mechanochemistry of a viral dna packaging...

Mechanochemistry of a Viral DNA Packaging Motor

Jin Yu1, Jeffrey Moffitt1, Craig L. Hetherington1,Carlos Bustamante2 and George Oster3⁎1Department of Physics,University of California,Berkeley, CA 94720, USA2Department of Physics,Chemistry, Molecular and CellBiology, Howard HughesMedical Insititute, Universityof California, Berkeley,CA 94720, USA3Department of Molecular andCell Biology, EnvironmentalScience, Policy, & Management,University of California,Berkeley, CA 94720, USA

Received 16 February 2010;received in revised form30 April 2010;accepted 2 May 2010Available online7 May 2010

The pentameric ATPasemotor gp16 packages double-strandedDNA into thebacteriophageϕ29 virus capsid. On the basis of the results of single-moleculeexperimental studies, we propose a push and roll mechanism to explain howthe packaging motor translocates the DNA in bursts of four 2.5 bp powerstrokes, while rotating the DNA. In this mechanism, each power strokeaccompanies Pi release after ATP hydrolysis. Since the high-resolutionstructure of the gp16 motor is not available, we borrowed characterizedfeatures from the P4 RNA packaging motor in bacteriophage ϕ12. For eachpower stroke, a lumenal lever froma single subunit is electrostatically steeredto the DNA backbone. The lever then pushes sterically, orthogonal to thebackbone axis, such that the right-handed DNA helix is translocated androtated in a left-handed direction. The electrostatic association allows tightcoupling between the lever and the DNA and prevents DNA from slippingback. The lever affinity for DNA decreases towards the end of the powerstroke and the DNA rolls to the lever on the next subunit. Each power strokefacilitates ATP hydrolysis in the next catalytic site by inserting an Arg -fingerinto the site, as captured in ϕ12-P4. At the end of every four power strokes,ADP release happens slowly, so the cycle pauses constituting a dwell phaseduring which four ATPs are loaded into the catalytic sites. The next burstphase of four power strokes starts once spontaneous ATP hydrolysis takesplace in the fifth site without insertion of an Arg finger. The push and rollmodel provides a new perspective on how a multimeric ATPase transportsDNA, and it might apply to other ring motors as well.

© 2010 Elsevier Ltd. All rights reserved.

Edited by P. J. HagermanKeywords: bacteriophage; DNA packaging; molecular motor; mechano-chemistry; stochastic simulation

Introduction

Multimeric ring ATPases transform chemicalenergy into mechanical work by hydrolyzingnucleotides at multiple catalytic sites to generateforces. These motors translocate substrates alongtheir axes for various purposes, such as proteindegradation and DNA translocation.1,2 Two essen-tial questions arise in understanding the way theseATPases achieve their functions. (1) How do thesubunits of the ATPase couple the catalytic site tothe DNA (or protein substrate) and generate thetranslocation forces? (2) How are the individualsubunits coordinated around the ring during re-peating cycles of ATP hydrolysis? Most of the

known multimeric ring ATPases appear to coordi-nate hydrolysis sequentially,3–8 albeit exceptionsexist.9 Nevertheless, the exact mechanism used forcoordination is unclear. Moreover, the force-gener-ation mechanism between the ATPase subunits andthe substrate remains vague, in part due to theabsence of the DNA in structural studies. Toinvestigate these issues, we studied a model system:the DNA packaging motor in bacteriophage ϕ29.Packaging the genome into the viral capsid is a

key event in the life-cycle of many viruses. Thebacteriophage ϕ29 that infects Bacillus subtilis is oneof the simplest and most intensively studied viralpackaging systems.10,11 The genome of ϕ29 is madeof a linear double-stranded DNA (dsDNA) of about19.3 kb, encoding 20 proteins. Packaging the longpiece of dsDNA into a near-crystalline state inside avirus capsid ∼50 nm in diameter generates a highback pressure due to the entropic barriers, electro-static repulsions and bending energies of DNA.12–15

The pressure can be utilized later on to eject the

*Corresponding author. E-mail address:[email protected] used: dsDNA, double-stranded DNA;

cryo-EM, cryo-electron microscope.

doi:10.1016/j.jmb.2010.05.002 J. Mol. Biol. (2010) 400, 186–203

Available online at www.sciencedirect.com

0022-2836/$ - see front matter © 2010 Elsevier Ltd. All rights reserved.

genome into the host cell during viral infection.16,17To surmount the high pressure, the packaging isdone by a powerful molecular motor that uses thechemical energy of ATP hydrolysis to translocate thedsDNA into the capsid. An image of the ϕ29 DNApackaging system obtained from the cryo-electronmicroscope (cryo-EM) density map is shown inFig. 1a.18 It consists of three multimeric rings: anATPase (gp16), a 174 base RNA (pRNA), and adodecameric portal connector (gp10). These ringsare located at a unique five-fold vertex of theicosahedral capsid (the prohead).19 Cryo-EM recon-struction shows that the ATPase and pRNA formpentamers.18 A similar pentameric organization ofthe ATPase has been found recently in the DNApackaging motor of bacteriophage T4.20

Despite early suggestions of a variety of DNApackaging mechanisms,21–24 involving mostly theportal connector rotation, other work25–27 and mostrecent experimental studies28 show that the activeforce generation component of the motor is the gp16ATPase. In this work, therefore, we studied theDNA packaging ATPase of ϕ29. The system hasbeen investigated intensively in experiments usingsingle-molecule manipulation techniques.28–33 Wepresent a mechanochemical model based on currentexperimental knowledge, homolog structural infor-mation, and a few necessary, but generic, assump-tions. The model provides a physical picture of howthis multimeric motor system translocates dsDNA.

Constructing the Model

Experimental basis for the model fromsingle-molecule studies

Our model is built upon experimental studies ofthe ϕ29 DNA packaging system using single-

molecule optical tweezer measurements.29–32 Theseexperiments indicated that DNA translocation doesnot occur during ATP binding to the motor, but islikely to be associated with inorganic phosphate (Pi)release after ATP hydrolysis.30 Also, experimentsshowed that the motor affinity for the DNA is highin the ATP-bound state but low in the ADP-boundor empty (apo) state.30

Further high-resolution optical tweezer studiesexamined the coordination between subunits andthe DNA translocation step size.31 Measurements athigh load forces showed that the packaging pro-ceeds in bursts of 10 bp steps, each composed of four2.5 bp substeps. Data analysis suggested that adwell phase following the burst phase is composedof four ATP-binding events and several non-ATP-binding events (Fig. 1b).Recent experiments on packaging-modified DNA

substrates indicates that the motor is promiscuousbecause it can package a variety of chemicalmoieties.32 In particular, experiments showed thatthe motor can package a DNA substrate withcharge-neutral inserts, and that the packagingprobability decreases with the length of the insert.Statistical analysis of neutralized DNA packagingsuggested that the motor maintains specific contactsduring the dwell phase with successive 10th and11th phosphate charges. Many different contacts aremade during the burst phase of 10 bp, when theDNA is actually translocated.32 These burst phasecontacts are not made with unique nucleic acids,suggesting that much of the force that drivestranslocation is mediated by steric interactions.Importantly, preliminary studies suggest that theDNA can be negatively rotated (in the under-winding direction) during packaging.33 Here, wedevelop a mechanochemical model of the DNApackaging based mainly upon the recent high-resolution31 and DNA rotation measurements.33

Fig. 1. The ϕ29 DNA packaging system and essential optical tweezer measurements. (a) Images (courtesy of M.Morais) from cryo-EM studies of the packaging system.18 The dodecameric connector (gp10, green), pRNA (magenta) andATPase pentamer (gp16, blue) with the DNA modeled for visualization.18 (b) The essential experimental results fromhigh-resolution optical tweezer measurements.31 DNA translocation proceeds in bursts of four power strokes of 2.5 bpseparated by dwell phases wherein four ATPs are loaded into four catalytic sites. There are also multiple rate-limitingevents in the dwell phase that do not involve ATP binding.

187Mechanochemistry of a Viral DNA Packaging Motor

Borrowing structural and mechanochemicalfeatures from the P4 packaging motor

Currently, there is no atomic structure for thegp16 packaging ATPase, so we used structuralfeatures from similar systems where the structure isknown. Comparative genomic studies34 suggestthat ϕ29-like ATPases belong to the HerA/Ftsksuperfamily.35 However, homology modeling ofthe gp16 (see Supplementary Data and Fig. S1)shows that it is unlikely that gp16 utilizes therotary inchworm mechanism proposed for Ftsk.36

As we examined candidate reaction schemes for thegp16 packaging consistent with the optical tweezermeasurements,31,32 one particular scheme lent itselfnaturally to the structural features and coordina-tion mechanisms characterized in the ϕ12-P4motor.6,37,38

The P4 protein is a hexameric ATPase closelyrelated to the superfamily 4 helicases in the RecA-like ASCE division of the P-loop NTPases.34 Itpackages single-stranded (ss) RNA into the capsidof ϕ12 by amplifying small changes in the ATP-binding site (∼1 Å) into a larger translocationmovement (∼6 Å).37,38 The structural entity in P4that directly drives the ssRNA translocation is amoving lever composed of the superfamily 4 heli-case motifs H3 and H4 and loop L2 (helix–loop). Thelever is in the up position in the ATP-bound state,and the lever moves downward after ATP hydroly-sis and Pi release

37,38 (note that in P4, by convention,the packaging direction is down; the packagingdirection is up for gp16 in this work). A positivelycharged residue (Lys241) lies at the tip of the leverand binds to the RNA backbone during thetranslocation. Also, the movement of the lever inone subunit might facilitate hydrolysis in theadjacent subunit by inserting an Arg finger(Arg279) into the neighboring catalytic site tostabilize the transition state.6,38 Finally, the forcegeneration step for P4, as in gp16, is likely to beaccompanied by Pi release after ATP hydrolysis.36

In the current model of gp16, we have borrowedthe lever motion and its sequential and cooperativehydrolysis mechanisms from P4,6,37,38 which arelikely to be general features of many ring translo-cases, but they have been elucidated most promi-nently in P4. The cryo-EM studies of gp16 haveidentified some density associated with the ATPasein the central channel of the motor in proximity tothe DNA, suggesting a structure there that activelytranslocates DNA.18 Interestingly, one can findseveral candidate amino acid residues in the vicinityof, and including, Arg122 that are likely to bind tothe DNA (see Supplementary Data Fig. S1).39 Thislocation also seems coincident with the centraldensity mentioned above. Therefore, the modelhere for gp16 utilizes the lumenal loops, or motorlevers, to drive DNA packaging in ϕ29 by movingup and down in each subunit. This choice is logicalbecause the lumenal loops emanate from the centralβ-sheet from which also emanates the P-loop thatgrasps the nucleotide during ATP binding in P4.37

Besides, an Arg finger has been identified in gp16(Arg146) on the basis of sequence alignments fromcomparative genomic studies.34,40 Thus, the hydro-lysis coordination mechanism using Arg fingerinsertion in P46,38 is also used in building thechemical reaction path in our gp16 model.

The motor is a closed planar pentameric ring

Currently, the detailed structure of gp16 is notknown. At relatively low resolution, the cryo-EMreconstruction18 of gp16 and the structural studiesof ϕ12-P437 and the T4-gp17 packaging motor20 donot show any large structural deviation of the motorfrom a symmetrical multimeric ring. To build amotor model with minimal geometric complexity,we assume that, on average, the gp16 motorsubunits form a closed planar ring with five-foldsymmetry.18,20 Thus, our model differs from sys-tems that deviate significantly from a planargeometry, such as the hexameric transcription factorRho7 and the E1 helicase.8

In what follows, we first present a kinematicmodel explaining how the DNA can translocate30,31

and possibly rotate33 when it is packaged throughthe motor ring. Next, we examine the forces thatdrive or assist the DNA along the kinematictrajectory. These forces arise from several sources.The proximal energy source for driving transloca-tion is the binding of ATP to the catalytic site. Themotion of the lever is delivered to the DNA by stericinteractions between the DNA backbone and thelever that transmits the packaging force from thecatalytic site to the DNA. The electrostatic interac-tions between the DNA phosphates and the levercharges couples the lever movements to the DNA,stabilize the DNA–lever association and facilitatethe DNA to roll from one subunit to the next. Wecomplete the mechanochemical model by construct-ing a dominant reaction pathway deduced from theexperimental measurements on the dwell–burstbehavior of the packaging cycle.30,31 We used thismodel to conduct stochastic simulations of thepackaging dynamics. These simulations fit quanti-tatively the optical tweezer measurements undervarious substrate concentrations and load forces.30

Further questions are addressed separately, such ashow the 2.5 bp step size arises and its relatedregistry problem, which strand the motor pushes on,and how the gp16 packaging motor compares withthe relatively well characterized F1-ATPase motor.

Results

Kinematics of the push-and-roll model

For simplicity, we begin by examining thekinematics of the motor DNA packaging systemthat geometrically fit with experimental measure-ments. The experiments have found that, duringpackaging, DNA translocates in substeps of ∼2.5 bp(∼0.85 nm)31 and rotates negatively in the DNA

188 Mechanochemistry of a Viral DNA Packaging Motor

underwinding direction.33 Below, we use the ϕ12-P4lever structure to illustrate the gp16 model, asshown in Fig. 2a (note that he packaging directionis up).In this kinematic model, the DNA movement

consists of two parts: a push by the motor leverperpendicular to the phosphate ridge (see Fig. 2aand below), followed by a rolling of the DNA whenthe lever releases/withdraws from the DNA at theend of the power stroke. This rolling motion carriesthe DNA from the current lever to the next (i.e. fromsubunit S2 to subunit S3) as shown in the lowerpanel of Fig. 2b.The lever driving the power stroke in gp16 was

assumed to be similar to that in ϕ12-P4 (see alsoSupplementary Data Fig. S1). It has a positivelycharged residue at the tip of the lumenal loop thatextends from the central β-sheet that engages theDNA by moving toward the DNA backbone andpushing upwards in the DNA groove perpendicularto the DNA backbone (see Fig. 2a). As a result, thelever moves the DNA both vertically, along thepackaging direction, and horizontally, in the tan-gential direction of the DNA cross-section. The

horizontal motion causes the DNA to rotate aboutits axis by an angle θpush. For each 2.5 bp ofpackaging distance (modeled as the swing distanceof the lever tip), the lever rotates the DNA by anangle θpush≈−30°, due to the tilting angle of theDNA strand (Fig. 2b; Supplementary Data Fig. S2c).Note that the DNA translocates in the directioncoming out of the page in the top view, and thisconvention applies to all other figures in this work.θpush depends only on the DNA geometry, which weassume is in the B-form. The negative sign indicatesa clockwise DNA rotation with respect to the viewin the figure, which in this case coincides with theDNA underwinding direction.Upon completion of the power stroke, the lever

releases the DNA, allowing the DNA to roll(without sliding) from the current subunit towardsthe next subunit located clockwise at φ=−2π/5=−72°. This rolling motion also leads to a DNArotation, θroll, in a counterclockwise (positive)direction. The magnitude of θroll depends on therelative size of the motor ring lumen to that of theDNA. The DNA radius in its cross-section is RN,and the motor lumen radius is RP, measured in the

Fig. 2. The push and roll model of the DNA packaging in ϕ29. (a) A molecular view of the packaging lever and theDNA substrate. This view was generated from VMD60 using the structure of the P4 motor from ϕ1237 for the purpose ofillustration. The packaging direction here is up. Themotor lever is shown inmagenta, with a positively (+) charged residue(Lys241 in P4) at the tip. The rest of the motor subunit is in green. The dsDNA is modeled as the substrate of gp16. TheDNA is colored according to atom-type: oxygen, red; phosphorus, brown; carbon, cyan; nitrogen, blue. (b) One exampleshowing theDNAmovement upon a power stroke. Left: a schematic top view of the DNAand themotor ring. Subunits aredenoted by Si, i=1,… 5, and the hydrolysis state of each is in parentheses: T, ATP-bound; D, ADP-bound. For each powerstroke (T→D),30 the motor lever pushes the DNA up by 2.5 bp (out of the paper) and rotates it by θpush≈–30°. The DNAthen rolls to the next subunit when the lever releases the DNA, leading to an additional DNA rotation of θroll=+18° (whenRP/RN=5/4). Thus, each power stroke can produce a total DNA rotation of –12°. Right: coordinate and trace of a point onthe DNA during packaging corresponding to the special case: the DNA is rotated by θpush(1→2)+θroll(2→3)=θtotal≈–12°for every 2.5 bp translocated. The actual trajectory is likely to be more like the red line where the rolling commences beforethe push has completed. (c) The overall DNA rotation predicted for each power stroke versus the relative size of the motorring cross-section to that of the DNA (RP/RN): a negative (underwinding) rotation of the DNA ifRP/RN b1.4, no rotation atRP/RN ∼1.4, and a positive (overwinding) DNA rotation for larger sizes of the motor ring RP/RN N1.4.


cross-section at the position of the protein lever (forexact descriptions, see Methods and SupplementaryData Fig. S2). Overall, for each 2.5 bp packaged, theDNA rotates by:

utotal = upush + uroll

Fig. 2c shows that, if the motor ring is small (RP/RNb1.4), the DNA rotates negatively (underwinding);otherwise (RP/RN N1.4), it rotates positively (over-winding). When RP/RN ∼1.4, the DNA does notnecessarily rotate. Preliminary single-moleculeexperiments have shown negative rotations ofthe DNA during packaging,33 suggesting that theDNA fits rather snugly within the protein ring(RPb1.4 RN).Fig. 2b (right) also shows an example packaging

trajectory (with RP/RN=5/4) as a trace of a point onthe DNA along z (the packaging length) and θtotal(the total DNA rotation), two observables in thesingle-molecule optical tweezer experiments. Notethat we have ideally separated the push and roll ofthe DNA in a sequential way in presenting themodel; the actual motion probably combines thetwo motions, as suggested by the red line in the firstpower stroke in Fig. 2b.

Essential forces in the push-and-roll model

Since the motor can package a variety of chemicalmoieties, including neutralized DNA,32 we suggestthat the packaging forces pushing on the DNA aremainly steric. In particular, the push would beperpendicular to the DNA strand if the van derWaals surface of the DNA backbone appearsuniform relative to the DNA–lever contact, andelectrostatic interactions between the DNA andmotor lever also affect the packaging.The generic electrostatic interactions we consider

include only those between the negative charges onthe DNA phosphate groups and the five positivecharges at the tip of the motor levers (e.g.corresponding to Lys241 in ϕ12-P4). These interac-tions are calculated as Coulomb potentials betweenthe charges screened by an exponential damping ofthe Debye length (see Methods and SupplementaryData Fig. S3). In the Supplementary Data Fig. S3bwe show that electrostatic interactions dictate that itis energetically more stable for the DNA to adhereto the periphery of the motor lumen than at thecenter. Thus, packaging the DNA is accompaniedby rolling and sliding along the lumen edge. Toexamine the generic electrostatic effects during thepower stroke we computed the electrostatic inter-actions (Vel) as the DNA translocates along z androtates by θ. In Supplementary Data Fig. S4 weshow Vel (z, θ) at the beginning and at the end ofthe power stroke, as well as Vel (θroll) for DNArolling thereafter. The results show that the genericelectrostatic interactions between the DNA andlever charges facilitate and steer the DNA packag-ing, ensuring tight coupling between the move-ments of the motor lever and DNA. The overall

interactions between the motor lumen and the DNAare not known, so we cannot assess the rolling orsliding friction. However, the rolling of the DNAfrom one lever site to the next seems energeticallyefficient because the attraction of the DNA to thenext lever subsidizes the escape of the DNA fromthe current lever.Consideration of both the steric and the electro-

static interactions leads to the following features ofthe model.

• The driving levers can be steered electrostati-cally toward the DNA backbone, approachingthe nearest phosphate charge on the DNA.

• By pushing sterically on the DNA backbone,each lever motion moves the DNA verticallyup (2.5 bp) and rotates the DNA laterally.

• Electrostatic interactions between the DNAand the levers facilitate the DNA motion tofollow closely that of the lever during the stericpush.

• The local electrostatic attraction between theDNA and the lever decreases towards the endof the power stroke, producing a circumferen-tial electrostatic gradient around the motorring that assists the DNA rolling to the nextlever.

Notice also that the 2.5 bp substep along with theDNA rotation would move the lever–DNA contactslightly differently each time as the DNA rolls to thenext subunit. Because of thermal fluctuations, andallowing for protein plasticity, the lever can adjustfor this “mismatch” (see Discussion). Thus, we canassume that the DNA faces approximately the sameinteraction potentials for every power stroke cycle.

The dominant chemical reaction pathway andmechanochemical coupling

Finally, we turn to the coupling between themotor lever and the catalytic site. On the basis ofexperiments,30,31 we propose the dominant reactionscheme shown in Fig. 3a. This scheme addresses twomajor questions arising from the experiments: (i)why does the pentameric motor package the DNAin four,24 but not five, substeps per reaction cycle;and (ii) what are the likely rate-limiting steps in thedwell phase at a saturating concentration of ATP?

A three-state hydrolysis cycle and the forcegeneration step

The chemical states of an ATPase motor generallycomprise six transitions between six catalytic statesat each site.41 This can be written as:

E X T X T4 XD�P X D4 XD XE

where E is the empty (apo) state, T is the weaklybound ATP docking state, T⁎ is the tight ATP


binding state, D ·P is the product state right afterhydrolysis and before Pi release, D⁎ is the tightlybound ADP state, and D is the weakly bound ADPstate. However, since this detailed descriptiondemands much more information than currentexperiments can provide, we have utilized a simplerthree-state description of the ATP hydrolysis cycle ateach catalytic site, which we write as:

E X T X D X E

where E is the empty state, T is the ATP-bound state(includes the T, T⁎ and D ·P states underlined

above), and D is the ADP-bound state (includesthe D⁎ and D states).Experiments show that the force generation, i.e.

translocation step, is likely to accompany Pi release(i.e. the T→D step in the three-step kinetic model),although ADP release (i.e., from tight to loose ADPbinding D⁎→D), cannot be ruled out.30 The experi-ments show also that the DNA affinity of a subunit ishigh in the T state, but low in the D and E states.30 Inthe model, we adopt a force generation stepaccompanying Pi release in the T→D step, becausethe DNA affinity decrease after the power strokeis crucial for the DNA to roll to the next subunit

Fig. 3. A dominant reaction scheme. (a) The reaction path (right), the schematics of the pentameric motorconfiguration (upper left) and the power stroke firing direction within each cycle versus the direction the first subunitshifts in subsequent cycles (lower left) proposed in the model. (b) The cooperative ADP release mechanism connects fourADP release and ATP loading events sequentially around the motor ring. The reaction cycle is divided into two phases.The burst phase contains four sequential power strokes at four consecutive catalytic sites. Each power stroke generates a2.5 bp substep.31 The dwell phase contains four consecutive ATP loadings and several non-ATP-binding events.31 Eachpower stroke commences during the T→D transition (Pi release)

30 when the subunit is attached to the DNA (indicated bya thick continuous line). Each power stroke requires the next subunit (thin continuous line) to be in the T state to receivethe DNA when it rolls following the completion of the power stroke. After four contiguous power strokes in the burstphase, the motor pauses because the next subunit has been left in the low DNA affinity D state, and the system enters thedwell phase. During the dwell phase, the first ADP release is slow, but the following ADP releases (second to fourth D→Etransition) proceed faster as ATP binds quickly (at high [ATP]) and accelerates ADP release at the next site (shown in b).The waiting time for the first power stroke is another rate-limiting non-ATP-binding event during the dwell phase afterthe four ATPs are loaded. The ensuing power strokes (second to fourth T→D) happen very quickly in the next burstphase. A related hydrolysis cooperative mechanism is the insertion of an Arg finger from the preceding subunit, driven bythe hydrolysis/power stroke in that subunit (see also Fig. 6).


(cf. SupplementaryData Fig. S4c).However, assigningtranslocation to T→D does not explicitly differentiatebetween the ATP hydrolysis step (T→D·P) and Pirelease (D·P→D). (Some complications consequenton this are discussed in the Supplementary Data).

A scenario for four substeps, but not five

We have assumed that the gp16 motor is planarwith five-fold symmetry. Our explanation for thefour-substep packaging burst around the penta-meric ring is attributed not to the geometry but tothe following dynamical and kinetic constraints. (1)Each packaging substep requires the coordination oftwo neighboring subunits (thus four continuouspairs are available around the pentameric ringduring each cycle). In order for the T→D (Pi release)step to take place in the current ATP-bound subunitwith a high level of DNA affinity, the next subunitmust also be in the high DNA affinity ATP state tobe ready to “receive” the DNA as it rolls to the nextsite when the current power stroke ends. Thecoordination between the two neighboring subunitsmay depend on stress communicated through theDNA, or through the subunit interface where thecatalytic site resides. (2) Un-assisted ADP release(the first D→E) is very slow so that after fourcontinuous power strokes the fifth one cannot takeplace because the next catalytic site has been left inthe low DNA-affinity ADP state, and is not ready tobind DNA stably.Notice that for each cycle the “fifth” subunit shifts

its identity around the ring (counterclockwise inlower left Fig. 3a, i.e., after four power strokes in onecycle S1S2S3S4, the fifth subunit is in S5, while for thenext cycle, the four power strokes would take placein the order S5S1S2S3, so that the fifth subunit movesto S4). That is to say, the “first” subunit shiftscounterclockwise around the ring in consequentcycles (with S1 in cycle I, S5 in cycle II, S4, S3 andS2 inlater cycles, etc.; see Fig. 3a, lower left ), opposite tothe clockwise “firing” direction of the four powerstrokes in one cycle. This behavior suggested fromour model can be tested through high-resolutionpackaging experiments with ATPase rings thatinclude a single catalytically inactive subunit.

Slow non-ATP-binding events in the dwell phase

Experiments show that the dwell phase containsmore than three rate-limiting events other than thoseassociated with awaiting ATP binding.31 We pro-pose that these slow non-ATP-binding eventsinvolve the sequential release of four ADPs, fol-lowed by waiting for the first power stroke, or thefirst ATP hydrolysis. The reasons for this choice areas follows.First, ATP hydrolysis (T⁎→D·P) can be very slow

in multimeric ATPases if the Arg finger is notaligned in the correct position to stabilize thetransition state.42,43 ADP release (D⁎→D→E) canalso be slow because it requires breaking most of thehydrogen bonds formed during ATP binding.4 With

no assistance from neighboring subunits, thesespontaneous (thermally activated) processes arelikely to proceed slowly. If we assume that the firstATP hydrolysis and the first ADP release are bothspontaneous, then they can account for two of therate-limiting dwell events. To avoid complications,we did not consider ATP weak to tight-binding(T→T⁎) as being slow in the current model.Second, along with the four ATP loading events

for a 10 bp packaging cycle,31 there should be fourATP hydrolysis events and four ADP release events.Invoking the Arg finger cooperative mechanism inthe ϕ12-P4 packagingmotor,6,38 the second to fourthATP hydrolyses in the burst phase occur veryquickly after the first ATP hydrolysis, so they donot contribute to the slow dwell events. However, inour model the additional (more than two) slowdwell events can be attributed to the second tofourth ADP releases, even though they can beaccelerated (but less significantly than that in Argfinger catalysis) compared to the first ADP release.Overall, the four ADP releases along with the firstATP hydrolysis (the first power stroke followsimmediately), contribute to the rate-limiting eventsmeasured at high [ATP], which has a minimumbetween 3 and 4.31

A simple mechanism for ADP release satisfyingthe above experimental constraint is that the ADPreleases are sequential and the second to fourthare assisted by ATP binding at the previous site.Figure 3b shows schematically how ATP bindingprovides strain energy to encourage ADP release atthe next site. The first ADP release is spontaneousand thus slow, while the second to fourth ADPreleases are relatively faster due to ATP binding atthe preceding site. That is, once the first ADP hasbeen released, ATP binds into and tightens thecurrent catalytic site, which helps to open the nextcatalytic site and releases the second ADP. Thissequence is repeated continuously around the motorring for the third and fourth ADP releases. Thismechanism can explain why tight binding of ATPappears to be coupled with next-site ATP docking(immediately after ADP release).31 This explainsalso why only one site is available for ATP dockingat any given time in the dwell phase (because theother sites are bound with ADP).31 (Details ofimplementing this mechanism are presented in theSupplementary Data and Fig. S6.)

The dominant reaction scheme

The dominant reaction scheme we propose is builtupon the constraints set by the experimental resultsas shown in Fig. 3a.30,31 In summary, the reactioncycle is divided into a burst phase and a dwellphase. The burst phase is composed of fourcontiguous 2.5 bp substeps, thereby adding up to10 bp per reaction cycle. Each 2.5 bp substepcorresponds to a power stroke accompanied by aquick Pi release. The dwell phase contains four ATPloading events, as well as several non-ATP-bindingslow events, suggested above as the four ADP


releases, and waiting for the first power stroke. Thefirst ADP release is spontaneous and slow, while thefollowing are accelerated by ATP binding at thepreceding site. Besides, the first power stroke needsto be inhibited until all subunits are loaded withATP, which is consistent with the experimentalobservation that no 2.5 bp substep was visible evenat very low [ATP] under low load forces.31 Theinhibition mechanism is not clear, but it could arisebecause the first ATP hydrolysis (or Pi release)requires a circumferential stress that is present onlywhen all the subunits are loaded with ATP.Upon loading four ATPs, the first ATP hydrolysis

takes place spontaneously and slowly. The follow-ing hydrolyses happen much faster because of theArg finger insertion from the preceding subunit (seeDiscussion). These facilitated ATP hydrolysis eventstake place before the first power stroke or they couldproceed one at a time before each power stroke inthe burst phase.Table 1 gives a summary of the proposed “rules”

in constructing the reaction scheme shown in Fig. 3a.Note that the reaction scheme cannot be determineduniquely from current experimental data becauseseveral key chemical steps, such as the ADP releaseand ATP hydrolysis, have not been resolved in thepackaging experiments. Nevertheless, our proposedreaction scheme is a reasonable choice that inte-grates the current knowledge of multimeric ringATPases.

Themechanochemical profile of onepower stroke

Combining the features discussed above, we candescribe one power stroke at a subunit as follows.First, ATP binding to the current catalytic site drivessliding of the P-loop over the nucleotide.44 Assum-

ing similarity to F1-ATPase, this deforms the centralβ-sheet from which the P-loop emanates.45 Theelastic energy stored in the β-sheet deformation istransmitted to the lumenal loop that comprises thedriving lever. Driven by ATP binding, the lever“cocks” onto the DNA backbone, steered there bythe electrostatic attraction between the positivelycharged residue at the lever tip and the negativephosphate charges on the DNA. The DNA, havingjust rolled to the current site after the previouspower stroke, is now captured by the lever. Thepower stroke (or ATP hydrolysis) of the previoussite also drives the insertion of its Arg finger into thecurrent catalytic site to accelerate ATP hydrolysis.6,38

The hydrolysis thus triggered, Pi is promptlyreleased from the current site. The input from ATPbinding energy in the catalytic site allows the elasticenergy stored in the β-sheet45 to be released to drivethe “recoil” translocation stroke. The lever pushesperpendicularly against the DNA backbone, movingit sterically upwards by 2.5 bp, and laterally toinduce DNA rotation.Without electrostatic steering and stabilization,

the packaging would take place with reducedefficiency, and even with frequent DNA back-slipping. The price of the electrostatic association,however, is that at the end of the stroke the lever isstuck in the electrostatic grip of the DNA. Therefore,energy stored from the ATP binding is also used to“withdraw” the lever from the DNA by weakeningits affinity with the DNA. This is “compensated” bythe attraction of the DNA to the next subunit (in theT state with high DNA affinity), which facilitatesrolling of the DNA to the next packaging lever. Afterthe fourth power stroke, the system waits for theADP releases, starting from the fifth site. Once theADP is released from a site, ATP binds and moves

Table 1. Summary of the reaction rules implemented in the model

Reaction rule Experimental source and reason

Power stroke: T→D · Force generation likely coincident with Pi release30

· DNA affinity high in T, low in D and E30

T→D requires the next subunit to be in the T state · Four substeps/cycle for pentameric gp1631

· DNA affinity high in T, low in D and E;30 the lever is downin T and up in D37,38

The first power stroke requires all subunits bound with ATP · The first power stroke happens only after four ATPs areloaded (no substep of 2.5 bp visible in low [ATP] at low force)31

The first power stroke is slow (in waiting time), while thesecond to the fourth are largely accelerated

· Power strokes generated in bursts after four ATPs are loaded31

· At least three to four non-ATP binding events rate-limiting31

· Arg finger insertion-assisted hydrolysis6,38

The first ADP release is slow, while the second to thefourth are accelerated

· Four substeps/cycle for pentameric gp1631

· Three to four non-ATP binding events rate-limiting31

· Hints from F1-ATPase 41,54 on ATP binding-assistedADP-release next site

Four ADP releases proceed sequentially · Cooperative ADP releases suggested above· Randomness parameter values (nmin versus [ATP] inFig. S6)31

The first ATP docking relatively slow, while the following arefaster; ATP docking and previous site ATP tight-binding coupled

· Two rate-limiting events at low [ATP]31 and Michaelis–Mentenpackaging velocity versus [ATP]30

· Cooperative ATP dockings connected by irreversibletight-binding suggested in Ref. 31· Cooperative ADP releases suggested above

For explanations, see Results and the mechanochemical analysis in the Supplementary Data.


the lever into position ready to receive the DNA inthe next cycle. ATP binding also weakens thebinding of ADP in the next site so that the ADP isreleased quickly.Thus, the push-and-roll combining with the ATP

cycles allows the coordinated, sequential transloca-tion of the DNA by the motor subunits in anefficient, tightly coupled manner so that, on average,each ATP cycle may drive one translocation substepof 2.5 bp.

Simulation of DNA packaging dynamics

Using the proposed reaction scheme in Fig. 3a, weconducted stochastic simulations on the translation-al movement of the DNA. The simulations take intoaccount frictional forces, the motor driving force, theexternal load force from the optical trap, andthermal fluctuations. We generated stochastic tra-jectories of the DNA packaging based on theMarkov-Fokker-Planck method.46 The unknownkinetic constants were tuned self-consistentlyunder experimental conditions, such as at saturating[ATP] or close to stall force (∼70 pN; see Methods),

and kept constant later on in the simulation. Thevalues of the computational parameters are sum-marized in Supplementary Data Table S2. Therationale setting these parameters is also discussedin Supplementary Data.

Low load force

The simulation results under low external loadforce conditions (FL ∼5 pN) are given in Fig. 4. Weshow the DNA packaging velocities at variousconcentrations of ATP, ADP or Pi as described.30

In Fig. 4a, the sample trajectories of the DNApackaging are shown under various [ATP] (5–250 μM) while keeping [ADP] and [Pi] at 5 μM.The computed trajectories are similar to thosedescribed experimentally,31 revealing 10 bp steps(inset). There are occasional backward steps at low[ATP], which reflect transitions that are not builtinto the dominant packaging pathway, i.e. off-pathway, but become detectable due to the stochas-tic nature of the system. Note that these trivialbackward steps are different from the significantback-slipping of the packaging reported in Ref. 30,

Fig. 4. Stochastic simulation results under low load force (FL∼5 pN). (a) Sample trajectories of DNA packaging (bp) atATP concentrations of 5–250 μM ([ADP]=[Pi]=5 μM). The inset shows the step size distribution peaked at 10 bp. (b) Thepackaging velocity versus [ATP] at ADP concentrations of 5–1000 μM ([Pi]=5 μM). In the legend, E5 representsExperiment at [ADP]=5 μM, M denotes the Model and MM denotes the Michaelis–Menten fit to the experimental data.30

(c) Histograms of the dwell time distribution for the packaging trajectories at [ATP] of 5–250 μM, in the same colorcorresponding to each case in a. (d) The inverse of the randomness parameter (n=1/r) versus [ATP], calculated from thedwell time distributions produced from the experiments31 and from the model.


which may come from occasional loose couplingsbetween the motor and DNA that our model doesnot taken into account. Figure 4b shows the velocityversus ATP concentration at various [ADP] andcompares the results with the experiments. Thevelocities demonstrate Michaelis–Menten-like de-pendence on [ATP], and fit well with the experi-mental data. The trend that increasing [ADP]decreases the velocities significantly at low [ATP](i.e., increases KM) indicates that ADP competes withATP for access to binding sites.30 Our results aretuned in accord with the experimental observation30

that increasing [Pi] (up to 1000-fold) has nodiscernible effect on the packaging velocities (datanot shown). Figure 4c shows the dwell time (i.e.,pause time in the dwell phase) distribution atvarious [ATP], corresponding to the cases in Fig.4a. The peaked shape of the distribution indicatesthat multiple rate-limiting events exist at both highand low [ATP]. We use the randomness parameter,r, to characterize the relative variation of thedistributions.47,48 The inverse of the randomness

parameter, n=1/r, is an effective measure of theminimum number of rate-limiting events.31,47 Fig-ure 4d shows that n versus [ATP] produced fromour model fits well with the experiments.31 Thevalue n=3–4 at saturating [ATP] corresponds to thenumber of slow non-ATP-binding events, whichinclude four ADP releases (one slow and three fast)and waiting for the first power stroke. The value ofn of ∼2 at low [ATP] points to the cooperative ATPdocking events connected by irreversible tight-binding steps as suggested.31 The effects wereadopted by modulating the kinetic rate (kT≥E) tomimic the tight-binding ATP state once the nextsite is bound with ATP (see Supplementary DataTable S2).

High load force

The simulation results under the higher loadforces are presented in Fig. 5. In Fig. 5a, a sampletrajectory from simulation of the DNA packaginglength is shown under a load force of FL ∼40 pN at

Fig. 5. Stochastic simulation results under high load forces. (a) A sample trajectory of the DNA packaging trace takenunder FL ∼40 pN at [ATP]=250 μM ([ADP]=[Pi]=5 μM). The inset shows the step size distribution for each cycle,revealing both the 2.5 bp substeps and the dominant 10 bp steps. (b) Schematics of the packaging trajectories under a low(FL ∼5 pN) and a high load force (FL ∼40 pN),31 both at a high concentration (250 μM) of ATP. The burst phase is coloredblue and the dwell phase is colored red. Our proposed reaction scheme suggests that the dwell phase is always rate-limited by ADP releases and waiting for the first power stroke when ATP loadings are fast (at high [ATP]). (c) Thepackaging velocity versus load force at [ATP] of 10–500 μM ([ADP]=[Pi]=5 μM). In the legend (right) M500 representsModel at [ATP]=500 μM (curves linking the model data are for smooth fitting), and E represents Experiment.30 (d) Theforce-dependent chemical rate constants used in our model (mechanism II), shown in values relative to that at low force(FL ∼5 pN), for the first T→D (red, left axis) and the second to fourth T→D (blue, right axis), respectively.


high [ATP] (250 μM), and low [ADP] and [Pi] (5 μM).This trajectory shows the 2.5 bp substeps while10 bp is still the dominant step-size for a reactioncycle (Fig. 5a, inset). To compare the basic feature ofthe packaging trajectories under the low and highload conditions, Fig. 5b schematically shows theexperimentally measured trajectories under lowload (FL ∼5 pN) and high load (40 pN) forces bothat high [ATP] (250 μM). It is clear that high loadforces lengthen the pause time between the 2.5 bpsubsteps, which were invisible at low load force.31

Under our current reaction scenario, the rate-limiting non-ATP-binding events in the dwellphase are composed of four ADP releases andwaiting for the first power stroke. If the reactionscheme is the same under different load forces, slowADP release after the burst phase would ensure thatthere are always four, but not five, substeps perreaction cycle. There are two possible mechanisms(I and II) for the force response of ADP releases (seeSupplementary Data and Fig. S8a). Here, weillustrate only the simpler mechanism (II), whichassumes no force-dependence of the ADP releasechemical rates. In this case, the first ADP release isprohibited during the burst phase (under all loadforces) until being activated, by some “allosteric”event; e.g. at the end of the burst phase. The burstphase is lengthened under the high load force andthe beginning of the ADP release is, therefore, alsodelayed under the load force. Our analysis suggestthat the average waiting time for the first powerstroke has not been affected much by the load force(cf. the Supplementary Data).Figure 5c shows a plot of velocity versus load force

under various [ATP] and a comparison with theexperimental data.30 To keep the reaction scenario thesame under various load conditions, we found itnecessary to introduce force dependence into some ofthe chemical rate constants (see Supplementary DataTable S2). Figure 5d shows that the chemical rateconstant for the first power stroke (1st T→D) isincreased with increased load force, while the rateconstant for the second to the fourth power strokes(which are accelerated relative to the first T→D) isdecreased as the load force is increased. The chemicalrate constant for T→D captures the rate of thechemical (but not the mechanical) part of thetransition; i.e. the ATP hydrolysis and Pi release, butnot the packaging stage.Weassumed that in the T→Dtransition the hydrolysis is rate-limiting with the Pirelease following immediately. The rate constant thuscharacterizes the rate of hydrolysis. Therefore, theresults indicate that the load force increases the rate ofthe spontaneous ATP hydrolysis (for the first T→D)but decreases the rate of the accelerated hydrolyses(for the second to the fourth T→D),which are assistedby the Arg finger insertion mechanism.

Discussion

To explain the experimentally observed packagingbehavior, we built a mechanochemical model of the

ϕ29 packaging motor gp16 by utilizing structuralinformation from another similar nucleic acid pack-aging motor and examining generic interactionsbetween the motor and DNA. The model providestestable answers for several questions. (i) Should theDNA rotate during packaging? (ii)Whydoes amotorof five subunits package DNA in four substeps percycle, and what happens during the 10 bp pauses?(iii) How does the load force affect individual stagesof packaging? (iv) Is there an out-of-registry problembetween the motor and the DNA with 2.5 bpsubsteps? (v) What are the essential forces drivingthe packaging, andwhich DNA strand is pushed on?

Does DNA rotate?

Using only the basic geometry and genericelectrostatic interactions of the motor with theDNA, we have constructed a simple push-and-rollmodel. During the push phase, the right-handedhelical geometry of the DNA allows it to be movedup as well as to be rotated in a left-handed direction.The amount of rotation is determined by thegeometry of the DNA helix, which we assumed tobe in the B-form. After (or coupled with) the leverpush, the DNA detaches from one subunit and rollsdown an electrostatic gradient to the next. Therolling of the DNA from one site to the next isefficient; that is, the energy cost to release the DNAfrom the electrostatic grasp at one site is partiallycompensated by the attraction of theDNA to the nextsite. The rolling rotates the DNA in the directionopposite to the rotation caused by the push, with amagnitude depending on the relative size of themotor ring to that of the DNA cross-section (Fig. 2c).Geometric analysis shows that if the motor ring isabove a certain size, the DNA will rotate in theoverwinding direction. Preliminary experimentalresults suggest that the DNA rotation is negative inthe underwinding direction,33 implying a relativelysmall motor ring size. If the periodic phosphatecontacts implied in Ref. 32 are considered, onlycertain DNA rotation values are allowed, and theseare determined by the size and the symmetry of theDNA and the motor. This constrains the motorlumen size (RP) to certain values such that the pushand roll conspire to align the DNA phosphate with aDNA-binding lever at the end of a 10 bp burst.In summary, our push-and-roll model provides a

mechanism for DNA rotation during packaging. Itsuggests that the push and roll rotate the DNA inopposite directions, resulting in a combined effectthat depends on the motor ring size. Indeed, a netDNA rotation is not energetically costly because thepacking energy inside the capsid appears to bedominated by electrostatic self-interaction of theDNA and its bending energy.49,50

Note that the current model deals only with anidealized system in which the motor is treated as aplanar ring composed of five identical and evenlyspaced subunits, and the DNA is in the B-form. Wehave attributed all the DNA motion to the action ofthe motor. However, other effects, such as coiling of


the packaged DNA inside the capsid, could affectthe rotation of the DNA. Further studies are neededto clarify these issues.

The mechanochemical framework

Question (ii) asked why the pentameric motorpackages DNA in four 2.5 bp substeps and whathappens during the 10 bp pause. These issues can beaddressed together using some general features ofthe ring motor that have been captured clearly in theϕ12-P4 packaging motor.6,37,38 The major structuralfeature we borrowed from P4 is the motor lever.This consists of a helix–loop region with a positivelycharged residue at the lever tip that moves along themechanical reaction coordinate. As illustrated byFig. 6 for the ϕ29-gp16 motor, the lever position isdown in the ATP-bound state (T) and up in theADP-bound state (D) after ATP hydrolysis (again,the packaging direction is up in our ϕ29-gp16 modeland down in ϕ12-P4). Experiments indicated thatthe DNA affinity of the motor subunit is highest inthe T state and lower in the D and E states.30In constructing the model, we used a three-state

hydrolysis cycle: (E→T→D→E), where the T→Dtransition includes hydrolysis and the Pi release(coupled with the power stroke), but these stepshave not been distinguished experimentally. Thepower stroke begins in the high DNA affinitycondition, and the affinity decreases as the powerstroke progresses. In order to avoid the motor losingits grip on the DNA between power strokes, the nextsubunit needs to be in the high affinity T state. If thenext subunit is in the D state, the transition T→Dwould be discouraged in the current subunit

because the next lever is up and in the low-affinitystate, not ready for stable DNA binding. As ADPrelease is slow, after four continuous power strokes,the fifth one cannot take place until the D→E→Ttransition is complete in all the subunits for the nextreaction cycle. Thus, the model explains the four-stroke burst phase followed by a dwell phase.Another key functional feature we borrowed

from the P4 motor is its proposed mechanismfor triggering ATP hydrolysis in a coordinatedfashion,6,38 which might be shared by other ringmotors. In this mechanism, insertion of an Argfinger into the next catalytic site is concomitant withATP hydrolysis and/or the lever movement (powerstroke) at the current subunit (see Fig. 6). The Argfinger insertion greatly facilitates ATP hydrolysis atthe next site by stabilizing the transition state.42,43

Indeed, the Arg finger in ϕ29 has been clearlyidentified in comparative genomic studies of theϕ29-like ATPases with the Ftsk-clade proteins.34

This inter-subunit cooperative hydrolysis mecha-nism fits our proposed reaction scheme in which thefirst T→D transition is slow, while the second tofourth T→D transitions follow rapidly. Without theArg finger from the previous site, the first hydrolysisstep (T→D) happens spontaneously and slowly;while the second to fourth T→D transitions aremuch faster because the Arg finger acceleration is atwork. In our current model, the accelerated secondto fourth hydrolysis can either take place immedi-ately before each power stroke (Pi release), or alltogether during thewaiting for the first power stroke(see the Supplementary Data). Either explanationrequires that Pi is released rapidly after hydrolysis.Further experiments are required to resolve the

Fig. 6. The ATP hydrolysis co-operative mechanism proposed forϕ29 DNA packaging. The mecha-nism along with the cartoon issimilar to that used for the ϕ12-P4packaging motor (in particular, Fig-ure 5 in Ref. 38), adapted here forour push-and-roll model.6,37,38 Themodel assumes that ϕ29-gp16 usesa molecular lever similar to that inϕ12-P437 to package the DNA. Thepackaging direction is up. The leveris down in the T (ATP-bound) stateand up in the D (ADP-bound) state,while the DNA affinity of the lever/subunit is high in the T state but lowin the D state.30 The lever configu-ration also explains why a powerstroke, T→D, requires the nextsubunit to be in the T state. Theessential idea of the cooperativehydrolysis mechanism is that ATPhydrolysis/Pi release in the current

subunit triggers the Arg finger insertion into the next catalytic site, accelerating the (otherwise slow) ATP hydrolysis.6,38

This implies that the first power stroke (T→D) happens slowlywhile the second to the fourth power strokes aremuch faster(see the text). As the lever moves up during the power stroke, the DNA is pushed up by ∼2.5 bp, rotates, and rolls to thenext subunit. ADP release is supposed to be slow in the ϕ29 DNA packaging cycle so that, after four continuous powerstrokes (∼10 bp), the system pauses, waiting for the ADP releases and the ATP loadings to start a new cycle.


individual steps. Under our proposed reactionscheme, therefore, the major chemical events hap-pening during the 10 bp pause consist of: (i) fourADP releases; (ii) four ATP bindings; and (iii)waiting for the first power stroke. In particular, tobe consistent with experimental measurements,31

the first power stroke (either due to the firsthydrolysis or Pi release) must be inhibited until allsubunits are loaded with ATP. The underlyingmechanisms need further study.The slow ADP releases contribute mainly to the

long pause after the four substeps. Also, to describethe dwell time characteristics of the 10 bp pauses31

in our model correctly, the four ADP releases needto proceed sequentially around the motor ring(Fig. 3; Supplementary Data Fig. S6). The first ADPrelease happens slowly, while the following can beaccelerated by ATP binding to the precedingsubunit. That is, ATP binding at one site helps toopen the next catalytic site with ADP bound, andthis process goes around the ring sequentially foreach ATP loading and subsequent ADP release.Therefore, this ADP release mechanism leads also tothe time-ordered ATP binding, which has beenidentified experimentally.31

Load-velocity behavior

In single molecule optical tweezer measurements,the packaging of DNA can be opposed by a loadforce from an optical trap.When the load force is low(∼5 pN), its effect on packaging is negligible. As theload force increases to∼40 pN, however, sub-pausesbecome visible within the 10 bp bursts to split eachinto four 2.5 bp power stroke substeps.31 If weassume that the same dominant reaction scheme asthat shown in Fig. 3a applies to different loadconditions, we can identify the force effects on thechemical rate constants by fitting our model toexperimental data (Fig. 5c and d). However, inter-pretations of these force-dependent rate constantsare not straightforward. For example, our calcula-tions show that the chemical rate constant of the firstpower stroke (first T→D) increases with load force,while the chemical rate constant of the subsequentones (second to fourth T→D) decreases. Since thethree-state (E, T and D) description does notexplicitly distinguish between the ATP hydrolysisand Pi release steps in the T→D transition, we cannotdeterminewhether the chemical rate change is due tothe ATP hydrolysis, the Pi release or both. However,if we assume that the Pi release is always fast, we canexplain the force-rate effect: the load force promotesspontaneous ATP hydrolysis in the first T→D but, atthe same time, inhibits more significantly the inter-subunit cooperativity imposed on the second tofourth T→D through the Arg finger insertionmechanism. These force responses can be interpretedphysically through a stress pathway:

DNAYLeverYh�SheetYCatalytic site

But understanding exactly how the force responsesare generated will require further study.

Furthermore, a basic hypothesis here is that thefirst ADP release is always a slow event, ensuringfour substeps per reaction cycle regardless of loadforce conditions. Consequently, one mechanismconsistent with the hypothesis is that the first ADPrelease is inhibited, as in a tight ADP-bound state,until the end of the burst phase, regardless of theload force. Therefore, the ADP releases would bedelayed to start under a high load force because theend of the burst phase is delayed, while the D↔Erate constant is not force-dependent. However, inthis case, an “allosteric” effect seems necessary totrigger the first ADP release; that is, to change theADP-bound state from tight to loose. This allostericeffect can be caused by, for example, the binding ofDNA to the fourth subunit, which just precedes thesubunit where the first ADP is about to be released(see the reaction scheme in Fig. 3a). An alternativemechanism (labeled I) for ADP release that involvesa force-dependent rate constant is discussed in theSupplementary Data. Figure S8c shows additionalsimulated force–velocity data that might differenti-ate these two mechanisms at very high concentra-tions of ADP.)

The 2.5 bp substep and the registry problem

We used the experimentally measured substepsize of 2.5 bp in ourϕ29 DNApackagingmodel.31 Atthe current level of description, 2.5 bp is adopted asthe “swing distance” of the lever tip that is regulatedby the mechanochemical couplings between theATP-binding site and the lever.In the sequential packaging model, the DNA rolls

after each power stroke so that the DNA moves tothe next subunit. However, the protein–DNAcontact, ideally through DNA phosphate charges,would be slightly different from one subunit to thenext, i.e. slightly out of “registry”, after the 2.5 bppackaging distance as well as the rotation of theDNA. This small mismatch accumulates with eachpush and roll, giving rise to the question of howdoes this registry issue affect the packaging? Toanswer this question, one needs to understand howthe packaging force is generated. As supported bythe recent experimental measurements,32 the forcegeneration during the translocation step of packag-ing is likely to be achieved via steric interactions. Asmismatches with the DNA phosphates do not affectthe steric push significantly, the out-of-registry doesnot affect the local packaging. However, the overallpackaging efficiency can suffer due to weakerDNA–motor electrostatic associations caused bythe mismatch.There are ways of ameliorating the out-of-registry

issue. Most importantly, like all proteins thepackaging system is not rigid. The residues andstructural elements are flexible and bulky, and suffersignificant thermal fluctuations. Thus, small spatialmismatches in the DNA–lever contact before thepower stroke can be accommodated by fluctuationsof residues and the lever. Before binding, the leverfluctuates while it searches for the nearest DNA


phosphate. However, fluctuations do not interferemuch with the 2.5 bp lever movement during thepower stroke: once engaged with the DNA, the leverbecomes mechanically strained, and thus rigid, sothat the 2.5 bp power stroke is subject to only smallfluctuations.31

The uniformity of the non-integer step size31

requires that the lever be rigid throughout thepower stroke until the next subunit has engaged theDNA. Thus, the DNA-binding levers are not entirelyflexible; they are pliable when not bound to DNA,but firm when undergoing the power stroke andtransferring the DNA to the next subunit. If the10 bp periodic contacts are made as proposed,32

then in order to accommodate the 2.5 bp substepsize, the levers on adjacent subunits should beguided by electrostatic interactions with the phos-phates to adopt alternating conformations, eithershifting its tip 0.5 bp upper or lower, when theyengage the backbone at the beginning of the powerstroke.Moreover, to generally avoid accumulating large

mismatches between the lever and the nearest DNAphosphate, the motor subunit should be flexibleenough to adjust its position along the packaging(translational) direction or around (rotational) themotor axis, while the connector between the ATPaseand the viral capsid is axially and radially flexible.51

In this way, the system can be reset by the motormovements when the mismatch becomes large, sothat the motor lever can still approach the nearestphosphate for a stabilized binding configuration.Nevertheless, there is no evidence that the motorassembly moves relative to the capsid.In brief, the motor system is flexible enough to

compensate for small mismatches, but rigid enoughto execute highly repeatable 2.5 bp substeps. Beingout-of-registry is not a problem for packaging, but itaffects the overall packaging efficiency. As long asthe steric push generates a large enough packagingforce, the motor will be robust enough to packagethe DNA at least intermittently. Plasticity in thepackaging system can make the overall packagingprocess simultaneously smooth and efficient.

The role of electrostatic and steric interactions

In the model we have examined explicitly thegeneric electrostatic interactions between the DNAand the lever charges during packaging. We showthat the electrostatic interactions: (i) keep the DNAon the inner surface of the motor ring (cf. Supple-mentary Data and Fig. S3b); (ii) steer the levercharge toward the phosphate group on the DNAbackbone at the beginning of the power stroke; (iii)couple the DNA movement with the lever pushduring the power stroke; and (iv) facilitate therolling of the DNA to the next lever at the end of thepower stroke (cf. Supplementary Data Fig. S4a to c).The steric interactions between the motor lever andthe DNA are essential in generating the packagingforce: the lever pushes sterically on the DNAbackbone, i.e. the upper edge of the groove, and

the force generated accounts for most of the motorpackaging force,32 which can exceed 60 pN.29,52

Note that during the power stroke, the electrostaticminimum follows the lever as it sterically pushesonto the DNA strand, so the power stroke is nothindered by electrostatic interactions. Indeed, theelectrostatic force ensures tight coupling betweenthe motions of lever and the DNA. At the end of thepower stroke, weakening of the electrostatic inter-actions between the DNA and the current subunitfacilitates the DNA rolling from the current subunitto the next. This is consistent with the experimentalfindings that the packaging force is likely generatedafter ATP hydrolysis, during which the DNA–subunit affinity changes from high (T state) to low(D state).30

The roles of the electrostatic and steric interactionsbecome more distinguishable from the recentexperimental study on ϕ29 packaging of modifiedDNA substrates.32 It was found that the DNA can bepackaged when electrostatically neutralized dsDNAinsert is added, although the packaging efficiencydecreases as the length of the insert increases.32Supplementary Data Fig. S5 shows that a neutralinsert also introduces an electrostatic energy barrieralong the packaging direction. The longer the insert,the larger the barrier, and the lower the packagingprobability. The fact that the motor can still packagethe insert supports the idea that steric interactionsplay a major role in generating the packaging force.However, losing local electrostatic associationsbetween the lever and the DNA phosphate chargescan lead to a high-energy configuration of the DNAand, therefore, back-slipping. Recent moleculardynamics studies also demonstrated that attractiveinteractions between the DNA and a ring ATPaseare essential for the unidirectional movements.53 Weconclude, therefore, that as the steric interactionsdrive the DNA directly in bursts of actions, theelectrostatic interactions always steer the lever closeto a nearby DNA phosphate, in one way providingrelatively stable associations of the DNA with themotor to prevent DNA from back-slipping ordissociation, in the other way coupling the move-ment of the lever tightly with that of DNA duringthe push and facilitating DNA rolling toward thenext subunit.

Which strand to push?

In an experimental study of ϕ29 packaging usingmodified DNA substrates,32 it was shown thatneutralizing the DNA phosphate groups on the 5′to 3′ strand (∼30 bp) abolished the packaging, whileneutralizing the same length of phosphate groupson the 3′ to 5′ strand does not affect the packagingvery much.32 Because the backbone of the 5′ to 3′ (3′to 5′) strand forms the upper edge of the DNAmajor(minor) groove, we infer that during the powerstroke the motor lever pushes more effectively onthe upper edge of the major groove than on theminor groove. Indeed, the major groove is ∼12 Åwide, whereas the minor groove is ∼6 Å in B-form


DNA. Hence, the minor groove might be too narrowfor the lever to produce an effective steric push asthe lever approaches the groove edge from below,due either to steric hindrances or to an entropicbarrier of locating a “right” position. Using electro-static analyses (data not shown), we found thatneutralizing the charges on the 5′ to 3′ strand wouldenergetically lead the lever to approach the upperedge (3′ to 5′ strand) of the minor groove and push,which might not generate sufficient force to sustainpackaging. Nevertheless, when the motor is pack-aging a normal DNA substrate, there is no largeelectrostatic energy barrier like the one that exists inthe neutralized case. Thus, the lever can approacheither the 5′ to 3′ or the 3′ to 5′ strand and push onwhatever steric elements it encounters, although weexpect that the packaging force is generated moreeffectively when the motor pushes on the 5′ to 3′strand.

Summary

Building on currently available single-moleculeexperimental information28–33 and borrowing somestructural and functional features from a similarsystem, the P4 motor in ϕ12,6,37,38 we haveconstructed a mechanochemical framework forunderstanding how ϕ29 DNA packaging can takeplace. We propose a push-and-roll mechanism anda dominant chemical reaction scheme. The gp16motor subunits actively package the DNA usingthe lumenal loops as mechanical levers. The leverpushes and rotates the DNA with each powerstroke as ATP hydrolyzes and releases Pi. At theend of the power stroke the protein–DNA affinityis decreased and the DNA rolls to the next subunit,while both hydrolysis and products release seemcooperative and sequential around the motor ring.For every four power strokes, the packagingpauses and waits for the four ADP products tobe replaced by four ATPs to start the next reactioncycle.In each ATP hydrolysis cycle, the energy input

into the system takes place during the ATP-bindingstep that possibly drives a hinge-bending motion ofthe domain like that in the F1 motor.41,54 Althoughsequence similarities between the F1-ATPase andgp16 are not significant, structure similarities couldbe high around their catalytic regions as in other P-loop NTPases.55,56 The difference between the gp16motor and the F1 motor is that ATP binding doesnot drive the translocation stroke directly in gp16 asit does in F1.41,54 In gp16, the ATP-binding energy isstored in the protein as elastic energy, probably inthe central β-sheet as in F1, and is released afterATP hydrolysis as a recoil power stroke that drivesDNA translocation. Supplementary Data Fig. S9compares the free energy changes during an ATPhydrolysis cycle for the F1 and gp16 ATPase.Interestingly, from an evolutionary perspective,57

the rotary F1-ATPase has been proposed to haveevolved from membrane translocases, which origi-

nated from nucleic acid translocases in ancient cellsfrom which the packaging ATPase like gp16 wasdescended.In summary, we have provided a working model

for understanding ϕ29 DNA packaging. The modelprovides detailed mechanisms and is subject toexperimental testing. The model was built directlyupon the experimental observations28–33 and it fitsthe current experimental data.22,24While themodel isself-consistent, it relies on a series of assumptions thatdeserve further investigation. Structurally, we havemade twomajor assumptions. First, we assumed thatthe gp16 motor is a planar ring with five-foldsymmetry. This assumption meant that the four(but not five) packaging substeps per reaction cycleare due not to motor geometry but to the dynamicsand kinetics of the system. The other importantassumption involves using structural features iden-tified from the ϕ12-P4 packaging motor subunit todescribe the force generation of gp16. Thismeant thatin gp16 there are corresponding structural features:the motor levers and the lever charges. Theseassumptions can be verified or not as soon as ahigh-resolution structure of gp16 becomes available.We have assumed also that the non-ATP-binding

dwell events are chemical transitions involvingchanges of nucleotide state rather than pure proteinconformational changes. The dwell events are rate-limited mainly by the (unassisted) first ADP releaseand the (spontaneous) first ATP hydrolysis. Insubsequent reactions ADP release can be facilitatedby ATP binding, and ATP hydrolysis is coordinatedby Arg finger insertion as in ϕ12-P4. The validity ofthese assumptions can be examined by perturbingthe packaging with different chemical analogs andresolving more intermediate states.Our model provides the following predictions,

which can be tested in future experiments.

(1) According to the push-and-roll mechanism, theDNA rolls around the motor lumen duringpackaging rather than staying at the center ofthe ring. The DNA rotates in a direction thatdepends on the relative size of the motor ring. Asnug fit of the DNA inside the ring seems likely tobe because preliminary measurements shownegative DNA rotations.26

(2) Our dominant reaction scheme implies that thefirst power stroke of each 10 bp packaging cyclestarts with a different subunit and proceedssequentially around the ring for subsequentcycles. In Fig. 3a, cycle I starts with subunit 1,cycle II starts with subunit 5, and later cycles startwith subunits 4, 3 and 2 and then repeats thesequence. This sequence is opposite to the powerstroke sequence. During the dwell phase, ADPrelease alternates with ATP binding, and the nextburst phase of the packaging cannot begin until allsubunits are loaded with ATP: Only then can thefirst spontaneous ATP hydrolysis (or Pi release)take place. In addition, the model predicts how


the rates of relevant chemical transitions changeunder different load forces, so long as thedominant reaction path is not altered by the load.

(3) Proteins are not rigid bodies operating likemachine parts. The whole packaging apparatus,including the ATPase, can be flexible bothlongitudinally along the packaging directionand circumferentially around the pentamer. Thisallows the motor to sustain a high level ofpackaging efficiency. Otherwise, electrostaticmismatches would prevent the packaging frombeing tightly coupled throughout each mechano-chemical cycle.

Each of the above propositions is amenable toexperimental investigation by combining single-molecule manipulations with labeling and imagingtechniques. Overall, the work presented here pro-vides a detailed working model for a multimericring motor that can package dsDNA in a wayconsistent with all current experimental measure-ments. The model might apply also to a moregeneral class of multimeric ring motors that trans-port nucleic acids.

Methods

Defining the DNA coordinates

We used the following set of independent coordinates todescribe DNA movements during its packaging. (i) TheDNA packaging length z (measured in base pairs) and (ii)the DNA self-rotation about its axis, θ. These are the twoobservables in the optical tweezers experiments. We usedtwo additional coordinates to describe the relativeposition of the DNA inside the motor: (iii) φ is the angularposition of the DNA in the lumen with respect to thesymmetry axis of the motor, and (iv) rd is the radialdeviation of the DNA from the motor center (for thegeometry, see Supplementary Data Fig. S2a). In definingthese coordinates, we assumed that the motor subunits arearranged in a planar pentameric ring and the DNA passesthrough the ring perpendicularly. (More detailed descrip-tions are provided in Supplementary Data.)The DNA can both roll and slide on the surface of the

lumen. For a pure rolling motion of the DNA around thelumen surface, rd is a constant and φ and θ are dependent.We can describe the rolling motion θroll as:

uroll = − B RP =RN − 1ð ÞwhereRP is the protein ring radius, i.e. the radius of a circleconnecting the tips of the five motor levers, and RN is theradius of the DNA cross-section. Note that θroll and φ haveopposite sign so that as φ increases clockwise, θ increasescounter clockwise (see Supplementary Data and Fig. S2b).By considering the generic electrostatic interactions be-tween the DNA and the motor lever charges, we show thatthe DNA favors the rolling configurations along theperiphery of the motor ring with a constant rd=RP–RN(see SupplementaryData and Fig. S3b). Hence,we used thethree independent coordinates (z,φ, θ) to describe theDNAmovement, with φ describing the rolling, and θ describingthe DNA rotation, excluding the rolling component.

Calculating the electrostatic interactions between theDNA and the motor lever

In order to examine the generic electrostatic effects, weput five positive point charges on the motor levers andcalculated the screened Coulomb interactions between allthese positive charges and the negative phosphate chargeson the DNA backbone through:

Vel = −Xij

Qiqje2exp −Erij� �

4kq0qrrijð1Þ

with e2/4πɛ0ɛr estimated as ∼4 kBT·bp in the dielectricenvironment of the lumen (if ɛr estimated is∼40) and withan exponential damping term characterized by the Debyelength 1/λ, describing the overall solution ionic effect.Long stretches of DNA charge pairs were included in thecalculations to preserve the periodicity of the dsDNA (fordetails, see Supplementary Data and Fig. S3).

Constructing the dominant chemical reaction scheme

In Table 1 we summarize our proposed reaction rulesindividually, followed by the experimental basis for eachrule. Detailed explanations are provided in both the maintext and in Supplementary Data. The aim was to constructa reasonable reaction scheme that is consistent withcurrent experimental observations.

Stochastic simulation of DNA translocation

Here, we treat the motor as the source of translocatingforces (generated by the electrostatic and steric interactionsbetween the DNA and the motor), and write equations ofmotion of the DNA in terms of its packaging distance z(t)only. Because every power stroke is regarded identical inthe model, we can treat z as a periodic variable. TheLangevin equation describing the DNA translocation is:

~ :z = −BVj

eff zð ÞBz|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

Translocation forces

− FL|{z}Load force

+ f̃ tð Þ|{z}Brownian forces

; j = E;T;D; 0bzV2:5 ð2Þ

Here, z is the DNA packaging length (in bp), ζ is a dragcoefficient, and the random thermal fluctuations f̃ (t) satisfy⟨f̃ (t) f̃ (t′)⟩=2kBT ζδ(t-t′).58,59 Veff

j (z) is the potential of meanforce exerted on the DNA by the motor in chemical state j.The exact form of this potential is unknown; however, itsessential features can be modeled directly using thefollowing constraints.

(1) As an approximation Veffj (z) depends only on the

chemical state (E, T, D) of the subunit attached to theDNA.

(2) The energy minimum in the D state is 2.5 bp from thatof the T state in the T↔D transition.

(3) The slope of the potential (i.e. the translocation force) inthe D state can provide a stall force ∼70 pN (betweenthe 57 pN average stall force measured earlier,29 andthe higher value estimated recently52).

(4) The potentials repeat every 2.5 bp as the DNAmoves tothe next subunit.

In a thermal activation process, the exact mathematicalform of the potentials is not important, and we have used


smooth sin2(·) functions representingVeffj (z). These are 2AE

sin2(πz/10), 2AT sin2(πz/10), and 2AD sin2[π(z-2.5)/10] for

the E, T and D states, respectively (see SupplementaryData Fig. S7). The amplitudes obey ATNAE, AD (i.e. the Tstate has the highest DNA affinity), and AD≈15 kBT(70 pN·2.5 bp from (3)). For convenience, we set AT=2ADand AE=AD.Eq. (2) corresponds to a Fokker–Planck equation,58 a

probability description equivalent to the Langevin Eq. (2):

BUj

Bt=

DkBT

B

BzB

BzVj

eff zð Þ + FL

� �Uj

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Translocation and load forces

+ DB2Uj

Bz2|fflfflffl{zfflfflffl}Brownian motion

+X

ikji zð ÞUi|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

Chemical reactions

ð3Þ

where ρj(z,t) is the probability density of the system inchemical state j being at position z (with packagingdistance z) at time t.

Pi kji zð Þqi describes the Markov

transitions among different chemical states (with theconstraint: kjj zð Þ = −

Pi p j kij zð Þ). The diffusion constant

D is related to the drag coefficient ζ by D=kBT/ζ, andshould take into account effects on the DNA from thesolvent and from the rest of packaging apparatus. The valueof D was self-consistently tuned with the chemical rateconstants under the velocity constraints (see SupplementaryData). Note that for a motor with five subunits there are 35possible chemical states because each subunit can adopt oneof three states (E, T and D). When DNA is present, itattaches to one of the subunits at any given time and theoverall number of chemical states is then 5×35=1215.In Supplementary Data, we provide procedures de-

scribing how to simulate the DNA packaging trajectoriesand how to establish the parameters of the model. Thisallows us to compare the quantitative results of the modelwith the experimental data.

Acknowledgements

J.Y. was supported by a UC Berkeley Chancellor'sPostdoctoral Fellowship. G.O. was supported byNSF grant DMS 0414039.

Supplementary Data

Supplementary data associated with this articlecan be found, in the online version, at doi:10.1016/j.jmb.2010.05.002

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