coupling translocation with nucleic acid …coupling translocation with nucleic acid unwinding by...

Coupling Translocation with Nucleic Acid Unwindingby NS3 Helicase

Jin Yu1⁎, Wei Cheng2, Carlos Bustamante1,3,4,5 and George Oster4,6⁎1Department of Physics, University of California, Berkeley, CA 94720, USA2Department of Pharmaceutical Science, University of Michigan, Ann Arbor, MI 48109, USA3Department of Chemistry, University of California, Berkeley, CA 94720, USA4Department of Molecular and Cell Biology, University of California, Berkeley, CA 94720, USA5Howard Hughes Medical Institute, University of California, Berkeley, CA 94720, USA6Department of Environmental Science, Policy and Management, University of California, Berkeley, CA 94720, USA

Received 23 June 2010;received in revised form27 August 2010;accepted 20 September 2010Available online29 September 2010

Edited by A. Pyle

Keywords:molecular motor;helicase;DNA/RNA unwinding;diffusion;stochastic simulation

We present a semiquantitative model for translocation and unwindingactivities of monomeric nonstructural protein 3 (NS3) helicase. The model isbased on structural, biochemical, and single-molecule measurements. Themodel predicts that the NS3 helicase actively unwinds duplex by reducingmore than 50% the free energy that stabilizes base pairing/stacking. Theunwinding activity slows the movement of the helicase in a sequence-dependent manner, lowering the average unwinding efficiency to less than1 bp per ATP cycle. When bound with ATP, the NS3 helicase can displaysignificant translocational diffusion. This increases displacement fluctua-tions of the helicase, decreases the average unwinding efficiency, andenhances the sequence dependence. Also, interactions between the helicaseand the duplex stabilize the helicase at the junction, facilitating the helicase'sunwinding activity while preventing it from dissociating. In the presence oftranslocational diffusion during active unwinding, the dissociation rate ofthe helicase also exhibits sequence dependence. Based on unwindingvelocity fluctuations measured from single-molecule experiments, weestimate the diffusion rate to be on the order of 10 s−1 . The generic featuresof coupling single-stranded nucleic acid translocation with duplexunwinding presented in this work may apply generally to a class ofhelicases.

© 2010 Elsevier Ltd. All rights reserved.

Introduction

Helicases are enzymes essential for every aspect ofDNA or RNA metabolism.1–3 Like most motorproteins, helicases use the free energy from NTPhydrolysis to translocate along single-stranded (ss)

DNA or RNA, unwind double-stranded (ds) duplex,or displace proteins along the nucleic acid track.4,5There are six helicase superfamilies classified accord-ing to their signature sequence motifs.1,6 Amongthem, the helicase superfamily 1 and 2 (SF1 and SF2)are the two largest groups, consisting mainly ofmonomeric and dimeric helicases,7 with each mono-mer being a single polypeptide chain containing twoRecA-like folds.8,9 Other families of helicases oftenassemble inmultimeric forms, such as hexamers.3,10,11

Hepatitis C is a disease that affects about 180million people worldwide. The C-terminal portionof nonstructural protein 3 (NS3) from the hepatitis Cvirus (HCV) forms a three-domain enzyme and isclassified as an SF2 helicase,2,12 while the N-terminal

*Corresponding authors. 216 Wellman Hall, University ofCalifornia, Berkeley, CA 94720, USA. E-mail addresses:[email protected]; [email protected] used: NS3, nonstructural protein 3; ss,

single-stranded; ds, double-stranded; SF, superfamily;HCV, hepatitis C virus.

doi:10.1016/j.jmb.2010.09.047 J. Mol. Biol. (2010) 404, 439–455

Contents lists available at www.sciencedirect.com

Journal of Molecular Biologyj ourna l homepage: ht tp : / /ees .e lsev ie r.com. jmb

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

portion of the NS3 forms a protease domain. Thenonstructural proteins are responsible for replica-tion and packaging of the viral genome. The helicaseportion of NS3 is the first crystallized HCV proteinknown to hydrolyze ATP and to unwind DNA orRNA;13–15 its exact role in viral replication is unclear,although essential. Identifying inhibitors that targetthe NS3 helicase would help resolve the functionalissue and possibly develop anti-HCV drugs.16 In itsmonomeric form, the NS3 helicase can translocatealong ss nucleic acids in the 3′ to 5′ direction,17–22

unwind duplex regions, and displace other boundproteins on the nucleic acid.23 The helicase shareswith other monomeric translocases the two RecA-like domains (1 and 2),7 with an ATP binding sitelocated in between the two domains. Figure 1 shows

the NS3 helicase in complex with a 6-nt piece ofssDNA obtained from previous high-resolutionstructural studies.14 Essential residues involved inATP and ssDNA binding are also depicted. ThessDNA threads through a groove between domain 3and the two translocase domains (1 and 2) of theenzyme. In the lower right portion of Fig. 1, astructural alignment is shown between domains 1and 2 of the NS3 helicase and the two RecA-likedomains 1A and 2A of an SF1 helicase PcrA.25,26

According to the alignment algorithm,24,28 theoverall PcrA structure is not very similar to that ofthe NS3 helicase; however, the two translocasedomains in these two helicases are structurallysimilar.Recently, high-resolution structures of the NS3

helicase have been obtained bound with ATPanalogs.29 These structures provide conformationalsnapshots suggesting that two nucleic acid bindingsurfaces (NABS2 and NABS1), located on domains 1and 2, respectively, are required for the helicase tomove unidirectionally along the ss.29 Upon ATPbinding between the two translocase domains andrelease of hydrolysis products (ADP and Pi), the twobinding sites alternate moving forward 1 nt at atime. Themovements are coordinated through intra-and interdomain motions. Overall, the motor walksalong the ss in an ‘inchworm’ fashion. Thismovement is similar to that proposed for the SF1PcrA helicase,25,30 although the exact domain move-ments and detailed structural characteristics differsomewhat.PcrA and monomeric Rep and UvrD1,7 are among

the best studied SF1 helicases. ATP also binds intothe cleft between the two RecA-like translocasedomains (1A and 2A) of PcrA. As in NS3, the cleft‘closes’ as ATP binds and ‘opens’ upon release of thehydrolysis products.25,29 According to structuralanalyses and simulation studies,25,30,31 the unidirec-tional movement of the PcrA helicase along ssDNAis achieved by coordinating the ATP hydrolysiscycle with individual translocase domain motions.In addition, PcrA domain 1B (and/or part of 2B)corresponds to domain 3 in NS3 (see Fig. 1). Basedon crystal structure, it appears that domain 2B maydestabilize the duplex DNA via conformationalcoupling to the ATP binding region in PcrA25 [seeFig. S1 from Supporting Information (SI)]; however,the exact role of this domain remains controversialas deletion of the entire 2B domain in Rep results in afully functional monomeric helicase.32 In this work,we assume that the NS3 helicase shares importantfeatures with PcrA in its ss translocation activity;that is, the helicase coordinates movements ofdomains 1 and 2 with the ATP hydrolysis cyclessuch that each domain steps forward 1 nt at a time.The unidirectional translocation model we developfor the NS3 helicase is, therefore, based mainly oncomputational work on PcrA.30,31

Fig. 1. The structure of HCV-NS3 helicase and itsstructural alignment with PcrA in two translocasedomains. (Left) The NS3 helicase domain with a boundoligonucleotide obtained from a high-resolution structure(Protein Data Bank ID: 1A1V).14 Domains 1, 2, and 3 ofNS3 are in green, red, and blue, respectively. The poly(dU)oligonucleotide was crystallized with six residues visible,in orange. The ATP binding site is located in betweendomains 1 and 2, with residues essential for ATP bindingand hydrolysis in purple (these include K210, D290, E291,and H293 from domain 1 and Q460 and R467 fromdomain 2). The oligonucleotide threads through NS3 inbetween domain 3 and the other two domains, withimportant residues that interact with the oligonucleotidein brown (from left to right, they include W510, T269,E493, R393, T411, T450, and H369).12 (Right) Structuralalignment24 between the two translocase domains of theNS3 helicase and that of the PcrA helicase.25,26 The PcrAhelicase is shown in semi-transparent representation, withdomains colored the same as their corresponding domainsin the NS3 helicase: 1A to domain 1 (green), 2A to domain2 (red), 1B (2B) to domain 3 (blue). The molecular imageswere generated by VMD.27

440 Coupling Translocation with Duplex Unwinding

Recent measurements on the ss translocationactivity of NS320–22 show that the translocation canbe affected by sugar and base moieties of the ss suchthat the helicase translocates faster on ssRNA thanon ssDNA.22 Nevertheless, structural studies haveshown that the helicase interacts with the ssdominantly via backbone interactions.14,25,29–31

Thus, the base effect on the translocation, if any, islikely indirect, arising from base–backbone cou-plings in the ss. In this work, we assume that the sstranslocation of the helicase is sequence indepen-dent as suggested for PcrA,25,30,31 and regard thebackbone–base coupling as a higher-order effect.Previous biochemical measurements on NS3 heli-

case also show that its binding affinity to ssDNA issignificantly lowered when the helicase is boundwith ATP (or an analog).33 Moreover, the helicaseappears to unwind base pairs at the junction simplyby binding to the duplex.34 These results suggestthat the NS3 helicase unwinding/translocation iscoupled with its Brownian fluctuations asfollows:33,34 ATP binding loosens the grip of thehelicase on the ss so that the helicase may transientlymove randomly and bidirectionally (i.e., diffuse)along the ss; ATP hydrolysis and product releasere-induce tight binding of the helicase to the nucleicacid, resulting in a biased forward movement of 1 ntthat leads to directional translocation33 and duplexunwinding.34 However, the structural basis and themolecular mechanism of the directional movementof the NS3 helicase have remained unclear.Here, we model the translocation and unwinding

activities of NS3 helicase by combining itsstructural14,29 and biochemical properties.33,34 Forthe directional movement of the helicase, we adoptan alternating domain stepping model similar tothat used for PcrA,30,31 consistent with the structuralstudies of NS3.14,29 In addition, we included in ourmodel the diffusive property of NS3 helicase assuggested from Ref. 34. This means that the helicasecan diffuse or hop forward and backward along thess when it is bound with ATP. Thus, our modelcombines the inchworm-like directional steppingfeature25,30 with nontrivial Brownian diffusion.34In modeling NS3 unwinding activity, we have

incorporated properties of T7 helicase unwinding35,36

that seem to apply to NS3. Studies suggested that theunwinding of the T7 helicase is not associated withdTTP binding but with hydrolysis or productsrelease.36 We assume in this work that binding ofATP induces only the closing of the two domainspromoting the stepping of domain 1 (the trailingdomain of the enzyme) forward by 1 nt. Uponhydrolysis and release of products, the two translo-case domains must separate again. This separation isachieved by the stepping forward of domain 2 (theleading domain) by 1 nt. This promotes the unwind-ing of the duplex at the fork and completes themechanochemical cycle of the motor. Recent experi-

ments supported this assumption and suggested thatPi release triggers the power stroke for NS3unwinding.37

The unwinding and translocation model proposedhere incorporates many more structural and bio-chemical properties of the motor than previousstudies.38–40 Importantly, the sequence effect ofhelicase unwinding has not been addressed previ-ously but is explicitly modeled in this work. Tocalibrate and quantify the current model, we havefitted our numerical results with experimental dataon RNA unwinding of NS3 from single-moleculeoptical tweezer measurements.18,41

However, there are additional structural andmechanistic features of NS3 that we have notexplicitly taken into account in the current model,which are likely related to the protease domain of thefull-length NS3. In addition to the ss nucleic acidbinding sites revealed from the crystal structure,14

single-molecule fluorescence resonance energy trans-fer experiments42 and structural-based modeling13,43

suggest that there are also secondary RNA bindingsites in NS3. Moreover, the protease domain in NS3plays an important role in RNA helicase activity andgreatly facilitates binding of substrate RNA byNS3.44

Although these features have not been explicitlyconsidered here, it is possible that the enzymecontacts the dsRNA region upstream of the junctionthrough its protease and/or its domain 3. Thesecontacts anchor the enzyme on the ds region,destabilizing the fork and permitting the translocasedomains to open the fork. Such a scenario is notinconsistent with the detailed mechanism describedhere and may explain the 11-bp step size observed insingle-molecule optical tweezers18 and bulk kineticexperiments.43

In the following, we first describe how weconstructed the model and then show how severalexperimental observations can be understood interms of the model.

Results

Constructing the model

In this part, we describe how we model thetranslocation of the NS3 helicase along ssDNA orssRNA and how it subsequently unwinds the RNA/DNA duplex when it encounters a fork. We assumequalitatively that the NS3 helicase behaves the sameon the DNA and RNA, although the magnitudes ofthe activities may differ.

Translocation along the ss nucleic acid

Previous biochemical measurements suggest thatNS3 helicase translocates along the ss nucleic acid in

441Coupling Translocation with Duplex Unwinding

two alternating states: (a) a high-affinity state as themotor binds tightly to the ss when the ATP bindingsite is empty (so-called ‘apo’ state) and (b) a low-affinity state as themotor loosely binds to the ss whenthe catalytic site is occupied by ATP.33 Henceforth: ‘a’refers to the apo state, and ‘b’ refers the ATP-boundstate. In each state, in principle, the helicase (domains1, 2, and 3 together) can move back and forth by 1 ntalong ss. The movements are characterized by aforward rate, ri

+, and a backward rate, ri− (i=a, b). The

motor has no directional bias within each state—thedirectionality arises from transitions between thestates via ATP hydrolysis cycles. Therefore, wedenote ra

+=ra−=ra and rb

+=rb−=rb. To be consistent

with the motor–ss affinities, we require rab rb, so thathigh (low) ss affinity corresponds to low (high)translocation mobility.30 Since NS3 helicase binds thess tightly in the apo state, we let ra→0, whilemaintaining rbN0. That is, the low affinity of themotor to the ss in the ATP state allows it to moverandomly and birectionally (diffuse) as suggestedfrom experimental work.33,34 Though direct observa-tion on the helicase diffusion has not been made, thediffusive character is realistic as long as the diffusionhappens faster than dissociation of the helicase fromthe ss; that is, rb is larger than the dissociation rate koff

b

in the ATP-bound state. Later, we will show that thisis indeed true for our NS3 model (rb∼10 s−1 andkoffb ∼1.7 s−1).The high and low motor–ss affinities are also

supported by structural studies showing that thelow motor–ss affinity upon ATP binding is due tostructural rearrangements that lose several hydrogen-bond interactions between the NS3 helicase and the 3′ssDNA segments.29 Similar observations have beenreinforced by computational studies on PcrA showingthat hydrogen-bonding interactions between the heli-case and the 3′ ssDNA segment are weaker in theATP-bound state than in the apo state.31

Alternating domain stepping for directionality

Like the PcrA helicase, monomeric NS3 helicasecan be regarded as a two-domain translocase23 (seeFig. 1). We have assumed, therefore, that thetranslocation mechanisms are similar for these twohelicases. In particular, in both helicases, the twotranslocase domains are relatively separated fromeach other in the absence of ATP, but move closertogether when ATP binds in between, and resumetheir original separation after hydrolysis products arereleased. We show below that this cycle of displace-ments causes the two domains to alternate movingforward 1 nt at a time as the helicase transits betweenthe two states a and b. These properties are supportedfrom structural studies of both helicases.14,25,29In Fig. 2a, we illustrate the helicase translocation

cycle of NS3. In this scheme, domain 1 (trailing/rear) and domain 2 (leading/front) each moves

along the ss nucleic acid subject to their ownpotential of mean force whose period is 1 nt: greenfor domain 1 and red for domain 2. We denote byx1, x2 the location of the rear and front domain,respectively, along the ss. The position of thehelicase as a whole is defined as the average positionof the two domains, denoted by x=(x1+x2)/2, andindexed by j (state a) or j+1/2 (state b) located at thepotential minima. The two domains are coupled toeach other by a linear spring whose rest length isdifferent in the two states. Before ATP binds, themotor is in state a at position j (configuration 1), andthe rest length of the spring is l0. In this configura-tion, both domains bind tightly to the ss via theirindividual potentials. Both potential barriers arehigh, but domain 1 sees a lower-energy barrier thandomain 2. This property persists as ATP initiallybinds between the two domains (configuration 2)and the spring length shrinks from l0 to l0−1.Because of its lower-energy barrier, domain 1 hopsforward faster than domain 2 hops backward. Thisleads to a stabilized ATP-bound configuration 3,with the motor in state b at the position j+1/2, andthe spring length equilibrated at l0−1. The transitiona→b (configuration 1→3) is assumed to be ratelimited by ATP binding (rate ωab

I =kET · [ATP]). Thisis because both domain potential barriers decreaseduring the ATP binding process (i.e., ss affinitydecreases as ATP binds), and the coupling springcan pull the individual domains together (configu-ration 2) strongly enough such that the domain 1movement occurs very fast.The low affinity of the NS3 helicase to the ss as ATP

stably binds (configuration 3) allows the helicase todiffuse forward and backward along ss in state b.Essential conformation changes may take place uponstable ATP binding, so that the relative mobilities ofthe two domains can switch and domain 2 becomesmore mobile than domain 1. Upon ATP hydrolysisand product (Pi and ADP) release (configuration 4),the two domains are driven apart as the couplingspring expands to its original rest length l0. Duringthis recoil power stroke, domain 2's lower barrierallows it to move forward 1 nt as the motor relaxesback to its apo state at configuration 5. Thus, thehelicase has advanced forward 1 nt from from 1. Therate-limiting event during the transition b→a (con-figuration 3→5) is assumed to be domain 2'smovement, which is coupled with product release(at rate ωba

II =kcat).37 The rate-limiting domain move-

ment is affected by the spring force. The spring forceis relatively weak compared with the domainpotentials (configuration 4; compare 2 and 4), whichgrow as the motor–ss affinity increases toward theapo state. Note that the rate-limiting steps were alsoarranged to be compatible with the unwindingproperties assumed for the NS3 helicase. That is, theunwinding force is generated upon product release(i.e., the transition b→a is rate limited mechanically)


but not upon ATP binding (i.e., the transition a→b ischemically rate limited).36,37 The exact details of themechanochemical cycle of NS3 helicase translocationneed to be elucidated in future experiments.

Unwinding duplex nucleic acid

Figure 2b shows the binding configuration of theNS3 helicase to the RNA/DNA junction when itabuts a duplex junction. In this configuration, theend base pair of the duplex at the junction is locatedat x0 and the helicase covers the ss tail region of thejunction for 7 nt, with its front domain 2 (red)labeled at x2=x0−1 and the rear domain 1 (green) atx1=x0−7. The ‘front edge’ of the helicase (fromdomain 2 and/or domain 3) extends into the duplex(backbone) region for an extra 3 nt. The 7-nt ssdistance covered by the helicase was chosen to beconsistent with its structure14 (see the 6-nt oligonu-cleotide in between domains 1 and 2 in Fig. 1) andbiochemical properties. Experimentally, the helicasebinds to the junction and unwinds the duplex when

the 3′ ss tail linked to the duplex is less than 7 ntlong.34 The extra 3 nt represents the backbone regionof the duplex that is closely associated with thehelicase front domain 2 and/or domain 3. Theassociation stabilizes the helicase at the junction(discussed below) and, during each ATP cycle,facilitates base pair unwinding through backbonedistortions, likely similar to PcrA (see Fig. S1). Theoverall 10-nt distance of the helicase, projected ontothe nucleic acid track, is consistent with experimen-tal measurements showing that the helicase binds toa DNA junction with a 10-nt ss tail as fast as it bindsto a pure ssDNA of 10 nt long.34 Detailed analysesare provided in the SI and Fig. S2.

Duplex effects during unwinding

The basic feature of duplex unwinding in thecurrent model is that, as the helicase translocatesalong ss, it interacts with the RNA/DNA duplex atthe junction where the base pairing/stacking blocksits progress. The hydrogen-bonded base pairing,

Fig. 2. The translocation and unwinding scheme of the NS3 helicase. (a) The ss translocation is modeled as a two-stateprocess, based on the PcrA model.30 State a is the apo state with the catalytic site empty. In state a, NS3 has a high affinityto the ssDNA/RNA. State b is the ATP-bound state. In state b, NS3 has a low affinity to the ss33 and can hop (diffuse) toadjacent sites at rate rb. The red disk represents front/leading domain 2 (at position x2), and the green one represents rear/trailing domain 1 (at position x1). The individual domain potentials along the ss are shown in the corresponding color.30

Upon ATP binding (at rate ωabI ; configuration 1 to 2), the two domains are drawn closer to each other in comparison with

the apo state; domain 1 steps forward 1 nt as it faces a lower potential barrier than domain 2. After ATP stably binds(configuration 3), domain 2 experiences a lower potential barrier than domain 1, such that upon hydrolysis productrelease, domain 2 steps forward 1 nt (at rate ωba

II ; configuration 3 to 4) as the two domains separate from each other. In thisway, the helicase translocates forward 1 nt over an ATP hydrolysis cycle (compare 5 to 1). (b) The schematics of NS3helicase duplex unwinding. Shown are a binding configuration of the apo NS3 helicase at the ds–ss junction/fork, theindividual domain potentials in the ss translocation [as in (a)], and effective potentials due to the presence of the duplex. Inthis helicase binding configuration, the duplex end is located at x0, the front domain 2 is labeled at x2=x0−1, and the reardomain 1 at x1=x0−7. The bulk of the NS3 helicase (gray) covers a 7-nt ss tail region of the junction. Additionally, the frontedge of the helicase (some region on domain 2 or 3) extends into the duplex backbone region for 3 nt. The base-pair-unwinding potential is located between x0−1 and x0 to prevent domain 2 from moving forward. The potential is resultedfrom helicase active unwinding, which reduces the base pair stabilization free energy from ΔG0 to λΔG0 (0≤λ≤1) bysupplying energy (1−λ)ΔG0. δA is the activation energy separating the base pair open to the closed state. The junction-stabilizing potential, depicted relative to domain 2 in this diagram, has its minimum located at current position x2=x0−1,reaching a high plateau U0 at 3 nt further, preventing the helicase from moving away from the junction. This potentiallikely results from the helicase front edge associating favorably with the duplex backbone for 3 nt. The associationfacilitates active unwinding and prevents the helicase from dissociating.


however, fluctuates between ‘open’ and ‘closed’,biased toward the closed state at equilibrium. Thisassumption about the base pair ‘open↔ close’equilibrium is reasonable since its fluctuationfrequency (105–107 s−1 )38,45 is much faster than thehelicase moving rate (10–102 s−1). The energy ofATP hydrolysis driving ss translocation can alsoallow the helicase to actively ‘tilt’ the potential sothat the base pair equilibrium is biased away fromthe closed state and toward the open state. The workto accomplish this is provided by exerting force onthe hydrogen bonds of the base pair via the frontdomain movement and possibly also by perturbingthe duplex structure with associated domain 2 and/or domain 3. Otherwise, the helicase would need towait passively for transient spontaneous base pairopening to move forward,46 which also requires thatthe helicase does not dissociate from the junction toofast. Additionally, the NS3 helicase can be stabilizedat the junction (see SI text and Fig. S2). Thisstabilization has been demonstrated experimentallysince the NS3 helicase shows a higher affinity to aduplex substrate with a 3′-ss tail than to a pure sssubstrate of the same tail length (of 7 to 10 nt ss).34

The stabilization can be achieved via helicaseinteractions with the duplex that also facilitateunwinding. Moreover, the stabilization can preventthe helicase from dissociating from the junctionbefore unwinding proceeds.

Effective potentials of unwinding

To model the above effects during unwinding, weintroduce two effective potentials at the junctionalong the helicase translocation coordinate x (seeFig. 2b) while keeping the same translocationpotentials for domains 1 and 2 (as in Fig. 2a). Bothunwinding potentials are shown as acting at theposition of domain 2, and the corresponding energyminimum is located at x2=x0−1 where domain 2 islabeled in Fig. 2b.

Base-pair-unwinding potential. The base-pair-un-winding potential is located between x0−1 and x0when the NS3 helicase actively unwinds the duplexat its front domains (2 and 3). This potential tiltsthe open↔close equilibrium of the end base pairaway from the closed state and toward the openstate. Suppose the original free-energy differencebetween the base pair open and closed state, that is,the base pair stabilization free energy, isΔG0≡Gopen−Gclose≥0, so that the ratio between the base pairopening rate (kopen

0 ) and the closing rate (kclose0 )is kopen

0 /kclose0 =e−ΔG0. The helicase activity some-

how decreases the base pair stabilization freeenergy from ΔG0 to λ ·ΔG0 where 0≤λ≤1 (λ=0corresponds to the maximum active case and λ=1corresponds to the minimum active or passivecase). Here, we use λ as a phenomenological

parameter to measure how active a helicase isduring its unwinding activity. Note that theenergy reduction in the base pair stabilizationfree energy occurs when the helicase supplies freeenergy ɛint= (1−λ) ·ΔG0 to the base pair. To modelthe interaction, we used a one-step linear potential,which is a special case of more general interactionpotentials discussed in Ref. 38. As a result of thisinteraction, the ratio between the opening andclosing rate of the end base pair changes to:

kopenkclose

=k0openk0close

eeint =k0openk0close

e 1−Eð Þ�DG0 = e−E�DG0 ð1Þ

On the other hand, the forward movement rate(k+) of the helicase at the position just before thepotential, as well as the backward movement rate(k−) just after, also changes from their originalvalues (k±

0) such that

kþk−

=k0þk0−

e− eint =k0þk0−

e− 1−Eð Þ�DG0 ð2Þ

Here, the forward rate applies specifically to the‘catalytic’ transition rate from b to a (kcat) thatcouples with helicase domain 2 movement atx2=x0−1, or to the forward diffusion rate (rb

+) ofthe helicase bound with ATP. The backward rateapplies to the backward diffusion rate (rb

−) as thehelicase domain 2 moves onto the duplex end atx2=x0. To specify how much the forward andbackward rates are individually affected, we useda factor 0bξb1, such that k+=k+

0e−ξ(1−λ)ΔG0 andk−=k−

0e(1− ξ)(1−λ)ΔG0; this ensures the detailed bal-ance condition in Eq. (2).Notice that in the passive unwinding case sug-

gested in Ref. 46, the helicase waits for spontaneousbase pair opening to move forward without affectingthe equilibrium. That is, when λ=1, the base-pair-unwinding potential vanishes, and the forward andbackward rates of the helicase are not affected via thepotential. Indeed, we consider passive unwinding asan extreme case when λ=1 (minimal active unwind-ing). Although the open–closed equilibrium of theend base pair is not affected when λ=1, the modelallows an extra probability of base pair opening if thehelicase has an additional energy λ ·ΔG0+δA thatimmediately activates a ‘forced’ closed→open tran-sition (see Fig. 2b). Note that δA ranges about 0–10 kBT (see SI on parameter tuning and Table S1). Theextra probability is detectable if δA→0. WhileδA→10 kBT, the extra probability approaches zeroand the minimal active unwinding converges to thepassive unwinding.

Junction-stabilizing potential. The junction-stabiliz-ing potential comes from attractive interactionsbetween the front edge of the helicase and the duplex(see Fig. 2b). The depth of this attractive potential is


denoted U0N0. The attraction likely arises fromhelicase front edge interacting with the backbone ofthe duplex, similar to that observed in PcrA (see SIand Fig. S1). Interestingly, recent experimentalstudies indicate that bases immediately adjacent tothe junction are less stacked than ssDNA baseselsewhere.47 Hence, it is also possible that the partialunstacking of the bases at the junction contributes tothe preferential helicase–duplex association.As shown in Fig. 2b, the potential reaches a high

plateau of U0 when x2 moves to x0−4 or when therear end (on domain 1) of the helicase in the apostate is 10 bp away from the duplex end (x1=x0−10)on the ss. As mentioned above, it has been measuredthat the NS3 helicase binds to the duplex DNA witha 10-nt ss tail as fast as it binds to a 10-nt ssDNA,34

while the binding rate to the duplex with an ss tail ofless than 10 nt (≥7 nt) is larger than that to the sssubstrate of the same tail length. That is, when therear end of the NS3 helicase is within 10 nt from theduplex end, the helicase can sense the duplexthrough its front-edge association with the duplex;when the rear end is at or beyond 10 nt, the duplexhas no effect on the helicase.Furthermore, the NS3 helicase has a higher

affinity for the duplex with an ss tail than to thepure ss substrate of the same ss length (7–10 nt).34

Thus, the magnitude ofU0 can be estimated from themeasured affinities (see SI). Note that U0 variesbetween the apo and the ATP-bound state as thehelicase–duplex interactions switch between the twostates (cf. structures of PcrA–DNA complexes in Fig.S1). We denote the junction stabilizing strengthassociated with these two states by Uo

a and Uob

accordingly.Another characteristic of this stabilizing potential

is that it deepens along the translocation coordinateas the helicase approaches the junction. The poten-tial energy along the strand is minimum when thefront edge of the helicase extends into the duplexbackbone region for 3 nt. At this energy minimumconfiguration (in the apo state), x2=x0−1, andcorrespondingly, the rear part of the helicasereaches 7 bp distance from the duplex end atx2=x0−7. As implied in the experiments,34 ashorter distance than 7 nt between the helicaserear end and the duplex results in base pairunwinding. Overall, the junction-stabilizing poten-tial is made consistent with the binding configu-ration of the NS3 helicase at the junction (see alsoFig. S2). In general, the backward and forwardrates of the helicase would also be affected by thispotential. By specifying a factor 0bζb1 similar to ξfor the base-pair-unwinding potential, the back-ward rates are decreased by e−ζU0/3, while theforward rates are increased by e(1− ζ)U0/3. In ourmodel, only the backward diffusion rate in theATP-bound state can be reduced somewhat by thispotential (see details in SI).

Results on translocation

Translocation velocity

Using the two-state representation and transloca-tion scheme in Fig. 2a, one can derive via a chemicalmaster equation the average translocation velocityin the steady state (see SI). By assuming that (i) theforward transition a→b is rate limited by ATPbinding (ωab

I =kET · [ATP]) while the forward transi-tion b→a is rate limited by domain 2movement thatcouples with hydrolysis product release (ωba

II =kcat),(ii) the reverse backward transitions are slowenough (ωba

I ≪ωabI , ωab

II ≪ωbaII ), and (iii) the diffu-

sion rate in the apo state is negligible (ra→0), whilethe diffusion rate in the ATP-bound state issignificant (rbN0), we obtain the following expres-sion for the average translocation velocity:

mtrans = l0kcat � ATP½ �kcatkET

+ ATP½ � ð3Þ

Here l0=1 bp is the periodicity of the helicasemovement. Equation (3) has a form of a Michaelis–Menten-like dependence on ATP concentration andis independent of the diffusion rate. We usedkcat∼200 s−1 for ssRNA translocation estimatedfrom single-molecule experiments41 and an ATPbinding rate constant kET∼1 μM−1 s−1 (see SI onparameter tuning and Table S1). In Fig. 3a, we plotthe average translocation velocity versus [ATP], withνmaxtrans = l0kcat ∼200 bp/s and Km = kcat

kETf200lM. We

will see later that the Michaelis–Menten form of thevelocity is maintained during NS3 unwinding,while both νmax

trans and Km become smaller andsequence dependent. Notice also that the ratekcat∼ 200 s−1 used above can be regarded as anupper bound for the NS3 translocation.41 It is muchlarger than that measured recently from bulkexperiments,20–22 in which different protein statesare likely mixed.We can define a measure of the ‘stepping

efficiency’, ef, as the number of nucleotides translo-cated by the helicase in one ATP cycle.48 Based onstructural studies, we have assumed that the heli-case domain steps 1 nt at a time.29 With the use ofthe average velocity expression Eq. (3), the averagestepping efficiency of the helicase translocation,according to Eq. (4), is always 1 nt/ATP:

efumtrans � tnATP

= mtrans � s = mtrans

� 1kET � ATP½ � +

1kcat

� �ð4Þ

where nATP is the number of ATPs consumed in timet and τ is the average time for one ATP cycle. Notethat ef is indeed the inverse of the ATP-couplingstoichiometry (ATP/nt) commonly measured. As a


matter of fact, if a higher-order ‘sequence effect’ ofthe ss translocation is considered22 (as mentioned inthe Introduction), the average stepping efficiencycan drop slightly lower than 1 nt/ATP. Later on, wewill show that during duplex unwinding, theaverage efficiency ef drops well below 1 bp/ATPdue to a ‘wall’ effect: the duplex end base pair, whenclosed, transiently blocks the helicase translocationand leads to futile ATP cycles. A similar effectapplies also to sequence-dependent translocation:when the next sequence barrier is higher than thecurrent one, ef drops below 1 nt/ATP due to futileATP cycles; otherwise, ef remains at 1 nt/ATP.Overall, the stepping efficiency would be slightlylower than 1 nt/ATP if higher-order sequenceeffects are detectable during helicase translocation.This measure of the stepping efficiency can be testedexperimentally.Currently, direct measurements of NS3 transloca-

tion at the single-molecule level have not beensystematically conducted. The results providedhere—such as the average velocity and efficiencypresented above, as well as the effective diffusionand processivity properties discussed below—aretestable, qualitatively and semiquantitatively, infuture experiments. A similar form of translocationvelocity had been derived in Ref. 40 based on the‘Brownian motor’ (or ‘flashing ratchet’) scheme,34 inwhich the helicase does not step when a→b, butjumps 1 nt forward when b→a. In our current

scheme, however, the helicase as a whole effectivelymoves 1/2 nt at a time because domains 1 and 2move forward 1 nt, respectively, for a→b and b→a.The two schemes give similar kinetic results, but thecurrent scheme has a clearer structural and mech-anistic basis.29,30

Translocational fluctuations

In SI, we derive effective diffusion constantDeff

49,50 for NS3 ss translocation:

Deffuhx2i − hxi2

2t=

kcat kcatkET

� �2+ ATP½ �2

� �ATP½ �

2 kcatkET

+ ATP½ �� 3

+rb ATP½ �

kcatkET

+ ATP½ � ð5Þ

where the average is taken over an ensemble oftrajectories. Deff characterizes the randomness aris-ing from both chemical kinetics (the first term inthe above equation) and spatial diffusion (thesecond term) and is proportional to velocity fluc-tuations measured over a certain time interval(r2mu

var xð Þt2 = 2Deff

t , where σν is the standard deviationof the velocities). The results show thatDeff increasesas [ATP] or any of the parameters, kET, kcat, or rb,increases (see Fig. S3a). Hence, the width of the

Fig. 3. Translocation properties ofNS3 computed from the model. (a)The average translocational velocityandprocessivity length versus [ATP].For the velocity, νmax is 200 bp/s (anupper bound value) and Km is about200 μM (when kET ∼1 μM−1 s−1 ) inthis case. The processivity length(PL) is close to its maximum(∼120 bp) over a large range of[ATP]. (b) Distributions of ss trans-location velocity and stepping effi-ciency. The velocity distribution istaken over an ensemble of trajecto-ries for 1 s each at [ATP]=1 mM,generated from numerical simula-tion. The average velocity is∼165 nt/s, while the standard devi-ation is ∼12 nt/s. The width of thedistribution, that is, the standarddeviation, can also be derived fromthe effective diffusion constant (seethe text). The measure of steppingefficiency, defined as steps (nucleo-tides) translocated per ATP cycle,averages at 1 nt/ATP during the sstranslocation. A larger diffusion raterb would broaden the distributions(shown in Fig. S3b).


velocity distribution also increases with [ATP], kET,kcat, and rb. Notice that the spatial diffusion (rb) ofthe helicase is not evident in the average velocity[Eq. (3)] but can be revealed from measuring Deff orthe velocity fluctuations (see Fig. 3b). In our model,we used rb=10 s

−1 , which was estimated to an orderof magnitude by examining the velocity fluctuationsduring unwinding in comparison with experimentaldata18,41 (see next section and SI). A more straight-forward way of measuring rb experimentally is tomeasure Deff or the velocity fluctuations duringhelicase translocation.Additionally, we conducted kinetic Monte Carlo

simulations51,52 under the above translocationscheme and calibrated the simulations with analyticderivations [see SI and Eqs. (S1) to (S3)]. Withrb=10 s−1 , the distributions of the translocationvelocity (measured for 1000 trajectories of 1 s each at[ATP]=1 mM) and the stepping efficiency (nucleo-tides per ATP cycle) are shown in Fig. 3b. A larger rb(100 s−1 ) leading to broader distributions of thesequantities are shown in Fig. S3b.

Translocation processivity

We also examined processivity of NS3 duringtranslocation by taking into account dissociationevents of the helicase as it translocates. We definekoffa and koff

b as two dissociation (off) rates of the NS3during translocation, in the apo and ATP-boundstate, respectively. Since the affinity of the NS3 to thess is much weaker in the ATP-bound state than inthe apo state, we set koff

a bkoffb . In this way, the

processivity length (PL), that is, the average distancethe helicase has traveled along the ss beforedissociation, can be written as (see SI):

PLtransumtrans

koff=

kcatkboff

� ATP½ �kcatkET

� kaoffkboff

+ ATP½ �ð6Þ

where koff is an average (between states a and b)dissociation rate during translocation. PLtrans versus[ATP] also has a ‘Michaelis–Menten’ form, with amaximum value koff

b . The dissociation rate koffb is

estimated ∼1.7 s−1 using PLtrans≤120 bp (unpub-lished data). The value of koff

a is unknown but can beobtained by measuring PLtrans at a low ATPconcentration. Using koff

a =0.1 s−1 in Fig. 3a, itshows that PLtrans reaches its half-maximum at arelatively low Km

PL (∼ 12 μM), while above[ATP] ∼200 μM, PLtrans varies little with [ATP].

Results on unwinding

Michaelis–Menten form of the unwinding velocity

We simulated the unwinding kinetics of thehelicase when it translocates or binds to the duplex

region (Fig. 2b and Fig. S2), taking into account thebase-pair-unwinding potential and junction-stabi-lizing potential to modulate the forward andbackward rates. Under our simulation scheme, wederived approximate solutions that describe well thesimulated unwinding behaviors (see details in SI).Our results show that the average unwindingvelocity can be approximated as:

muwd = efl0kcate−j�DG0 ATP½ �kcate − j�DG0

kET+ ATP½ �

ð7Þ

Here, ɛf is an average efficiency factor (0bɛfb1) thatwe will discuss below, l0=1 bp is the periodicity,and kET and kcat are the ATP binding and productcatalysis/release (in coupling with the front domainmovement) rate of the helicase. ΔG0 is the base pairstabilizing free energy (the free-energy differencebetween the base pair open and closed states), andκ≡ξ(1−λ), a constant depending on how active thehelicase is (0≤λ≤1) and how large the base-pair-unwinding potential reduces the forward rate(0bξb1).Except for a lowered average efficiency (ɛfb1), the

above unwinding velocity has a Michaelis–Mentenform similar to that of the translocation, in which onlythe effective ‘catalytic’ rate is reduced (kcate

−κ·ΔG0).This is because at the dominant configuration of theNS3 unwinding (x2=x0−1 and x1=x0−6 or 7 in the bor a state, respectively; see SI), the rate kcat thatdetermines how fast the front domainmoves forward,in coupling with hydrolysis products release, isreduced by the base-pair-unwinding potential(according to k+=k+

0e−ξ(1−λ)ΔG0). From the aboveunwinding velocity, one obtains a sequence-depen-dent Michaelis constant that is smaller than that beingmeasured in ss translocation: Kuwd

m = kcatkET

e−jDG0 ; themore stable the sequence (i.e., the larger the ΔG0), thesmaller the Km

uwd. In order to fit with experimentaldata, we set n = 1

3 logkcat

100kET

� �= 1 − kð Þ for the NS3

helicase as λ≠1, that is, jun 1 − kð Þ = 13 log

kcat100kET

� �. In

this way, we can getKmuwd∼100 μM forΔG0∼3 kBT, as

experimentally measured for NS3 unwinding on amixed (52% GC) RNA sequence.18

Notice that in a situation when ATP bindinggenerates the unwinding force rather than the ATPhydrolysis/product release step, kET can be reducedto kETe

−κ·ΔG0 upon the base-pair-unwinding poten-tial while kcat remains constant. In that situation,Kuwdm = kcat

kETejDG0should be larger than that being

measured under ss translocation [Km = kcatkET

f200lMfrom Eq. (3)]. Moreover, the more stable thesequence to be unwound (the larger the ΔG0), thelarger the Km

uwd. Hence, by comparing experimen-tally the Michaelis constant measured duringunwinding with that in ss translocation, and byexamining the sequence dependence of the Km

uwd, it


would be possible to test from future experiments ifthe unwinding force is generated after hydrolysisproduct release or during ATP binding.In Fig. 4a, we show the unwinding velocities of the

NS3 helicase versus [ATP] calculated for varioussequences using Eq. (7). The RNA sequences 100%AU, 100% GC, and mixed (52% GC) are taken from

experiments,18,41 with ΔG0AU = 1.5 ± 0.5 kBT,

ΔG0GC=4.0±1.0 kBT, andΔG0

mixed=3.0±1.0 kBT53 for

each sequence on average, at close to experimentalionic conditions (see SI). Note that the ΔG0 we usedcounts not only hydrogen-bonding interactionswithinthe base pair but also stacking interactions betweenneighboring base pairs.54,55 In our calculations, the

Fig. 4. The unwinding velocity and efficiency of NS3 computed from the model. (a) The velocity versus [ATP] of theNS3 as it unwinds different RNA sequences. The 100% GC sequence, the mixed (52% GC), and the 100% AU sequencewere adopted from the single-molecule experiments.18,41 The average values of the velocities were calculated from Eq. (7).The experimentally measured velocities of unwinding the mixed sequence along with the measured error bars (standarddeviations) are shown in red. The corresponding error bars calculated from our simulations are shown in light blue. Notethat velocity distributions, simulated on unwinding the mixed sequence, are shown in (d). The unwinding velocities areall significantly lower than the translocation velocities and show a Michaelis–Menten dependence on [ATP], withsequence-dependent νmax and Km. (b) The sequence dependence of the unwinding velocity at various helicase ‘activeness’(with rb =0 and [ATP]=1 mM). The sequence stability is measured by ΔG0, the base pair stabilizing free energy. Theactiveness of the helicase unwinding is characterized by λ(0≤λ≤1); the smaller the λ, the more active the helicase and thelarger the unwinding velocities. A passive unwinding is a special situation of λ=1 (see the text). (c) The sequencedependence of unwinding velocity in the absence and presence of translocation diffusion (rb=0 and rb=10 s

−1 , both withλ=0.2 at [ATP]=1 mM). The data points were taken from numerical simulations while the curves are drawn from theapproximate solution [Eq. (S10) in SI]. Experimental data for the AU (ΔG0∼1.5 kBT), the mixed (ΔG0∼3 kBT), and the GC(ΔG0∼4 kBT) sequences are shown in red.41 The inset shows the sequence dependence of the average unwindingefficiency (ɛf) with rb=0 (dark blue) and rb=10 s−1 (light blue). (d) The distributions of the unwinding velocity andunwinding efficiency (of the mixed sequence) from numerical simulations. The velocity distributions were taken at[ATP]=0.05, 0.1, and 1 mM in order to compare with experimental data.18 The average and standard deviation ofthe distribution, from both the simulation and experimental data, are shown in (a). The distributions show somedeviations from the Gaussian fit (curves). The standard deviation of each distribution has been shown in (a) atcorresponding [ATP] (light blue error bar). The unwinding efficiency is defined as the number of base pairs unwound perATP cycle. The histogram is collected from a long trajectory (t=100 s) with λ=0.2, rb=10 s−1 at [ATP]=1 mM. The samevalues of λ and rb are utilized also in (a); unless specified, these values are used by default through this work.


velocities are the same whether we used the aboveaverage ΔG0 for the sequence according to Eq. (7) orwe adopted variable ΔG0 for each base pair along thesequence and simulated the unwinding numerically.The results in Fig. 4a show that the larger the sequencebarrier (ΔG0), the lower the velocity curve and thelower the Km

uwd, which is ∼140 μM, 100 μM, and40 μM for the sequence AU, mixed, and GC,respectively. In particular, the experimental data(red)18 for unwinding the mixed sequence areshown, with our fitted results. The main parametervalues we used in the calculations are specified later(see all parameter values in Table S1 in SI).

Sequence dependence and active unwinding

In Eq. (7), the sequence dependence of theunwinding velocity resides not only in the reducedforward rate kcat but also in the average efficiencyfactor ɛf, which can be regarded as the probabilitythat the base pair opens when the helicase frontdomain reaches to the duplex end (x2=x0). In asimple case when rb=0, this efficiency factor can bewritten as:

ef = Popen + 1 − Popen� � Pswitch ð8Þ

where Popen = e − k�DG01 + e − k�DG0 accounts for the base pair

opening probability under the helicase action, withthe base pair stabilizing free energy reduced toλΔG0. Pswitch = e − kDG0 − dA

1þe − kDG0 − dAaccounts for an additional

probability of opening the base pair, by switchingthe base pair from the closed state to the open stateas the helicase uses an extra energy to immediatelycross the activation barrier λΔG0+δA (see Fig. 2b).Note that when δA is large, Pswitch is close to zero.Hence, when δA→∝ and λ=1, the unwindingbecomes passive; that is, the helicase proceeds onlywhen the base pair spontaneously opens,46 leadingto muwd

passive =e − DG0

1þe − DG0mtrans.

The sequence dependence of the unwindingvelocity (with rb=0 and δA=1 kBT at [ATP]=1 mM)is shown in Fig. 4b.One can see that the smaller theλ,that is, the more active the helicase, the larger theunwinding velocity, and the slower the velocityapproaches to zero as ΔG0 increases (i.e., smallersequence dependence). On the other hand, for ahelicase not active enough (λ→1), its velocity canapproach zero when ΔG0 is fairly small. In that case,the helicase would stall at the DNA/RNA junctionwithout being able to proceed further. This mayexplain why some helicases, like monomeric Rep,32

cannot unwind duplex DNA though it can translo-cate well along ssDNA. Notice that when ΔG0→0(base pair opening and closing are equally alike),with the activation barrier δA=1 kBT relatively small,the unwinding velocities for all active cases convergeto ∼111 bp/s (with kcat=200 s−1 and other default

parameter values in this model, see Table S1); whilefor the passive case, the average velocity is about88 bp/s atΔG0→0, which is half of the translocationvelocity at this ATP concentration, detectably lowerthan any active case.When rb≠0, the diffusional flux decreases ɛf. Since

the forward diffusion rate rb+ is reduced significantly

uponunwinding,while the backwarddiffusion rb− is

affected less, a negative velocity flux results [see theapproximate solution Eq. (S10) in SI]. Hence, ɛfdepends not only on the sequence stability (ΔG0) andhow active the helicase is (λ) but also on thediffusional property of the helicase along the nucleicacid track (rb). Figure 4c shows the unwindingvelocity and the efficiency factor versus ΔG0 for bothrb=0 and rb=10 s−1 , with δA=1 kBT and λ=0.2 at[ATP]=1mM.One can see that the larger theΔG0, rb,or λ, the smaller the velocity and the efficiency.When rb is relatively large (rb=10 s

−1 ), the sequencedependence of the unwinding efficiency (in basepair per ATP for ɛf times l0) is more significant thanwhen rb= 0 (inset in Fig. 4c). The sequencedependence of the unwinding efficiency can betested experimentally.

Unwinding characteristics and fluctuation properties

In Fig. 4c, we also show the experimental data(red)41 on NS3 unwinding of AU, mixed, and GCsequence at [ATP]=1 mM, which can be fitted wellwith parameters δA=1 kBT, λ=0.2, and rb=10 s−1

(see Table S1). The activation barrier δA for the RNAend base pair open⇒close transition is not knownexactly, but its upper bound value is about 10 kBT.

56

We tested a range of δA (from 0 to 10 kBT) and foundthat λ can always be tuned to 0.1–0.3 by fitting theunwinding velocity at a specific value of ΔG0.Further, by varying ATP binding rate constant kETwithin its range (0.1bkETb2 μM−1 s−1 , see SI andTable S1), λ varies between 0 and 0.5. In principle, λdoes not have to be the same for different sequences,but our results show that tuning λ for the AU,mixed, and GC individually led to similar oridentical λ. In addition, by considering the noise/fluctuations (second moment) of the unwindingvelocities besides the average values measured fromthe single-molecule experiments,18,41 we found thatrb=10 s−1 recovers about 50% to 90% of themeasured noise, while the upper bound of rb islower than 50 s−1 . Details on choosing appropriateparameters for the current model are provided in SI,Table S1, Table S2, and related text. These resultsindicate that NS3 is a very active helicase (λb0.5); onthe other hand, the diffusion/fluctuation characterof NS3 is significant (rb∼10 s−1 ) and adverselyaffects its unwinding efficiency.In Fig. 4d, we show distributions of unwinding

velocities of the NS3 helicase on the mixed RNAsequence, obtained from simulating an ensemble


(N=1000) of trajectories at [ATP]=1 mM (gray),0.1 mM (red), and 0.05 mM (blue). These distribu-tions are similar to the experimental measurementsobtained in Fig. 2c of Ref. 18. The standarddeviations of the velocity distributions calculatedat different [ATP] are shown in Fig. 4a (light blueerror bars), which are about 50–90% the experimen-tal values (red error bars), leaving part of experi-mentally measured fluctuations to other sources ofnoise. To be consistent with experimentalmeasurements,18 we sampled the velocities andfluctuations at every 11 bp (pauses were detected inexperiments18 for about every 11 bp) on thesimulated trajectories. Thus, for each sampling, arelatively small number of ATP cycles were includ-ed. This short-time velocity sampling, together withthe diffusive character of the NS3 helicase(rb∼10 s−1 ), led to relative broad distributions ofvelocities. Similar to the translocation case in Fig. 3b,the width of the distribution increases with [ATP]because a higher concentration of ATP leads to ahigher probability of the helicase being in the ATP-bound state. This introduces a larger stochasticity(i.e., more diffusive) into the model [see also Eq. (5)].The distributions also show detectable deviationsfrom a Gaussian distribution (the fitting curve in Fig.4d), which would otherwise be a good approxima-tion if a large number of ATP cycles (long time) weresampled.In the inset of Fig. 4d, we also show the distribution

of the ‘unwinding efficiency’, that is, the number ofbase pairs unwound per ATP cycle for a longtrajectory unwinding a mixed RNA sequence(t=100 s without considering pauses or helicasedissociation). This is comparable to the ‘translocationefficiency’, that is, the translocation steps per ATPcycle shown in Fig. 3b. Notice that the efficiency isindependent of [ATP] [cf. Eqs. (3), (4) and (7)].Importantly, compared with translocation, helicaseunwinding has a significant number of zero (futilecycles) or even negative steps (under small but non-zero backward rates ωba

I and ωabII , and under

backward diffusion when rbN0) accumulated as netmovements for an ATP cycle, leading to an averagestepping (unwinding) efficiency ef lower than 1 bp/ATP (the average efficiency is proportional to theefficiency factor ef: ef≡ɛf · l0). That is to say, theaverage stepping efficiency changes from 1 nt/ATPin the ss translocation to a sequence-dependent valuesmaller than 1 bp/ATP during unwinding (see ɛf inFig. 4c). As mentioned before, this is mainly due tothe presence of the duplex that interrupts thetranslocation and leads to occasional futile ATPcycles. The average efficiency also decreases duringunwinding when the spatial diffusion increases(rbN0). During ss translocation, the diffusion doesnot affect the average efficiency but only expands thedistribution of the efficiency measure, that is, stepsper cycle (see Fig. 3b and Fig. S3b).

Sequence-dependent dissociation

NS3 is not a highly processive helicase2 due tofrequent dissociations from the nucleic acid track.Modeling studies had suggested that active un-winding of the helicase leads to sequence-dependentdissociation.39 Single-molecule measurements havealso identified this behavior.41 Below, we examinemore closely the NS3 dissociation in the two-statemodel and in the presence of helicase diffusion.As discussed above, the dissociation rates of the

helicase in the ATP-bound and the apo states duringtranslocation are koff

b (∼1.7 s−1 ) and koffa (≪koff

b ),respectively. At the DNA/RNA junction, NS3 isstabilized and the dissociation rate decreases corre-spondingly to kaff

a,be−Ua,b0

due to the attractive potentialUa,b

0 . During unwinding, the dissociation rate in theindividual state would be further modulated to koff

a ,b

e−Ua,b0 + (1−λ)ΔG 0, as the helicase front domain reaches

the duplex end (x2=x0) where the base-pair-unwind-ing potential (1−λ)ΔG0 peaks (see Fig. 2b).In Fig. 5a, we show simulation data on the average

dissociation rate koffuwd of the NS3 helicase as it

unwinds the AU, the mixed, and the GC sequence atvarious [ATP]. From the diagram, one can see that ahigher sequence stability leads to a larger koff

uwd, dueto the higher sequence barrier (1−λ)ΔG0. For the GCsequence at the high [ATP] (1 mM), we fit theexperimental data koff

uwd = 0.5 s−1. For the AUsequence at the same condition, we obtained koff

uwd

about half of that for GC, which agrees very wellwith experimental results.41 For each sequence, thelower the [ATP], the lower the koff

uwd. This effectoccurs because at low [ATP], the helicase spendsmost its time in the apo state, in which the helicasehas a high affinity to the substrate than in the ATP-bound state (see below), and dissociates slower fromthe nucleic acid than in the ATP-bound state.Note that the junction stabilization strength in the

apo state U0a ∼4.4 kBT was estimated from biochem-

ical data34 (see SI). By fitting koffuwd=0.5 s−1 on the GC

sequence from single-molecule measurements,41 weobtained U0

b ∼1.8 kBT in the ATP-bound state (U0b

ranges 1.3–2 kBTwhen parameter kET varies). Hence,both the larger junction stabilization (U0

aNU0b) and

the higher ss affinity (smaller ss dissociation ratekoffa bkoff

b ) in the apo state contribute to the lowerdissociation rate in this state and, consequently, tothe lower average koff

uwd at low [ATP].The curves of dissociation rate in Fig. 5a also

demonstrate Michaelis–Menten-like forms, with the‘Michaelis constant’ fit to 78 μM, 70 μM, and 58 μMfor the AU, mixed, and GC case, respectively. Thatis, the Michaelis constant for the dissociation ratealso decreases with increasing sequence stability,similarly as that for the unwinding velocity. There-fore, if one calculates the processivity length PLu v

koff,

one obtains PL relatively insensitive to [ATP] over alarge range (see Fig. S4b). Further, from force effects


on the velocity and the dissociation rate (see Fig.S4a), the processivity is expected to rise significantlywith the unwinding force; this is consistent withexperimental results.18,57 Nevertheless, NS3 spendsa significant amount of time pausing duringunwinding,18 which has not been taken intoaccount here. Thus, the experimentally measuredprocessivity length should be much smaller than thevalue calculated above.In Fig. 5b, we plot the sequence dependence of

koffuwd at three different values of rb. At rb=10 s

−1 , wesee that koff

uwd consistently rises asΔG0 increases over

a large range of values. However, over ΔG0=7–10 kBT, koff

uwd changes little and then drops; it drops abit further at ΔG0=10–13 kBT and increases again(data not shown). The non-monotonic behavior ofkoffuwd versus ΔG0 is due to competition between twoeffects: (i) the sequence barrier ΔG0 increases thedissociation rate by e(1 −λ)ΔG0, which happensoccasionally at x2=x0; (ii) ΔG0 also decreases theforward rate (kcat or rb) of the helicase by e−κΔG0 atx2=x0−1, reducing the chances that the helicasereaches the sequence barrier at x2=x0. Combiningthese two effects, ΔG0 increases the dissociation rateto some extent and then inhibits the rate fromgrowing further. This property can be testedexperimentally using sufficiently stable sequences.In addition, the sequence dependence of the

dissociation rate is also affected by the diffusivecharacter of the helicase. When rb=0, there is only aslight increase in koff

uwd within ΔG0=0–10 kBT (koffuwd

can become bigger for very large ΔG0, data notshown); when rb=30 s−1 , the change of koff

uwd overthe same range of ΔG0 is much bigger than that forsmaller rb. The reason is that, in the absence of thediffusion, the only events through which the heli-case reaches the sequence barrier are forwardmovements of domain 2 upon the hydrolysisproducts release (in kcat); the presence of thediffusion (in rb) brings more opportunities for thehelicase to reach the sequence barrier. Thus, thelarger the rb, the larger the sequence effect on koff

uwd.Overall, the average dissociation rate of the

helicase is affected by several factors during theactive unwinding: the probability the helicase beingbound with ATP, the sequence stability, and thehelicase diffusion rate. The different affinities of thehelicase for the ss and ds forms of the nucleic acid,with and without ATP, make the dissociation ratediffer between the two states. The dominantsequence effect comes from the base-pair-unwind-ing potential, that is, the sequence barrier, whichmay also destabilize the helicase at the junction,fostering dissociation. On the other hand, thesequence barrier resists the helicase approachingthe duplex end, additionally affecting the dissocia-tion. Lastly, the diffusive character of the helicasecan enhance the sequence effect on helicase disso-ciation by increasing the frequency of the helicaseapproaching to the sequence barrier.

Discussion

We have modeled the NS3 helicase activity basedon structural14,29 and biochemical properties.33,34 Asingle catalytic site binds ATP between the twotranslocation domains of the helicase and modulatesboth interdomain associations and individualdomain affinities to the nucleic acid substrate. Inparticular, the overall affinity of the NS3 to the ss

Fig. 5. The dissociation rate of NS3 during activeunwinding computed from the model. (a) The dissocia-tion rate versus [ATP] for different sequences. The datapoints were taken from numerical simulations of anensemble of trajectories (N=1000), and the curves are fitto the data points. The dissociation rates show aMichaelis–Menten-like dependence on [ATP] withsequence-dependent characteristics. (b) The sequencedependence of the dissociation rate at different diffusionrates: rb=0, rb=10 s−1, and rb=30 s−1 . In the absence ofthe translocation diffusion (rb=0), the sequence depen-dence is hardly discernable. In the presence of thediffusion (rb=10 s−1 and rb=30 s−1 ), the dissociationrate increases with the sequence stability (ΔG0) over alarge range; the larger the diffusion rate, the larger thesequence dependence.


nucleic acid is significantly reduced in the ATP-bound sate,33,34 allowing the helicase to diffusealong the ss. When the helicase reaches the RNA/DNA duplex, the forward movement of the helicasefront domain induces base pair unwinding. Byfitting data on RNA unwinding velocities measuredfrom single-molecule experiments,18,41 our studiessuggest that the NS3 helicase actively shifts the basepair open↔close equilibrium away from the closedstate toward the open state. It does this by reducingthe free energy stabilizing the base pairing/stackingat the end of the duplex to less than half of itsoriginal value.During active unwinding, the NS3 helicase

interacts with the RNA/DNA duplex, perturbingthe structure of the duplex as well. The interac-tions lead to two detectable effects. First, a base-pair-unwinding potential at the duplex endreduces the stepping velocity of the helicase byinhibiting forward movement of the front domainand reducing the average stepping efficiency incomparison with ss translocation. Both of thesereductions depend on the DNA/RNA sequenceencountered by the front domain. Second, thehelicase preferentially associates with the duplexand, thus, is stabilized at the junction. Thepreferential association may come from interac-tions of the duplex backbone with the helicase assuggested for PcrA25,30 and may also be contrib-uted by partial base unstacking at the junction.47

The junction stabilization reduces the dissociationrate of the helicase during unwinding but does notaffect the translocation velocity much. Neverthe-less, both the base pair-unwinding and thejunction-stabilization effects arise from the helicaseinteracting with the duplex. It is likely, therefore,that the strengths of the two effects are correlated;that is, the larger the junction stabilization duringthe ATP hydrolysis cycle, the more active thehelicase will be in base pair unwinding. Moredetailed studies are needed to clarify this issue.When external forces that assist unwinding are

applied to the DNA/RNA duplex, the base pairstabilizing free energy should drop. The modelshows how the applied force affects the unwindingvelocity and dissociation rate (see Fig. S4). However,the force effects were not detected for NS3 unwind-ing of RNA in single-molecule experiments.18,41 Thisforce insensitivity could have been explained duringtranslocation, as some chemical step other thanduplex unwinding is rate limiting in the overallmechanochemical cycle of the enzyme. This inter-pretation is, however, inconsistent with the slowingdown of the enzyme observed experimentally whenit encounters a sequence rich in GC content.41 Onepossible explanation is that the enzyme possessesadditional nucleic acid binding sites such that theforce does not act directly on the junction or fork ofthe duplex, but at a different RNA–protein contact.

This would prevent the externally applied forcefromweakening the base pair at the duplex junction.An important characteristic of the NS3 helicase

in this model is that translocational diffusion is notnegligible when the helicase is bound with ATP.During ss translocation, the diffusion affects thefluctuation properties of the helicase withoutimpacting the average translocation velocity orthe tight coupling between hydrolysis and step-ping that produces an average stepping efficiencyof 1 nt/ATP. During unwinding, however, thediffusive character does affect both the fluctuationproperties and the average velocity/efficiency ofthe helicase. Even in the absence of diffusion, theunwinding efficiency is lower than 1 bp/ATP dueto occasional futile ATP cycles upon the sequencebarrier. The diffusion further lowers the averageefficiency. The larger the diffusion rate, the morethe sequence barrier decreases the efficiency, andthis is achieved by increasing the frequency ofbackward jumps (−1 nt at a time) duringunwinding. In the model, significant backwardjumps are present when the diffusion rate is non-zero. Our numerical results suggest that the ratiobetween backward and forward jumps duringunwinding increases with sequence stability, forexample, from ∼10% in unwinding AU to ∼30%in unwinding GC (at 1 mM [ATP] with defaultparameters in Table S1). In future single-moleculeexperiments, the backward jumps can be bettercharacterized. Hence, the experiments are expectedto substantiate and quantify the diffusive characterof the helicase.The interplay between the diffusion and sequence

effects is also displayed in helicase dissociationduring active unwinding. The helicase's dissociationrate usually increases with increasing sequencestability/barrier. However, a very large sequencebarrier may prevent the helicase from reaching thebarrier so that the dissociation rate cannot growmonotonically with the strength of the barrier.According to our numerical results, the sequencedependence of the dissociation rate is detectableonly in the presence of diffusion; the larger thediffusion rate, the more pronounced the sequencedependence.To quantify the diffusive character of the NS3

helicase, we estimated the diffusion rate to an orderof 10 s−1 based on unwinding velocity fluctuations(standard deviation)measured from single-moleculeexperiments.18,41 The diffusion rate can also beestimated by measuring the average unwindingefficiency (or the ATP stoichiometry), that is, howmany base pairs unwound for each ATP consumed,or vice versa. More straightforwardly, one couldobtain the diffusion rate from measuring velocityfluctuation, or effective diffusion rate, during sstranslocation of the helicase in single-moleculeexperiments.


Coupling ss nucleic acid translocation with duplexunwinding is a generic feature of most helicases. Inour current unwinding model, the monomeric NS3helicase shares some common mechanochemicalfeatures with the hexameric ring-shaped T7helicase.36 For example, the force generation duringunwinding for both helicases is likely associatedwith hydrolysis product (Pi) release, not ATP/dTTPbinding.36,37 Hence, even though their structures arevery different, the sequence-dependent effect duringunwinding could be similar for these two helicases,as shown in previous experiments.35 It would beinteresting to establish whether the diffusive char-acter of the NS3 helicase is shared by other nucleicacids motors and to investigate how diffusion mayimpact their function.According to most recent structural studies,29 the

fundamental or physical step size of the NS3helicase should be 1 nt at a time. This propertyshould be measurable in future single-moleculeexperiments with sufficiently high resolution. Notethat this 1-nt physical step leads to an averagestepping efficiency less than 1 bp per ATP cycle inour model during unwinding and to a broadeneddistribution involving several base pairs forward orbackward per ATP cycle (i.e., the net or totalmovement of the helicase within one ATP cycle).Nevertheless, our current work cannot explain the 3-bp periodic steps detected in single-molecule fluo-rescence experiments for NS3 in DNA unwinding42

or the regular pauses displayed by NS3 about every11 bp during RNA unwinding in single-moleculeoptical tweezer experiments.18

The 11-bp step may correspond to the binding ofthe third domain of the protein to the duplex regionsome 10 or 11 bp away from the fork. As suggestedby single-molecule experiments,18,41 the third do-main may be used both as an anchor for the enzymeand as a way of locally destabilizing the duplexahead of the fork. This would facilitate inchwormingof the other two domains of the protein duringunwinding. Furthermore, additional binding sites ofNS3 to nucleic acids have been suggested,43,44 andthis could explain the force insensitivity of theunwinding rate observed in single-moleculeexperiments.18 In particular, positively chargedareas seem to extend from the exterior side ofdomains 3 and 2 to the protease part of NS3,43,44

providing attractive interactions for the DNA/RNAstrand. Hence, the associations between the helicaseand the nucleic acid may block an externally appliedforce from affecting the stability of the duplex at thefork, and it may also explain the pausing andstepping behaviors displayed by the helicase. It isalso possible that the helicase employs a ‘scrunch-ing’ mechanism as has been suggested for UvrDhelicase.58 This could also lead to complex pausingand stepping behaviors. Further studies will beneeded to clarity these issues.

Materials and Methods

In the SI, we show in detail (a) how we derive thehelicase binding configuration at the junction, (b) how weuse master equation approach to obtain translocationproperties of the helicase, (c) how we simulate helicaseunwinding dynamics and derive approximate solutions,and (d) how we tune parameters in this model.

Acknowledgements

J.Y. was supported by University of CaliforniaBerkeley Chancellor's postdoctoral fellowship. G.O.was supported by National Science FoundationGrant DMS 0414039.

Supplementary Data

Supplementary data associated with this articlecan be found, in the online version, at doi:10.1016/j.jmb.2010.09.047

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1

Supplementary Information I. PcrA-DNA complex with and without ATP bound (Figure S1) II. NS3 helicase binding at the junction and stabilization effect (Figure S2) - NS3 helicase docking size 10-nt - NS3 helicase stabilized with 7-nt on ss and 3-nt into duplex III. Master equation approach to the two-state translocation model (Eq. S1)

- Average translocation velocity (Eq. S2) - Effective diffusion and velocity fluctuation (Eq. S3) (Figure S3) - Dissociation and processivity (Eq. S4)

IV. The sequence-dependent duplex unwinding in the two-state model

- Simulation scheme for duplex unwinding - Approximate solutions of the unwinding (Eq. S5 – S11) - The unwinding force effect (Fig. S4)

V. Tuning parameters of the translocation and unwinding model (Table S1) - The appropriate values of ATP binding constant kET and related parameters - Calculating bp stabilizing free energies for different RNA sequences - Choosing the parameters for bp unwinding (Table S2) - Estimating parameters related to dissociation and processivity

I. PcrA-DNA complex with and without ATP bound PcrA had been crystallized with a short piece of DNA duplex 1, while the structure of

NS3 in complex with duplex DNA/RNA has not been available. In Figure S1 we show

the structures of the PcrA-DNA complexes from molecular dynamics (MD) simulations 2,

with the PcrA bound with ATP (left) or in apo state (right), and with both ssDNA and

duplex part present. It can be seen that the duplex region is significantly distorted when

the PcrA is in the apo state without ATP. This property can also be true for NS3 when it

is in complex with the DNA/RNA duplex. In our model, the property is demonstrated as

that the NS3 helicase has a stronger association with the junction in the apo state than in

the ATP bound state (

!

U0

a>U

0

b ). See more discussions later.

2

Figure S1 PcrA-DNA complex with and without ATP bound, obtained from MD simulations 2 of respective structures 1. The PcrA is shown in wide ribbon representation, with domain 1A, 2A, 1B and 2B colored in green, red, yellow and blue. The DNA is shown in licorice representation and its backbone also shown in narrow ribbons. The ATP, bound in between domain 1A and 2A (on the left), are shown in small vdW spheres. The duplex region of the DNA is significantly distorted when the PcrA is not bound with ATP (on the right). II. NS3 helicase binding at the junction and stabilization effect A schematic model was developed on (i) how the NS3 helicase binds to the DNA/RNA

junction, with the duplex end linked with a piece of 3′ single strand (a ds-ss substrate),

and (ii) how the helicase is stabilized at the junction. The model was built upon

biochemical measurements of the NS3 helicase 3, and is illustrated in Figure S2.

NS3 helicase docking size 10-nt Experiments had revealed that when NS3 binds to a

ds-ss with the ss tail ranging from 7 to 10 nt, the helicase does not melt the bp in the

duplex; however, the helicase shows a higher affinity to the ds-ss than to a pure ssDNA

of the same ss length 3. The affinity is characterized by the dissociation constant of the

helicase to the nucleic acid substrate:

!

Kd = koff /kon . In particular, when the ss length is

10-nt, the binding rate

!

kon

is the same as the helicase binds either to the ss or to the ds-ss 3. Based on these measurements, we propose that the NS3 helicase has a 10-nt ‘docking

3

size’ (see upper panel of Figure S2) when it initially binds to the ds-ss with a long tail

(longer than 10-nt). This directly explains why

!

kon

is the same for both the ds-ss and the

ss of 10-nt. Further, tighter association of the helicase with the ds-ss (than the ss) must be

due to the duplex effect, such that the unbinding rate

!

koff of the helicase is smaller on the

ds-ss than that on the ss, leading to a lower

!

Kd (higher affinity). The duplex stabilization

effect of the ds-ss to the helicase can be quantified by

!

U0 = logK

d

ss10

Kd

ds"ss10, here

!

Kd

ss10 and

!

Kd

ds"ss10 are dissociation constants of the helicase with the 10-nt ss and with the ds-ss of a

10-nt ss tail, respectively.

Figure S2 A schematic representation of NS3 binding at DNA/RNA junction. The junction is composed of a duplex region and a single stranded 3’-tail (ds-ss substrate). First, the helicase docks onto the ss tail, with a projected size of 10-nt. Next, the helicase is stabilized at the junction, likely through favorable associations with the duplex backbone. In the stabilized configuration, the helicase covers 7-nt ss tail while nudges its front end into the duplex region for 3-nt. We represent the stabilization potential of strength U0 as the front end of the helicase moves along the 3’ strand.

4

NS3 helicase stabilized with 7-nt on ss and 3-nt into duplex The experimental

measurements 3 also showed that, for the helicase binding to the ss and to the ds-ss with a

ss tail of the same length (7 or 10 nt),

!

logK

d

ss7

Kd

ds"ss7~ log

Kd

ss10

Kd

ds"ss10# log80 ~ 4.4 ; that is to

say, the ds-ss stabilization effect is almost the same when the ss tail is 7-nt or 10-nt, and

the stabilization strength is U0~ 4.4 kBT (labeled

!

U0

a later, for the helicase in apo state).

More importantly, the experiments had indicated that when the ss tail is shorter than 7-nt,

the duplex part of the ds-ss can be partially unwound by the NS3 helicase just through

binding 3. Therefore, we inferred that in the stabilized configuration of the helicase at the

ds-ss junction (with the ss tail 7-nt or longer), the helicase covers 7-nt size on the ss part.

Since the overall docking size of the helicase ‘projected’ on the ss is 10-nt, as discussed

above, the left 3-nt would ‘nudge’ into the duplex region, likely, by binding to the

backbone of the duplex (see the lower panel of Figure S2). It is right because of the 3-nt

association between the helicase and the duplex, the helicase is stabilized at the junction,

‘trapped’ in the potential energy well

!

U0. In current model,

!

U0 can be modulated through

ATP hydrolysis cycles, hence we used

!

U0

a and

!

U0

b to differentiate its values in the two

states: apo (a) and ATP bound (b). Our numerical fitting showed that

!

U0

a>

!

U0

b , i.e., the

stabilization effect is stronger in the apo state than in the ATP bound state. This is

consistent with the observation on PcrA-DNA complexes (see Figure S1 and the

previous section) as the DNA duplex is more strongly associated with, thus distorted by

the helicase in the apo state than in the ATP bound state.

III. Master equation approach to the two-state translocation model Once we have built the translocation scheme in Figure 2a, we can write down the master

equation describing the change of probability

!

P( j, t) of the NS3 helicase at position x = j

(along the ss nucleic acid), at time t, where x is the average of x1 and x2 for the position of

domain 1 and 2:

5

dPa( j,t)

!t= r

a

+Pa( j "1,t)+ r

a

"Pa( j +1,t)+#

ba

I Pb( j +

1

2,t)+#

ba

II Pb( j "

1

2,t)

"ra

+Pa( j,t) " r

a

"Pa( j,t) "#

ab

I Pa( j,t) "#

ab

II Pa( j,t)

dPb( j +

1

2,t)

!t= r

b

+Pb( j "

1

2,t)+ r

b

"Pb( j +

3

2,t)+#

ab

I Pa( j,t)+#

ab

II Pb( j +1,t)

"rb

+Pb( j +

1

2,t) " r

b

"Pb( j +

1

2,t) "#

ba

I Pb( j +

1

2,t) "#

ba

II Pa( j +

1

2,t)

(S1)

At steady state (left hand side set to zero), one can solve the probability distribution over

the two states a and b. Summing over j in Eq. (S1), the system is translational

invariable:

Pa,b

j

! ( j,t) = Pa,b

j

! ( j +1,t) . One can then obtain the steady-state distribution

Pa! P

a

j

" ( j,t) and

Pb! P

b

j

" ( j +1

2,t) as:

!

Pa

="

ba

I+"

ba

II

"ab

I+"

ab

II+"

ba

I+"

ba

II and

!

Pb

="

ab

I+"

ab

II

"ab

I+"

ab

II+"

ba

I+"

ba

II. In condition that

!

"ba

I# 0 and

!

"ab

II# 0 in our model,

!

Pa"

#ba

II

#ab

I+#

ba

II and

!

Pb"

#ab

I

#ab

I+#

ba

II.

Average translocation velocity The average translocation velocity of the helicase can

be written as

vtrans=

dx

dt= l

0[ jP

a( j,t)

j

! + ( j +1

2)P

b( j +

1

2,t)] (with l0 the basic stepping

size 1-bp). At the steady state, it can be expressed:

vtrans

= l0[1

2(!

ab

IPa"!

ba

IPb+!

ba

IIPb"!

ab

IIPa)+ (r

a

+" r

a

")P

a+ (r

b

+" r

b

")P

b]

# l0

!ab

I !ba

II+!

ab

I(rb

+" r

b

")

!ab

I+!

ba

II

(S2)

The approximation in Eq. (S2) is taken under the condition

!

"ba

I# 0 ,

!

"ab

II# 0 , and

!

ra

±" 0 , which applies no matter in ss translocation or duplex unwinding. In the ss

6

translocation,

!

rb

+= r

b

"= r

b, it gives vtrans = l

0

!ab

I !ba

II

!ab

I+!

ba

II, and further, the Michaelis-Menten

form Eq. (3) in main text. The derivation above shows that the ‘diffusive character’

(

!

rb

> 0) does not affect the average translocation velocity.

Effective diffusion and velocity fluctuation Fluctuational properties of the helicase,

however, would be affected by

!

rb

> 0 . This can be demonstrated through an effective

diffusion coefficient

!

Deff "var(x)

2t 4; 5, which generally quantifies stochastic effects

arising from both chemical reaction and spatial diffusion. By solving x2 from its time

derivative

!

d x2

dt= l

0

2j2 dPa ( j,t)

dt+" ( j +1/2)

2 dPb ( j +1/2,t)

dt through the master equation

(S1), we get

!

Deff =x2" v

2t2

2t#

$ab

I $ba

II

4($ab

I+$ba

II)

+$ab

I $ba

II($ab

I"$ba

II)2

4($ab

I+$ba

II)3

chemical

1 2 4 4 4 4 4 4 3 4 4 4 4 4 4

+$ab

Irb

$ab

I+$ba

II

spatial

1 2 4 3 4

, (S3)

where v is the average translocation velocity obtained in Eq. (S2). Again, the

approximation in Eq. (S3) is taken under the condition

!

"ba

I# 0 ,

!

"ab

II# 0 ,

!

ra

±" 0 , and

!

rb

+= r

b

"= r

b for the ss translocation. By using

!

"ab

I= k

ET# [ATP] and

!

"ba

II= k

cat assumed in

our model, one can plot

!

Deff vs. [ATP] in Figure S3a, which shows that

!

Deff always

increases as the [ATP] increases. Increases of either kcat or kET also leads to increase of

!

Deff . Actually, by measuring the effective diffusion Deff

* at a specific ATP concentration:

[ATP]* = kcat / kET (so that !ab

I=!

ba

II ), one can estimate rb as rb = 2Deff

*! kcat / 4 , directly

from Eq. (S3).

If one estimates the velocity fluctuation in a certain constant time t, then the variance of

the translocation velocity can be written as! v

2"var(x)

t2

=2Deff

t. Hence, !

v

2 also increases

7

with [ATP] and

!

rb, due to its relation with

!

Deff . By measuring the velocity fluctuation

during the translocation, therefore, one can also estimate

!

rb. In Figure S3b, we show a

large value of

!

rb (100 s-1) gives a broad distribution of translocation velocity (

!

"v~ 18

nt/s) and stepping efficiency (nts per ATP cycle), in comparison with that shown in main

Figure 3b with

!

rb =10 s-1 (

!

"v~ 12 nt/s) while other conditions the same.

Figure S3 Effective diffusion and fluctuation properties during translocation. (a)

!

Deff vs. [ATP] (dark solid line), with a larger kET (2 µM-1 s-1; gray solid), a larger kcat (400 s-1; gray broken), or no diffusion (

!

rb=0; dark dotted). (b) The velocity distribution and stepping

efficiency distribution. The same conditions as in Figure 3b, except that

!

rb =100 s-1.

Dissociation and processivity Consider that the helicase can dissociate from the ss

nucleic acid during its translocation, and suppose that the dissociation rate is

!

koffa in the

apo state a,

!

koffb in the ATP bound state b (

k

off

a< k

off

b since the affinity is lower in the

ATP bound state), thus, the average dissociation rate

!

koff in the translocation is:

!

koff = Pakoffa

+ Pbkoffb

="ba

IIkoffa

+"ab

Ikoffb

"ab

I+"ba

II (S4)

Correspondingly, the average dissociation time is

!

toff "1/koff . The processivity, i.e., the

distance that the helicase travels on average before it dissociates from the ss track, is

8

PL ! vtrans

" toff = vtrans

/ koff =#ab

I #ba

II

#ba

IIkoffa+#ab

Ikoffb=

kcat

koffb[ATP]

kcat " koffa

kET " koffb+ [ATP]

, shown in main Eq. (6).

IV. The sequence-dependent duplex unwinding in the two-state model The unwinding scheme presented Figure 2b was built upon the two-state translocation

model under effective unwinding potentials. Based on this scheme, we numerically

simulated the duplex unwinding of the NS3 helicase. The simulation applies kinetic

Monte-Carlo or Gillespie’s algorithm 6; 7 for transitions, with the transition rates

modulated by the effective potentials at different configurations. Below, we show first the

detailed simulation scheme. Next, we derive approximate solutions for the unwinding

velocity under specific conditions.

Simulation scheme for duplex unwinding In our model, the configuration of the

helicase is determined by its front and rear domain positions, x2 and x1, relative to the

position of the DNA/RNA duplex, x0 (the first bp in front of the helicase). Under the bp-

unwinding and junction stabilization potentials (Figure 2b), the kinetic rates vary

according to rules below:

1) When x2 = x0 – 1, the helicase has its forward movement rates decreased exponentially

upon the bp-unwinding potential, by a factor e!" (1!# )$G0 , with ΔG0 the original bp

stabilization free energy at the x0, λ ∈ [0,1] the activity parameter (activeness) of the

helicase, ξ ∈ (0,1) the pre-factor balancing changes of the forward (

!

k+

= k0

+e"# (1"$)%G0 at x2

= x0 – 1) and backward (

!

k"

= k0

"e(1"# )(1"$)%G0 at x2 = x0) rates such that a detailed-balance

related relationship (

!

k+

k"

=k0

+

k0

"e"(1"#)$G0 ) is preserved 8.

2) When x2 = x0, the helicase takes a trial accessing the duplex end. There is a probability

of Popen =e!"#G

0

1+ e!"#G

0

that the bp at x0 opened under the helicase action λ, plus a probability

9

(1- Popen)⋅Pswtich that the bp at x0 is initially closed, but is forced to open immediately as

the helicase has an extra energy ΔE = λ ΔG0+ δA to jump the barrier (the activation

energy from bp closed to open, see Figure 2b; Pswitch

=e!"E

1+ e!"E

=e!#"G

0!$A

1+ e!#"G

0!$A

). Note that

when δA is large, this extra effect is negligible (Pswitch → 0). Anyhow, once the first bp at

x0 duplex is sensed ‘open’ in this trial, the duplex end ‘moves’ forward with x0 ⇒ x0+1.

Still, there is a left probability (1- Popen)(1- Pswtich) that the bp at x0 keeps closed in this

trial, then the helicase sets back to x2 = x2 -1 and x1 = x1 -1, keeps at the same original state

a/b. This way a futile ATP cycle is generated.

3) When x2 = x0-2, the helicase is not supposed to act on duplex unwinding. On the other

hand, the separated duplex end likely rewinds immediately at x0-1, with probability

Pn=

1

1+ e!"G

0

so that one can set x0 ⇒ x0 -1. There is a slight probability for x2 < x0-2, but

we neglected the probability so that the helicase front is always kept close enough to the

duplex end (x2 >= x0-2).

Indeed, both configurations 2) and 3) are transient (due to the fast equilibrium the bp

reaches) and would transform soon to configuration 1). Hence, the dominant

configuration in the helicase unwinding is x2 = x0 – 1. In this configuration, the reduced

forward rates include the catalytic rate kcat, which is coupled to domain 2 moving

forward, and the forward diffusion rate at the ATP bound state rb. The forward transition

involving only domain 1 moving forward (upon ATP binding) is not supposed to affect

the bp unwinding in current model.

In principle, the junction stabilizing potential U0 can similarly affect the forward and

backward rates according to, e.g., the relative position of x2 (or x1) to x0:

4) When x0-4 < x2 < x0-1 (x0-10 < x1 < x0-7 in state a or x0-9 < x1 < x0-6 in state b), the

backward rates are decreased by e!"U0 /3 , while the forward rates are increased by

e!("!1)U0 /3 ), with ζ ∈ (0,1) the similar pre-factor balancing between forward and backward

rates as ξ for the bp-unwinding potential. When x2 = x0-1, only backward rates are

affected, while when x2 = x0 - 4, which is not really allowed (see above), only forward

10

rates are affected. The dominant configuration of the helicase is at x2 = x0 – 1. In the apo

state, there is negligible backward movement of the helicase (both ra and !ab

II about zero).

Therefore, only the backward diffusion rate rb- in the ATP bound sate can be affected a

bit by the junction stabilization potential. Overall, the junction stabilization does not

affect much the translocation rates.

Approximate solutions of the unwinding Under the above scheme, the dominant

configuration of the helicase is x2 = x0 – 1 and x1 = x0 – 6 or 7 (in b or a state) in the

simulation. The simulations showed that the unwinding velocity is insensitive to the

junction stabilization potential U0. One can then write down an approximate solution of

the unwinding velocity, in a form of

!

vuwd

= " f# v , based on the translocation velocity

obtained in Eq. (S2). In case rb=0, the expression of

!

" v is:

!v = l0

"ab

I "ab

IIe#$ (1#% )&G0

"ab

I+"

ab

IIe#$ (1#% )&G0

= l0

kcate#$ (1#% )&G0 [ATP]

kcate#$ (1#% )&G0

kET

+ [ATP]

(S5)

as the forward rate

!

"ab

II= k

cat is exponentially reduced by the bp-unwinding potential at x2

= x0 – 1. Upon reaching x2 = x0, there is an probability

!

" f (0<

!

" f <1 even when rb=0) that

the bp opens upon the helicase activity:

! f = Popen + (1" Popen ) # Pswtich

=e"$%G0

1+ e"$%G0

+1

1+ e"$%G0

#e"$%G0 "&A

1+ e"$%G0 "&A

(S6)

which includes Popen , the probability that the front bp is already opened when x2 reaches

x0, plus the probability that the helicase with an extra energy switches the bp from closed

to open at x2 =x0, as explained previously. Note that when λ=1, there is no bp-unwinding

interaction potential,

!

" v recovers to

!

vtrans; additionally, if

!

"A is large, then

!

" f = Popen and

the unwinding velocity reduces to the passive case, in which the helicase only moves

forward when the bp spontaneously opens:

11

vpassiveuwd

= l0

e!"G0

1+ e!"G0

kcat[ATP]

kcat

kET+ [ATP]

. (S7)

However, when λ=1 but

!

"A is small (minimum active case), the unwinding velocity can

be detected larger than the passive one: when

!

"G0=0, the passive velocity has a value of

!

1

2vtrans, while in all active cases (λ ∈ [0,1]) the unwinding velocities converge to

1

2(1+

e!"

A

1+ e!"

A

) # vtrans at

!

"G0= 0, approaching

!

3

4vtrans as

!

"A →0.

The approximate forms of

!

" v and

!

" f are more complicated in case rb > 0, since now the

forward diffusion rate in the ATP bound state is reduced to

!

rb

+= r

be"# (1"$)%G0 , while the

backward diffusion rate is also reduced a bit to

!

rb

"= r

be"#U

0

b/ 3 . When we chose

!

"

relatively small (e.g.

!

" =0.1),

!

rb

"~ r

b, so the backward diffusion rate is often larger than

the forward one, leading to additional negative velocity flux based on the average

velocity form in Eq. (S2). The first part of the unwinding velocity, thus, can be written

approximately as:

!v(1)= !v (r

b= 0)+ l

0

"ab

I(rb

+# r

b

#)

"ab

I+"

ab

IIe#$ (1#% )&G0

= l0

[kcate#$ (1#% )&G0 + (e

#$ (1#% )&G0 #1)rb] ' [ATP]

kcate#$ (1#% )&G0

kET

+ [ATP]

(S8)

with

!

" v (rb

= 0) from Eq. (S5). The corresponding probability factor is the same as that

from Eq. (S6):

!

" f(1)

= " f (rb = 0). The second part of the unwinding velocity comes purely

from the backward diffusion in case that the helicase is waiting at the junction in the ATP

bound state,

12

!v(2)= l

0

"ab

I(rb

+# r

b

#)

"ab

I+"

ab

IIe#$ (1#% )&G0

= l0

#rb' [ATP]

kcate#$ (1#% )&G0

kET

+ [ATP]

(S9)

now with

!

rb

+= 0 and

!

rb

"= r

b. The corresponding probability factor is

!

" f(2)

=1#" f(1) . By

summing up the above two parts to a more compact form

!

vuwd(rb " 0) = # f (rb " 0) $ % v & #(1) % v

(1)+ #(2) % v

(2), keeping the same form of

!

" v as that in Eq.

(S5), we obtain:

! f (rb " 0) = Popen[1+(e

#$ (1#% )&G0 #1) ' rb

kcate#$ (1#% )&G0

]

+(1# Popen ) ' Pswtich # (1# Popen )(1# Pswtich )rb

kcate#$ (1#% )&G0

(S10)

In this way we always write vuwd = ! f " #v . From Eq. (S10), one can see that the larger the rb, the smaller the ! f . When r

b= 0 , Eq. (S10) reduces back to Eq. (S6).

In case of the passive unwinding (λ=1 and

!

"A large, or

!

Pswtich

" 0) in the presence of

diffusion, it gives

vpassiveuwd

(rb ! 0) = Popenvtrans

" (1" Popen )rb

kcatvtrans

=e"#G0

1+ e"#G0

$kcat[ATP]

kcat

kET+ [ATP]

"1

1+ e"#G0

$rb[ATP]

kcat

kET+ [ATP]

(S11)

The unwinding force effect In principle, forces assisting duplex unwinding can directly

affect the unwinding properties, such as velocity and processivity, of the helicase. The

force effect can be described as decreasing the original bp stabilization free energy from

!

"G0 to

!

"G0# F $ "X

F, where

!

"XF

is the change of the length of the substrate under the

force F, satisfying approximately a worm-like-chain relation

13

!

F =kBT

Lp

[1/4

(1"#XF

L)2

"1/4 +#XF

L] 9, where Lp is the persistent length (for ssRNA it is

about 10 Å), and L the contour length (for 2 nt released from separating 1-bp RNA, it is

about 12 Å). By replacing

!

"G0 with

Figure S4 Unwinding force effect and processivity. (a) The unwinding velocity and the dissociation rate vs. the force. The dark solid curve is for the velocity that was derived from the approximate solution (see Eq. (S5) and (S10)). The dissociation rate was obtained from simulations as data points (circles), fitted by a gray broken line. Note that in experiments 10; 11 the NS3 helicase was not detected with the force effect, see main text. (b) The processivity length vs. [ATP] for unwinding different RNA sequences. Note that the 11-bp regular pauses were not taken into account so that the obtained values of PL are much larger than measured ones. The PL for translocation (gray line) is also shown for comparison.

!

"G0# F $ "X

F, one can obtain the velocity and dissociation rate vs. force relations. In

Figure S4a, we show

!

vGC

uwd and

!

koffuwd vs. F for the NS3 helicase unwinding the GC

sequence (

!

"G0= 4 kBT). Since the sequence barrier

!

(1" #)$G0 decreases upon the force,

the velocity increases with the force while the dissociation rate decreases. The force

effects are consistent with experimental results on T7 helicase unwinding. However, in

order to detect the above force effects, experimental forces need to be effectively applied

to assist the duplex unwinding; otherwise, the force effect may not reveal.

Additionally, in Figure S4b, we show the processivity length (distance) vs. [ATP] during

unwinding when the 11-bp pauses are not taken into account. See main for descriptions.

14

V. Tuning parameters of the translocation and unwinding model Most of parameters utilized in this model adopt specific values and have been tuned

appropriately based on known information. The parameters along with their default

values and appropriate ranges are listed in Table S1 below. The rational for specifying

and tuning the parameters is presented in the following.

parameter value note parameter value note

kET ~1 µM-1 s-1 0.1 < kET < 2

µM-1 s-1 (see text

below)

rb 10 s-1 fit to an order

kcat 200 s-1 c.f. 11 ξ 0.3 fit to 0.1 ~ 0.3

!

"G0

GC 4.0 kBT calculated from

mfold 12

!

koffa 0.1 s-1 estimated to

0.001~ 1.0

!

"G0

mix 3.0 kBT same above

!

koffb 1.7 s-1 fit by ss

processivity

!

"G0

AU 1.5 kBT same above ζ 0.1 arbitrary

δA ~ 1 kBT estimated to an

order 13

!

U0

a 4.4 kBT calculated from

affinities 3

λ 0.2 fit to 0.1 ~0.5

!

U0

b 1.8 kBT fit to 1.3 ~2.0

kBT

Table S1 List of parameters used in the model, with their default values and appropriate ranges or information sources.

The appropriate values of ATP binding constant kET and related parameters In

main, we have defined

!

" # $(1% &), with 0≤

!

"≤1 the activeness of the helicase, and

0<

!

" <1 the pre-factor balancing the forward and backward rates upon the bp unwinding

potential. Hence one has 0≤

!

"≤1.

15

From Eq .(7) in main or Eq. (S5), one gets

!

KM

uwd=kcat

kET

e"#$G

0 . As mentioned in the main

text, to fit with experimental measurements such that unwinding the mixed RNA

sequence (

!

"G0~ 3k

BT ) gives

!

KM

uwd~ 100µM 10, we need

!

" =1

3log(

kcat

100kET

) . By using

kcat = 200 s-1 11 and 0<

!

"<1, one obtains, therefore, 0.1 < kET < 2 µM-1 s-1.

In fitting data with experimental measured unwinding velocities, we obtained that λ ≈

0.1, 0.2, 0.3, 0.4 and 0.5 with kET = 0.8, 1.0, 1.2, 1.5 and 1.8 µM-1 s-1. Accordingly, we

got

!

" about 0.03 to 0.3 and ξ =

!

" /(1# $) about 0.1 to 0.3. Also, we tuned

correspondingly the value of

!

U0

b to fit the dissociation rate 11 during unwinding, under

different values of kET and λ. As a result

!

U0

b is in between 1.3 (λ=0.5 and kET = 1.8 µM-1

s-1) to 2.0 kBT (λ=0.1 and kET = 0.8 µM-1 s-1). Other parameters values are either

irrelevant or insensitive to the values of λ or kET.

Calculating bp stabilizing free energies for different RNA sequences We used the

two-state hybridization server from mfold 12 to calculate the free energy that stabilizes

each bp, i.e., the free energy difference between the bp open and the closed state on the

RNA duplex. Indeed, this free energy consists of both hydrogen bonding interactions of

the bp and stacking interactions with a neighboring bp 14; 15. The calculations were

performed for RNA sequences of 100% GC, 100% AU and the mixed (52% GC). These

sequences were utilized in the single molecule studies of RNA unwinding in NS3 11. The

energy values were first determined under the temperature of T= 22 °C for RNA at a

default ionic condition ([Na+] = 1M and no Magnesium). These values were then

adjusted for the experimental ionic condition 10 ([Na+] = 30 mM and [Mg+] = 0.75 mM),

by timing a constant (~ 0.8) which was estimated from comparing the DNA free energy

values at the two ionic conditions. We assumed that counterions affect the RNA bp

stability similarly as that for the DNA in the same ionic conditions.

The values of

!

"G0

GC ,

!

"G0

AU and

!

"G0

mix listed in Table S1 are average values of the RNA-

bp stabilizing free energies along the GC, AU and the mixed sequences; the standard

deviations are about 1 kBT for the GC and the mixed sequences, and about 0.5 kBT for the

AU sequence. We found that the unwinding velocities of the helicase were calculated the

16

same by either using the average value of

!

"G0 for each sequence through Eq. (7), (S5)

and (S10), or using variable bp-by-bp free energies along the sequence to simulate

unwinding for a long time (over tens of seconds e.g.).

Choosing the parameters for bp unwinding Here we show how we chose appropriate

values for parameters ξ, λ, and δA (with a certain value of kET), which are essential for bp

unwinding. Since rb is also closely related to unwinding velocity and fluctuations, we

tuned its value simultaneously.

From Eq. (S5), one sees that the unwinding velocity maintains a Michaelis-Menten form

with

!

KM

uwd=kcat

kET

e"# (1"$)%G0 . As mentioned above, if one writes

!

" # $(1% &), it then requires

!

" =1

3log(

kcat

100kET

) in order to fit with experimental measurements:

!

KM

uwd ~ 100 µM when

!

"G0~3 kBT 10. Thus, using a certain value of kET, one can get a constant

!

" . At the same

time, λ can be tuned and one has

!

" =# /(1$ %) (λ≠1) fully depending on λ. Also

mentioned above, kET can adopt values in between 0.1 and 2 µM-1s-1. Below we discuss

only the case when one has kET = 1µM-1s-1 as a constant.

As the parameters λ, δA and rb act together to determine the helicase unwinding velocities

(see Eq. (S5) and (S10)), we tested a range of parameter values for {δA, rb, λ}. It had been

calculated before that a single bp opening at the center of a DNA duplex requires an

activation energy ~ 7 kcal/mol 13. Since the parameter δA counts for the pure activation

part of the bp opening energy at the end of a DNA/RNA duplex, we expect it smaller than

7 kcal/mol. Hence, we chose δA between 0 to 10 kBT.

The parameter rb can affect the unwinding velocity: as it increases, it reduces the

coupling efficiency ! f (see Eq. (S10)). Yet, more significantly, larger values of rb lead to

larger velocity fluctuations (see Eq. (S3) in translocation). Since experimental

measurements include all sorts of noises, the measured standard deviations of the

unwinding velocities 10; 11 can be used to estimate an upper bound of rb. Depending on the

17

value of δA, we found that the upper range of rb varies between 20 to 40 s-1 (smaller δA

gives a larger upper bound rb).

For a certain {δA, rb}, we tuned the parameter λ to fit experimentally measured

unwinding velocities for the RNA sequence 100% GC, AU or the mixed 10; 11. In

principle, it is not necessary that λ is kept the same for difference sequences. But our

calculations showed that in most cases, one could use a similar or identical λ to fit with

experimental velocities for the different sequences. Overall, the results show that λ

ranges from 0 to 0.3 (as kET = 1µM-1s-1).

δA kBT 0 10 5 2 1 data rb s-1 10 40 10 20 10 20 10 20 10 30 λ ∈ [0,1] 0.3 0.1 0.1 0 0.1 0 0.2 0.1 0.2 0.1 vGC bp/s 25.9 26.2 24.6 26.4 25.0 26.4 21.3 22.3 26.0 21.9 22 σv

GC 11.2 16.4 11.0 13.0 11.4 13.0 9.8 11.8 11.4 13.6 17 λ ∈ [0,1] 0.3 0.1 0 0 0 0 0.1 0.0 0.2 0.1 vmix 50.9 53.2 49.1 45.1 50.5 44.9 49.1 51.2 47.5 45.4 51 σv

mix 19.2 25.9 18.8 20.4 19.0 20.4 19.3 21.7 18.1 21.5 26 λ ∈ [0,1] 0.5 0.4 0 0 0 0 0.2 0.1 0.2 0.1 vAU 69.6 65.3 62.9 58.9 62.8 60.7 60.0 62.5 69.9 69.9 62 σv

AU 24.4 30.1 22.9 23.8 23.1 24.7 21.7 24.9 24.6 29.4 26

Table S2 Sequence-dependent velocities and fluctuations (v, the average velocity; σ, the standard deviation of the velocities) generated from our model under different parameters. The right column list the data measured from experiments 10; 11.

In Table S2 we list representative values of {δA, rb, λ} and corresponding unwinding

velocities (averages and standard deviations) calculated from current model for different

RNA sequences. To an order of magnitude, if we estimate δA as 1 kBT and rb as 10 s-1,

then we get λ =0.2. In current model, these values of {δA, rb, λ} are used by default.

Further, one can vary kET value while keeping {δA, rb} the same, then a range of λ along

with

!

" can be obtained accordingly, which have been shown in Table S1.

Estimating parameters related to dissociation and processivity When helicase

dissociation is considered during ss translocation, one needs

!

koffa and koff

b to describe the

18

dissociation rates at state a and b. While the dissociation is considered during duplex

unwinding, the junction stabilization potentials

!

U0

a and U0

b in state a and b are also

needed, such that the individual dissociation rates are exponentially decreased:

koff !uwda,b

= koffa,be!U

0

a ,b

.

From Eq. (S4) one can derive a ‘Michaelis-Menten’ form of the processivity during

translocation (as in Eq. (6) in main), with the maximum PL as kcatkoffb

. Using an upper

bound value of PL=120 bp from experiments (unpublished), one obtainskoffb! 1.67s

"1 .

One can also estimate koffa to an order of magnitude in between 0.001 to 1.0 s-1, which

lead to a relative flat PL vs. [ATP] curve as [ATP] is relatively large; yet the curve drops

sharply to a much lower value at low [ATP]. These results are consistent with

experimental observations 10 (and unpublished data). By default we used koffa~ 0.1s

!1 ;

other parameter values are insensitive to the exact value of koffa .

The strength of the junction stabilization potential

!

U0

a was estimated ~ 4.4 kBT in the

previous section. The value of U0

b affects significantly the overall dissociation rate of the

helicase unwinding a certain sequence. From the experimental value

koff !uwd = 0.5 ± 0.2s!1 measured for the NS3 helicase unwinding 100% GC RNA sequence

11, we tuned U0

b ~ 1.8 kBT (when kET = 1µM-1s-1 and λ~0.2). Varying kET in appropriate

ranges gives U0

b in between 1.3 to 2.0 kBT, always smaller than

!

U0

a .

There is an additional parameter 0<ζ<1 that determines how much the backward

(forward) rates are affected by the junction stabilization potential, similar to ξ for the bp

unwinding potential. We used an arbitrary value ζ=0.1 such that the backward rate is

reduced slightly by the junction stabilization potential.

19

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