 West Taylor Street, Chicago, Illinois 60607

2Center for Studies in Physics and Biology, The Rockefeller University, 1230 York Avenue, New York, New York 10021~Received 28 April 2000; revised manuscript received 22 August 2000; published 27 February 2001!

The enzymes that pass DNA through DNA so as to remove entanglements, adenosine-triphosphate-hydrolyzing type-II topoisomerases, are able to suppress the probability of self-entanglements~knots! andmutual entanglements~links! between;10 kb plasmids, well below the levels expected, given the assumptionthat the topoisomerases pass DNA segments at random by thermal motion. This implies that a 10-nm type-IItopoisomerase can somehow sense the topology of a large DNA. We previously introduced a ‘‘kinetic proof-reading’’ model which supposes the enzyme to require two successive collisions in order to allow exchange ofDNA segments, and we showed how it could quantitatively explain the reduction in knotting and linkingcomplexity. Here we show how the same model quantitatively explains the reduced variance of the double-helix linking number~supercoiling! distribution observed experimentally.

DOI: 10.1103/PhysRevE.63.031909 PACS number~s!: 87.14.Gg, 82.70.2y, 68.15.1e, 02.40.2k

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I. INTRODUCTION

In many biological processes, chemical reactions betwspecific molecules~e.g., specific amino acids in the caseconstruction of proteins! must be tightly regulated and sequenced with low rates of error. At the molecular level reglation depends on the binding of biomolecules to particutargets. Hopfield@1# and independently, Ninio@2# consideredthe problem of explaining how many biological processachieve error rates below those expected on the basis ofree-energy differences between ‘‘correctly’’ and ‘‘incorectly’’ matched reactants.

Recognizing that all such cases used external sourceenergy@e.g., in the form of adenosine triphosphate~ATP!,which can be hydrolyzed to liberate 10kBT of free energy#,Hopfield and Ninio showed how byrepeatinga reversiblerecognition step one could reduce the probability of ancorrect match of reactants below that expected on thermonamic grounds. For this reduction of the error rate toachieved, the repeated recognition steps need to be sepaby irreversible processes. Hopfield called such mechani‘‘kinetic proofreading’’ reactions@1#. Both Hopfield@1# andNinio @2# imagined them as applied to primarily energe~enthalpic! interactions, where differingoff rateswould pro-vide the main means of distinguishing correctly from incorectly matched reactants.

In this paper we apply ideas similar to those of Ninio aHopfield to a model for the mechanism of DNA toposomerases, the enzymes that catalyze topology changeDNA molecules in cells@3#. Recent experiments of Rybenkov et al. @4# showed that certain ATP-hydrolyzing toposomerases are able to suppress the probability of knotspercoils, and links of large DNA molecules, far below tlevel expected in thermodynamic equilibrium. To do this, ttopoisomerases must sense the differences inon rates, i.e.,differing rates of collision between DNA segments, betweentangled and disentangled states. Here we show thatnetic proofreading model, previously introduced@5# to de-scribe suppression of knotting, also can explain the sucoiling suppression experiments. This model is based o

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two-step process that provides more sensitivity to the diffences in collision rates between entangled and disentanDNA molecules than is possible in thermal equilibrium.

A. Topoisomerases catalyze passages of DNA through DNA

For simplicity, we shall only consider topoisomerasesteracting with circular double-stranded DNA moleculewhich have well-defined topologies. The topology of circuDNA can be changed in two distinct fashions. First, we cimagine changing the ‘‘internal’’ linking number Lk of thedouble helix itself@Fig. 1~a!#. Second, we can imagine pasing one double-helix segment through another so as to knsingle circular DNA, or to link two~or more! circles @Figs.

FIG. 1. Topology changes of circular DNA molecules.~a! Inter-nal linking number of a double helix can be changed by temporabreaking one of the sugar-phosphate backbones along a dostranded DNA.~b! Knotting of a circular double-helix DNA can bechanged by passing the double helix through itself.~c! Catenation~linkage! of two DNA molecules can be changed by passing odouble helix through another.

©2001 The American Physical Society09-1

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JIE YAN, MARCELO O. MAGNASCO, AND JOHN F. MARKO PHYSICAL REVIEW E63 031909

FIG. 2. ~Color! Two-gate model of type-II topoisomerase function. The topoisomerase reversibly binds to aG ~‘‘gate’’ ! segment. AboundG segment can then be reversibly cleaved by the enzyme. A secondT ~‘‘transported’’! segment can then fluctuate in and out of tgate. ATP binding irreversibly triggers enzyme conformational change which traps theT segment; theT segment is eventually transporteout the other side of the enzyme. Finally, the products of ATP hydrolysis are released to reset the enzyme to its initial state.

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1~b,c!#. Note that interlinking between two circular DNAmolecules is often calledcatenationto distinguish it from theinternal linking of the two strands in the DNA double heli

There are two separate families of enzymes in all cthat are specialized to accomplish these two jobs. Typtopoisomerases transiently cutoneof the double-helix back-bones to allow changes in internal duplex linking numbType-II topoisomerases pass one double-helix segmthrough another, to allow changes in knotting of one mecule, or interlinking of double helices.

Most type-I topoisomerases are simple catalysts and athermal equilibrium; they thus allow internal linking numb~and usually twisting stress! in a DNA to relax to its thermo-dynamic equilibrium value. If a type-I topoisomerase islowed to equilibrate with circular DNA molecules in solution, a thermal distribution@6# of double helix linkingnumbers will eventually be obtained@7#.

Most type-II topoisomerases hydrolyze ATP, thus releing stored energy, during their reaction cycle of passagedouble helix through double helix@8#. When first binding toDNA, they cut the molecule to form a double-helix-prote‘‘gate’’ through which a second double-helical DNA sement can pass. The segment that is cut is called theG seg-ment, while the segment that is transported through theis called theT segment. This transport process is unidiretional, and requires ATP binding to occur@8#. The enzymenormally binds two ATPs, hydrolyzing both of them durinits reaction cycle@8–10#, but the details of how ATP bindingand hydrolysis are coupled to enzyme function are incopletely understood.

Experiments to date support a ‘‘clamp’’ model of hoATP binding and hydrolysis are coupled to the enzymmechanical function~Fig. 2! @11,12#. Supposing that aGsegment is already bound, in the absence of bound ATPenzyme is in an ‘‘open clamp’’ state, allowing theT segment

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to freely enter and exit the gate entrance. ATP bindcauses the clamp to close, and if there is aT segment caughin the clamp, it will be transported through the gap in theGsegment to pass out through the exit on the opposite sidthe enzyme. ATP hydrolysis occurs during this step. Afrelease of the product of ATP hydrolysis, the enzyme retuto the open-clamp form, still bound to theG segment. At thispoint, the enzyme–G-segment complex has completedentire DNA-passage cycle, and may either repeat~with adifferent T segment!, or, following ligation of the cut in theG segment, the enzyme may dissociate from theG segment.The essential point is that ATP binding may be consideredbe an irreversible step, triggering enzyme conformatiochange and eventual ATP hydrolysis.

In fact, ATP hydrolysis is not actually required foG-segment cleavage or religation: experiments on nonhydlyzable ATP analogs show that the reaction proceeds allway toT-segment passage@11#. Instead, ATP hydrolysis andproduct release appear necessary only for the final reseof the enzyme to its initial state, i.e., reopening of the claand repair of the brokenG segment.

While all type-II topoisomerases have this basic functiothey can be divided into two subfamilies based upon detof their biological functions and differences in DNA interation. Enzymes in one subfamily primarily decatenate intlinked DNA, and include all eukaryotic type-II topoisomerases as well as bacterial topoisomerase IV and pT4 topoisomerase II. This family of DNA disentanglintype-II topoisomerases is the focus of this paper. The secsubfamily is comprised of bacterial gyrases that use ATPnegatively supercoil~i.e., unwind! DNA. Gyrases clearly uti-lize the energy released during ATP hydrolysis to applyternal torsional stress to DNA, increasing its free energy

On the other hand, how the energy liberated by ATP hdrolysis is used by enzymes in the first subfamily is n

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KINETIC PROOFREADING CAN EXPLAIN THE . . . PHYSICAL REVIEW E63 031909

clear. The basic two-gate model described above doesdescribe how the DNA is affected by the irreversibility of thone-way strand transfer process. In fact, before 1997question of whether type-II topoisomerase incubated wDNA and ATP would generate a thermal distributionknot, catenation or supercoil topoisomers was unanswer

B. Experiments on topological simplification by type-IItopoisomerases

In 1997, Rybenkov and co-workers@4# definitively estab-lished that in fact type-II topoisomers use ATP hydrolysisstrongly suppress knotting, catenation, and supercoilingDNA molecules in solution to levels far below those epected from thermal fluctuation of topology. Thermal distbutions of topoisomerase have long been studied, andbe prepared by the use of enzymes that change topologthermal equilibrium~i.e., without ATP hydrolysis or otheirreversible steps! such as topoisomerase I. Equilibrium ditributions of knots and catenanes are usually generatedusing DNA molecules that which have breaks or ‘‘nicksalong their sugar-phosphate backbones, which allow themintermittently open in thermal equilibrium. After equilibration these nicks can be sealed up and the resulting topocal distribution of the circular DNA molecules can thenanalyzed.

For double-stranded DNA, the thermal knotting probabity, given some equilibrium pathway to topology change,well characterized experimentally and theoretically, and fofew kb DNA molecules, the probability of having a nontrivial knot is a few percent, with almost all the knots simptrefoils. This small knotting probability expresses the larentropic cost of constraining a few kb DNA to cross itselfa particular fashion. Rybenkovet al. @4# found that the prob-ability of knotting of 10 kbP4 DNA is reduced from itsequilibrium value of Pknot

eq 50.031, to only Pknottopo II

50.000 62 when a type-II topoisomerase and ATP wpresent. Likewise, the knotting probability for 7 kbPAB4DNA was reduced fromPknot

eq 50.017 toPknottopo II50.0002.

Similar experiments were done to examine DNA catetion; again, the comparison was for a DNA concentratwhere the probability of linkages occurring by thermal flutuation of topology was a few percent. Although this typeexperiment is somewhat more complicated than knottisince it involves multiple molecules and therefore depeon the DNA solution concentration, the result was analogto that for knotting. The probability for interlinkage of tw10-kbP4 DNA molecules was reduced from the equilibriuvalue Plinkage

eq 50.064 for certain concentration and solutioconditions, to the much smallerPlinkage

topo II 50.004.Rybenkov et al. also examined fluctuation of interna

double-helix linkage number@Fig. 1~a!#, or supercoiling ofcircular DNA molecules, comparing thermal equilibratiovia topoisomerase I, with the steady state obtained usingpoisomerase II and ATP. The change in linking number frthe value energetically preferred by an undistorted douhelix ~i.e., one turn per 10.5 bp!, or DLk is the relevantquantity to consider, sinceDLk&eq50.

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It is crucial to note that type-II topoisomerase is ablechange double helix linking number, but only by steps62. This can be best illustrated by taking a thin paper sand noting that the linkage number of its edges is changed62 when the strip is passed through itself. Rybenkovet al.found that the variance of the linking number distributiowas reduced from its equilibrium value^DLk&eq

2 53.1 to theexperimental valueDLk& topo II

2 51.7, for bacterial topoiso-mer IV.

C. Kinetic proofreading model and plan of this paper

In each of the three cases of knotting, catenation,supercoiling, Rybenkovet al. thus demonstrated that toposomerase II reduced the topological complexity below wwould be expected for thermal fluctuation of topology, i.by Brownian segmental motion and collision@4#. Somehow,type-II topoisomerases recognize DNA molecules with coplex topology and preferentially make strand passagesthose molecules. This is only possible because type-II topsomerases are machines~or motors! that use an externasource of energy, i.e., ATP, to give them an irreversibcycle~Fig. 2!. However, given ATP hydrolysis, it is not cleahow the small~10 nm diameter! enzyme determines the topological state of a. 1000 nm length of DNA.

The purpose of this paper is to show how a modelpreviously proposed@5# to explain the knot and catenatiodata can also simply explain the result for the reductionsupercoiling fluctuations. This model was based on the rization that the experimental results corresponded closelthe topological distributions in the presence of topsomerases II being on the order of thesquaresof the distri-butions obtained thermally. We further realized that thezyme could use its irreversible cycle to be sensitive tocollision rate of DNA segments, and that this informatiocould plausibly provide the guidance necessary to reducetopological complexity to the observed levels. The minimmodel required to do this is one where two successive DNDNA collisions must occur within a short time window tachieve DNA strand passage. We call this general typemodel a kinetic proofreading model, in reference to relamodels for molecular recognition processes@1,2#.

In the remainder of the paper, we first briefly review omodel for reduction of knot and catenation complexitySec. II. The discussion is particularly simple for these cabecause for the few kb DNA molecules of interest, expementally, transitions between only two topological sta@unknot↔ trefoil knot, and unlink↔ single link, as in Figs.1~b,c!# need be considered. By contrast, for the case ofpercoiling, ‘‘ladders’’ of topological states must be treatewith transitions driven between states with Lk’s differing b62.

In Sec. III we review established results for thermal flutuation of Lk , and for the free-energy cost of perturbing Lfrom the relaxed~lowest free energy! state where there is onlink per 10.4 bp of DNA. In Sec. IV we discuss how the freenergy differences between different supercoils can be uto estimate the rates of collisions that lead to transitionstween the different states. This prepares us to present

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JIE YAN, MARCELO O. MAGNASCO, AND JOHN F. MARKO PHYSICAL REVIEW E63 031909

two-collision kinetic proofreading model applied to the caof supercoiling fluctuations in Sec. V. This section includequantitative comparison of the model and the experimedata.

We finally note that while we were preparing this papZhanget al. @13# discussed a similar application of our knetic proofreading model to supercoiling topoisomeraseThe results of their analytical work are rather similar to ouThey also present some kinetic data from simulationssupport some of the assumptions we have made abouconnection between collision rates and the free energytopoisomers. These points will be discussed in Sec. V.

II. KINETIC PROOFREADING MODEL OF TYPE-IITOPOISOMERASES

The results of Rybenkovet al. @4# suggested to us that thapparentsquaringof the probability distribution of topologi-cal states might be the result of a similar kinetic proofreadmechanism, but applied instead to the differingon rates, ortopology-dependent collision ratesthat are the kinetic originof the thermodynamic equilibrium distribution of toposomerasease. Thus, we arrived at a two-collision modetype-II topoisomerases@5#, which could be rather simply applied to the knotting and catenation experiments of Rybkov et al. @4#, since for the,10 kb DNA molecules studiedin equilibrium, the only appreciably likely topological statewere a trivial state~unknot or unlink! and a nontrivial state~trefoil knot or single link!.

We review the two-collision model for knot-unknot transitions~Fig. 3!. Starting from either knots or unknots withtopoisomerase bound to aG segment~Fig. 3, state 1!, a‘‘synapse’’ between theG and a secondT segment can reversibly occur~Fig. 3, 1↔2; this corresponds to the reverible G-T collision step of Fig. 2!. For these states, note ththe numbers ‘‘1’’ and ‘‘2’’ indicate how many DNA segments are bound by the topoisomerase. A key point ispolymer-conformational fluctuations drive the forward areverse transitions at this step. Since theT segment must‘‘find’’ the bound topoisomerase by random fluctuation, asince the polymer dynamics of the polymer will certaindepend on whether it is knotted or not, the on rate will dpend on the topological state.

The off rate will depend only on the dynamics of a festatistical segments of polymer near the synapse, and peron local interactions between the topoisomerase and thTsegment. Both of these effects are going to be independetopology so long as the knots are not so small that there ielastic distortion of the DNA that can strongly affect its dnamics from the synapse state. Since we are conside.5 kb DNA molecules, and since a statistical segmen300 bp, we are not in the elastic regime. We thus assumethe on rates~for knots k, and for unknotsy) depend ontopology but that the off rates (l for both knots and unknots!do not.

If these collisions were allowed to pass DNA throuDNA, they would generate topological changes. Underassumption that all the collisions lead to topology chanthe steady state would satisfyyPunknot

eq 5kPknoteq . Here,Pknot

eq

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and Punknoteq are the equilibrium probabilities for the ‘‘phan

tom’’ case where DNA segments are permitted to freely pthrough one another. With the additional assumption thattopoisomerase bound to DNA does not strongly influenceglobal conformation or its intermolecular collision rates, wthen obtainy/k'Pknot

eq /Punknoteq .

This identification is shown as approximate since cosions that do not change topology will also contribute toyandk. However, it is reasonable that the total self-collisirates differ strongly for few kb knots and unknots. Prelimnary results from our own molecular dynamics simulatioof knots and unknots verify this hypothesis, and will be pulished separately.

The next step in our model takes place from the synastate 2. We suppose that an irreversible process can ofrom 2, to a state 1* with the topoisomerase in an ‘‘activatstate’’ ~denoted by *!, and with only one DNA segmenbound ~denoted by the 1!. We imagine this to occur by aconformational change of the enzyme, combined withlease of theT segment captured in the previous step. Irversibility could be enforced via ATP binding, possibcoupled to topoisomerase conformation change and cleaof the bound DNA. As no overall DNA conformationachange is involved in this step, the ratea of this step isinsensitive to DNA topology. The function of this step isprovide the irreversible step separating two recognitsteps; for a second recognition step to occur, the firstT seg-ment must be released. The first collision (1↔2) is used asa ‘‘first pass’’ of recognition, after which the topoisomerais put into the activated ‘‘*’’ state which is able to effect aDNA strand passage.

The state 1* can irreversibly ‘‘decay’’ back to 1 at atopology-independent rateg, giving spontaneous ‘‘deactivation’’ of the topoisomerase with no change in DNA topoogy. This decay process includes hydrolysis and releasthe bound ATP, and enzyme conformational change bacthe original state. However, before this decay can occunew synapse might reversibly form between a new DNsegment and the activated topoisomerase-DNA comp(1*↔2*), that will lead to strand transfer. As before, thoff-ratesl8 are assumed insensitive to DNA knotting. As ware only concerned with topology-changing events, we massumey8/k8'Pknot

eq /Punknoteq .

If a recollision does occur, then irreversible transitiofrom 2*→1 achieve strand passage, DNA religation, arelease of the passed segment. At the end of the 2*→1transitions, the topoisomerase is reset and ready for anocycle. ATP hydrolysis and product release could be involvin this step. This irreversible transition should be controllby local topoisomerase-DNA interactions, and should ocat a ratem that is independent of DNA topology.

The chemical rate equations can easily be solved for tsteady state, since all the reactions are unimolecular andrate equations therefore linear. The steady-state knot/unratio takes a simple form due to the symmetry between mof the rates on the knot and unknot sides of the reaction

Pknot

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KINETIC PROOFREADING CAN EXPLAIN THE . . . PHYSICAL REVIEW E63 031909

FIG. 3. ~Color! Two-collision model for type-II topoisomerase that accomplishes kinetic proofreading of DNA topology. Self-color ‘‘synapsing’’ occurs twice along knotting and unknotting pathways. The labels ‘‘1’’ and ‘‘2’’ indicate the number of DNA segmbound to the topoisomerase; * indicates states where the topoisomerase is ‘‘activated’’ to be able to pass DNA through DNA~see text!. Thetopoisomerase itself is shown in blue when inactive and red when active. By cascading two synapsis events separated by irtransitions, the first synapsis~1↔2! delivers an excess of knots over unknots to the second synapsis (1*↔2*); the second reaction thus‘‘proofreads’’ the first. Proofreading can reduce the knot-to-unknot ratio to below (Pknot

eq /Punknoteq )2. Most of the transitions do not depend o

knottedness~green arrows!; all discrimination of topology is based on synapse formation rates~red arrows!. The many topology-independenrate constants~e.g., dependent on DNA enzyme unbinding kinetics, which will be energetically and locally determined! do not contribute tothe final knot-to-unknot steady-state ratio of this model.

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The value of the decay rateg plays a crucial role in Eq.~1!. If g→0, or the activation of the enzyme is permaneand therefore always eventually leads to a strand transthen the steady-state knot/unknot ratio becomesy8/k8,which we argued above to be just the equilibrium ratio. Toccurs because without the decay pathway active, the tstep process simply delays the eventual approach to themodynamical equilibrium ratio. Put another way, in thlimit, the topoisomerase has no internal time referenagainst which to measure the difference between collisrates on a DNA, and therefore will make random strand psages and will arrive at the thermodynamical knot/unkdistribution.

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On the other hand, ifg is large enough so thatg@l81m#@k8m ~and therefore also, so thatg@l81m#@y8m),then the decay pathway more rapidly depletes unknots tknots arriving at 1* , and there occurs a reduction of thknot/unknot ratio to

Pknot

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kk8. ~2!

Thus, assuming that the ratio of the total self-collisiratesy/k is roughly equal to that for the topology-changinrate ratioy8/k8, we find that the reaction of Fig. 3 can re

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JIE YAN, MARCELO O. MAGNASCO, AND JOHN F. MARKO PHYSICAL REVIEW E63 031909

duce the ratio of the knot and unknot probabilities to tlevel of thesquareof Pknot

eq /Punknoteq . This was the main resul

of our previous paper@5#. In theg→0 limit, all the topology-independent rate constants drop out of the final knot prability, leaving a result with no adjustable parameters, trepresents theoptimal reduction of knot and link complexitypossible by this proofreading reaction.

Linking-unlinking of circular DNA molecules can be discussed in exactly the same way as the knotting-unknotsteady state discussed above, for the dilute solution cotions of Ref. @4#. In all the experiments, the unknot/unlinprobabilities were nearly 1, with the knot/link probabilitiesthe level of a few percent, allowing the use of a two-stkinetic model. The two-step proofreading process, if opmized, could be expected to reduce the probability of knand links in those experiments to a level as low as thesquareof the probabilities expected in thermodynamic equilibriui.e., to the'1024 level.

The ‘‘squared law’’ derived above agrees well with thexperimental results of Rybenkovet al. @4# for knotting andlinking probabilities for DNA molecules incubated wittype-II topoisomerases and ATP~Fig. 4!. In our previouspaper@5# we noted that the same model could also explanalogous results obtained by Rybenkovet al. @4# for super-coiling topoisomers, and that this provided additional suport for our hypothesis that the type-II topoisomerases ustwo-collision process.

In the next sections we discuss the thermodynamicaltribution of supercoiling topoisomers, the kinetic rates thcan be expected for collisions that interconvert those topsomerasease, and the generalization of the model discuin this section to the case of supercoiling topoisomers.

III. THERMAL EQUILIBRIUM LINKING NUMBERDISTRIBUTION

Thermal fluctuations of linking number Lk of a circuladouble-helix DNA are well characterized both experime

FIG. 4. Experimental knotting and linking probabilities of~sym-bols!. Horizontal axis shows thermal equilibrium entanglemeprobability; vertical axis shows the steady-state result in presenctype-II topoisomerases and ATP. The kinetic proofreading modethe text is able to reduce knotting probability to levels belowline, which corresponds to(steady state)5(equilibrium)2.

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tally @6,7# and theoretically@14–16#. The minimum free en-ergy state has Lk5Lk0, where Lk0 is the DNA length inbase pairs, divided by about 10.4, reflecting the energeticfavored rate of winding of one strand around the other inrelaxed double helix. The free energy cost of supercoilingessentially harmonic:

F~Lk!

kBT5

G

2~Lk2Lk0!25

G

2~DLk!2. ~3!

There are some deviations from this harmonic behavwhen F@kBT, but since we are primarily interested in ththermal fluctuation regimeF'kBT, Eq. ~3! is simple andaccurate. In the experiments of Rybenkovet al. @4#, the ini-tial state for all type-II topoisomerase supercoiling expements was this thermal distribution.

The stiffnessG is numerically a bit less than that whicwould be assumed from a thermal fluctuation argumentplied to DNA twisting, or 4p2 times the torsional stiffness oDNA in kBT units ~i.e., the twist persistence lengthC'100 nm'300 bp!, divided by the total DNA lengthL. Thereduction ofG below this value occurs because ofwrithing,or chiral bending, of the DNA backbone@16#. The scalingwith L is expected so that the free energy is intensive~notethat Lk is extensive!. For DNA under physiological solutionconditions, the stiffness observed experimentally is abouG'(2200 bp)/L5(750 nm)/L @16#.

If Lk is allowed to thermally fluctuate, it will equilibrateto a Boltzmann distribution:

PLkeq}e2F(Lk)/kBT, ~4!

which leads to a Gaussian distribution:

PLkeq}expF2

G

2~DLk!2G . ~5!

The thermal fluctuations have amplitude^(DLk) 2&eq51/G'L/2.2 kb; the scaling follows from independent twistinand writhing fluctuations occurring in persistence-lengthgions along the molecule@16# For the 7 kb DNA studied byRybenkovet al. @4#, G'0.3, and^(DLk) 2&eq'3.

IV. KINETICS OF Lk TRANSITIONS GENERATEDBY TYPE-II TOPOISOMERASES

A. Odd- and even-Lk subpopulations are not mixed by type-IItopoisomerases

Type-II topoisomerases can only change the linking nuber of circular DNA by 62. Therefore, after introducingtopoisomerase II plus ATP into a solution of initially themally equilibrated circular DNA molecules~assuming thatthe topoisomer II is the only available mechanism for topogy change!, Lk for any particular molecule will always beeven or odd. The steady state distribution of Lk’s obtainby the action of topoisomerase II is actually a sum of indpendent even-Lk and odd-Lk distributions.

The parity-conserving nature of type-II topoisomerameans that a molecule with linkage Lk5k can undergo onlytwo transitions,k→k12 andk→k22. The kinetics of the

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KINETIC PROOFREADING CAN EXPLAIN THE . . . PHYSICAL REVIEW E63 031909

FIG. 5. ~Color! Freely self-crossing passage reaction model of a type-II topoisomerase distinguishes two DNA-DNA ‘‘synapse’’On isolated circular DNA molecules, transitions occur from Lk5k to Lk5k12 at a ratekk,k12l, and back again at a ratekk12,kl. For atopoisomerase that does not consume stored energy,kk12,k /kk,k12 must be equal to the ratio of the thermal equilibrium probabilities ofstates Lk5k andk12.

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combined topoisomerase-II–DNA system are therefore smarized by the transition ratesRk,k62, for all integersk. Thetransitions thus are organized into two decoupled odimensional chains of reactions, for the odd- and evenstates~Fig. 5, top!.

B. Detailed-balance-like condition for the steady state

The steady-state distribution attained by the type-II topsomerase,PLk, satisfies

Rk,k12Pk5Rk12,kPk12 ~6!

for all integersk, since otherwise there would be a net floof probability along the one-dimensional ladders of sta~Fig. 5, top!. Neighboring states in each ladder therefore sisfy a ‘‘detailed balance’’-like condition in the steady stat

Pk12

Pk5

Rk,k12

Rk12,k. ~7!

This type of relation in general does not hold in nonequilrium systems, but is forced here by the one-dimensionalture of the transitions of Lk .

C. Random-collision model

The simplest kinetic model for topoisomerase II activitythe ‘‘random-collision’’ or ‘‘freely-self-crossing’’ model,where the topoisomerase passes theT segment with some

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fixed probability after it hits the enzyme–G-segment com-plex. By construction, this model must reach thermal eqlibrium for the even- and odd-Lk groups of molecules, sinall that is being introduced is a reversible and unbiased trsition path between states with different Lk . Another waysee that thermal equilibrium must be reached by this mois to realize that its statics must be identical to those o‘‘phantom’’ polymer that is allowed to occasionally freelpass through itself.

If the topology distribution is initially thermally equili-brated, then introduction of a topoisomerase II that mastrand passages after random collision will thereforechange the Lk distribution. Since this was the mannerwhich the experiments of Rybenkovet al. @4# were carriedout, topoisomerase II was not carrying out strand passafollowing simple random collisions. However, it will be useful for the discussion of the kinetic proofreading modelanalyze the random-collision model in slightly more deta

Thek→k12 andk12→k transitions~Fig. 5, top! can bedecomposed into collision~‘‘on’’ ! steps that form a ‘‘syn-apse’’ between theT andG segments, and exit~‘‘off’’ ! stepsleading away from the synapses. We label the on ratekk,k12 and kk12,k . These transition rates will depend onksince the rate at which collisions leading to particular topogy changes will obviously depend on details of the proxiity of DNA segments to one another, which depends ondegree of supercoiling.

On the other hand, the kinetics of segments exiting a sapse are essentially independent ofk. This is because a seg

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JIE YAN, MARCELO O. MAGNASCO, AND JOHN F. MARKO PHYSICAL REVIEW E63 031909

ment exits at a rate determined by its interaction withenzyme~e.g., its local unbinding kinetics! and by its fluctua-tion dynamics on the scale of a fraction of a DNA perstence length, which arek independent. Thus we label all thsynapse exit process with a single off ratel, giving ratesRk,k125kk,k12l andRk12,k5kk12,kl.

Since the steady state of this model is the thermal elibrium of the odd- and even-Lk populations, the ratio of tk ’s is thus constrained:

Pk12

Pk5

Pk12eq

Pkeq

5kk,k12

kk12,k~8!

for every integerk. This identification of the on rates withproperties of the equilibrium distribution will be an impotant tool in the next section. Note thatPLkÞPLk

eq since thesteady-state distribution has separate even- and odd-Lkmalizations; the total numbers of even- and odd-Lk mecules cannot be changed by type-II topoisomerases.

D. A simple model for the on rates

We will need a bit more information about the on ratkk,k62 in order to completely compute the probability distbution resulting from our proofreading model. The main feture that we want the dynamics to exhibit is a tendencythe collision rates to go up withDLk , since as the moleculeis made more supercoiled, it will writhe, and therefore secollide more frequently. Perhaps the simplest kinetic mothat does this and is consistent with Eq.~8! are simple ther-mally activated kinetics:

kk,k625reF(k)/kBT. ~9!

This model has rates that satisfy Eq.~8! and are thereforeconsistent with thermal equilibration of the freely-crossimodel; that increase with the free energy of the initial st(Lk5k); and that do not depend on the final state (Lk5k62). This independence on the final state means that in sk, transitions occur with equal probability to the two adjacestates. This will be very nearly true in the random-coil flutuation regime relevant to the experiments, where the linknumbers are less than or equal to the total number of petence lengths along the molecule.

The common factorr is proportional to the characteristicollision rate of one site on a molecule with other sites alothat molecule times a factor associated with segment capby the topoisomerasease. The characteristic rater will bevery roughly the DNA segment orientational relaxation rakBT/(hb3)'105 sec21 since the collisions experienced bysegment in a long polymer are mainly with adjacent sments along the molecule. The rater will not actually appearin our final results because we need only ratios of rates sas Eq.~9! with nearby values ofk.

V. KINETIC PROOFREADING MODEL APPLIED TOSUPERCOILING

Our double-collision model for type-II topoisomerases@5#can be placed into each step of the chain of Lk transiti

03190

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-

i-

or--

-r

-l

e

tet-e

is-

gre

-

ch

s

~Fig. 6!. In comparison to the random collision model of Fi5, the double collision proofreading model contains irreveible steps and reaction cycles. As was the case for knotslinks, this model is a kind of machine that can supresspercoiling complexity. The transition rates between Lk5kand Lk5k12, Rk,k12 andRk12,k are different from those inthe random-collision model. Steady states established forodd- and even-Lk distributions that differ from the thermdistributions. However, Eq.~7! still holds for the steady-statedistribution:

Pk12

Pk5

Rk,k12

Rk12,k. ~10!

The ratesRk,k12 and Rk12,k are determined by the reaction of Fig. 6, bottom. The analysis carried out previously@5#~Sec. I! gives

Rk,k12

Rk12,k5

~g@l81m#1mkk12,k!

~g@l81m#1mkk,k12!

sk

sk12

kk,k12

kk12,k. ~11!

As before, the only topologically dependent transitirates are thes andk collision, or on rates. The ratesk is thetotal rate of all possible collisions. We have already seen tcollisions are of two types, based on our analysis ofrandom-collision model~Sec. III!, corresponding to the pathways for Lk changing by62. Thussk5kk,k121kk,k22.

As for the application of our model to removal of knoand links, we assume a large enough decay rateg so thatg@l81m#@k8m andg@l81m#@n8m. This limit leads to

Rk,k12

Rk12,k5

sk

sk12

kk,k12

kk12,k5S kk,k121kk,k22

kk12,k1kk12,k14Dkk,k12

kk12,k~12!

in exact analogy to our previous results for the effectivetwo-state knotting and linking problems@5#.

The topology-dependent ratesy, y8, k, andk8 are deter-mined by thermal fluctuations. Note that in contrast to tknotting and catenation problems, all collisions correspoto a topology-changing pathway for Lk . Thus, since we aconsidering theg→` limit, we may assume the primed anunprimed rates to be essentially the same.

Equation ~8! connects the topology-changing collisiorate-ratio to thethermal equilibriumdistributions:

kk,k12

kk12,k5

Pk12eq

Pkeq

. ~13!

Therefore,

Pk12

Pk5S kk,k12

kk12,kD 2 11~kk,k22 /kk,k12!

11~kk12,k14 /kk12,k!

5S Pk12eq

Pkeq D 2

11~kk,k22 /kk,k12!

11~kk12,k14 /kk12,k!. ~14!

The steady-state odd- and even-Lk distributions are driaway from the equilibrium distributions to be the square

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KINETIC PROOFREADING CAN EXPLAIN THE . . . PHYSICAL REVIEW E63 031909

FIG. 6. ~Color! Double collision model for type-II topoisomerase. Top: transitions occur from Lk5k to Lk5k12 at a net rateRk,k12,and back again at a rateRk12,k . Bottom: the net transition rates follow from the double collision model applied between the states LLk12. For a large activated state decay rateg, Rk,k12 /Rk12,k5(sk /sk12)(kk,k12 /kk12,k)'(kk,k12 /kk12,k)

2 ~see text!.

ceus

iherinn

l-

nis

the equilibrium distribution times a correction factor. Sinkk,l is l independent the remaining rate ratios are jkk,k22 /kk,k1251 andkk12,k14 /kk12,k51, giving simply

Pk12

Pk5S Pk12

eq

Pkeq D 2

. ~15!

Thus, the steady-state shape of the Lk distributions wstill be nearly Gaussian, but with half the variance of tequilibrium distribution, roughly what was observed expementally @4#. The squaring of the whole distribution is iexact analogy to the squaring of the knot-to-unknot aunlink-to-link ratios discussed previously@5#.

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Finally, given the ratio of the number of odd-Lk moecules to the even-Lk moleculesNodd/Neven, we can writethe steady-state distribution for the odd-Lk molecules:

Pk5A

11Neven/Nodd~Pk

eq!2, ~16!

and for the even-Lk ones:

Pk5A

11Nodd/Neven~Pk

eq!2, ~17!

up to an overall constantA determined by the normalizatiocondition (Pk51. If a type-II topoisomerase experiment

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JIE YAN, MARCELO O. MAGNASCO, AND JOHN F. MARKO PHYSICAL REVIEW E63 031909

started from a thermally equilibrated sample~as done in Ref.@4#!, one will haveNeven'Nodd for few kb DNA molecules.

A. Comparison with experiment

We now use the results~16! and ~17! to compare thedistribution predicted by our model with the experimenresults for 7-kb DNA molecules@4#. Lk0 is the moleculelength divided by the helix repeat of 10.4 bp, or Lk05673.The experimentally observed thermal fluctuation of linkinumber is^DLk&eq

2 51/G53.1. Thus the equilibrium distri-bution is Gaussian, with a width ofDLk51.7.

To compute the distribution generated by our kineproofreading model we need to know the fractions of mecules present initially with even and odd Lk, i.e.,Neven/NandNodd/N, whereN5Neven1Nodd. This follows from thefact that the experimenters thermally equilibrated the supcoiling of their sample before adding type-II topos and ATThus from~5! we know

Neven

Nodd5

(k50,62,64, . . .

PLkeq

(k561,63,65, . . .

PLkeq

. ~18!

For the parameters relevant to the experiments@4# this ratiois Neven/Nodd51.000 001, very nearly unity.

The result of the proofreading model for the experimencase is therefore very close to the square of the equilibrdistribution, i.e.,Pk5A(Pk

eq)2 whereA is the normalization.Thus the width of the proofreading steady-state distributis half the width of the thermal distribution. To see hoclose this is to the experimental data, we present our dibution in the form of Fig. 2 of Ref.@4#. Instead of plottingthe distribution, those authors plotted

2ln~PLk!/ ln~PLk0

!

Lk2Lk0. ~19!

For the equilibrium distribution~5! with ^(DLk) 2&53.1 thisshould plot as a line through the origin with slopeG/2'0.16; the experimental results can be seen to followlaw ~Fig. 7, stars!.

The proofreading model predicts a steady-state distrtion that is Gaussian with half the width of the equilibriudistribution, i.e.,^(DLk) 2&51.55, which plots as a straighline with slope 0.32 in Fig. 7~diamonds!. This can be seen tobe just slightly steeper than the experimental data forsteady-state distribution@4# obtained using a type-II topoisomer ~E. coli topoisomerase IV, Fig. 7, crosses!. The experi-mental data fit to a line of slope 0.29, corresponding tovariance^(DLk) 2&51.7.

Rybenkovet al. also published the reduction of the varances of (DLk) 2& obtained using a number of type-II toposomeraseases, relative to the equilibrium varia^(DLk) eq

2 &. For E. coli topoisomerase IV, the variance wareduced by a factor 3.1/1.751.8, which is slightly below theratio 2.0 predicted by the model of this paper. Other top

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somerases gave reductions in variances from 1.35 totimes. This suggests that type-II topoisomerases may‘‘leaky,’’ perhaps by occasionally becoming ‘‘activated’’ bthermal conformational fluctuation of the enzyme. Somlevel of ‘‘errors’’ in the operation of the machine of Fig. 6and some variation of this error rate from enzyme to enzymis a plausible explanation for the variation in the varianreduction observed experimentally.

One way to add such error pathways to the model ispermit additional transitions between the Lk5k and k12states of Fig. 6. These error pathways provide a transibetween adjacent topoisomers of the same parity that iparallel to the proofreading transition. If these error trantions are thermally activated, their rates must satisfy detabalance. This is perhaps most easily seen if one imaginesATP-hydrolyzing part of the reaction to be turned off: thethe steady state will be achieved by the thermal error trations alone, and must be the thermal equilibrium distributiIf the error rates are sufficiently large, the error transitiowill drive the steady state to the equilibrium distribution.

B. Comparison with theory of Zhang et al.

While this paper was being prepared, Zhanget al. @13#made calculations similar to those presented here. Zhet al. also used our two-step model@5# applied to the chainof transitions between different-Lk states. The main diffeence between the works appears to be the use of a MCarlo simulation of the dynamics of circular DNA to estmate the rates that we estimated analytically in Sec. III D

While Zhanget al. did not present their results in precisely the form of Fig. 7, they also computed the probabildistribution for the steady-state model using their Mon

FIG. 7. Comparison of the linking number distributions obtainin thermal equilibrium~stars!, in the type-II topoisomerase-drivesteady state of Ref.@1# ~crosses!, and from the two-collision kineticproofreading model of the present paper~diamonds!. What is plot-ted is the logarithm of the probability distribution divided byDLk,and therefore a Gaussian distribution will yield a straight line, wslope equal to (2DLk&2)21. The proofreading model and type-Itopoisomerase distributions are in close agreement, each shoroughly double the slope of the equilibrium distribution.

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KINETIC PROOFREADING CAN EXPLAIN THE . . . PHYSICAL REVIEW E63 031909

Carlo calculation. They therefore had to assign numervalues to the rates of Fig. 6. They studied the steady-sprobability distribution as a function of the decay rateg forthe activated state, and found that for sufficiently largeg, thesteady-state value of^(DLk) 2& was driven below its thermaequilibrium value. They also noted that by tuning the vaof g the degree of reduction in variance of the Lk distribtion could be tuned so as to fit data for the different enzymstudied experimentally@4#.

Zhanget al. thus provided a valuable verification of thcorrectness of the assignment of the collision rates for mecules with different Lk we made in Sec. III D. More geneally they have demonstrated via explicit computer simulatthe feasibility of the basic topological proofreading concediscussed previously from an analytical perspective@5#. Themain limitation of the simulations of Zhanget al. is the useof Monte Carlo dynamics to draw conclusions about retively short-distance and short-time collision dynamicsDNA. It would be worth repeating their study using a morealistic molecular dynamics simulation, e.g., Brownian dnamics as developed by Vologodskii and co-workers forstudy of DNA-site juxtapositions@17#.

VI. CONCLUSION

We have described application of our two-collision mod@5# for simplification of the topology of DNA molecules below equilibrium levels by type-II topoisomerases, to doubhelix linking number Lk . Our model effectively senses tself-collision rates of molecules, using the informatigained from the two collision steps — information thatstoredin irreversible conformational change of the enzymDNA complex — to push the probability of knots, links, ansupercoils below their equilibrium levels.

We have presented a straightforward generalization ofprevious result of knotting and linking probabilities attainwith topoisomerases beingsquaredrelative to the thermaequilibrium levels, namely, that the entire distribution of L

.

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becomes squared:PLktopo}(PLk

eq)2. Since the equilibrium dis-tribution is a nearly Gaussian distribution centered on L0,the type-II topoisomer1ATP distribution is also essentiallyGaussian, but with half the width, i.e.,(DLk) 2& topo50.5 (DLk) 2&eq.

The Gaussian shape of the steady-state distribution anreduction in variance relative to the equilibrium distributioreproduce corresponding effects observed experimentallya number of ATP-hydrolyzing type-II topoisomerases@4#. Inaddition our theoretical distribution is in quantitatively gooagreement with the distribution obtained withE. coli topoi-somerase IV, which is the most potent entanglemeremoving type-II topoisomerase. Thus our basic twcollision model can explain diverse results for supressionknotting, linking, and now supercoiling.

Of course our model cannot be considered as ‘‘proveuntil the chemical-mechanical cycle of a type-II toposomerase is completely dissected. From a chemical stpoint, there has been recent progress in demonstratingthe two ATP hydrolysis steps occur at different times durithe topoisomerase cycle@9,10#. Progress has also recentbeen made in studying the mechanical operation of a typtopoisomerase directly, via micromanipulation of the DNtemplate@18#. These techniques may allow the questionwhether or not two DNA-topoisomerase collisions precestrand transfer to be answered.

ACKNOWLEDGMENTS

We acknowledge helpful discussions with D. ChatenN.R. Cozzarelli, G.B. Mindlin, V. Rybenkov, E.D. SiggiaA.V. Vologodskii and E.L. Zechiedrich; J.F.M. and J.Y. aknowledge support of the NSF through Grant No. DM9734178, Research Corporation, the Petroleum ReseFund, the Whitaker Foundation, and the University of Ilnois Foundation. M.O.M. acknowledges support of the SloFoundation and the Mathers Foundation.

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