Mechanochemistry of T7 DNA Helicase

doi:10.1016/j.jmb.2005.04.051 J. Mol. Biol. (2005) 350, 452–475

Mechanochemistry of T7 DNA Helicase

Jung-Chi Liao1†, Yong-Joo Jeong2†, Dong-Eun Kim2, Smita S. Patel2*and George Oster1*

1Departments of Molecular andCell Biology and ESPMUniversity of CaliforniaBerkeley, CA 94720-3112USA

2Department of BiochemistryRobert Wood Johnson MedicalSchool, 675 Hoes LanePiscataway, NJ 08854, USA

0022-2836/$ - see front matter q 2005 E

† J.-C.L. & Y.-J.J. made equal contrPresent addresses: D.-E. Kim, De

Biotechnology and Bioengineering,Busan 614-714, South Korea. Y.-J. Kiand Nanochemistry, Kookmin UnivChongnung-dong, Songbuk-gu, SeoAbbreviations used: ssDNA and

stranded and double-stranded DNAE-mail addresses of the correspon

[email protected]; goster@nature.

The bacteriophage T7 helicase is a ring-shaped hexameric motor proteinthat unwinds double-stranded DNA during DNA replication andrecombination. To accomplish this it couples energy from the nucleotidehydrolysis cycle to translocate along one of the DNA strands. Here, wecombine computational biology with new biochemical measurements toinfer the following properties of the T7 helicase: (1) all hexameric subunitsare catalytic; (2) the mechanical movement along the DNA strand is drivenby the binding transition of nucleotide into the catalytic site; (3) hydrolysisis coordinated between adjacent subunits that bind DNA; (4) the hydrolysisstep changes the affinity of a subunit for DNA allowing passage of DNAfrom one subunit to the next. We construct a numerical optimizationscheme to analyze transient and steady-state biochemical measurements todetermine the rate constants for the hydrolysis cycle and determine the fluxdistribution through the reaction network. We find that, under physiologi-cal and experimental conditions, there is no dominant pathway; ratherthere is a distribution of pathways that varies with the ambient conditions.Our analysis methods provide a systematic procedure to study kineticpathways of multi-subunit, multi-state cooperative enzymes.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: helicase; ring; ATPase; sequential; pre-steady state kinetics
*Corresponding authors
Introduction

Helicases are motor proteins that translocatealong nucleic acid chains using the energy of NTPhydrolysis. The ability to translocate unidirection-ally enables them to carry out processes involvingnucleic acid metabolism, especially those thatrequire the separation of duplex nucleic acids intotheir component single strands.1–3 Helicases drivecritical biological processes such as DNA replica-tion, repair, and recombination; hence, mutations ina helicase protein lead to many human diseasesincluding cancer and/or premature aging.4

Bacteriophage T7 gene 4 helicase is a hexamericring helicase that translocates along DNA during

lsevier Ltd. All rights reserve

ibutions to this work.partment ofDong-Eui University,m, Department of Bioersity, 861-1,ul 136-702, Korea.dsDNA, single-, respectively.ding authors:berkeley.edu

DNA replication and recombination and unwindsthe complementary DNA strands.1,5 Althoughcrystal structures for heptameric helicases havebeen reported,6 we will focus here on the hexamericspecies. However, our analyses can easily beapplied to the heptameric ring with some modifi-cations. The energy source for the T7 helicase isdTTP (deoxythymidine triphosphate), whose func-tion is equivalent to that of ATP in other motorproteins. The chemical energy of dTTP hydrolysis isconverted into mechanical work to move the heli-case unidirectionally along the DNA. dTTPs (orother nucleotides) also stabilize the formation of thehexameric ring structure in T7 helicase.7 Similarly,we have shown that T7 helicase binds DNA tightlyin the presence of dTTP or dTMPPCP but not in thepresence of dTDP.7 Thus, dTTP binding andhydrolysis allow DNA bind–release cycles but thestep in the dTTPase cycle that triggers DNA releaseis not known. Translocation of T7 helicase proceedsat a rate of w130 nt/s along ssDNA, hydrolyzingone dTTP per three-base movement.8 On unwind-ing forked duplex DNA, the movement is slower,where strand separation catalyzed by T7 helicaseproceeds at an average rate w15 bp/s.9

Figure 1(a) shows two subunits of the ring

d.

Figure 1. Structural arrangement of T7 helicase. (a) The model of DNA strand separation by T7 helicase built fromelectron microscopy13 and the structures of two helicase subunits and their interaction with an ssDNA.5 Mutationexperiments suggest that motif II (residues 424–439, shown in green) and motif III (residues 464–475, shown in pink)interact with DNA.5,27,28 (b) Topological diagram of the secondary structures of two adjacent subunits using the samecolor coding as in (a).11 A Mg-dTTP molecule is shown adjacent to subunits of the helicase. The conserved b-sheetstrands are numbered in the usual order: 5,1,4,3,2. The network of hydrogen bonds and salt-bridges that anneal theMg2C and the nucleotide to the catalytic site are shown broken and color coded according to their donor sites. Thesebonds were determined using a 3.3 A cutoff; a larger cutoff would reveal more, but weaker, bonds. The suspected ssDNAbinding motifs reside in the loops from the b-strands adjacent to the b-strands that form the dTTP catalytic site.

Mechanochemistry of T7 DNA Helicase 453

helicase from the crystal structure,5 and a possiblesingle-stranded DNA (ssDNA) configuration in thecentral channel of the ring. The structure of thecatalytic site in each subunit belongs to the class of“RecA-like” NTPases,10,11 with the highly con-served five-stranded parallel b-sheet as shown inFigure 1(b). Two loops emanating from the centralb-sheet are suggested in the crystal structure tocontact DNA.5 These two loops emerge fromstrands 3 and 4, which are adjacent to the energy-converting P-loop (Figure 1(b)), and this arrange-ment may facilitate the energy transfer from theMg$NTP hydrogen bond network to the DNAmovement.

To understand the mechanochemical coupling forT7 helicase, it is first necessary to determine thekinetic pathways of dTTP hydrolysis, so that therelationship between the kinetics and the mechan-ical steps can be identified. Previous studies havedetermined the dTTP hydrolysis pathway of T7helicase in the absence of DNA.12 Those studiesshowed that only one of the six sites in the hexamerhydrolyzed dTTP at a fast rate and not all six sitesparticipated in the dTTPase turnover. Here, weinvestigate the pathways of dTTP hydrolysiscatalyzed by T7 helicase in the presence ofssDNA. We find that up to four sites in the hexamerhydrolyze dTTP at a fast rate and that all six sites

454 Mechanochemistry of T7 DNA Helicase

likely participate in dTTPase turnover. Under highdTTP concentration, both in the absence and in thepresence of the DNA, the rate-limiting step is therelease of the product, Pi. The number of possiblekinetic states of the six catalytic sites is extremelylarge, and so to analyze the kinetics of dTTPhydrolysis a numerical method is developed todetermine the kinetic rate constant for each step andto determine the main reaction pathways. Using thecombined approach of pre-steady-state kinetics andcomputational kinetic modeling, we dissect thedTTPase mechanism of T7 helicase in the presenceof ssDNA and propose a multi-subunit, multi-statekinetic model that indicates that dTTP hydrolysisand power strokes occur sequentially around thehexameric ring. This model is mechanically con-sistent with the electron microscopy observationsthat showed only one or two subunits bind DNA.13

Also, recently Mancini et al. proposed a similarsequential translocation mechanism for the f12 P4hexameric packaging motor, which is structurallyrelated to T7 helicase.14 However, additional datawill be required to determine if the power-strokesare partially or strictly sequential around the T7helicase ring.

The methodology we use augments thekinematic† notion of “conformational coupling”with the mechanical concept of stress, or force perunit area. This allows us to bring to bear mechanicalconcepts to the phenomenology of cooperativitybetween different regions of a protein.

Results

To understand the mechanisms of translocation,it is necessary to dissect the dTTPase pathway in theabsence and in the presence of the DNA anddetermine when DNA binds and releases from thesubunit. In addition to dissecting the dTTPasepathway, it is important to determine how thedTTPase cycles are coordinated among the hexamersubunits. Previously, using pre-steady-state kineticapproaches, we have determined the kinetic path-way of dTTP hydrolysis in the absence of DNA.12,15

We found that, in the absence of DNA, only onesubunit of the hexamer hydrolyzed dTTP at a fastrate and that several of the hexamer subunitsappeared to be non-catalytic.12,15 The dTTPaseturnover rate was much slower at the non-catalyticsubunits compared to the observed dTTPase kcatvalue. Here, we use a combination of pre-steady-state kinetic methods and computational modelingto determine the dTTP hydrolysis pathway in thepresence of DNA. We use these approaches toaddress the following issues: (a) the number ofactively participating hexamer subunits; (b) the ratelimiting step(s) in the presence of DNA; (c) the

† The branch of mechanics that studies the motion of abody or a system of bodies without consideration given toits mass or the forces acting on it.

specific steps at which DNA binds and releasesfrom the helicase subunit; (d) the coordination ofdTTP hydrolysis among the hexamer subunits.

Pre-assembly of the helicase hexamer on DNA

DNA binds within the central channel of the T7helicase hexamer and previous studies have shownthat DNA binding is a slow process.16 To measurethe pre-steady-state kinetics of dTTP hydrolysis, itis therefore important to pre-assemble the helicasehexamer on the DNA to assure that DNA binding orhexamer assembly does not limit the observeddTTP hydrolysis rate. Since the T7 hexamer requiresthe presence of dTTP to bind DNA,7 it wasparticularly challenging to set up conditions tomeasure the pre-steady-state kinetics of dTTPhydrolysis in the presence of DNA. Our findingthat T7 helicase can be pre-assembled on the DNAwith dTTP without Mg2C provided us with initialconditions to pre-assemble the hexamer on theDNA prior to reaction start. In the absence of Mg2C,dTTP hydrolysis is insignificant (8.5!10K3 M/Mper second at 25 8C; w5000-fold slower than thedTTPase rate in the presence of Mg2C).17 The pre-steady-state dTTP hydrolysis experiments werecarried out using the long circular M13 ssDNA asthe nucleic acid substrate.8 We determined exper-imentally that the optimal concentration of M13ssDNA was 15 nM per mM of T7 helicase hexamer(see Appendix A). The pre-steady-state kinetic setup was as follows: T7 helicase was incubated withDNA in the presence of dTTP (without Mg2C) forthree minutes, which was determined experimen-tally as an optimum pre-incubation period thatprovides sufficient time for assembly and to loadthe pre-incubated sample in the quench-flowinstrument, and reactions were initiated byaddition of Mg2C.

Chase-time kinetics indicates all subunits arecatalytic

The chase-time experiments provide informationabout the number of dTTPs that are tightly bound tothe helicase hexamer and competent in hydrolysis.In the chase-time setting, T7 helicase was incubatedwith dTTP, [a-32P]dTTP, and DNA in the absence ofMg2C. After three minutes, excess non-radiolabeledMg-dTTP was added as a chase and the chase-timewas varied before acid-quenching. During thechase-time, the dTTP molecules that were boundinitially to the hexamer subunits can get hydrolyzedor may dissociate into solution. Once the radio-labeled dTTP dissociates from the helicase intosolution, it is diluted by excess non-radiolabeleddTTP chase and hence not observed in ourreactions. Thus, the chase-time kinetics measuresthe hydrolysis reaction only from the tightly bounddTTPs. In this setting, and also later in the acid-quench experiment, the measurements includedthree different species: dTDP in solution; dTDPbound to enzyme; and dTDP-Pi bound to enzymes.


Detection of dTDP$Pi permits tracing the hydrolysisevents.

Figure 2(a) shows the chase-time kinetics of dTTPhydrolysis at various dTTP concentrations. Thedata fit to the single-exponential equation (4),shown in Experimental Procedures, that providesthe observed amplitudes shown in Figure 2(b). Theamplitude provides an estimate of the number oftightly bound dTTP molecules that are hydrolyzedupon binding Mg2C. Figure 2(b) shows that theamplitude increases with increasing concentrationof radiolabeled dTTP, reaching a maximum value ofabout four dTTPs/hexamer. These chase-timeexperiments, in addition to showing the numberof catalytic sites, provide information about theinitial conditions. The amplitude in Figure 2(b)gives an estimate of the ensemble average dTTPoccupancy for all helicases in solution. Later we willuse this information as the initial conditions in ourkinetic simulations. Similar chase-time experimentsin the absence of DNA have shown that only onesite per hexamer hydrolyzes dTTP at a fast rate anda second one at a slower rate, which is consistentwith product release at the first site limitinghydrolysis at the next.12 Such biphasic kinetics inthe chase-time experiments of dTTP hydrolysis wasnot observed in the presence of DNA.

The reason that less than six sites appear catalyticin the chase-time experiment might be that theinitially bound dTTP dissociates, and the chase-time experiment can only detect the hydrolysisevents of those subunits whose dTTPs do notdissociate from the catalytic sites. However, weconsider this unlikely because direct dTTP anddTMPPCP binding experiments have shown aboutfour nucleotides bound per hexamer.17,18 A morelikely explanation is that negative cooperativity indTTP binding biases the preferences of the helicaseto bind no more than four dTTPs in each hexamer.The basis for the negative cooperativity may bemechanical because the binding of dTTP introducesmechanical stress around the enzyme ring. Thiscircumferential stress becomes large when accom-

Figure 2. Time trajectories of chase-time experiments and tdTDP-Pi, dTDP inside the catalytic sites and dTDP in solution500 mM dTTP) fit approximately to equation (4). (b) The amplproduction (A in equation (4)) in (a). This amplitude representinitially occupied by dTTP.

modating more than four dTTPs in the ring. Similarnegative cooperativity in nucleotide binding hasbeen reported for other ring-shaped hexamerhelicases.18–21

The chase-time experiments showing that theaverage number of catalytic sites is greater thanthree for T7 helicase is crucial, because without thispiece of information it is difficult to distinguish athree catalytic site model and a six catalytic sitemodel. Based on mechanochemical considerations,we argue as follows that it is most likely that all sixsites are catalytically active in the hexameric T7helicase. Figure 3 shows the three different possi-bilities for a four catalytic site model ((a), (b) and (c))and the single possibility for a five catalytic sitemodel (d). A site is considered non-catalytic only ifit remains non-catalytic at all times (at least muchlonger than the time-scale of the other kineticevents). All of the cases shown in Figure 3 arevery unlikely, according to the following reasoning.The only communication mechanism between

subunits is via the stresses developed at the catalyticsites during the hydrolysis cycle and the conse-quent conformational couplings.As each catalytic site passes through its occu-

pancy cycle, it simultaneously passes through itsstress cycle as well.Any site that loses its catalytic activity at all times

must do so because the stress cycles of other sitescan bypass its nucleotide binding site.We will show later that the T7 helicase translo-

cates DNA sequentially. Stress considerations makeit very unlikely that this homo-oligomeric enzymecan maintain the catalytic site arrangements shownin Figure 3 when hydrolysis cycles take placesequentially. That is, it would require very unusualand asymmetric stresses to keep some sites non-catalytic at all times while other sites cycle.The five catalytic site model shown in Figure 3(d)

is possible if there is an active open ring, where oneof the sites cannot form the hydrogen bond networkrequired for hydrolysis. However, this wouldreduce the processivity dramatically, which is not

he steady-state amplitudes. (a) The total [dTDP] includes. Time trajectories of chase-time experiments (60, 110, 250,itudes are obtained from the steady-state values of dTDPs the lower bound of the average number of catalytic sites

Figure 3. Configurations for a four or five catalytic sitemodel. The black ellipses represent sites that are catalyticat any moment, while the white ellipses are non-catalyticsites; the non-catalytic site in any hexamer must be non-catalytic at all times. (a), (b), and (c) are the three possibleconfigurations for a four catalytic site model. (d) The onlypossible configuration for a five catalytic site model.


consistent with the observed high processivityalong single-stranded DNA.8

Thus, the occupancy states continuously cycletheir state of stress, so no single site can remainpermanently inactive while subject to the time-varying stresses.

Thus, we conclude that in the presence of DNA, itis most likely that all subunits are catalytic.

Note that in a sequential mechanism, it is possibleto have a cyclically symmetric three-site or six-siteenzyme. For a three catalytic site model, the stressesbypass the adjacent site and affect the site twosubunits away. Electron microscopy shows a trimerof dimer structure for other ring-shaped helicases,22

which could support a three-site proposal. The non-catalytic sites observed previously in the absence ofDNA may belong to this three catalytic site species.

Pre-steady-state kinetics show biphasicbehavior in dTTP hydrolysis

To characterize the rate-limiting step in the dTTPhydrolysis pathway, acid-quench pre-steady-statekinetics experiments were performed. In the acid-quench setting, T7 helicase was pre-incubated withM13 ssDNA, dTTP, and [a-32P]dTTP and after athree minute incubation, the reaction was initiatedby rapidly mixing with a solution of Mg2C. Thereactions were acid-quenched at various times(milliseconds to seconds). The acid-quench settingdiffers from the chase-time setting in that excessnon-radiolabeled dTTP is not added with theMg2C.Thus, the acid-quench experiment exhibits the

kinetics of dTTP hydrolysis in the first turnover aswell as those of the subsequent ones. If the firstturnover is faster than the subsequent ones, thekinetics will show a “burst” phase. The burst kineticsin our experimental setting indicates that a step afterdTTP hydrolysis is rate limiting, because our initialconditions involved pre-incubating the helicase withdTTP that bypasses the dTTP binding step.

Figure 4(a) shows the time trajectories of dTTPhydrolysis in the acid-quench setting at differentinitial [dTTP]0 that clearly shows the burst kinetics.For each dTTP concentration, the kinetics of dTDPformation was biphasic and can be fit by a singleexponential followed by linear growth (cf. equation(5) in Experimental Procedures). The initial expo-nential phase (burst phase) reflects the fast hydroly-sis step, while the linear phase is the steady-statecontrolled by the rate-limiting step. Figure 4(b)shows that the slope of the linear phase increaseswith increasing dTTP concentrations in a hyper-bolic manner. Thus, from low to high concen-trations of dTTP the rate-limiting step shifts fromdTTP binding to product release. The burst ampli-tude, similar to the chase-time amplitude, providesinformation about the average number of catalyticsites undergoing fast dTTP hydrolysis. In theabsence of DNA we previously observed a burstamplitude about one dTTP/hexamer; here, in thepresence of DNA, about four dTTPs are hydrolyzedper hexamer at saturating dTTP.

Pi release is the rate-limiting step in thehydrolysis cycle

To investigate which product release (Pi or dTDP)is the rate-limiting step, a coupled assay was usedto measure the real-time kinetics of Pi release.23

A mixture of T7 helicase, M13 ssDNA, and dTTPwas rapidly mixed with Mg2C and PBP-MDCC in astopped-flow instrument, and the fluorescencechange was measured as a function of time.A standard curve was used to convert the magni-tude of fluorescence increase into Pi concentration.Figure 4(c) shows the time trajectories of Pi releaseat different initial concentrations, [dTTP]0. Unlikethe acid-quench experiments there are no clearbiphasic characteristics. The linear steady-statephase has approximately the same slope as in theacid-quench experiment for each corresponding[dTTP]0, because the rate-limiting step is the samefor both experiments. In Figure 4(d) comparison ofthe acid-quench and Pi-release experiments show adelay in Pi release relative to dTTP hydrolysis.These results indicate that the hexamer subunitshydrolyze dTTP at a faster rate than they release theproduct Pi. The kinetics were modeled to determinethe rate constants and, as shown in Table 1, we findthat at high dTTP concentration Pi release is therate-limiting step.

dTDP$ Pi is weakly bound to DNA

Figure 4(d) shows the delay in Pi release relative

Figure 4. Experimental data and simulation results for [dTDP] and [Pi] time trajectories. (a) The total [dTDP] includesdTDP-Pi, dTDP inside the catalytic sites and dTDP in solution. The simulations (continuous lines) agree well with thetransient and steady-state experiments for four different initial dTTP concentrations. (b) The steady-state hydrolysis rateper hexamer from slopes of the linear phases in (a). It shows a hyperbolic trend toward saturation with increasing[dTTP]. (c) [Pi] in solution is measured and simulated under the same four initial dTTP concentrations. The continuouslines represent the simulation results. (d) Comparison between dTTP hydrolysis and Pi release at 500 mM dTTP. Thedelay in of Pi release in the transient phase is caused by the slow Pi release compared to dTTP hydrolysis.


to dTTP hydrolysis by the hexamer subunits. Thisalso provides insight into the DNA binding affinityof the dTDP-Pi state. As subunits start hydrolysisand produce dTDP and Pi, Pi is not immediatelyreleased when dTTP is hydrolyzed in other sub-

Table 1. Rate constants and corresponding valid ranges for s

Step Reaction Rate con

E/T* dTTP docking 5.0!104 MT*/N$T* DNA binding 951 sK

N$T*/N$T Power stroke 778 sK

N$T/DP dTTP hydrolysis 66 sK

DP/D Pi release 15 sK

D/E dTDP release 159 sK

T*/E dTTP release 1.5 sK

N$T*/T* DNA release 70 sK

N$T/N$T* Recoil power stroke 6.2 sK

DP/N$T dTTP synthesis 0.03 sK

D/DP Pi binding 1.1!103 ME/D dTDP binding 1.0!106 M

The rate constants are obtained by least-squares fit to the data. The loone parameter at a time but keeping all others fixed so that the residutolerance is roughly equivalent to 10% prediction error, on average. Nuof magnitude as the best-fit rate constants.

units. For [dTTP]0Z500 mM, more than two sub-units per hexamer are delayed in releasing Pi

relative to the subunits hydrolyzing dTTP. If thedTDP-Pi state has a high affinity for the DNA and Pi

release is more than two subunits behind the

teps in the hydrolysis cycle of T7 helicase

stant Lower bound Upper bound

K1 sK1 4.8!104 MK1 sK1 5.7!104 MK1 sK1

1 380 sK1 O1!106 sK1

1 311 sK1 1.17!104 sK1

1 53 sK1 297 sK1

1 14 sK1 18 sK1

1 103 sK1 O1!106 sK1

1 !1!10K5 sK1 2.5 sK1

1 !1!10K5 sK1 1.35!103 sK1

1 !1!10K5 sK1 120 sK1

1 0.028 sK1 0.041 sK1

K1 sK1 n/a n/aK1 sK1 n/a n/a

wer and upper bounds of the rate constants are found by varyingal errors of the prediction are within tolerance (see the text). Thismbers in bold are those upper or lower bounds in the same order


hydrolysis subunit, then at least three subunits mustbe attached to the DNA. This contradicts the protein–DNA cross-linking experiments and electronmicroscopy observations showing that single-stranded DNA interacts with only one or twosubunits of the hexamer.13 Thus, we conclude thatthe dTDP-Pi bound state must have a low affinity forDNA.

Mechanical considerations also suggest that thedTDP-Pi state should have a low affinity for DNA. Ifthree or more subunits in the dTDP-Pi state bindDNA, and only one of them is responsible fortranslocating DNA, then DNA will be distortedsignificantly. The bending of ssDNA at this lengthscale requires energy, not favorable for the motor toproceed. If three or more subunits attempt to moveDNA together, all these subunits must movesimultaneously about the same distance so thatthe DNA strand is not ruptured. Moreover, uponcompletion of the power stroke, other subunitsmust bind DNA to commence the next powerstroke, and this transition will bend ssDNAconsiderably. Taken together, we find this scenariohighly unlikely.

Another possibility is that all subunits worksimultaneously. In this scenario, after one powerstroke, all subunits must reset to their originalpositions so that the next power stroke can begin.This reset transition requires the subunits have alow affinity for DNA so that DNAwill not bemovedback to its original position. However, this synchro-nous low affinity for DNA corresponds to reducingthe duty ratio of the motor, which would allowDNA to diffuse away. Furthermore, if w2–3 nt/dTTP are translocated, then the power strokedistance would have to be w3.6–5.4 nm, which isimpossible given the lengths of the possible DNAbinding loops, and very unlikely if all subunits gothrough a dramatic rotation synchronously alongthe z axis in one chemical transition.

Taken together, we conclude that DNA will notbind to three or more subunits simultaneously, andthat the dTDP-Pi state has low affinity for DNA.

dTMPPCP promotes synthesis of dTTP

To determine if the dTTP hydrolysis step isreversible at the helicase-active sites, we carriedout the [18O]Pi medium exchange experimentsdeveloped by Boyer and co-workers that havebeen used to analyze the reversibility of thehydrolysis step of many ATPases.24 Briefly, highconcentrations of dTDP (6 mM) and [18O]Pi

(20 mM) were mixed with T7 helicase in thepresence of M13 ssDNA. The dTDP concentrationused here is saturating as determined by a dTDPinhibition experiment (Ki value of dTDP w200 mM;data not shown). The time-course of incorporationof unlabeled oxygen from water into Pi due toreversible dTTPase reactions was monitored.The loss of 18O label in Pi was monitored by 31PNMR experiments, as the 18O bound to thephosphorus atom shifts the 31P NMR by about

0.02 ppm up-field compared that of the 16O species.Figure 5(a) and (b) show that [18O]Pi to H2Oexchange was not observed even after 12 hours ofreaction, with and without DNA. The absence ofmedium [18O]Pi exchange indicates that dTTPhydrolysis is almost irreversible in the helicase-active sites or that Pi binding is weak.

Surprisingly, Figure 5(c) shows that when a smallamount of dTMPPCP, a non-hydrolyzable versionof dTTP, is included in the reaction with ssDNA,dTDP, and [18O]Pi, considerable exchange wasobserved. That is, in the presence of dTMPPCPand ssDNA, Pi can rebind to the catalytic site andeven participate in re-synthesis of dTTP. However,as shown in Figure 5(d), in the presence ofdTMPPCP, but without ssDNA, no exchange wasobserved. One explanation is that the reverse stepdTDPCPi/dTTP can only take place if DNA ispresent in the central channel of the ring. Recall thatthe dTMPPCP-bound state has a high affinity forDNA, while the dTDP-bound state has a lowaffinity.7 Because dTMPPCP helps recruit ssDNAinto the ring by its high affinity, other subunits areaccessible so that the transition dTDP/dTDP. Pi/dTTP is possible during the incubation. In theexperiment of Figure 5(b), although DNA, dTDP,and Pi are all added, DNA cannot be recruited to thecentral channel. Therefore, the DNA–dTDP statecannot be achieved, and so neither can the DNA–dTDP-Pi or DNA–dTTP states. This scenario alsoexplains why exchange was not observed withdTMPPCP in the absence of DNA (Figure 5(d)), andis consistent with the conclusion that the dTDP-Pi-bound state has low affinity for DNA. There areother possible explanations for the results of theoxygen exchange experiments, but they requiremore sophisticated models that include the effectsof multi-subunit interactions (see Appendix A).

A DNA binding-deficient mutant inhibits theactivity of T7 helicase

To distinguish between random and sequentialmechanisms of power strokes (see below), westudied the activity of mixed helicase hexamersmade with wild-type T7 helicase and an inactive T7helicase mutant protein. Previous studies haveshown that mixed hexamers can be made by mixingmutant and wild-type T7 helicase proteins.25

Previous studies were carried out with mixedhexamers made with a dTTPase-defective mutantprotein, and these studies showed that both thedTTPase and helicase activities decreased steeply asa function of mutant protein concentration.25,26

These results support a sequential power strokemechanism. However, since the dTTPase-defectivemutant was able to bind ssDNA, the dramaticdecrease in the activity of the mixed hexamer couldbe due to ssDNA binding to the mutant subunit inthe hexamer, thereby blocking the rest of thesubunits in the hexamer from binding or translocat-ing ssDNA. Therefore, we chose to carry out mixedhexamer studies with a DNA binding-deficient


mutant, which would not adhere to the DNA in amixed hexamer. Previous studies have shown thatR487C mutation in T7 helicase greatly reduces theDNA binding and helicase activities of T7 helicase,but the R487Cmutant still retains the ability to form

Figure 5. 31P NMR spectra of different species of [18O]Pi unand the unshaded 16O). (a) T7 helicase (1 mM), 6 mM dTDP, 20and Pi. (b) T7 helicase (1 mM), 6 mM dTDP, 20 mM [18O]Pi, 15observed in the presence of dTDP, Pi, and ssDNA. (c) T7 helica50 mMdTMPPCP. dTMPPCP significantly enhances the synthehelicase (1 mM), 6 mM dTDP, 20 mM [18O]Pi, 50 mM dTMPPCthe presence of dTMPPCP. Thus, it is necessary to have both

hexamers, bind dTTP, and hydrolyze dTTP at theDNA unstimulated rate.27,28 We mixed R487C andwild-type T7 helicase proteins in different ratios,but the final concentration of the protein mixturewas kept constant at 100 nM hexamer, and the

der different conditions (the shaded circles represent 18OmM [18O]Pi. No synthesis event was observed with dTDPnM M13 ssDNA. The only minimal synthesis event was

se (1 mM), 6 mMdTDP, 20 mM [18O]Pi, 15 nMM13 ssDNA,sis of dTTP in the presence of dTDP, Pi and ssDNA. (d) T7P. No synthesis event was observed without DNA even inssDNA and dTMPPCP to promote synthesis of dTTP.


dTTPase activity of the mixed hexamers wasmeasured in the presence of ssM13 DNA. Figure 6shows the measured hydrolysis rates under differ-ent [R487C] versus [wild-type] ratios.

To test whether the power stroke mechanism issequential or random, we conducted simulationsfor both cases and compared them with theexperimental results. Figure 6 shows that when allsubunits are made by the R487C mutant speciesð½R487C�=½R487C�C ½wt�Z1Þ under our experimen-tal conditions, hydrolysis occurred at about 20% ofthe wild-type rate. If DNA translocation is carriedout by hexamer subunits randomly, then after apower stroke the probability that the next powerstroke is performed by a wild-type subunit or amutant subunit depends on the ratio of the twospecies. This assumes that both species formhexamers equivalently. Thus, the overall hydrolysisrate will be proportional to the average of the wild-type fast rate and the mutant slow rate weighted bycorresponding species amounts. While the ratios ofthe two species change linearly, the hydrolysis ratesshould also change linearly, as shown in Figure 6.This is different from the experimental observationsthat as [R487C] increases in the protein mixture, thessM13-stimulated dTTP hydrolysis rate decreasessharply. The non-linear decrease in dTTP hydrolysisrate rules out the random power stroke mechanism.We also carried out simulations for a sequentialpower stroke mechanism and it shows a non-lineardecrease in dTTP hydrolysis rate as shown inFigure 6. The detail of how the simulation wasconducted is shown below after we introduce thealgorithm of the kinetic model.

In summary, the experiments described in this

Figure 6. Effect of the R487C mutant on the dTTPaseactivity of T7 helicase. The ssDNA-stimulated dTTPaserate at steady-state was measured for a mixture of R487Cand wild-type T7 helicase. The proteins were mixed indifferent ratios keeping the final protein constant at100 nM hexamer. The plot shows the decrease in thedTTPase rate as a function of increasing [R487C]. Thecontinuous line shows the predicted behavior of asequential power stroke mechanism, while the brokenline shows the predicted behavior of a random powerstroke mechanism.

paper show that (i) at saturating dTTP concen-trations, up to four dTTPs are hydrolyzed in a singleexponential phase. (ii) At saturating dTTP concen-trations, the rate-limiting step in a single dTTPasecycle is the release of the product Pi. dTTPhydrolysis is faster at the hexamer subunits, withthe result that more than two subunits of thehexamer lag behind in Pi release. (iii) We haveshown that DNA binds to the hexamer in the dTTP-bound state,7 and experiments described hereindicate that DNA releases immediately afterdTTP is hydrolyzed. (iv) The mutant poisoningexperiments rule out a random power strokemechanism and support a sequential power strokemechanism.

Using the available data, we develop below akinetic model that globally fits all the data, andconstruct a numerical optimization method todeduce the rate constants of the individual steps.We also develop a computational approach toidentify the main pathways by which a multi-siteenzyme system such as T7 helicase coordinates theactivity at each subunit to drive translocation alongDNA.

The DNA translocation power stroke occursduring dTTP binding

Previous studies have shown that the affinity ofDNA for helicase is high in the dTTP-bound states,but low in both the dTDP-bound state and theempty state.7 Here, we show that the dTDP-Pi-bound state also has low affinity for DNA. In orderto translocate DNA, the subunit must be in the high-affinity state for DNA. Thus, the DNA translocationstep can only take place in the dTTP-bound state.

In order to describe the translocation of DNA asthe hydrolysis cycle proceeds in each subunit werequire the following minimal kinetic steps: (1)weak dTTP binding (or “docking”); (2) strong dTTPbinding; (3) dTTP hydrolysis; (4) Pi release; and (5)dTDP release. In addition, (6) DNA binding and (7)unbinding must also be considered. Here, weassume that, like many other motors (e.g. F1-ATPase, kinesin, and myosin) Pi is released beforedTDP. This is because Pi has many fewer hydrogenbonds than dTDP, and additionally dTDP hashydrophobic interactions with the catalytic site, sothat the energy required for Pi release is smallerthan that for dTDP release.

Thus, the kinetic paths in each subunit can bewritten in terms of the states of its catalytic site:

(1)

Here, E is the empty state, T* is the weakly bounddTTP state, T is the tightly bound dTTP state, DP isthe dTDP-Pi state (after hydrolysis), and D is thedTDP state (after Pi release). The lower row (with Nprefix) corresponds to DNA-bound states. Since


N$DP is a short-lived state, the energy well for thisstate is not very deep and so we omit this state.Mathematically, it is always possible to lump twostates together into one state and use rate constantsequivalent to the collective jump probabilities of theoriginal states.

Thus, we can write the principal pathway for asubunit as:

E4T*4N,T*4PowerStroke

N,T4DP4D4E (2)

Structural considerations suggest two possiblemechanisms for the translocation power stroke.First, the crystal structure suggests that the relativerotational angles of adjacent subunits of thehexamer are different for the empty state and thedTTP-bound state. In this “subunit rotation” model,when dTTP binds to the catalytic site, it triggers thealignment of the positive charge groups constitut-ing the “arginine finger” with the negativelycharged g-phosphate of dTTP.29 This alignmentdrives the relative rotation between neighboringsubunits, and so the DNA contact residues rotatewith it to translocate the DNA. A second possibilityis that dTTP binding directly deforms the local b-sheet structure, as in F1-ATPase.

30,31 In this “directdrive” model, the deformation induced in the b-sheet propagates to the DNA binding loop that alsoemanates from the b-sheet, and thence to DNAcontact residues to translocate the DNA strand.

Recent crystal structures of the f12 P4 hexamericpackaging motor show its RNA contact residuesundergo a large conformational change from anATP-bound state to an ADP-bound state.14 Thetranslocation stroke is coupled to a different kineticstep from the helicase studied here, presumablydue to the difference in the high-affinity state to thesubstrate RNA or DNA. However, the idea beingthat subunit deformation induced by nucleotidebinding and release is the same. These twomechanisms differ in how stress radiates outwardfrom the catalytic site to drive the translocationpower stroke. Both mechanisms begin with thedTTP binding site undergoing the “binding zipper”transition: the formation of the hydrogen bondnetwork during dTTP binding.32,33 In both cases,the power stroke of each subunit is driven by thebinding zipper transition of dTTP from its weaklybound to the tightly bound state, N$T*/N$T. Theconformational changes accompanyingNTP bindinghave been shown for F1-ATPase

32,33 and myosin.34

The translocation power strokes take placesequentially around the ring

One of the most important questions that relate tothe mechanism of the ring helicases is the sequenceof the power strokes that drive the movement of thering along the DNA strand. Knowing the order ofthe power strokes helps clarify how the hexamersubunits communicate with each other. We addressthe following questions: are the translocation powerstrokes in the hexamer subunits simultaneous,

random, or sequential? Four power strokesequences must be considered.

Simultaneous

Experiments indicate that the helicase trans-locates an average of 3 nt/dTTP hydrolyzed.8

Since nucleotides in DNA are separated byw0.3 nm, the distance translocated per dTTP isw0.9 nm. If all six subunits work simultaneously(i.e. the power strokes act in parallel), then thetranslocation distance for each subunit would haveto be 6!0.9Z5.4 nm. Since the channel height of theT7 hexamer is only w5 nm, this is very unlikely.Furthermore, we have mentioned that the resettransition requires the subunits have a low affinityfor DNA, and this synchronous low affinity forDNA corresponds to reducing the duty ratio of themotor, which would allow DNA to diffuse away.Therefore, we discard this possibility.

Paired sequential

Based on the “dimerof trimers” arrangement in thecrystal structure, Singleton et al. suggested that thepower strokes progress sequentially around the ring,butwith diametrically opposing subunits in the samestate.5 For example, subunit 4 is opposite subunit 1and so the two pass through the hydrolysis cyclesynchronously. In this scheme, two dTTPs areconsumed during each translocation step, and sothe translocation step should be w1.8 nm. This isinconsistent with the 0.6–0.7 nm distance of subunitrotation in the possible DNA contact regions deter-mined from the crystal structure.5 Thus, we considerthe paired power stroke sequence unlikely; indeed,Singleton et al.5 pointed out that, in the presence ofDNA, asymmetry may occur and the paired sequen-tial mechanism could be invalid.

Random

There are two possible random power strokemechanisms. (1) Random in time: the power strokeof each subunit starts and finishes at random timesindependent of other subunits. This mechanism isnot consistent with the very high processivityobserved in experiments because there is anappreciable probability that none of the subunitsbind DNA at the same time. Thus, we can discardthis mechanism. (2) Random in sequence: the powerstrokes are sequential in time (i.e. each subunit canonly commence after another subunit finishes), butthe order of the power strokes around the ring israndom. That is, the sequence of the power strokesis not 1-2-3-4-5-6, but random, as in rolling a die.After subunit 1 binds and moves the DNA strand,subunits 2–6 have equal probabilities of bindingDNA and moving it. In terms of binding DNA, it isstill sequential (one after the other), but in terms ofthe power stroke order, it is random. The mutantpoisoning experiment does not support the randommechanisms.


Sequential

This mechanism carries out power strokes instrict sequential order: 1-2-3-4-5-6. After one sub-unit finishes a power stroke, the adjacent subunitbinds DNA and executes the next power stroke.This scheme requires cooperativity between adja-cent subunits to coordinate the sequence. Weconsider this mechanism most likely, for it isconsistent with the structural features that permitadjacent subunit coupling discussed below.

It is also possible that the power strokes arepartially sequential; that is, sequential but withsome random variation. For example, after trans-location by subunit 1, there is a probability ofcontinuing the next translocation in subunit 2 andanother probability to continue in subunit 3.Currently, there is no evidence that can unequi-vocally demonstrate that the order of translocationpower strokes is strictly sequential or mixed. Theonly thing known is that power strokes cannot takeplace simultaneously. Here, we use the purelysequential mechanism to illustrate how “one afterthe other” power strokes can work. This sequentialmechanism is consistent with the electronmicroscopy observations and the mechanismshown for the P4 packaging motor.13,14 We will seethat this fits all of the experimental observationssatisfactorily.

Mechanical stress coordinates the hydrolysiscycles between subunits

Based on experimental data and structuralconsiderations we develop a set of rules that governsequential hydrolysis of dTTP and processive DNAtranslocation. If the power strokes are sequential,the translocation step N$T*/N$T can take placeonly when the DNA is bound to only one subunit.Otherwise, each translocation step would beimpeded by the DNA being tethered to an adjacentsubunit. After completing one power stroke, thenext subunit must bind DNA to continue transloca-tion. Using the kinetic notation in equation (2), theDNA binding step in the next subunit, T*/N$T*,commences when the previous subunit in thesequence has completed its power stroke and is inthe N$T state. Geometrically, this is possible if thepower stroke of the previous subunit brings theDNA strand into a position where it can quicklyfluctuate to the next subunit.

Since hydrolysis enables release of the DNAstrand, in order to ensure high processivity theunbinding of DNA in one subunit must take placeafter the binding of DNA to the next subunit. Thus,the transition N$T/DP in one catalytic site mustfollow the binding of nucleotide to the next site, i.e.state T*/N$T*. (Recall: N$DP is omitted from thekinetics in equation (2) because it is a short-livedstate.) The most likely explanation for this is thatefficient hydrolysis is greatly enhanced when thetwo subunits that are bound to DNA form amechanical stress loop through the arginine finger

(see Figure A1 in Appendix A).5,29 The effect of thistransmitted stress is to ensure the catalytic residuesand the catalytic water molecules are constrained inthe right configurations to form the transition statefor hydrolysis.

To summarize, three cooperative steps arerequired for sequential DNA bind–release cyclesthat also assure processivity of translocation (theyare shown in Figure 8):

(1)
The power stroke, N$T*/N$T, takes placewhen the DNA is bound to one subunit.
(2)
The transition T*/N$T* requires the previoussubunit to be in the N$T state.
(3)
The transition N$T/DP requires the nextsubunit to be in the N$T* state.
We also examined the necessity of each of thesethree steps, and the simulations showed that thesethree steps are essential (see Appendix A).

In order to model the three cooperative steps, weapplied a rate enhancement factor to each of theabove steps. For all other transitions the rateconstant is independent of the states of othersubunits. To ensure a detailed balance, which isnecessary for the kinetic scheme to obey the SecondLaw of Thermodynamics, if a forward rate isenhanced then the same enhancement must beapplied to the reverse rate constant. The standardfree energy of 12.5 kBT is used for the hydrolysiscycle, so one of the reverse rate constants ispredefined by this energy relationship. This simplemodel with cooperativity only in the above threetransitions reduces the total number of fittingparameters to six forward rate constants, fivereverse constants, and three enhancement factors,for a total of 14 unknown parameters. The forwardrate constants are kE/T*, kT*/NT*, kNT*/NT, kNT/DP,kDP/D*, and kD/E. The backward rate constants,including the one obtained from the free energyconstraint, are kT*/E, kNT*/T*, kNT/NT*, kDP/NT,kD/DP, and kE/D. The three enhancement factorsare fNT*/NT, fT*/NT*, fNT/DP*, corresponding to thethree enhancement rules stated above. With theseparameters, we can construct the rate equations andsolve them.

Kinetic pathways

As shown above in equation (2), there are sixnucleotide ligation states in each subunit of thehexamer, so that the total number of states is 66Z46,656. Each hexamer state can be denoted byEEEEEE (all sites empty), EEDDDD (four sitesoccupied by dTDP), etc. The rate equation for allstates can be written as pseudo-first-orderequations:

dc

dtZK,c (3)

where c(t) is a concentration vector of all thepossible kinetic states. Thus, c(t) is a concentration

Box 1. Sketch of the computational algorithm

The solution to equation (3) determines the rateconstants for the pre-steady-state kinetic data shownin Figure 4. The flow chart for the computationalalgorithm is shown in Figure 10 in Appendix A,where more details can be found. The underlyingprinciple is to first solve the rate equations with atrial set of rate constants, and then to use theoptimization algorithm to find a set of best-fit rateconstants. The best fit is defined as the least-squareserror of the prediction compared with the exper-imental data. First, a set of 14 estimated parameters(11 rate constants and three enhancement factors) areassigned to the corresponding kinetic steps aselements of the rate matrix K. For example, theelement in the matrix K is the rate constant, since itcorresponds to the transition T*/E in the lastsubunit. Another example: the element is assignedto, since it corresponds to the mechanical couplingenhancement for the hydrolysis step, with the nextsubunit ready to take over the next power stroke.Once the matrix is constructed, the rate equationdc/dtZK$c can be solved using an iterative method(the quasi-minimal residual method inMATLABe).38 Simply guessing the set of initialparameters gives a poor fit, so we used the simplexoptimization method.39 An initial 14 dimensionalsimplex is constructed consisting of 14C1 vertices.Each vertex contains a set of 14 fitting parameters.For each vertex (i.e. each set of rate constants), a ratematrix K is constructed and the rate equation issolved. To compare the predictions with themeasurements, the computed concentrations mustbe converted to the corresponding experimentalconcentrations. For each vertex, we obtain the sum ofthe square of the difference between predictions andmeasurements:, where cp is the predicted concen-trations and cm is the measured concentrations. Theprediction error, E, is to be minimized. Afterprediction errors are obtained for all vertices, thevertices of the simplex are moved according to theNelder–Mead method so that the size of the simplexgradually shrinks corresponding to convergence to aminimum.39 The best-fit parameters are obtainedwhen the size of the simplex is smaller than apredetermined cutoff.


vector with 46,656 elements, and K is the 46,656!46,656 rate matrix. For example, k12 is the rateconstant for the change of state [EEEEEE] (i.e. theconcentration of state EEEEEE) as a function of[EEEEET*]:kEEEEET*/EEEEEE. Equation (3) corre-sponds to the experimental conditions where theconcentrations of dTTP, dTDP, and Pi do not varytoo much over the course of an experiment, so thattheir concentrations can be incorporated into therate matrix, K. (However, the algorithm we use isnot confined by the linear approximation.)

Notice that the complexity of this large system ofequations is purely computational: the underlyingchemistry includes only the three cooperative steps,similar to the classical treatment of subunit inter-actions.35–37 These three cooperative steps devel-oped from structural and mechanochemicalconsiderations impose constraints on the systemsuch that many of the 46,656 states are either totallyinaccessible or have very low probability. Thecomputational results will report these inaccessiblestates as well as the time-dependent concentrationsof those accessible states. The purpose of thealgorithm is to find the main pathway(s) by fittingthis equation to the data. The difficulty arises fromthe large number of coupled differential equations.However, the rate matrix, K, while large, is sparse,so that an efficient numerical method can be used tointegrate the equations.38

The initial conditions can be inferred from thechase-time experiments. The optimization algor-ithm is used to obtain 14 parameters needed for thecomputation. The details of obtaining initial con-ditions and defining the necessary parameters aregiven in Appendix A.

The results of the calculations are shown inFigure 4, and the computed rate constants are givenin Table 1. The fit to the data is quite good in boththe transient and steady-state phases. The freeenergy diagram calculated from these best-fit rateconstants is given in Appendix A.

A sensitivity test was conducted to find the validranges of the rate constants obtained from the fittingalgorithm. To determine the range of one rateconstant, we varied that constant while keepingall other constants fixed, and then computed thetime trajectories and the corresponding residualerror of the new parameters. The residual error isScase i St DtðCðtÞi;computeKCðtÞi;measureÞ

2, summingthe concentration differences of all cases and alltime steps. When that constant is different from thebest-fit rate constant, the residual error is greaterthan the best-fit residual error. We chose a toleranceresidual error roughly equivalent to 10% predictionerror on average. The upper and lower bounds ofeach rate constant reported in Table 1 are identifiedwhen the corresponding residual error equals thetolerance residual error. The Table shows that somerate constants are sensitive while others are not.Thus, our experiments can determine some rateconstants with good accuracy, leaving other con-stants underdetermined. The bold-face numbers inTable 1 are those whose best-fit rate constants are of

the same order of magnitude. For example, ourexperiments can determine the dTTP docking rate,the Pi release rate, and the dTTP synthesis rate.Some rates have well-defined lower bounds whileothers have well-defined upper bounds. A sensi-tivity test of the Pi and dTDP binding rates is notpossible because dTDP and Pi concentration-dependent experiments are not available.With the best-fit rate constants and their corre-

sponding ranges being considered, Table 1 showsthat when the initial concentration, [dTTP]0, is low,the E/T* step is rate-limiting. When [dTTP]0 ishigh, the E/T* step and the DP/D steps are of thesame order, and a combined effect must beconsidered. Changing [dTTP]0 shifts the rate limit-ing step. The slow DP/D step is consistent withthe phase delay between hydrolysis and Pi release.The mechanical power stroke step, N$T*/N$T, is


fast compared to other kinetic steps. A mechanicalstep is generally not equivalent to a kinetic step. If amechanical step is rate limiting, the dynamics of thesystem is dominated by spatial motion. A kineticprocess is an instantaneous jump transition with adistribution of waiting times in the source state;therefore, it is generally not a good model for amechanical step. On the other hand, if the mechan-ical step is fast compared to other kinetic steps, thedynamics of the system are dominated by thekinetic process. In this case, it is appropriate to use akinetic step to approximate the mechanical step,with equivalent mean passage time chosen for thekinetic approximation.

We used the same approach for simulations of theDNA binding-deficient mutant, as illustrated inFigure 6. First, we assume the mutant species havethe same property as the wild-type species informing hexamers. Thus, different proportions ofwild-type versus mutant species give differentdistributions of various mixed hexamers, such asWWWWWW, WWWWWM, etc., where W rep-resents the wild-type subunit while M representsthe mutant subunit. There are a total of 64 species ofmixed hexamers and their initial distributions canbe computed from the [R487C] versus [wild-type]ratio. For each species, the kinetic equations areformulated and the steady-state hydrolysis rate canbe obtained. Different species have a different ratematrix. For a mutant subunit, because it is unable tobind DNA, the states N$T* and N$T are notreachable. This subunit, however, is still able tohydrolyze dTTP as observed in the all mutanthexamers. Thus, we assign a detour state for mutantsubunits: E4T*4T4DP4D4E. We also make apreliminary assumption that all rate constants,except the one of the hydrolysis step, are the sameas those for the wild-type subunits. The powerstroke here T*/T has the same rate constants asNT*/NT. The hydrolysis step T/DP, however,becomes the rate-limiting step because of the lack ofthe subunit–subunit coordination as describedabove. This slow hydrolysis rate can be easilyfound by fitting to the data of all mutant species(MMMMMM) shown on the right in Figure 6. Withthis rate constant and all others known, weconstruct the rate matrix for each species and obtainits corresponding steady-state hydrolysis rate.Knowing the distributions of all species under acertain [R487C] versus [wild-type] ratio, we thenweight them accordingly and obtain the averagehydrolysis rate for each case. The simulated resultsfor this sequential power stroke mechanism areshown as the continuous line in Figure 6. The trendof the simulation is similar to the trend of theexperimental data, while in the middle range ofmixing the simulation underestimates the rate. Onepossible explanation for this discrepancy is that themixing of the mutant and the wild-type species maynot be uniform, so that it may have a higherprobability to form homogeneous wild-type ringsthan calculated. Another possibility is that thesequential power stroke is not as strict as we use

to compute, but follows the partially sequentialmechanism we describe above.

What is the “main” kinetic pathway?

One goal of the calculation is to find the “mainpathway(s)” amongst the very large number ofpossible pathways. One quantitative definition of amain pathway is that, at steady-state, it carries themaximum kinetic flux. Using the best fit rateconstants, the steady-state concentrations of allstates can be determined, and the fluxes of allkinetic steps computed (see, e.g. Hill40). Manypathways transmit negligible flux and can beignored. Those kinetic pathways with a fluxthreshold above 3% of the total flux were identified.Figure 7 shows the steady-state kinetic flux networkunder two different concentration conditions. Thethickness of the connections between any two statesis proportional to the kinetic flux between them.Figure 7 shows that the distribution of pathwaysdepends on the initial conditions. For physiologicalconditions (assumed to be [dTTP]Z100 mM,[dTDP]Z10 mM, and [Pi]Z1 mM), the flux passesmostly through one or two pathways with lowoccupancy, while under experimental conditions of[dTTP]0Z500 mM, the kinetics spread over manypathways. In the latter case, it is not possible todefine a single principal pathway. With a 3% fluxthreshold, the net flux in the partial network shownin Figure 7(a) is about 91% of the total flux, while inFigure 7(b) the net flux is only about two-thirds ofthe total flux. Figure 7(b) also shows the shifting ofkinetic fluxes toward high-occupancy states com-pared to Figure 7(a), and this is consistent with theexperiments where increasing [dTTP]0 correspondsto increasing average occupancy. The calculation ofthe kinetic flux network shown here can be appliedto any cooperative multiple-protein kinetics.

Translocation along single-stranded DNAinvolves multiple kinetic pathways

Here, we summarize how the subunits of thehexamer coordinate their actions to translocatealong DNA. The three required cooperative stepsin the reaction network are shown schematically inFigure 8(a). Previous results have shown that T7helicase translocates on an average 3 nt per dTTPhydrolyzed; therefore, we can assume that after onedTTP hydrolysis cycle, the ssDNA is translocatedby 3 nt. Figure 8(b) shows the pathways that wereextracted from the high-flux pathways shown inFigure 7(b). Within this subset, the translocation cantake place by several pathways. For example, thefirst step in Figure 8(a) is the hydrolysis step; thisstep occurs when DNA is bound to two subunitsand, as shown in Figure 8(b), one in which the thirdsubunit is E (in step 1a) and the other where thethird subunit is DP (in step 1b). After dTTPhydrolysis the DNA dissociates from that subunitand the N$T* state goes to N$T, which is the powerstroke step that translocates the DNA. Depending

Figure 7. The steady-state kinetic flux networks for hexameric T7 helicase. The mechanism behind part of these fluxnetworks is illustrated in Figure 8(b). Here, each hexagon represents a kinetic state of the hexamer. Each subunit of thehexamer is colored with the corresponding occupancy state shown in the key: white, empty state (E); cyan, dTTPdocking (weakly bound) state (T*); pink, dTTP docking with DNA bound (N$T*); red, dTTP tightly bound to DNA(N$T); yellow, the dTDP-Pi state (DP); and green, dTDP bound (D). The reaction cycle proceeds across the columns withthe possible reaction pathways linked by lines whose thickness reflect the fractions of the total flux carried through thatreaction branch. Thus, the magnitude of the flux (line thickness) illustrates the relative probabilities of each pathway. Therightmost column is the same state as the first column but shifted one subunit to form a repeated kinetic cycle. Linesrepresenting the power stroke driving DNA translocation are colored red; all other transitions are colored blue. Thelower rows have higher occupancy (increasing ligand concentration). The pattern of flux pathways depends on thesolution concentrations. For physiological conditions ([dTTP]0Z100 mM, [dTDP]0Z10 mM, [Pi]0Z1 mM), there is adominant kinetic pathway with low occupancy, while for [dTTP]0Z500 mM, the kinetics spread over many pathways. Inthe latter case, the concept of a “principal pathway” is not well-defined. Along different pathways the order of thereaction steps is different. The states shown here are only the most populous of the 66Z46,656 possible states, i.e. thoseabove a threshold flux of 3% of the total flux.


on what is bound to the third subunit, the powerstroke step result in (DP, N$T, E) or (DP, N$T, DP)states, shown as states after step 2a and step 2b,respectively. In order to pass the DNA to the thirdsubunit for the next power stroke, the third subunitmust be in the T* state. The (DP, N$T, E) state canreach this state via two pathways: through steps 3a,4a, and 5a to (E, N$T, T*) state, or through steps 4band 5b to (DP, N$T, T*). The cycle is then reset afterthe subunit in the T* state binds DNA to become theN$T* state through steps 6a or 6b, where twosubunits once again bind DNA. In this simplifiednetwork, (DP, N$T, E) is the bottleneck that everypathway must pass through. If the networkincludes all six subunits there are many morepathways and the bottleneck may not exist forthere are multiple kinetic pathways for T7 helicase

forming a reaction network to accomplish thesequential translocation task processively.

Discussion

Based on a combination of experimental obser-vations and computational methods, we propose aconsistent mechanochemical model for ssDNAtranslocation by the hexameric T7 helicase. Weanalyzed data from (i) DNA and nucleotide bindingaffinities, (ii) pre-steady-state dTTP hydrolysiskinetics in the presence and absence of DNA, (iii)medium oxygen exchange, and (iv) a mutantpoisoning experiment. We combined these obser-vations with the structural arrangement of thecatalytic sites to deduce a cooperative mechanism

Figure 8. Cooperative steps and multiple pathways of the T7 helicase kinetic network. (a) Schematic plot of thesequential steps in a typical translocation cycle. The vertical bars represent ssDNAwith the red intervals representing thebinding phosphate groups three nucleotides apart. The orientations of DNA contacting loops are illustrated on bothsides of DNA by the “lever arms” (see Figure 1). The contacts between these loops emanating from subunits 1, 2, and 3(number in the upper left corner) are shown on both sides of the ssDNA. DNA is translocated downward in thisreference frame by power strokes driven by dTTP binding to the catalytic site. Only three steps are necessary to assure ahighly processive sequential mechanism. First, hydrolysis takes place when the next subunit binds to DNA ensuring thatDNA will not diffuse away. When only one subunit binds to DNA, the power stroke can proceed without constraint.After each power stroke, the next subunit must proceed to the N$T* state (weak binding to DNA) in order for the wholecycle to repeat. (b) Multiple pathways extracted from the high-flux pathways shown in Figure 7(b) with matching colorsof states. The helicase is shown “unwrapped” to form a band of subunits, and the position of each nucleotide binding siteis shown. The diagonal orange line represents phosphate groups of ssDNAwhich are translocated downward. The bluedots represent the positions of ssDNA binding residues. Only three out of six subunits adjacent to the DNA bindingsubunit are shown; the other three subunits can be in several possible states (E, T*, DP, or D). The network shown here ismeant only to represent a typical sequence amongst multiple pathways. Some kinetic steps are projected onto thecorresponding steps shown in (a).


that translocates ssDNA sequentially. In the absenceof DNA, we observed several sites that turned overdTTP slowly. The studies and analyses in thepresence of DNA presented here show different

behavior and support the view that all subunits ofthe T7 helicase hexamer are catalytic. Although weobserved that less than six sites are hydrolytic atany one time, this is likely due to the negative


cooperativity in NTP binding that has beenobserved in other hexameric helicases.18–21 Theexperimental data show that the rate-limiting stepchanges from dTTP binding to Pi release as thedTTP concentration increases. The hydrolysis ofdTTP constitutes an important step in inter-subunitcommunication and in modulating the DNA bind-ing affinity. The dTTP bound state has a high affinityfor DNA, whereas the dTDP-Pi, dTDP, and emptystates have low affinity for DNA. This informationallowed us to isolate themechanochemical couplingstep that transduces dTTP binding energy to DNAtranslocation. DNA binds to a subunit when thatsubunit binds dTTP; the power stroke that drivesDNA translocation must occur as the dTTP annealsinto a tight binding configuration in the catalyticsite. Fast dTTP hydrolysis dissociates the DNAfrom the driving loop, but allows the next subunitin the dTTP bound state to bind DNA and continuethe cycle of DNA translocation.

Sequential mechanisms in which subunits of theprotein bind and release DNA successively havebeen proposed for ring-shaped5,15,20,21 and non-ring-shaped helicases3 as well as proteins such asthe f12 P4 hexameric packaging motor.14 Thesemodels explain processive translocation of thesemotor proteins along nucleic acids. The modelsbased on crystal structure data for ring-shapedhelicases are based on structures in the absence ofthe nucleic acid bound in the central channel.5,41,42

Only two ring helicases, Rho protein and T7helicase, have been characterized by detailedbiochemical and kinetic studies in the presence ofnucleic acid. Sequential hydrolysis of ATP and RNAbinding and release around the hexamer ring hasbeen proposed for the Rho protein where three siteshave been invoked in the mechanism,20,43 which isdifferent from the six-site mechanism that wepropose here for T7 helicase in the presence ofDNA.

Recently, Gai et al.44 have suggested that thehexameric helicase of simian virus 40 (SV40) large Tantigen works by a concerted mechanism. Thismodel is also based on structural data obtained inthe absence of DNA. The structures of SV40 large Tantigen in the absence of nucleic acid demonstratemany important aspects of the protein function.They show conformational changes inside a subunitand between adjacent subunits when in differentnucleotide binding states and these conformationalchanges are proposed to move DNA. It is possiblethat when DNA is present the resulting asymmetricforces will induce only part of all subunits to changeconformations and to move DNA. The shapes andthe charge distributions are highly asymmetric, notsurprising considering the major and minorgrooves of DNA, as well as the base and thephosphate backbone in any cross-section of ssDNAand dsDNA. These asymmetric properties of theDNAwhen bound in the central channel introduceasymmetric forces on the surrounding subunits.Without the imposition of these asymmetric forces,the ring will show symmetric arrangements, as

observed in the crystal structures. Although, theconcerted mechanism cannot be ruled out, it is noteasy to see how all subunits can reset in concertwithout losing contact with DNA. If the motor doesnot hold DNA most of the time (high duty ratio),DNAwill diffuse away or at least slip several base-pairs.We developed a computational algorithm to

determine the rate constants for the NTP hydrolysiscycle and to determine the kinetic fluxes throughthe reaction network, which is a priori quite large.This allowed us to deduce a reduced set of states tocharacterize the helicase. The results reveal thatdTTP hydrolysis can occur by multiple pathways,particularly at saturating dTTP concentrations.Multiple pathways provide more degrees of free-dom for the system to find the “low impedance”sequence, as shown in Figure 8(a). Parallel kineticpathways also exist in other multi-subunit motorssuch as myosin V.45 In multimeric proteins theexistence of multiple pathways reduces the neces-sity for strict cooperativity among subunits at eachkinetic step. For some required sequential steps thecooperativity must be strong enough to ensure thatthe motor proteins can work processively, but forother kinetic steps many pathways are possible. Thepresence of both high and low cooperative inter-actions ensure robustness of these processive motorproteins.The elucidation of the kinetic pathways described

here has been influenced by reasoning aboutmechanical stresses to identify and quantify poss-ible cooperative interactions and to eliminateunacceptable ones. We still do not have a completeknowledge of the detailed interactions between thesubunits and how these interactions change as thehelicase moves along the DNA. However, becausethe only possible mechanism for cooperativeinteractions is via mechanical stress, the number ofpossible coupling mechanisms is limited. We triedseveral possible mechanisms and the one shownhere is the only one we found that could explain thedata. There may be other complicated mechanicalinteractions that we did not consider, and ourmethods cannot rule out those possibilities. Thedetailed distributions of the kinetic pathways mayrequire revision as future studies reveal moredetailed information about conformational changesin the protein.The large matrix analysis we have employed here

to model the data has several advantages over theusual Michaelis–Menten analysis. Michaelis–Men-ten analysis may provide insights into the individ-ual steps, including the binding events, inhibitioneffects, and a lumped rate for steps other than thebinding step. However, it is based on the assump-tions that the system is in the steady-state and thereis only a single kinetic pathway, which is almostnever the case. Departure from these assumptionsproduces non-hyperbolic enzyme kinetics andrequires more complicated kinetic equations,especially when considering the effects of subunitinteractions. The method we have employed to


analyze our experiments is a modification ofclassical models for subunit interactions, includingthe Monod–Wyman–Changeaux (MWC) and theKoshland–Nemethy–Filmer (KNF)-type models.35,46

Since these methods were proposed, computationalpower has advanced to the point that we couldinclude all six subunits in the simulation. This largematrix simulation allows us to identify the distri-bution of pathways for the T7 helicase (see Figure7), and enables fitting to both transient and steady-state phases of the data. Because the matrix is large,it might appear that the results are not reliable.However, the situation is simpler than the dimen-sionality might suggest, since all six subunits haveexactly the same kinetic cycle, and there are onlythree cooperative steps. Our method is similar tothe free energy of activation treatment proposed byRicard et al.36,37 The complexity lies only incomputation, not in the underlying physics; thatis, our method provides a way, based on simplephysics, to attack multimeric time-dependent kin-etics problems.

The algorithm developed here for T7 helicase canbe applied to any multimeric, multi-state enzyme,including ring-shape enzymes such as the virusportal protein, helicases, GroEL, and F1 ATPsynthase. It is also easy to implement for walkingmotors such as kinesins or double-headed myosins,although the total number of states for these motorsis within the purview of the analytical method ofFisher & Kolomeisky.47–49 Our method is notrestricted to steady-state kinetics; indeed, pre-steady-state kinetics and mechanical measurements(e.g. force–velocity curves) can be incorporated, solong as the power stroke step can be modeled as athermally activated event. In a subsequent publi-cation this restriction will be lifted so that the powerstroke can be described by an arbitrary mechano-chemical potential. Within its limitations, thisalgorithm provides a systematic procedure tostudy the multiple pathways of cooperative, allo-steric enzymes.

Experimental Procedures

Protein, nucleotides, and buffer

T7 gp4A 0 protein (referred to as the T7 helicase) is aM64L mutant of T7 helicase–primase protein andR487C were over-expressed and purified asdescribed.18,50 The protein concentration was deter-mined both by absorbance measurements at 280 nm in8 Murea (theextinction coefficient is 76,100 MK1 cmK1) andby the Bradford assay using bovine serum albumin as astandard. Both methods provided similar concentrations.The M13 ssDNA (M13mp18) was purified as described.51

dTTP and dTDP were purchased from Sigma Chemicals,and [g-32P]dTTPwas obtained fromAmersham PharmaciaBiotech. Helicase buffer (50 mM Tris–HCl (pH 7.6), 40 mMNaCl, and 10% (v/v) glycerol) was used throughout theexperiments unless specified otherwise.

Pre-steady-state chase-time kinetics of dTTPhydrolysis

The experiments were conducted using a rapid quench-flow instrument at 25 8C. T7 helicase was mixed withdTTPC[a-32P]dTTP, M13 ssDNA, and EDTA in thehelicase buffer and immediately loaded into one syringeof the quench-flow instrument. After three minutes, 24 mlof the enzyme complex was rapidly mixed with an equalvolume of unlabeled dTTP and MgCl2 from a secondsyringe of the instrument. The final concentrations aftermixing were as follows: T7 helicase (1 mM hexamer),radiolabeled dTTP (60 mM–500 mM), chase dTTP (5 mM),MgCl2 (20 mM) and M13 ssDNA (15 nM). After variouschase-times, the reactions were quenched with 4 Mformic acid and products were analyzed as described.12

The chase-time kinetics were fit to equation (5) and thelinear phase of dTTP hydrolysis due to inefficient chasewas subtracted. The resulting chase-time kinetics was fitto a single exponential:

DðtÞZAð1KeKk1tÞ (4)

whereD(t) is dTDP at time t,A is the burst amplitude, andk1 is the exponential rate constant.

Pre-steady-state acid-quench kinetics of dTTPhydrolysis

The experiments were conducted at 25 8C using a rapidchemical quench-flow instrument (KinTek RQF3 soft-ware, State College, PA). T7 helicase was mixed withdTTPC[a-32P]dTTP, M13 ssDNA, and EDTA in thehelicase buffer and immediately loaded into one syringeof the quench-flow instrument. After three minutes, 24 mlof the enzyme complex was rapidly mixed with an equalvolume of MgCl2 from a second syringe of the instrument.The final concentrations after mixing were as follows: T7helicase (1 mM hexamer), dTTP (60–500 mM), MgCl2(20 mM) and M13 ssDNA (15 nM). The mixed reactionswere quenched after millisecond to second intervals with4 M formic acid and products analyzed as described.12

The acid-quench kinetics were fit to the burst equation:

DðtÞZAð1KeKk1tÞCmt (5)

where D(t) is dTDP at time t, A is the burst amplitude, k1is the exponential rate constant, and m is the slope of thelinear phase.

dTTPase activity of mixed wild-type and mutant T7helicase hexamers

R487C and wild-type T7 helicase proteins in differentratios (final concentration of protein mixture, 100 nMhexamer) were mixed in helicase buffer for 30 minutesand the dTTPase reaction was initiated by adding Mg2C,2 mM dTTP spiked with [a-32P]dTTP, and 2 nM ssM13DNA. The reactions were stopped after 15, 30, 45, 60, 120,180, 240 seconds with 4 M formic acid, and dTTP anddTDP were quantified as described above. The steady-state dTTPase rate was calculated from the time-course(slope) and plotted as a function of [R487C].

Stopped-flow kinetics of inorganic phosphate release

The A197C phosphate-binding protein (PBP) waspurified and labeled with MDCC as described.23

Inorganic phosphate release reactions were performed


at 25 8C using a stopped-flow instrument manufacturedby KinTek Corp. (State College, PA). The instrumentsyringes were treated with the phosphate mop (0.5 unit/ml of PNPase and 300 mM 7-methylguanosine) to removecontaminating Pi.A phosphate calibration curve was created as follows.

To obtain a baseline value of fluorescence intensity, C0,PBP-MDCC (10 mM) from one syringe of the stopped-flowinstrument was mixed with the helicase buffer from asecond syringe. The fluorescence intensity, whichremained unchanged with time for at least 300 ms, wasmeasured using a 450 nm long pass filter (Corion LL-450F) after excitation at 425 nm to obtain the C0 value. Thisbaseline fluorescence intensity value was determinedbefore each Pi standard reaction. Immediately after C0

determination, PBP-MDCC (10 mM) from one syringe wasmixedwith a known concentration of Pi (0.1–0.8 mM) froma second syringe. The fluorescence intensity increasedwith time due to Pi binding to the PBP-MDCC protein,and the maximumfluorescence intensity provided Ci. Thevalue Ci–C0 was plotted against the final Pi concentrationto generate the phosphate calibration curve.The Pi-release kinetics were measured in the stopped-

flow instrument by reacting 40 ml of solution A withsolution B. Solution A consisted of T7 helicase (0.2 mM,hexamer), EDTA (5 mM), Pi -mop (0.5 unit/ml of PNPasewith 300 mM 7-MEG), M13 ssDNA (3 nM), and dTTP(200–3200 mM), and solution B consisted of MgCl2 (45.2–48.2 mM), Pi-mop, and 10 mM PBP-MDCC. A total of fouror five kinetic traces were averaged and the fluorescenceintensity time-courses were converted to the kinetics of Pi

release using the phosphate calibration curve. We ensuredthat the phosphate mop did not compete with PBP-MDCCfor phosphate (kcat/Km for the PNPase reaction with Pi is3.2!106 MK1 sK1).52 A control experiment was carriedout for each dTTP concentration by mixing solution Awith solution B lacking Mg2C. The control experimentwas subtracted from the experiment with Mg2C.

Phosphate–water oxygen exchange experiments

[18O]Pi was prepared by reacting PCl5 (Sigma) with99% H2

18O (Aldrich Chemical) followed by ion-exchangechromatography (Bio-Rad; AG1-X4).53 The resultinginorganic phosphate had an enrichment of 96% 18O.The concentration of [18O]Pi was determined by themolybdate method.54 T7 helicase (1 mM hexamer) wasincubated with 10 mM MgCl2, 0.5 mM EDTA, 6 mMdTDP, 20 mM [18O]Pi, M13 ssDNA (15 nM), and 60%(v/v) 2H2O in 0.5 ml of helicase buffer at 22 8C for 2–12hours. The reactions were quenched by vortexing withchloroform to denature the enzyme, and aqueous phasewas extracted for 31P NMR analysis. The distribution ofP18Oj

16O4Kj species, where 0%j%4, was determined by31P NMR.55 The spectra were obtained with JEOL GX-400instrument at a frequency of 162 MHz for phosphorus.Five hundred scans were accumulated and then Fouriertransformed. TheP18Oj

16O4Kjpeakswereeachseparatedbyshifts of 0.02 ppm.55 Fractional contribution of each [18O]Pi

species to the spectrum was evaluated directly from thepeak area.

Acknowledgements

S.S.P. and Y.-J. J. were supported by NIH grant

GM55310 and J.-C.L. and G.O. were supported byNIH grant GM59875-02. The authors thank DaleWigley, Oleg Igoshin, Jianhua Xing, Michael Grabe,and Joshua Adelman for valuable discussions.

Appendix A

Determination of the optimal concentration ofM13 ssDNA

T7 helicase (2 mM, hexamer) was incubated withM13 ssDNA (0–50 nM, molecules) in the helicasebuffer containing 5 mM EDTA and 1 mM dTTP.After 15 minutes, radiolabeled 76-mer ssDNA (final6 mM) was added and the samples were filteredthrough a NC-DEAE membrane assembly. Themembranes were washed before and after filtrationof samples with the membrane wash buffer (50 mMTris–HCl (pH 7.5), 5 mM NaCl). After the sampleswere filtered, radioactivity on both NC and DEAEfilters was quantified using a Phosphoimager(Molecular Dynamics). The molar amount of T7helicase–76-mer complex was calculated from theratio of radioactivity on NC over the sum ofradioactivity on both membranes, as described.7

In the absence of M13 ssDNA, a large amount ofradiolabeled 76-mer–gp4A 0 complex was observed.As the concentration of M13 ssDNA molecules wasincreased in the reaction, the amount of helicase–76-mer complex decreased, and beyond 20 nM M13ssDNA, no helicase–76-mer complex was observed,indicating that no free helicase was present in thereaction to bind radiolabeled 76-mer. These experi-ments indicate that for each mmol of T7 helicasehexamer, we need at least 10 nmol of M13 ssDNAmolecule to saturate the binding of protein to DNA.We therefore used a ratio of 15 nM M13 ssDNA permM T7 helicase hexamer in all our experiments. In aseparate experiment, we varied the time of incu-bation of the helicase with DNA and dTTP. No freeprotein was detectable even after a short (twominutes) period of incubation (data not shown).Based on this experiment, we chose a three minutepre-incubation period for all experiments, whichalso gives us enough time to load the pre-incubatedsample in the quench-flow instrument.

The effect of adding Mg2C in the kinetic steps

In quench, chase, or Pi release experiments, Mg2C

is added after dTTP and the helicase is allowed toreach equilibrium. That is, an extra state should beincluded to distinguish the states with and withoutMg2C in the catalytic site. Experiments show thatthe hydrolysis rate is negligible before Mg2C isadded to the solution.17 Therefore, before Mg2C isadded, the subunits can only be in E, T**, NE, NT**states, where T** represents the dTTP docking statewithout Mg2C. After adding Mg2C, Mg2C candiffuse into the catalytic site to become the T* orNT* state, or Mg–dTTP can be formed in solutionand directly dock to the catalytic site. In the latter


case, the E (NE) state can directly go to the T* (NT*)state without passing through the T** (NT**) state.

Since Mg2C is much smaller than dTTP, itsdiffusion to the catalytic site is very fast. Therefore,T**/T* is very fast and so is NT**/NT*. Underthis assumption, the T** and NT** states can beomitted. The assumption may break down if, afterMg2C diffuses into the catalytic site, Mg–dTTPtakes a long time to rearrange its coordination sothat the correct configuration of T* state is achieved.In this case, the rate of E to T** is not equivalent to Eto T*, and states T** and NT** must be retained inthe model. In the main text, we lump states T** andT* as one state, T*.

Alternative explanations for oxygen exchangeexperiments

Considering multi-subunit interactions providesalternate explanations for the oxygen exchangeobservations. dTMPPCP is a non-hydrolyzableversion of dTTP. When dTTP is replaced bydTMPPCP at one site, it may delay, or block, the

Figure A1. Stress paths between subunits in the T7 helicasmain-chain connections between adjacent catalytic sites. (b) Dcatalytic sites, assuming motif III is the DNA contact residues.the stress loop passing through the nucleotide in the (N$Tcontinuous red lines represent the loops emanating from the

kinetic steps following dTDP-Pi (DP) or dTDP (D) atthe adjoining site, increasing the probability of thereverse step dTDP-Pi/dTTP. For example, if dTTPhydrolysis at one site affects Pi release at an adjacentsite, then non-hydrolyzable dTMPPCP will preventrelease of Pi, giving time for dTDP–Pi/dTTP.Mechanically, hydrolysis in one subunit can inducestress in adjacent subunits facilitating Pi release.DNA provides coordination between these sub-units through an additional stress loop shown inFigure A1(c), so in Figure 5(d) when DNA is absent,synthesis events are not observed. Under thishypothesis, in the presence of DNA, Pi is stuck inthe catalytic site unless dTTP hydrolyzes in anothersite.

Another possibility is that DNA binding, T*/NT*, in one site triggers the release of Pi in theadjacent site. Notice that the binding of DNArequires the subunit to be in its weakly bounddTTP state. When dTMPPCP is trapped in acatalytic site, that subunit is in its tightly boundstate and the corresponding DNA binding residuesare not in position to bind DNA. Therefore, Pi is

e. (a) Stereo view of the T7 hexamer showing (in red) theetail showing the main stress pathways between adjacent(c) Cartoon illustrating the circumferential stress path and, N$T*) state. The springs represent the b-sheet and them.

4 DP 4 D 4 E

h h h

4 N,DP 4 N,D 4 N,E

(A1)


trapped in its site waiting for dTMPPCP todissociate. This waiting time can be sufficient forsome synthesis events.

These hypotheses rest on cooperativity betweenthe hexamer subunits for dTTP hydrolysis. Theabsence of medium [18O]Pi exchange withdTMPPCP without DNA indicates that regulationor cooperativity among subunits occurs via DNA,but not because dTTP and dTMPPCP have differentbinding affinities.

The complete simulation

We have presented alternate explanations for theresults of the oxygen exchange experiments. Inorder to distinguish one explanation from the other,a complete kinetic cycle must be consideredexplicitly, including both D/DP and N$D/N$DP steps. Also, the effect of non-hydrolyzabledTMPPCP binding must be included. Theminimal kinetics required to describe each subunitis:

T 4 T� 4 E 4 T� 4 T

h h h h h

N,T 4 N,T� 4 N,E 4 N,T� 4 N,T

where T represents the dTMPPCP-bound state.Because dTMPPCP is not hydrolyzable, the statesin the left side are terminated before the hydrolysisevents. The complete kinetic network has 146Z7,529,536 states. Although our method applies inprinciple, this system is too large to be addressedcomputationally within current computer powerand memory limits.

Initial conditions

The concentrations of all states at the initial timeare required to solve the kinetic equations. Chaseexperiments provide information to deduce theinitial conditions. For this, a simulation was carriedout by considering only three possible states beforeMg2C is added (i.e. hydrolysis has not yetoccurred): E, T*, and NT*. Using these states, theinitial occupancy can be inferred from the chaseexperiments. In order for sequential translocation toproceed, only one subunit can bind tightly to DNAat a time. That is, at most one subunit can be in stateNT*; all other subunits are in states E or T*. The acidquench data show the transient phase takes aboutfour or five hydrolysis cycles to reach the steady-state slope. The hydrolysis speed (slope) in thistransient phase is greater than the steady-statespeed. This is because some subunits are already inthe T* state, ready to hydrolyze. The [dTTP]0dependence of steady-state slopes indicates dTTPbinding is the rate-limiting step. Because dTTPbinding is rate limiting, the subunits, which alreadyhave dTTP bound, bypass this slow step. The fouror five startup hydrolysis cycles imply that initial

dTTP binding events probably occur in almost allsubunits. Therefore, to obtain the initial concen-trations of all states we modeled a random dTTPbinding process with binding probability given bythe chase occupancy.

Alternative optimization methods

We used the downhill simplex method for theoptimization to obtain rate constants that fit theexperimental data well. Several other optimizationmethods have been devised, including thosespecifically designed for least-squares fitting,56

maximum likelihood methods,57,58 and geneticalgorithms.59,60 However, because the function tobe minimized is highly non-linear, none of thesemethods guarantee finding a global minimum. Wehave compared the simplex method with severalother non-linear solvers and none has providedany compelling advantage over the simplestapproach.

Flux methods

The flux diagram method developed by T. Hillhas been successful in computing states and fluxesfor small kinetic networks.40 However, the networkfor the T7 helicase is too large for the diagrammethod, and so a numerical scheme such as wedeveloped here is required. Recently, Fisher &Kolomeisky developed kinetic models withsequences and branches applied to motor pro-teins.47–49 In these models, analytical solutions forany number of sequential states with branches andjumps were obtained and used to fit data for myosinand kinesin dimers with two catalytic sites. Rajen-dran et al. used the analytical solution of asequential kinetic model for another ring helicaseDnaB.19 However, it is difficult to extend thesemodels to the large kinetic network that character-izes the T7 helicase, so the problem is betterapproached by numerical schemes rather thananalytical solutions. The mechanical considerationswe used to prune the possible kinetic pathways arealso an important element in distinguishing ourmethod from others.

Three cooperative steps are necessary

The fitting is based on the three requiredcooperative steps involving multi-subunit inter-actions. To examine the necessity of each of thesethree steps, we also carried out simulations relaxingone required step at a time. In each case either thepower strokes are not sequential, or DNA losescontact to all subunits of the helicase, thus

Figure A2. Flow chart illustratingthe computational scheme used toobtain the best-fit rate constants tothe kinetic experiments.


losing tight coupling between hydrolysis andtranslocation.

To further test the model, extra requirementswere also imposed in addition to the three requiredcooperative steps. For example, in F1 ATP synthaseADP release is enhanced by ATP binding onanother subunit.31 Therefore, an additional require-ment was imposed that dTDP release is triggered bydTTP binding at adjacent sites. Simulation based onthis additional cooperative requirement shows thatdTDP is released too slowly so that the occupancydoes not correspond to the measurements.18 Sincethe data also show Pi release has a phase delay afterhydrolysis, we also simulated the situation that Pi

release is triggered by dTTP binding or the powerstroke. In both cases Pi release becomes the rate-limiting step under all initial [dTTP]0, and fitting fordifferent slopes under different [dTTP]0 is impos-sible. We conclude that the phase delay of Pi releaseresults simply because the rate is intrinsically slow,not because this step is affected by multi-subunitinteractions.

Predictions for single molecule force–velocityexperiments

Based on the computed rate constants, we can

estimate how this protein might function under amechanical load. Assume a single molecule force–velocity measurement can be accomplished for T7helicase, analogous to the DNA pulling experimentcarried out for the f29 portal protein.61 As the forceopposing translocation increases, the translocationpower stroke, NT*/NT, becomes the rate-limitingstep as the hydrolysis cycle slows. Because thehydrolysis cycle is tightly coupled to translocation,as the stall force is approached, the rate of thehydrolysis cycle becomes too small to be measured.The prediction of force dependence is shown inFigure A3 based on the assumption that the powerstroke can be modeled as a thermally activatedevent, with the forward rate modified by aBoltzmann factor, exp(KFd/2kBT), and the back-ward rate modified by exp(Fd/2kBT). Here, F is theexternal load force and d is the power strokedisplacement, dw0.9 nm. The initial conditions arethe same as those shown in Figure 4. Physiologicalconditions are assumed to be [dTTP]Z100 mM,[dTDP]Z10 mM, and [Pi]Z1 mM, as for Figure 7.For low initial [dTTP]0, a force less than 5 pN doesnot have a large effect on the hydrolysis rate. This isbecause dTTP binding, rather than the powerstroke, is the rate-limiting step at low load. As theload force increases, the hydrolysis rates for all

Figure A3. Force and free energies of translocation. (a) Predicted force dependence for the steady-state dTTPhydrolysis rate under different initial conditions assuming a Boltzmann force dependence; i.e. the power stroke is athermally activated step. Since N$T*/N$T is the power stroke, this step is affected by the load force.When the load forceis large, this step becomes rate limiting and the hydrolysis cycle slows. The values at FZ0 correspond to the steady-stateslopes in Figure 4. The prediction for physiological conditions is also shown ([dTTP]Z100 mM, [dTDP]Z10 mM, [Pi]Z1 mM). The stall force is estimated to be about 30 pN. (b) The free energy levels for physiological conditions underdifferent load forces. The corresponding hydrolysis rates under these four different forces are marked in (a). The loadforce only changes the free energy of the power stroke step, N$T*/N$T. As the load force increases, this step eventuallybecomes endothermic, and the corresponding hydrolysis rate decreases.


curves drop until, around 30 pN, the hydrolysis rateis very small, and can be considered an estimate ofthe stall force. An actual mechanical potentialmodel, rather than the exponential Boltzmannmodel used here, can be implemented to obtain adifferent force–velocity relationship, especially nearthe stall force.31,33

As shown in Figure A3(a), under physiologicalconditions the hydrolysis rate at zero force is about13.5 sK1. The hydrolysis rate also decreases withincreasing load force; values of four different loadforces are shown: 0 pN, 10 pN, 20 pN, and 30 pN.For each of these forces, the computed free energylevel of each chemical state is plotted in FigureA3(b), illustrating the free energy drop in eachkinetic step. Of course, there is an activation freeenergy barrier in each kinetic step; however, sincethe barrier heights are not determined (i.e. thefrequency factor of each step is not known), we linkthe free energy levels by broken lines. Note thatonly the power stroke step is affected by the loadforce, the free energy differences of all other stepsremain unaltered. At physiological conditions, theconcentration of Pi is high, and so the Pi release stepis endothermic. This is consistent with the exper-imental observation that Pi release is slow. Brow-nian fluctuations drive the system through thisunfavorable kinetic step, which will be rectified bythe subsequent steps (D/E, E/T*, etc) with largefree energy drops. As the load force increases, thepower stroke step becomes endothermic, and thisstep becomes rate limiting.

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Edited by R. Ebright

(Received 24 January 2005; received in revised form 22 April 2005; accepted 22 April 2005)

Mechanochemistry of T7 DNA Helicase

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