Processivity, Velocity and Universal Characteristics of Nucleic Acid Unwinding by Helicases

Home , Processivity

bioRxiv preprint doi: https://doi.org/10.1101/366914; this version posted July 10, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Processivity, velocity and universal characteristics of nucleic acid unwinding by helicases

Shaon Chakrabarti1,2, Christopher Jarzynski3 and D. Thirumalai4 1Department of Biostatistics and Computational Biology, Dana-Farber Cancer Institute, Boston, MA 02115 2Department of Biostatistics, Harvard T. H. Chan School of Public Health, Boston, MA 02115 3Department of Chemistry and Biochemistry, Institute for Physical Sciences and Technology, University of Maryland, College Park, MD 20742 4Department of Chemistry, University of Texas at Austin, TX 78712 Abstract Helicases that act as motors and unwind double stranded nucleic acids are broadly classified as either active or passive, depending on whether or not they directly destabilize the double strand. By using this description in a mathematical framework, we derive analytic expressions for the velocity and run-length of a general model of finitely processive helicases. We show that, in contrast to the helicase unwinding velocity, the processivity exhibits a universal increase in response to external force. We use our results to analyze velocity and processivity data from single molecule experiments on the superfamily-4 ring helicase T7, and establish quantitatively that T7 is a weakly active helicase. We predict that compared to single-strand translocation, there is almost a two orders- of-magnitude increase in the back-stepping probability of T7 while unwinding double-stranded DNA. Our quantitative analysis of T7 suggests that the tendency of helicases to take frequent back-steps may be more common than previously anticipated, as was recently shown for the XPD helicase. Finally, our results suggest the intriguing possibility of a single underlying physical principle governing the experimentally observed increase in unwinding efficiencies of helicases in the presence of force, oligomerization or partner proteins like single strand binding proteins. The clear implication is that helicases may have evolved to maximize processivity rather than speed.

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INTRODUCTION

Helicases are enzymes that play an important role in almost every aspect of RNA and

DNA metabolism [1–7]. This class of molecular motors, which has been grouped into six

super families, utilize the free energy from ATP hydrolysis to translocate directionally over

hundreds of bases on single-strand (ss) nucleic acids. In this respect, they are similar to other

motors like kinesin and myosin, which also predominantly exhibit directional motion on mi-

crotubules and actin ﬁlaments, respectively [8, 9]. In addition, some helicases can also couple

ss translocation with unwinding of double-stranded (ds) nucleic acids, thus playing a crucial

role in cellular functions like replication, recombination and DNA repair. The “processive,

zipper-like” ds unwinding activity was ﬁrst demonstrated by Abdel-Monem and colleagues

in E. Coli based DNA helicase in 1976 [10]. Interestingly however, most helicases cannot

unwind double strands of nucleic acids by themselves, and require either oligomerization

or assistance from other partner proteins to increase the eﬃciency of processive unwinding

[11–13]. The fundamental principles governing this collaborative improvement of unwinding

eﬃciency have not been fully elucidated.

A helicase that uses some of the energy of ATP hydrolysis to destabilize the ss-ds nucleic

acid junction, is considered to be ‘active’. On the other hand, a helicase that utilizes the

thermal fraying of the ds and opportunistically steps ahead when the site ahead is open,

is referred to as ‘passive’ [2]. Considerable eﬀort has been devoted to classifying various

helicases into these two categories, as this general description gives an intuitive idea of the

mechanism used by a particular helicase to unwind nucleic acids [14–24]. Earlier works

used bulk kinetic assays or analysis of crystal structures to classify the activity of helicases

[15–17], while more recent studies have primarily resorted to single-molecule experiments

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coupled with theoretical frameworks to establish the mode of unwinding of various helicases

[20–23, 25–27].

Although ascertaining whether a helicase is active or passive is diﬃcult and remains a

controversial question for a number of helicases, we recently used this description in a quan-

titative framework to predict some surprisingly universal characteristics, which are inde-

pendent of the dsDNA sequence [28]. By generalizing the original mathematical framework

introduced by Betterton and J¨ulicher [29–31], we predicted that the processivity of active

and passive helicases should always increase in response to external forces applied to the

termini of dsDNA or dsRNA. Unlike the processivity, the velocity of unwinding, however,

exhibits no such universal behavior, and exhibit a variety of responses to external force

depending on whether the helicase is active or passive [28]. In our earlier work, we solved

for the velocity and processivity of a helicase (with both the step-size and interaction range

equal to one basepair) taking the sequence of DNA into account using numerically precise

solutions. Here, we use a physically motivated approximation scheme to derive analytical

expressions for both the velocity and processivity of a model that also accounts for a general

step-size and interaction range. Similar models accounting for a non-zero step-size and inter-

action range have been used to study the velocity of unwinding using numerical simulations,

without the beneﬁt of easy to use analytical expressions [23, 32] . These expressions allow

us to demonstrate the universality of the nature of the response of processivity to force,

enabling us to draw far reaching conclusions on the evolution of helicases to optimize the

number of base pairs that are unwound.

The analytic results derived here also allow us to analyze single-molecule experimental

data on helicases more precisely than attempted before, by simultaneously analyzing the

velocity and processivity of a particular helicase, instead of merely the velocity. The frame-

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work introduced by Betterton and J¨ulicher (and its variants), force or sequence dependence

of the unwinding velocity had previously been ﬁt to experimental data to determine the na-

ture of T7, T4 and NS3 helicases [20–23]. However, Manosas et al [32] pointed out that the

multi-parameter ﬁt of the Betterton and J¨ulicher model to velocity data is not robust, with

different parameter sets fitting the data equally well. More disturbingly, the various best-fit

parameter sets suggest completely diﬀerent unwinding mechanisms, and fail to provide any

conclusive results for the ﬁtting parameters. This is especially problematic when parameters

like the step size and the potential back-stepping rate have not been characterized experi-

mentally for a particular helicase, and need to be estimated from ﬁts of the theory to data.

As a consequence, the literature on helicases is rife with contradictory claims about the

mechanism of action of a speciﬁc helicase. For instance, steady-state and pre steady-state

kinetic assays [33] as well as studies of crystal structures [15] determined that NS3 helicase is

passive, while a single molecule experiment coupled with a mathematical analysis suggested

that NS3 is active [20]. Similarly, the T7 helicase was deemed to be an active helicase in

two previous works [21, 23], while simple physical arguments suggested that T7 is passive

[32]. Here, we show that our theory, which is used to analyze simultaneously velocity and

processivity data can be used to robustly obtain the values of all the parameters associated

with the unwinding activity of a helicase. Using data on the T7 helicase as a case study,

we use our theory to explain both the force response of velocity and processivity, in the

process quantitatively showing that T7 is very weakly active. Further, we recapitulate a

number of independent experimental results on T7, thus conﬁrming that the model is a

good description of the helicase. Our theory also shows that an externally applied force,

which in vivo could be generated by binding of single stranded partner proteins, results in

an increase in processivity in all helicases, regardless of whether they are active or passive.

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FIG. 1: Model of nucleic acid unwinding by a helicase. (a) Single-strand stepping kinetics of

the helicase. (b) Double-strand thermal ‘breathing’. (c) Modiﬁcation of stepping kinetics of the

helicase and breathing rates of the double strand. (d) The interaction potential that causes the

modiﬁcation of rates in (c).

This important ﬁnding prompts us to propose that helicases may have evolved to optimize

processivity rather than speed.

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THEORY

Model: The model, shown in Fig 1, is a generalization of the one introduced by Betteron

and J¨ulicher [29, 31] that accounts for the ﬁnite processivity of the helicase, allows for an

arbitrary step-size, and a general interaction range between the helicase and the double

strand. Fig 1a shows the helicase (red ﬁlled circles) as it translocates on the ss nucleic acid

strand (depicted as a bold black line). The position of the helicase on the nucleic acid track

is denoted by n. The helicase can exhibit pure diﬀusion, and hence can step to the right or

left with equal rate k+ = k− = k. When the NTP hydrolyzes, the helicase moves forward at

a rate h where h > k, thus resulting in a net forward rate h+k while the backward rate is k.

If the mechanical step size of the helicase is s, then every time the motor steps forward or

backward it does so by s nucleotides, resulting in an average velocity along the single strand

given by Vss = s h. The helicase could also disengage from the nucleic acid track with a

dissociation rate, γ.

Fig 1b represents the ss–ds junction at position m. The base pair at the junction can

rupture at a rate α (increasing m to m + 1) while a new base pair can form (decreasing m

to m − 1) at rate β such that α/β = exp(−∆G), where ∆G is the stability of the particular

ds base pair. All energies in this paper are in units of kBT .

Fig 1c shows how the rates change when the helicase and the junction approach each

other. Modiﬁcations of the original rates occur due to the interaction potential U(j), a

particular example of which is shown in Fig 1d. As the helicase and junction approach each

other, they interact. As a result, the helicase has to perform extra work, which is U0 per

base pair to step forward. This energy is provided from the hydrolysis of ATP. We follow

the description in [30] and deﬁne j ≡ m − n to be the diﬀerence in the positions of the

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FIG. 2: Diﬀerent scenarios for the interaction potential between the helicase and double-strand.

(a) Crystal structure of T7 helicase (Protein Data Bank (PDB) code 1E0J) and cartoon of ds DNA.

The helicase encircles one strand and excludes the other, suggesting that the interaction with the

junction must happen from a distance, presumably due to electrostatic forces. (b) Crystal structure

of the PcrA helicase bound to ds DNA (PDB code 3PJR), showing the overlap of the helicase

domains with bases beyond the junction. Below both the ﬁgures are shown the corresponding

interaction potentials U(j). Identical expressions for the unwinding velocity and run-length result

for both the potentials.

junction and the helicase. The rates are modiﬁed depending on j, and this is indicated

by the value of j as a subscript. For example, h−2 denotes the modiﬁed forward stepping

rate when j = −2 (Fig 1c). The motivation for allowing negative values of j and hence

a potential as shown in Fig 1c, comes from the workings of helicases like PcrA and NS3

(see Fig 2b). These helicases seem to physically interact with downstream base pairs of the

double strand [16, 20], possibly distorting and destabilizing a number of bases beyond the

junction. Therefore, for a general scenario, we let the helicase interact with the ds over a

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range r of base pairs, after which a hard wall exists at j = −r. For ring helicases like T7,

which encircle one strand of the DNA while excluding the other [34, 35], destabilization is

not likely to happen by physical overlap with bases downstream of the junction (see Fig 2a),

but might occur due to electrostatic interactions [36]. For such helicases, j would always be

positive with a hard wall at j = 0. Note that the exact position of the potential does not

matter: a shifted potential with the ﬁrst step at j = 4 and hard wall at j = 0 (Fig 2a) would

give identical results for the velocity and processivity as the potential shown in Fig 2b. Hence

this model for the interaction range of the helicase is general. The short range potential is

chosen to have a constant step height U0 and is deﬁned as follows:

  ∞ j ≤ −r   U(j) = (1 − j) U0 − r < j ≤ 0 (1)    0 j > 0.

The nucleic acid breathing rates are modiﬁed because of this interaction as follows:

−(f−1)(U(j)−U(j+1)) αj = α e ,

−f(U(j)−U(j+1)) βj+1 = β e , (2)

α such that j = e−(∆G−U0) for all values of j. In Eq. 2, f is a number between 0 and 1 βj+1 representing the fractional position of the free energy barrier between base-pair open and

base-pair closed states, from the open state. Nucleic acid opening takes the system from

state j to state j + 1 while closing takes the system from j + 1 to j. Hence the exponents in

Eq. (2) involve the term U(j)−U(j +1). The rates of the helicase also get modiﬁed because

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of the interaction, and change in the following way:

+ −f(U(j−s)−U(j)) kj = k e

−f(U(j−s)−U(j)) hj = he

− −(f−1)(U(j−s)−U(j)) kj−s = k e

U(j) γj = γe . (3)

+ With a step size s, the helicase cannot move to the right if j ≤ s − r, and hence kj = hj = 0

for all j ≤ s − r. Eq. (3) shows that as long as the helicase and the ss–ds junction are

+ separated by a distance j > s, the forward rates are independent of U0: kj = k and hj = h.

− For the backward rate, kj = k as long as j > 0. Notice that the exponents in Eq. (3)

contain the term U(j − s) − U(j) since the helicase jumps s nucleotides every time it steps

forward or backward.

Force eﬀects: To model the eﬀect of a constant external force F applied directly to the

F F ds junction, the ds opening and closing rates change from αj, βj to αj , βj such that:

αF α j j ∆GF F = e βj+1 βj+1 = e−(∆G−U0−∆GF )

≡ b eU0 , (4)

−(∆G−∆GF ) where we deﬁned b ≡ e , and ∆GF is the destabilizing free energy of the base pair

at the junction, due to the constant external force F . Assuming a Freely-Jointed-Chain

model for the single strand DNA segments, the expression for ∆GF is given by [37]:

L 1 ∆G = 2 log sinh(F l) , (5) F l F l

where L is the contour length per base and l is the Kuhn length. In writing Eq. (4),

we assume that all of the external force F is transmitted through the single strands and

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destabilizes the base pair at the junction. For ring-shaped helicases like T7, which encircle

one strand while excluding the other [34, 35], this model is a very good description. However,

this may not be an accurate description of other helicases like the NS3, where the domains

surround both the strands of the nucleic acid, and the junction may be protected to some

extent from external forces [5]. For such helicases, a more careful analysis is needed, and is

left for future work.

With the model thus deﬁned, we need to solve for the velocity and the processivity of

the helicase as it unwinds the ds nucleic acid. The velocity is deﬁned as the average number

of bases per unit time that the helicase moves to the right, in a binding event. For the

processivity, multiple deﬁnitions have been proposed [31]. The mean binding time hτi, the

translocation processivity hδni which gives the average distance moved by the helicase in

a binding event, and the unwinding processivity hδmi which gives the distance moved by

the ss–ds junction during a single binding event of the helicase–are used as measures of

helicase processivity. As shown numerically in [31], the latter two deﬁnitions of processivity

are almost identical when the helicase attaches close to the ss–ds junction. Since this is the

physically relevant situation, as we argued previously [28], we will not diﬀerentiate between

hδni and hδmi in this work, and refer to both as the run-length or processivity.

UNWINDING VELOCITY AND RUN-LENGTH: SOLUTION OF THE MODEL

To solve for the velocity and run-length of a ﬁnitely processive helicase, we ﬁrst note

that α and β are very large [38–40]–larger by orders of magnitude compared to any rate

describing the kinetics of the helicase. Therefore, even before the helicase takes a single step

(backward, forward or detach), the ss–ds junction would have opened and closed multiple

times. As a result, the probability Pj of observing the helicase and junction at a separation

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j would have reached a steady state distribution, long before the helicase moves. Since there

is a hard wall at j = −r, there is no probability current between j and j + 1 in this steady P state, for any value of j. This, along with the normalization condition, j Pj = 1, allows

us to solve for Pj:

  0 j ≤ −r   r+j−1 (r+j−1)U0 Pj = b e P−r+1 − r < j ≤ 1 (6)    r+j−1 r U0 b e P−r+1 j > 1 where, −1 " U 1+r # b1+rerU0 be 0 − 1 P−r+1 = + . (7) 1 − b beU0 − 1

α ∆GF where b as deﬁned before, is given by β e . The two-body problem of the helicase and

junction can now be recast in terms of a one-body problem involving only the helicase,

+ − + moving with renormalized rates that we denote with a tilde (ek , ek , eh and γe); ek is given by

+ P + ek = j kj Pj, and similar expressions describe all the other renormalized rates. Performing

+ − the sums, the ﬁnal lengthy expressions for ek , eh, ek and γe are given in the Appendix. Notice

that although k+ = k− = k, the helicase–junction interaction causes the renormalized rates

to become diﬀerent, hence ek+ 6= ek−.

The unwinding velocity is given by:

+ − vunw = s (eh + ek − ek ). (8)

For s = 1 and r = 1, vunw reduces to,

−(f−1)U0 fU0 1,1 e 1 + b e − 1 (b(h + k) − k) vunw = . (9) 1 + b (eU0 − 1)

The expression for one-step active unwinding derived by Betterton and J¨ulicher (Eq.(27) in

[30]) reduces to our expression in Eq. (9), when all the rates associated with the helicase are

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neglected compared to α and β.

The mean attachment time of the helicase hτi is given by the inverse of the renormalized

detachment rate: 1 hτi = . (10) γe

The processivity is given by hδmi = vunwhτi. Using Eq. (8) and Eq. (10), we obtain,

s (eh + ek+ − ek−) hδmi = . (11) γe

For s = 1 and r = 1, hδmi reduces to the following expression:

e−fU0 1 + b efU0 − 1 (b(h + k) − k) hδmi1,1 = . (12) γ

Eq. (8), Eq. (10) and Eq. (11) along with Eq. (16) are the important results in this paper.

Note that by deﬁning k− ≡ k and k+ ≡ k + h in Eq. (9) and Eq. (12), we obtain the same

model used in our previous work [28].

UNIVERSAL FORCE RESPONSE OF THE UNWINDING PROCESSIVITY

In our earlier work [28], we showed numerically that the unwinding velocity and proces-

sivity show contrasting responses to external force. We analyzed a model with step size of

one basepair and an interaction potential with a one basepair range. Below, we ﬁrst revisit

the earlier model to highlight the basic results and show quantitatively the diﬀerences be-

tween the responses of velocity and processivity to force. We then point out the eﬀects of

increasing the step-size and the interaction range of the helicase.

For s = 1 and r = 1, the expressions for velocity and processivity are given by Eq. (9)

and Eq. (12) respectively. As in our previous work, we choose ∆GF = F ∆X for illustrative

purposes. Choosing this simple form instead of the more accurate model based on the Freely

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Jointed Chain (Eq. (5)), does not qualitatively change any of the results [28]. We will look

at the limit k = 0, to simplify all the analytic expressions. For the more general case of

non-zero k, as long as k h, all the results are valid. Helicases are believed to satisfy this

criterion [32], and is supported by our ﬁtted values (discussed below) from data on the T-7

DNA helicase.

Setting ∆GF = F ∆X, and diﬀerentiating Eq. (9) with respect to F , we obtain the

following expressions for passive (U0 = 0) and optimally active (f = 0,U0 = ∆G) helicases.

Note that optimally active helicases are ones where f → 0 and U0 ≥ ∆G.

 e−∆G+F ∆X h ∆X (passive) dv1,1  unw = (13) dF e2∆G+F ∆X h ∆X  (optimally active). (e∆G + eF ∆X (−1 + e∆G))2 The equation above shows that for a passive helicase, the slope of the velocity-force curve

is not only always positive, it increase exponentially with F (orange curve in Fig. 3a). On

the other hand, for an optimally active helicase, the expression for the slope has the term

eF ∆X both in the numerator as well as the denominator–with a higher power of F in the

denominator, implying that the increase in velocity with force will be much less rapid than

an exponential. As can be seen from Fig. 3a, the curves (green, blue and red) have the

opposite curvature compared to the passive helicase (orange curve). In contrast, the force-

dependent behavior of the processivity shows a universal increase, as can be seen from the

following equation:

 e−∆G+F ∆X h ∆X  (passive) dhδmi1,1  γ = (14) dF e−∆G+F ∆X h ∆X  (optimally active).  γ Eq. (14) shows that the slopes are identical for both passive and optimally active helicases,

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a. b.

300 80 000

200 (bp) 1,1 (bp/s) 40 000 1,1 unw

v 100 <δm>

0 0 0 4 8 12 0 4 8 12 F(pN) F(pN)

FIG. 3: Response of (a) velocity and (b) processivity to external force, for a helicase with a 1 bp

step-size and a 1 bp interaction range. The orange curve corresponds to a passive helicase while

the red, blue and green curves correspond to optimally active helicases. The velocity curves are

plots of Eq. (9) while the processivity curves are plots of Eq. (12).

and increase exponentially with force. This is very clearly illustrated in Fig. 3b, where all

the curves almost superpose.

For a general step-size and interaction range of the helicase, the full expressions for

unwinding velocity (Eq. (8)) and processivity (Eq. (11)) are complicated, and hence we only

show a few representative plots in Figs. 4 and 5, for a variety of step-sizes and interaction

ranges. Fig. 4 shows that increasing the range of interaction (keeping the step-size ﬁxed)

aﬀects the velocity and processivity in diﬀerent ways. At a given force, while the velocity

of unwinding increases when the range is increased (Fig. 4a,b), the processivity decreases

(Fig. 4c,d). However, the universal behavior obtained in our previous work [28] remains

valid: the unwinding velocity can increase or remain almost constant with force, depending

on whether the helicase is active or passive. The processivity on the other hand, always

increases with external force. Fig. 5 shows the eﬀect of increasing the step-size while keeping

the interaction range ﬁxed, for optimally active (Fig. 5a,c) as well as very weakly active

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a. b.

300 300

200 200 (bp/s) (bp/s) unw unw

v 100 v 100

0 0 0 4 8 12 0 4 8 12 F(pN) F(pN)

c. d.

80 000 80 000

40 000 40 000 <δm> (bp) <δm> (bp)

0 0 0 4 8 12 0 4 8 12 F(pN) F(pN)

FIG. 4: Eﬀect of increasing the interaction range of the helicase on the unwinding velocity and

processivity, while keeping the step-size ﬁxed. The step size in all panels is ﬁxed at 1 bp and

−1 h = 322s , thus Vss = 322 bp/s. The interaction ranges are 1 bp (red), 2 bp (blue), 3 bp (green),

4 bp (pink), 5 bp (orange). (a) and (c): Optimally active helicase with U0 = 5 kBT and f = 0.01.

(b) and (d): Very weakly active helicase with U0 = 0.3 kBT and f = 0.01.

(Fig. 5b,d) helicases. At a given value of force, both the velocity and processivity decreases

with increase in the step size. For an optimally active helicase, the decrease in velocity is

rapid, becoming strongly negative at low forces (Fig. 5a). The reason for this phenomenon

is that the back stepping rate becomes larger as U0 is larger (more active helicase). The

helicase-junction interaction leads to the helicase imparting a force on the junction ( = U0

divided by the base-pair distance). This in turn causes an opposite force from the junction

on the helicase. This opposing force is larger for larger U0, thereby leading to larger eﬀective

back-stepping rates as U0 increases. For larger step-sizes (3 bp for the green line in Fig. 5a),

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the helicase needs more bases open downstream simultaneously and hence is less likely to

step forward. Coupled with a larger back-stepping rate, the net eﬀect is negative unwinding

velocities of the helicase.

To summarize, our analysis in this section shows that irrespective of the step-size, interac-

tion range, active or passive nature of the helicase, the unwinding processivity should always

increase with force. The unwinding velocity however does not exhibit the same universal

increase with external force.

a. b.

0 300

200

(bp/s) -2000 (bp/s)

unw unw 100 v v

0 -4000 0 4 8 12 0 4 8 12 F(pN) F(pN)

c. d.

80 000 80 000

40 000 40 000 <δm> (bp) <δm> (bp)

0 0 0 4 8 12 0 4 8 12 F(pN) F(pN)

FIG. 5: Eﬀect of increasing the step-size of the helicase on the unwinding velocity and processivity,

while keeping the interaction range ﬁxed. The interaction range in all panels is ﬁxed at 3 bp and

322 −1 h = s s , thus Vss = 322 bp/s. The step-sizes are 1 bp (red), 2 bp (blue), 3 bp (green). (a) and

(c): Optimally active helicase with U0 = 5 kBT and f = 0.01. (b) and (d): Very weakly active

helicase with U0 = 0.3 kBT and f = 0.01.

16 bioRxiv preprint doi: https://doi.org/10.1101/366914; this version posted July 10, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

UNWINDING MECHANISM OF THE T7 HELICASE

Simultaneous ﬁtting of DNA unwinding velocity and run-length data: Since

ﬁtting only the expression for the unwinding velocity to experimental data proves to be

insuﬃcient for estimating physically reasonable parameters, we reasoned that ﬁtting the

theory to the available data to the two observables simultaneously should signiﬁcantly limit

the parameter space and allow for better extraction of the important parameters of the

system. Using the velocity given in Eq. (8) and the processivity given in Eq. (11), we use

our theory to analyze data from the T7 helicase.

To apply our theory to analyze the data, we ﬁrst observe that the velocity as a function

of force has the following parameters: U0, k, f, s, r and h. It follows from Eq. (11) that the

processivity has one extra parameter, the dissociation rate γ. However, γ is a quantity that

is measured in bulk experiments [41, 42], while the relation Vss = s h allows us to use the

experimentally determined value of Vss, to reduce another free parameter. Thus, the number

of free parameters left to be determined from ﬁtting to experimental data is ﬁve—U0, k, f, s

and r. To test our method, we ﬁtted velocity and processivity data from single molecule

experiments on the T7 helicase (Fig 6b and Fig S6 respectively, of [23]). Kim et al [41]

reported γ = 0.002s−1 at 18◦C. Since the single molecule experiment was performed at

25◦C, we used the rough estimate that around room temperature, a number of chemical

rates increase by about a factor of 2–3 for every 10◦C increase [43], to estimate γ at 25◦C.

We therefore used γ = 0.003, 0.004 and 0.005s−1. We used h = 322/s (s is the step size)

since the ss velocity was measured to be 322 bp/s [23], which seems not inconsistent with

the bulk result of 132 bp/s at 18◦C [41]. We took ∆G = 2.25 since the DNA sequence had

48% GC content (supplementary information of [22]). Both the bulk and single molecule

17 bioRxiv preprint doi: https://doi.org/10.1101/366914; this version posted July 10, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

a. b. 3000 200

150 2000

100 1000

Velocity(bp/s) 50 Processivity(bp)

0 0 0 2 4 6 8 10 0 2 4 6 8 Force(pN) Force(pN)

FIG. 6: Simultaneous ﬁtting of velocity and run-length data. The blue and red circles are exper-

imental data for the T7 helicase from [23].The blue line in (a) is Eq. (8) ﬁtted to velocity data

while the red line in (b) is Eq. (11) ﬁtted to processivity data. The best ﬁt parameters are given

in Table II.

experiments were performed at 2 mM dTTP concentration.

For the force dependent destabilization of the double strand given in Eq. (5), the pa-

rameters L and l need to be chosen carefully, to reproduce the critical force Fc observed in

the experiment. Fc (the force where ∆G = ∆GF ) was observed to be around 13.6-13.7 pN

for the ds DNA sequence we have analyzed in this work (Fig 6b of [23]). The usual values

chosen for L and l are 0.56 and 1.5 nm respectively [37]. However, Fc for this choice of

parameters (and ∆G = 2.25) is about 15 pN, so we chose L = 0.63 nm and l = 1.5 nm to

reproduce the observed critical force. To check the robustness of our results, we also tried

a diﬀerent parametrization L = 0.56 and l = 1.95 nm, which results in Fc = 13.6 pN. Both

these parametrizations produce nearly identical results, hence we show results with only the

ﬁrst one (Table I and II).

Quality of ﬁts: The results of simultaneous ﬁtting are shown in Fig 6. Tables I and

II show the quality of ﬁts (χ2) for a variety of parameter sets with ‘similar’ ﬁts. To quan-

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titatively deﬁne ‘similarity’ of ﬁts, we used the Akaike Information Criterion (AICc) [44]

deﬁned as: 2p(p + 1) AICc = χ2 + 2p + , (15) N − p − 1

where N is the number of data points and p the number of free parameters. The usefulness

of this criterion is that the quality of the two sets of ﬁts can be quantitatively compared:

if two model ﬁts have AICc values of a1 and a2 respectively, with a1 < a2, then model 2

has a likelihood exp((a1 − a2)/2) of being the true interpretation of the data, relative to

model 1. Using this interpretation, we show in Tables I and II all ﬁts that are at least 0.5

times as likely as the best ﬁt among that set. Table I shows the result of ﬁtting to only

velocity data—multiple parameter regions can ﬁt the velocity data with similar quality of

ﬁts. Table II shows results of our simultaneous ﬁtting procedure. Clearly, Table II shows

that the simultaneous procedure allows a much narrower range of parameters to produce

similar ﬁts. In addition, the errors on the parameters are small, allowing all ﬁve parameters

to be extracted with great robustness. The best ﬁt parameters are U0 = 0.69 kBT , f = 0.19,

k = 0.6 s−1, s = 2 bp and r = 5 bp.

Comparison with experiments on T7 unwinding of DNA under zero force con-

ditions: An earlier bulk experiment [45], and a more recent FRET-based single molecule

experiment [46] on T7 DNA, were carried out under conditions of zero external force. Using

an ‘all-or-none’ assay at 18◦C, ﬁve DNA sequences (average GC content of 37%) were un-

wound with T7 in [45], resulting in an average unwinding velocity of 15 bp/s. Approximately

consistent with these results, the unwinding velocity at 23◦C of T7 on a 35% GC sequence

was found to be 8 bp/s [46]. Fig 6a shows our model prediction for the unwinding velocity

at zero force: vunw = 7.1 bp/s. Keeping all the parameters ﬁxed at the values shown in

Table II, but reducing ∆G to 1.9 to correspond to a DNA sequence comprising roughly 37%

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TABLE I: Fitting to only velocity data. Shown are just a few ﬁts, obtained by varying s and r,

from all those that are similar. The deﬁnition of ‘similar’ is that the worst ﬁt should be 0.5 times

as likely as the best ﬁt, according to AICc (see text for details).

Step Size s (bp) Interaction Range r (bp) χ2

1 20 3.04

2 10 1.65

2 20 1.72

2 50 1.74

3 3 2.3

3 4 2.8

GC basepairs, our model predicts an unwinding velocity of 18 bp/s at zero force. Taking

into account that our analysis is based on an experiment performed at a slightly higher

temperature (25◦C) compared to either of these two zero-force experiments, our results are

consistent with the two previous works.

Predictions for sequence dependence of detachment and back-stepping rates of

T7 while unwinding dsDNA at zero-force: The sequence dependence of the detachment

rate of a helicase is an aspect that can be directly measured in experiments [20, 47], and

can be an indicator of whether the helicase is active or passive. By ﬁxing the parameters in

our model to the best-ﬁt values of Table II, and changing only ∆G, we can predict how the

detachment and back-stepping rate of T7 will depend on the sequence composition, while

unwinding DNA. The results of this analysis is shown in Fig 7. As is evident, neither the

detachment rate, nor the back-stepping rate are very sensitive to ∆G, because the helicase

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TABLE II: Fitting simultaneously to velocity and run-length data. Compared to Table I, the

parameter space with similar ﬁts to the data has been drastically reduced. The deﬁnition of

‘similar’ is exactly the same as that used in Table I.

Step Size s (bp) Interaction Range r (bp) χ2

2 5 4.69 a

2 6 5.23

a This is the best fit among all the similar fits. The parameters for this fit along with errors are :

u = 0.69 ± 0.02, f = 0.19 ± 0.04, k = 0.6 ± 0.4. Clearly, each parameter can be robustly estimated.

is only very weakly active. Interestingly, this is akin to the observations made in a previous

experiment on the DnaB helicase [47], where it was shown that for sequences with 50-100%

GC composition, the detachment rate was almost constant. Since both DnaB and T7 are

very similar in structure and sequence, belonging to the superfamily-4 group of helicases,

our results suggest that the ring helicases of the superfamily-4 group might all use a similar

weakly active mechanism for unwinding DNA.

DISCUSSION

T7 is weakly active: From the best ﬁt parameters in Table II, we infer thatthe T7

helicase is a weakly active helicase, destabilizing the ds junction by about 0.69 kBT per

base. The value of U0 obtained here, is diﬀerent from the (1 − 2)kBT estimate reported

earlier in [21, 23]. Both these works analysed only velocity data – the former looked at

sequence dependence while the latter examined the force dependence of velocity. It was

21 bioRxiv preprint doi: https://doi.org/10.1101/366914; this version posted July 10, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

a. b. 0.09 1.90

1.88 16 0.08 )

14 ) 40 -1 1.86 12 -1

(s 10 0.07

(s 30

- 8 ˜ ˜ γ

k 20 1.84 6 4 0.06 1.82 10 1.5 2.0 2.5 3.0 1.80 0.05 1.5 2.0 2.5 3.0 1.5 2.0 2.5 3.0 1.5 2.0 2.5 3.0 ΔG(k BT) ΔG(k BT)

FIG. 7: Predictions for sequence dependence of T7 detachment and back-stepping rates while

0 unwinding ds DNA. (a) The back stepping rate while unwinding (k−) hardly changes as a function

of the sequence stability, as a result of the helicase being only weakly active. (b) Similarly, the

0 detachment rate while unwinding (kd) changes by only a factor of 1.5 with change in ∆G. The

insets in both ﬁgures show the hypothetical situation of a highly active T7, with U0 = 2.0 kBT .

Both the back-stepping and detachment rate show much more sensitivity to ∆G under highly active

circumstances.

originally pointed out [32] and veriﬁed by us speciﬁcally for T7 in this work (Table I), that

analyzing only the velocity data using the multi parameter Betterton and J¨ulicher model is

not suﬃcient for robust parameter estimates. There are other diﬀerences as well between

our model and the ones used previously [21, 23]. The parameter f was ﬁxed to 0.05 in both

those earlier studies, whereas we allow it to vary, given that f is a physical quantity which

could take on any value between 0 and 1. We also have the extra parameter k, which gives

the rate of pure diﬀusion. The presence of this parameter allows for back-steps, which was

neglected in the previous studies. It is important to include this parameter, especially in

light of recent experiments that directly observed back-stepping [24, 46].

Step-size of T7: Our prediction of a step-size of 2 bp (2 bases advanced for each ATP

22 bioRxiv preprint doi: https://doi.org/10.1101/366914; this version posted July 10, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

hydrolyzed) is in agreement with certain previous experimental results. Using a pre steady-

state analysis, it was found that one ATP molecule is consumed for every 2 − 3 basepairs

translocated by T7 on a ss DNA [41]. A crystal structure of the DnaB helicase bound to

ssDNA, showed that the step-size of DnaB is 2 bp [48]. DnaB and T7 are both members of the

superfamily-4 group of helicases, with very similar sequence and structure in the C-terminal

domains [19, 49]. These results suggest that the step size of T7 while unwinding ds DNA

may also be 2-3 bp, under the assumption that ss translocation and ds unwinding occur with

the same step-size. A recent smFRET-based unwinding assay using T7 observed stochastic

pauses after every 2-3 bp of G-C rich DNA unwound [46]. However, the waiting times of

these pauses were gamma distributed rather than exponentially distributed, suggesting the

presence of hidden steps within those pauses. Though the results do not prove a direct

association of these hidden steps with ATP consumption, it would not be surprising if the

helicase has a distribution of step-sizes with shorter steps of 1 bp while unwinding G-C bases.

Since our model does not distinguish between the step size during translocation, unwinding

or for diﬀerent sequences, it is likely that our result of 2 bp per ATP consumed is a reﬂection

of the average step-size over the entire ds sequence of the DNA being unwound. Further

experiments, including determination of crystal structures, would be able to shed more light

on this interesting conundrum.

Interaction range of T7: The interaction range of ∼ 5 bases that we obtain from

our ﬁts, is physically reasonable given the structure of the T7 ring helicase and its mode of

binding to dsDNA. While the DNA strand excluded from the T7 ring is negatively charged,

the C-terminal face of T7 is also negatively charged [36]. Replacement of the charged residues

on the C-terminal by uncharged ones leads to a reduction in eﬃciency of complementary

strand displacement [50]. These results strongly suggest that the moderately weak (0.69 kBT

23 bioRxiv preprint doi: https://doi.org/10.1101/366914; this version posted July 10, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

per base) interactions between the helicase and DNA that we predict, are electrostatic in

nature. Given that the Debye-H¨uckel screening length is ∼ 1 nm in physiological salt

concentrations [51], the range of electrostatic interaction of the helicase should be a few

nanometers. Our prediction of 5 bases (∼ 1.7 nm) therefore is physically reasonable. Notice

that ﬁtting to only velocity data would predict an interaction range greater than 20 bp,

which would be unphysically large.

Back-stepping rate of T7 while unwinding or translocating: The back-stepping

rate of helicases is usually very diﬃcult to measure directly, due to lack of suﬃcient res-

olution in single-molecule experiments. Hence, the back-stepping rate is usually assumed

to be negligible compared to the forward stepping rate or neglected completely [32]. Using

the parameters extracted from our ﬁts to T7 data (given in Table II), we now show that

though this assumption is valid when the helicase translocates on ssDNA, the back-stepping

probability is orders of magnitude larger while unwinding dsDNA. Note that in this discus-

sion and throughout this paper, we set the ATP concentration at 1mM, which is roughly

the physiological concentration of ATP. Experiments are usually performed at ATP concen-

trations of 1mM or higher. If the ATP concentration is made very low in an experiment,

naturally the back-stepping probability will be substantially larger [24] until it approaches

50% at 0 ATP, implying no directionality associated with the motion of the helicase.

The ratio of forward to backward stepping rates while translocating is (h + k)/k, hence

using k = 0.6 and h = Vss/s = 161, this ratio turns out to be 269. This result is interesting,

and suggests that T7 back-steps approximately as frequently as some of the other processive

molecular motors like Kinesin, which also has similar values for this ratio in the absence of

external force [52]. The back-stepping probability at every step is given by k/(k + (h + k)),

which is a mere 0.3% for the T7 parameters given in Table II. However, when the helicase

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unwinds dsDNA, the rates change, and the modiﬁed back-stepping probability is given by

kf−/(kf− + (eh + kf−)). For the same parameter set, this works out to be 26%, almost two

orders of magnitude larger than the back-stepping probability while translocating. Our pre-

diction regarding this enhanced back-stepping probability while unwinding, is similar to the

observations in a recent experiment on the XPD helicase, where it was shown that at 1mM

ATP the back-stepping probability is about 10% [24]. Our analysis therefore suggests that

XPD, which belongs to superfamily 2, may not be unique in this respect – the superfamily

4 helicase T7 also seems to back-step with relatively large probability while unwinding ds-

DNA. Further experiments on helicases belonging to the same super family are needed to

probe the extent of back stepping.

Universal nature of force response of helicase processivity, oligomerization

and partner proteins: Helicases often cannot unwind double strand nucleic acids by

themselves, but require partner proteins like single strand binding proteins or oligomerization

to increase the eﬃciency [11–13]. We use eﬃciency to mean that the helicase is highly

processive. While the mechanistic reasons for the increase in eﬃciency is likely to vary for

individual helicases, our work suggests the possibility of a single physical principle underlying

this eﬀect. Our theory shows that the increase in processivity in response to external force

holds for all helicases. In particular, unlike the unwinding velocity, the processivity increases

with external force irrespective of the active or passive nature of the helicase. Since the

external force destabilizes the double strand and decreases the free energy of the base pairs

at the junction, we predict that any perturbation that results in reduction of base-pair

stability should result in universal eﬀects, similar to the force response of helicases. There

is strong evidence that partner proteins, like single strand binding proteins (SSB), melt

nucleic acids by reducing the stability of base pairs (see a more detailed discussion in [28]).

25 bioRxiv preprint doi: https://doi.org/10.1101/366914; this version posted July 10, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Similar eﬀects may be achieved by oligomerization of helicase monomers. A recent study

on the UvrD helicase showed that dimerization leads to a closed conformation of the 2B

subdomain of the leading UvrD monomer[13], a conformation that contacts the junction

duplex and presumably destabilizes it [53, 54]. This implies the possibility that a universal

underlying principle governs the response of helicases to force, partner proteins as well as

oligomerization. An immediate prediction of this theory is that a weakly active helicase like

T7 should exhibit increased velocity and processivity in the presence of partner proteins.

This phenomenon has indeed been observed previously, but not explained [55, 56]. On the

other hand, partner proteins should increase only the processivity of an active helicase like

NS3. This prediction is borne out as well in experiments [57]. It is rare to ﬁnd universal

behavior, especially on the nm scale representing motors including helicases. That this

seems to be the case for helicases, is remarkable. The universal increase of processivity with

force, regardless of the nature of the helicase, also suggests that this class of motors may

have evolved to optimize processivity rather than speed.

CONCLUSIONS

Whether a particular helicase unwinds double stranded nucleic acids using an ‘active’ or

‘passive’ mechanism has been a subject of much debate. Here, we derived analytic expres-

sions for both the velocity and processivity of generic unwinding helicases, and showed that

only by simultaneously using velocity and run-length data can the active/passive nature of

a helicase, and hence the unwinding mechanism, be discerned. Simple expressions for the

processivity or run-length of helicases were previously unavailable, hence our work should

prove useful in the future analysis of helicase unwinding trajectories. Our results also quan-

titatively predict that the processivity should show a universal increase with force unlike

26 bioRxiv preprint doi: https://doi.org/10.1101/366914; this version posted July 10, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

the velocity, which has implications for in vivo unwinding of nucleic acids by the replisomal

complex. This result seems to be borne out in a variety of experiments, and further work will

help in verifying these remarkable predictions. Finally, we have also quantitatively shown

for the T7 helicase, that the backstepping rate while unwinding is orders of magnitude larger

than the backstepping rate while translocating on single-strand nucleic acid. This result is

under-appreciated, and if taken into account could should qualitatively change the conclu-

sions of the usual approaches of modeling, which assume that back stepping probabilities

are negligible.

Acknowledgments: This work was completed while SC was a graduate student at the

Institute for Physical Sciences and Technology, University of Maryland. We acknowledge

the National Science Foundation (CHE 16-36424 and DMR-1506969) and the Collie-Welch

Foundation (F-0019) for supporting this work.

APPENDIX

+ − The full expressions for ek , eh, ek and γe are given below:

−fsU 1+r s r+s rU e 0 y −y r (r−fs)U0 s + b e 0 ( ) b e (z −z) ek 1−b + y−1 + z−1 = , k+ b1+rerU0 y1+r−1 1−b + y−1 −fsU 1+r s br+serU0 e 0 (y −y ) bre(r−fs)U0 (zs−z) eh 1−b + y−1 + z−1 = , h b1+rerU0 y1+r−1 1−b + y−1

r rU e(1−f)sU0 y1+r−s−1 e(1−f)(s−1)U0 y1+r−s zs−1−1 − b e 0 ( ) ( ) ek 1−b + y−1 + z−1 = , k− b1+rerU0 y1+r−1 1−b + y−1 γ erU0 (y − 1) e = , (16) γ b (1 + brerU0 (y − 1) − (b − 1)eU0 yr) − 1

where y ≡ beU0 and z ≡ befU0 .

27 bioRxiv preprint doi: https://doi.org/10.1101/366914; this version posted July 10, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

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