doi:10.1016/j.jmb.2005.04.051
J. Mol. Biol. (2005) 350, 452–475
Mechanochemistry of T7 DNA Helicase
Jung-Chi Liao1†, Yong-Joo Jeong2†, Dong-Eun Kim2, Smita S. Patel2*
and George Oster1*
1
Departments of Molecular and
Cell Biology and ESPM
University of California
Berkeley, CA 94720-3112
USA
2
Department of Biochemistry
Robert Wood Johnson Medical
School, 675 Hoes Lane
Piscataway, NJ 08854, USA
The bacteriophage T7 helicase is a ring-shaped hexameric motor protein
that unwinds double-stranded DNA during DNA replication and
recombination. To accomplish this it couples energy from the nucleotide
hydrolysis cycle to translocate along one of the DNA strands. Here, we
combine computational biology with new biochemical measurements to
infer the following properties of the T7 helicase: (1) all hexameric subunits
are catalytic; (2) the mechanical movement along the DNA strand is driven
by the binding transition of nucleotide into the catalytic site; (3) hydrolysis
is coordinated between adjacent subunits that bind DNA; (4) the hydrolysis
step changes the affinity of a subunit for DNA allowing passage of DNA
from one subunit to the next. We construct a numerical optimization
scheme to analyze transient and steady-state biochemical measurements to
determine the rate constants for the hydrolysis cycle and determine the flux
distribution through the reaction network. We find that, under physiological and experimental conditions, there is no dominant pathway; rather
there is a distribution of pathways that varies with the ambient conditions.
Our analysis methods provide a systematic procedure to study kinetic
pathways of multi-subunit, multi-state cooperative enzymes.
q 2005 Elsevier Ltd. All rights reserved.
*Corresponding authors
Keywords: helicase; ring; ATPase; sequential; pre-steady state kinetics
Introduction
Helicases are motor proteins that translocate
along nucleic acid chains using the energy of NTP
hydrolysis. The ability to translocate unidirectionally enables them to carry out processes involving
nucleic acid metabolism, especially those that
require the separation of duplex nucleic acids into
their component single strands.1–3 Helicases drive
critical biological processes such as DNA replication, repair, and recombination; hence, mutations in
a helicase protein lead to many human diseases
including cancer and/or premature aging.4
Bacteriophage T7 gene 4 helicase is a hexameric
ring helicase that translocates along DNA during
† J.-C.L. & Y.-J.J. made equal contributions to this work.
Present addresses: D.-E. Kim, Department of
Biotechnology and Bioengineering, Dong-Eui University,
Busan 614-714, South Korea. Y.-J. Kim, Department of Bio
and Nanochemistry, Kookmin University, 861-1,
Chongnung-dong, Songbuk-gu, Seoul 136-702, Korea.
Abbreviations used: ssDNA and dsDNA, singlestranded and double-stranded DNA, respectively.
E-mail addresses of the corresponding authors:
[email protected]; [email protected]
DNA replication and recombination and unwinds
the complementary DNA strands.1,5 Although
crystal structures for heptameric helicases have
been reported,6 we will focus here on the hexameric
species. However, our analyses can easily be
applied to the heptameric ring with some modifications. The energy source for the T7 helicase is
dTTP (deoxythymidine triphosphate), whose function is equivalent to that of ATP in other motor
proteins. The chemical energy of dTTP hydrolysis is
converted into mechanical work to move the helicase unidirectionally along the DNA. dTTPs (or
other nucleotides) also stabilize the formation of the
hexameric ring structure in T7 helicase.7 Similarly,
we have shown that T7 helicase binds DNA tightly
in the presence of dTTP or dTMPPCP but not in the
presence of dTDP.7 Thus, dTTP binding and
hydrolysis allow DNA bind–release cycles but the
step in the dTTPase cycle that triggers DNA release
is not known. Translocation of T7 helicase proceeds
at a rate of w130 nt/s along ssDNA, hydrolyzing
one dTTP per three-base movement.8 On unwinding forked duplex DNA, the movement is slower,
where strand separation catalyzed by T7 helicase
proceeds at an average rate w15 bp/s.9
Figure 1(a) shows two subunits of the ring
0022-2836/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
Mechanochemistry of T7 DNA Helicase
453
Figure 1. Structural arrangement of T7 helicase. (a) The model of DNA strand separation by T7 helicase built from
electron microscopy13 and the structures of two helicase subunits and their interaction with an ssDNA.5 Mutation
experiments suggest that motif II (residues 424–439, shown in green) and motif III (residues 464–475, shown in pink)
interact with DNA.5,27,28 (b) Topological diagram of the secondary structures of two adjacent subunits using the same
color coding as in (a).11 A Mg-dTTP molecule is shown adjacent to subunits of the helicase. The conserved b-sheet
strands are numbered in the usual order: 5,1,4,3,2. The network of hydrogen bonds and salt-bridges that anneal the
Mg2C and the nucleotide to the catalytic site are shown broken and color coded according to their donor sites. These
bonds were determined using a 3.3 Å cutoff; a larger cutoff would reveal more, but weaker, bonds. The suspected ssDNA
binding motifs reside in the loops from the b-strands adjacent to the b-strands that form the dTTP catalytic site.
helicase from the crystal structure,5 and a possible
single-stranded DNA (ssDNA) configuration in the
central channel of the ring. The structure of the
catalytic site in each subunit belongs to the class of
“RecA-like” NTPases,10,11 with the highly conserved five-stranded parallel b-sheet as shown in
Figure 1(b). Two loops emanating from the central
b-sheet are suggested in the crystal structure to
contact DNA.5 These two loops emerge from
strands 3 and 4, which are adjacent to the energyconverting P-loop (Figure 1(b)), and this arrangement may facilitate the energy transfer from the
Mg$NTP hydrogen bond network to the DNA
movement.
To understand the mechanochemical coupling for
T7 helicase, it is first necessary to determine the
kinetic pathways of dTTP hydrolysis, so that the
relationship between the kinetics and the mechanical steps can be identified. Previous studies have
determined the dTTP hydrolysis pathway of T7
helicase in the absence of DNA.12 Those studies
showed that only one of the six sites in the hexamer
hydrolyzed dTTP at a fast rate and not all six sites
participated in the dTTPase turnover. Here, we
investigate the pathways of dTTP hydrolysis
catalyzed by T7 helicase in the presence of
ssDNA. We find that up to four sites in the hexamer
hydrolyze dTTP at a fast rate and that all six sites
454
likely participate in dTTPase turnover. Under high
dTTP concentration, both in the absence and in the
presence of the DNA, the rate-limiting step is the
release of the product, Pi. The number of possible
kinetic states of the six catalytic sites is extremely
large, and so to analyze the kinetics of dTTP
hydrolysis a numerical method is developed to
determine the kinetic rate constant for each step and
to determine the main reaction pathways. Using the
combined approach of pre-steady-state kinetics and
computational kinetic modeling, we dissect the
dTTPase mechanism of T7 helicase in the presence
of ssDNA and propose a multi-subunit, multi-state
kinetic model that indicates that dTTP hydrolysis
and power strokes occur sequentially around the
hexameric ring. This model is mechanically consistent with the electron microscopy observations
that showed only one or two subunits bind DNA.13
Also, recently Mancini et al. proposed a similar
sequential translocation mechanism for the f12 P4
hexameric packaging motor, which is structurally
related to T7 helicase.14 However, additional data
will be required to determine if the power-strokes
are partially or strictly sequential around the T7
helicase ring.
The methodology we use augments the
kinematic† notion of “conformational coupling”
with the mechanical concept of stress, or force per
unit area. This allows us to bring to bear mechanical
concepts to the phenomenology of cooperativity
between different regions of a protein.
Results
To understand the mechanisms of translocation,
it is necessary to dissect the dTTPase pathway in the
absence and in the presence of the DNA and
determine when DNA binds and releases from the
subunit. In addition to dissecting the dTTPase
pathway, it is important to determine how the
dTTPase cycles are coordinated among the hexamer
subunits. Previously, using pre-steady-state kinetic
approaches, we have determined the kinetic pathway of dTTP hydrolysis in the absence of DNA.12,15
We found that, in the absence of DNA, only one
subunit of the hexamer hydrolyzed dTTP at a fast
rate and that several of the hexamer subunits
appeared to be non-catalytic.12,15 The dTTPase
turnover rate was much slower at the non-catalytic
subunits compared to the observed dTTPase kcat
value. Here, we use a combination of pre-steadystate kinetic methods and computational modeling
to determine the dTTP hydrolysis pathway in the
presence of DNA. We use these approaches to
address the following issues: (a) the number of
actively participating hexamer subunits; (b) the rate
limiting step(s) in the presence of DNA; (c) the
† The branch of mechanics that studies the motion of a
body or a system of bodies without consideration given to
its mass or the forces acting on it.
Mechanochemistry of T7 DNA Helicase
specific steps at which DNA binds and releases
from the helicase subunit; (d) the coordination of
dTTP hydrolysis among the hexamer subunits.
Pre-assembly of the helicase hexamer on DNA
DNA binds within the central channel of the T7
helicase hexamer and previous studies have shown
that DNA binding is a slow process.16 To measure
the pre-steady-state kinetics of dTTP hydrolysis, it
is therefore important to pre-assemble the helicase
hexamer on the DNA to assure that DNA binding or
hexamer assembly does not limit the observed
dTTP hydrolysis rate. Since the T7 hexamer requires
the presence of dTTP to bind DNA,7 it was
particularly challenging to set up conditions to
measure the pre-steady-state kinetics of dTTP
hydrolysis in the presence of DNA. Our finding
that T7 helicase can be pre-assembled on the DNA
with dTTP without Mg2C provided us with initial
conditions to pre-assemble the hexamer on the
DNA prior to reaction start. In the absence of Mg2C,
dTTP hydrolysis is insignificant (8.5!10K3 M/M
per second at 25 8C; w5000-fold slower than the
dTTPase rate in the presence of Mg2C).17 The presteady-state dTTP hydrolysis experiments were
carried out using the long circular M13 ssDNA as
the nucleic acid substrate.8 We determined experimentally that the optimal concentration of M13
ssDNA was 15 nM per mM of T7 helicase hexamer
(see Appendix A). The pre-steady-state kinetic set
up was as follows: T7 helicase was incubated with
DNA in the presence of dTTP (without Mg2C) for
three minutes, which was determined experimentally as an optimum pre-incubation period that
provides sufficient time for assembly and to load
the pre-incubated sample in the quench-flow
instrument, and reactions were initiated by
addition of Mg2C.
Chase-time kinetics indicates all subunits are
catalytic
The chase-time experiments provide information
about the number of dTTPs that are tightly bound to
the helicase hexamer and competent in hydrolysis.
In the chase-time setting, T7 helicase was incubated
with dTTP, [a-32P]dTTP, and DNA in the absence of
Mg2C. After three minutes, excess non-radiolabeled
Mg-dTTP was added as a chase and the chase-time
was varied before acid-quenching. During the
chase-time, the dTTP molecules that were bound
initially to the hexamer subunits can get hydrolyzed
or may dissociate into solution. Once the radiolabeled dTTP dissociates from the helicase into
solution, it is diluted by excess non-radiolabeled
dTTP chase and hence not observed in our
reactions. Thus, the chase-time kinetics measures
the hydrolysis reaction only from the tightly bound
dTTPs. In this setting, and also later in the acidquench experiment, the measurements included
three different species: dTDP in solution; dTDP
bound to enzyme; and dTDP-Pi bound to enzymes.
Mechanochemistry of T7 DNA Helicase
Detection of dTDP$Pi permits tracing the hydrolysis
events.
Figure 2(a) shows the chase-time kinetics of dTTP
hydrolysis at various dTTP concentrations. The
data fit to the single-exponential equation (4),
shown in Experimental Procedures, that provides
the observed amplitudes shown in Figure 2(b). The
amplitude provides an estimate of the number of
tightly bound dTTP molecules that are hydrolyzed
upon binding Mg2C. Figure 2(b) shows that the
amplitude increases with increasing concentration
of radiolabeled dTTP, reaching a maximum value of
about four dTTPs/hexamer. These chase-time
experiments, in addition to showing the number
of catalytic sites, provide information about the
initial conditions. The amplitude in Figure 2(b)
gives an estimate of the ensemble average dTTP
occupancy for all helicases in solution. Later we will
use this information as the initial conditions in our
kinetic simulations. Similar chase-time experiments
in the absence of DNA have shown that only one
site per hexamer hydrolyzes dTTP at a fast rate and
a second one at a slower rate, which is consistent
with product release at the first site limiting
hydrolysis at the next.12 Such biphasic kinetics in
the chase-time experiments of dTTP hydrolysis was
not observed in the presence of DNA.
The reason that less than six sites appear catalytic
in the chase-time experiment might be that the
initially bound dTTP dissociates, and the chasetime experiment can only detect the hydrolysis
events of those subunits whose dTTPs do not
dissociate from the catalytic sites. However, we
consider this unlikely because direct dTTP and
dTMPPCP binding experiments have shown about
four nucleotides bound per hexamer.17,18 A more
likely explanation is that negative cooperativity in
dTTP binding biases the preferences of the helicase
to bind no more than four dTTPs in each hexamer.
The basis for the negative cooperativity may be
mechanical because the binding of dTTP introduces
mechanical stress around the enzyme ring. This
circumferential stress becomes large when accom-
455
modating more than four dTTPs in the ring. Similar
negative cooperativity in nucleotide binding has
been reported for other ring-shaped hexamer
helicases.18–21
The chase-time experiments showing that the
average number of catalytic sites is greater than
three for T7 helicase is crucial, because without this
piece of information it is difficult to distinguish a
three catalytic site model and a six catalytic site
model. Based on mechanochemical considerations,
we argue as follows that it is most likely that all six
sites are catalytically active in the hexameric T7
helicase. Figure 3 shows the three different possibilities for a four catalytic site model ((a), (b) and (c))
and the single possibility for a five catalytic site
model (d). A site is considered non-catalytic only if
it remains non-catalytic at all times (at least much
longer than the time-scale of the other kinetic
events). All of the cases shown in Figure 3 are
very unlikely, according to the following reasoning.
The only communication mechanism between
subunits is via the stresses developed at the catalytic
sites during the hydrolysis cycle and the consequent conformational couplings.
As each catalytic site passes through its occupancy cycle, it simultaneously passes through its
stress cycle as well.
Any site that loses its catalytic activity at all times
must do so because the stress cycles of other sites
can bypass its nucleotide binding site.
We will show later that the T7 helicase translocates DNA sequentially. Stress considerations make
it very unlikely that this homo-oligomeric enzyme
can maintain the catalytic site arrangements shown
in Figure 3 when hydrolysis cycles take place
sequentially. That is, it would require very unusual
and asymmetric stresses to keep some sites noncatalytic at all times while other sites cycle.
The five catalytic site model shown in Figure 3(d)
is possible if there is an active open ring, where one
of the sites cannot form the hydrogen bond network
required for hydrolysis. However, this would
reduce the processivity dramatically, which is not
Figure 2. Time trajectories of chase-time experiments and the steady-state amplitudes. (a) The total [dTDP] includes
dTDP-Pi, dTDP inside the catalytic sites and dTDP in solution. Time trajectories of chase-time experiments (60, 110, 250,
500 mM dTTP) fit approximately to equation (4). (b) The amplitudes are obtained from the steady-state values of dTDP
production (A in equation (4)) in (a). This amplitude represents the lower bound of the average number of catalytic sites
initially occupied by dTTP.
456
Figure 3. Configurations for a four or five catalytic site
model. The black ellipses represent sites that are catalytic
at any moment, while the white ellipses are non-catalytic
sites; the non-catalytic site in any hexamer must be noncatalytic at all times. (a), (b), and (c) are the three possible
configurations for a four catalytic site model. (d) The only
possible configuration for a five catalytic site model.
consistent with the observed high processivity
along single-stranded DNA.8
Thus, the occupancy states continuously cycle
their state of stress, so no single site can remain
permanently inactive while subject to the timevarying stresses.
Thus, we conclude that in the presence of DNA, it
is most likely that all subunits are catalytic.
Note that in a sequential mechanism, it is possible
to have a cyclically symmetric three-site or six-site
enzyme. For a three catalytic site model, the stresses
bypass the adjacent site and affect the site two
subunits away. Electron microscopy shows a trimer
of dimer structure for other ring-shaped helicases,22
which could support a three-site proposal. The noncatalytic sites observed previously in the absence of
DNA may belong to this three catalytic site species.
Pre-steady-state kinetics show biphasic
behavior in dTTP hydrolysis
To characterize the rate-limiting step in the dTTP
hydrolysis pathway, acid-quench pre-steady-state
kinetics experiments were performed. In the acidquench setting, T7 helicase was pre-incubated with
M13 ssDNA, dTTP, and [a-32P]dTTP and after a
three minute incubation, the reaction was initiated
by rapidly mixing with a solution of Mg2C. The
reactions were acid-quenched at various times
(milliseconds to seconds). The acid-quench setting
differs from the chase-time setting in that excess
non-radiolabeled dTTP is not added with the Mg2C.
Thus, the acid-quench experiment exhibits the
Mechanochemistry of T7 DNA Helicase
kinetics of dTTP hydrolysis in the first turnover as
well as those of the subsequent ones. If the first
turnover is faster than the subsequent ones, the
kinetics will show a “burst” phase. The burst kinetics
in our experimental setting indicates that a step after
dTTP hydrolysis is rate limiting, because our initial
conditions involved pre-incubating the helicase with
dTTP that bypasses the dTTP binding step.
Figure 4(a) shows the time trajectories of dTTP
hydrolysis in the acid-quench setting at different
initial [dTTP]0 that clearly shows the burst kinetics.
For each dTTP concentration, the kinetics of dTDP
formation was biphasic and can be fit by a single
exponential followed by linear growth (cf. equation
(5) in Experimental Procedures). The initial exponential phase (burst phase) reflects the fast hydrolysis step, while the linear phase is the steady-state
controlled by the rate-limiting step. Figure 4(b)
shows that the slope of the linear phase increases
with increasing dTTP concentrations in a hyperbolic manner. Thus, from low to high concentrations of dTTP the rate-limiting step shifts from
dTTP binding to product release. The burst amplitude, similar to the chase-time amplitude, provides
information about the average number of catalytic
sites undergoing fast dTTP hydrolysis. In the
absence of DNA we previously observed a burst
amplitude about one dTTP/hexamer; here, in the
presence of DNA, about four dTTPs are hydrolyzed
per hexamer at saturating dTTP.
Pi release is the rate-limiting step in the
hydrolysis cycle
To investigate which product release (Pi or dTDP)
is the rate-limiting step, a coupled assay was used
to measure the real-time kinetics of Pi release.23
A mixture of T7 helicase, M13 ssDNA, and dTTP
was rapidly mixed with Mg2C and PBP-MDCC in a
stopped-flow instrument, and the fluorescence
change was measured as a function of time.
A standard curve was used to convert the magnitude of fluorescence increase into Pi concentration.
Figure 4(c) shows the time trajectories of Pi release
at different initial concentrations, [dTTP]0. Unlike
the acid-quench experiments there are no clear
biphasic characteristics. The linear steady-state
phase has approximately the same slope as in the
acid-quench experiment for each corresponding
[dTTP]0, because the rate-limiting step is the same
for both experiments. In Figure 4(d) comparison of
the acid-quench and Pi-release experiments show a
delay in Pi release relative to dTTP hydrolysis.
These results indicate that the hexamer subunits
hydrolyze dTTP at a faster rate than they release the
product Pi. The kinetics were modeled to determine
the rate constants and, as shown in Table 1, we find
that at high dTTP concentration Pi release is the
rate-limiting step.
dTDP$ Pi is weakly bound to DNA
Figure 4(d) shows the delay in Pi release relative
457
Mechanochemistry of T7 DNA Helicase
Figure 4. Experimental data and simulation results for [dTDP] and [Pi] time trajectories. (a) The total [dTDP] includes
dTDP-Pi, dTDP inside the catalytic sites and dTDP in solution. The simulations (continuous lines) agree well with the
transient and steady-state experiments for four different initial dTTP concentrations. (b) The steady-state hydrolysis rate
per hexamer from slopes of the linear phases in (a). It shows a hyperbolic trend toward saturation with increasing
[dTTP]. (c) [Pi] in solution is measured and simulated under the same four initial dTTP concentrations. The continuous
lines represent the simulation results. (d) Comparison between dTTP hydrolysis and Pi release at 500 mM dTTP. The
delay in of Pi release in the transient phase is caused by the slow Pi release compared to dTTP hydrolysis.
to dTTP hydrolysis by the hexamer subunits. This
also provides insight into the DNA binding affinity
of the dTDP-Pi state. As subunits start hydrolysis
and produce dTDP and Pi, Pi is not immediately
released when dTTP is hydrolyzed in other sub-
units. For [dTTP]0Z500 mM, more than two subunits per hexamer are delayed in releasing Pi
relative to the subunits hydrolyzing dTTP. If the
dTDP-Pi state has a high affinity for the DNA and Pi
release is more than two subunits behind the
Table 1. Rate constants and corresponding valid ranges for steps in the hydrolysis cycle of T7 helicase
Step
E/T*
T*/N$T*
N$T*/N$T
N$T/DP
DP/D
D/E
T*/E
N$T*/T*
N$T/N$T*
DP/N$T
D/DP
E/D
Reaction
Rate constant
Lower bound
Upper bound
dTTP docking
DNA binding
Power stroke
dTTP hydrolysis
Pi release
dTDP release
dTTP release
DNA release
Recoil power stroke
dTTP synthesis
Pi binding
dTDP binding
5.0!104 MK1 sK1
951 sK1
778 sK1
66 sK1
15 sK1
159 sK1
1.5 sK1
70 sK1
6.2 sK1
0.03 sK1
1.1!103 MK1 sK1
1.0!106 MK1 sK1
4.8!104 MK1 sK1
380 sK1
311 sK1
53 sK1
14 sK1
103 sK1
!1!10K5 sK1
!1!10K5 sK1
!1!10K5 sK1
0.028 sK1
n/a
n/a
5.7!104 MK1 sK1
O1!106 sK1
1.17!104 sK1
297 sK1
18 sK1
O1!106 sK1
2.5 sK1
1.35!103 sK1
120 sK1
0.041 sK1
n/a
n/a
The rate constants are obtained by least-squares fit to the data. The lower and upper bounds of the rate constants are found by varying
one parameter at a time but keeping all others fixed so that the residual errors of the prediction are within tolerance (see the text). This
tolerance is roughly equivalent to 10% prediction error, on average. Numbers in bold are those upper or lower bounds in the same order
of magnitude as the best-fit rate constants.
458
hydrolysis subunit, then at least three subunits must
be attached to the DNA. This contradicts the protein–
DNA cross-linking experiments and electron
microscopy observations showing that singlestranded DNA interacts with only one or two
subunits of the hexamer.13 Thus, we conclude that
the dTDP-Pi bound state must have a low affinity for
DNA.
Mechanical considerations also suggest that the
dTDP-Pi state should have a low affinity for DNA. If
three or more subunits in the dTDP-Pi state bind
DNA, and only one of them is responsible for
translocating DNA, then DNA will be distorted
significantly. The bending of ssDNA at this length
scale requires energy, not favorable for the motor to
proceed. If three or more subunits attempt to move
DNA together, all these subunits must move
simultaneously about the same distance so that
the DNA strand is not ruptured. Moreover, upon
completion of the power stroke, other subunits
must bind DNA to commence the next power
stroke, and this transition will bend ssDNA
considerably. Taken together, we find this scenario
highly unlikely.
Another possibility is that all subunits work
simultaneously. In this scenario, after one power
stroke, all subunits must reset to their original
positions so that the next power stroke can begin.
This reset transition requires the subunits have a
low affinity for DNA so that DNA will not be moved
back to its original position. However, this synchronous low affinity for DNA corresponds to reducing
the duty ratio of the motor, which would allow
DNA to diffuse away. Furthermore, if w2–
3 nt/dTTP are translocated, then the power stroke
distance would have to be w3.6–5.4 nm, which is
impossible given the lengths of the possible DNA
binding loops, and very unlikely if all subunits go
through a dramatic rotation synchronously along
the z axis in one chemical transition.
Taken together, we conclude that DNA will not
bind to three or more subunits simultaneously, and
that the dTDP-Pi state has low affinity for DNA.
dTMPPCP promotes synthesis of dTTP
To determine if the dTTP hydrolysis step is
reversible at the helicase-active sites, we carried
out the [18O]Pi medium exchange experiments
developed by Boyer and co-workers that have
been used to analyze the reversibility of the
hydrolysis step of many ATPases.24 Briefly, high
concentrations of dTDP (6 mM) and [18 O]Pi
(20 mM) were mixed with T7 helicase in the
presence of M13 ssDNA. The dTDP concentration
used here is saturating as determined by a dTDP
inhibition experiment (Ki value of dTDP w200 mM;
data not shown). The time-course of incorporation
of unlabeled oxygen from water into Pi due to
reversible dTTPase reactions was monitored.
The loss of 18O label in Pi was monitored by 31P
NMR experiments, as the 18O bound to the
phosphorus atom shifts the 31P NMR by about
Mechanochemistry of T7 DNA Helicase
0.02 ppm up-field compared that of the 16O species.
Figure 5(a) and (b) show that [18O]Pi to H2O
exchange was not observed even after 12 hours of
reaction, with and without DNA. The absence of
medium [18O]Pi exchange indicates that dTTP
hydrolysis is almost irreversible in the helicaseactive sites or that Pi binding is weak.
Surprisingly, Figure 5(c) shows that when a small
amount of dTMPPCP, a non-hydrolyzable version
of dTTP, is included in the reaction with ssDNA,
dTDP, and [18O]Pi, considerable exchange was
observed. That is, in the presence of dTMPPCP
and ssDNA, Pi can rebind to the catalytic site and
even participate in re-synthesis of dTTP. However,
as shown in Figure 5(d), in the presence of
dTMPPCP, but without ssDNA, no exchange was
observed. One explanation is that the reverse step
dTDPCPi/dTTP can only take place if DNA is
present in the central channel of the ring. Recall that
the dTMPPCP-bound state has a high affinity for
DNA, while the dTDP-bound state has a low
affinity.7 Because dTMPPCP helps recruit ssDNA
into the ring by its high affinity, other subunits are
accessible so that the transition dTDP/dTDP. Pi/
dTTP is possible during the incubation. In the
experiment of Figure 5(b), although DNA, dTDP,
and Pi are all added, DNA cannot be recruited to the
central channel. Therefore, the DNA–dTDP state
cannot be achieved, and so neither can the DNA–
dTDP-Pi or DNA–dTTP states. This scenario also
explains why exchange was not observed with
dTMPPCP in the absence of DNA (Figure 5(d)), and
is consistent with the conclusion that the dTDP-Pibound state has low affinity for DNA. There are
other possible explanations for the results of the
oxygen exchange experiments, but they require
more sophisticated models that include the effects
of multi-subunit interactions (see Appendix A).
A DNA binding-deficient mutant inhibits the
activity of T7 helicase
To distinguish between random and sequential
mechanisms of power strokes (see below), we
studied the activity of mixed helicase hexamers
made with wild-type T7 helicase and an inactive T7
helicase mutant protein. Previous studies have
shown that mixed hexamers can be made by mixing
mutant and wild-type T7 helicase proteins.25
Previous studies were carried out with mixed
hexamers made with a dTTPase-defective mutant
protein, and these studies showed that both the
dTTPase and helicase activities decreased steeply as
a function of mutant protein concentration.25,26
These results support a sequential power stroke
mechanism. However, since the dTTPase-defective
mutant was able to bind ssDNA, the dramatic
decrease in the activity of the mixed hexamer could
be due to ssDNA binding to the mutant subunit in
the hexamer, thereby blocking the rest of the
subunits in the hexamer from binding or translocating ssDNA. Therefore, we chose to carry out mixed
hexamer studies with a DNA binding-deficient
Mechanochemistry of T7 DNA Helicase
mutant, which would not adhere to the DNA in a
mixed hexamer. Previous studies have shown that
R487C mutation in T7 helicase greatly reduces the
DNA binding and helicase activities of T7 helicase,
but the R487C mutant still retains the ability to form
459
hexamers, bind dTTP, and hydrolyze dTTP at the
DNA unstimulated rate.27,28 We mixed R487C and
wild-type T7 helicase proteins in different ratios,
but the final concentration of the protein mixture
was kept constant at 100 nM hexamer, and the
Figure 5. 31P NMR spectra of different species of [18O]Pi under different conditions (the shaded circles represent 18O
and the unshaded 16O). (a) T7 helicase (1 mM), 6 mM dTDP, 20 mM [18O]Pi. No synthesis event was observed with dTDP
and Pi. (b) T7 helicase (1 mM), 6 mM dTDP, 20 mM [18O]Pi, 15 nM M13 ssDNA. The only minimal synthesis event was
observed in the presence of dTDP, Pi, and ssDNA. (c) T7 helicase (1 mM), 6 mM dTDP, 20 mM [18O]Pi, 15 nM M13 ssDNA,
50 mM dTMPPCP. dTMPPCP significantly enhances the synthesis of dTTP in the presence of dTDP, Pi and ssDNA. (d) T7
helicase (1 mM), 6 mM dTDP, 20 mM [18O]Pi, 50 mM dTMPPCP. No synthesis event was observed without DNA even in
the presence of dTMPPCP. Thus, it is necessary to have both ssDNA and dTMPPCP to promote synthesis of dTTP.
460
dTTPase activity of the mixed hexamers was
measured in the presence of ssM13 DNA. Figure 6
shows the measured hydrolysis rates under different [R487C] versus [wild-type] ratios.
To test whether the power stroke mechanism is
sequential or random, we conducted simulations
for both cases and compared them with the
experimental results. Figure 6 shows that when all
subunits are made by the R487C mutant species
ð½R487C=½R487CC ½wtZ 1Þ under our experimental conditions, hydrolysis occurred at about 20% of
the wild-type rate. If DNA translocation is carried
out by hexamer subunits randomly, then after a
power stroke the probability that the next power
stroke is performed by a wild-type subunit or a
mutant subunit depends on the ratio of the two
species. This assumes that both species form
hexamers equivalently. Thus, the overall hydrolysis
rate will be proportional to the average of the wildtype fast rate and the mutant slow rate weighted by
corresponding species amounts. While the ratios of
the two species change linearly, the hydrolysis rates
should also change linearly, as shown in Figure 6.
This is different from the experimental observations
that as [R487C] increases in the protein mixture, the
ssM13-stimulated dTTP hydrolysis rate decreases
sharply. The non-linear decrease in dTTP hydrolysis
rate rules out the random power stroke mechanism.
We also carried out simulations for a sequential
power stroke mechanism and it shows a non-linear
decrease in dTTP hydrolysis rate as shown in
Figure 6. The detail of how the simulation was
conducted is shown below after we introduce the
algorithm of the kinetic model.
In summary, the experiments described in this
Figure 6. Effect of the R487C mutant on the dTTPase
activity of T7 helicase. The ssDNA-stimulated dTTPase
rate at steady-state was measured for a mixture of R487C
and wild-type T7 helicase. The proteins were mixed in
different ratios keeping the final protein constant at
100 nM hexamer. The plot shows the decrease in the
dTTPase rate as a function of increasing [R487C]. The
continuous line shows the predicted behavior of a
sequential power stroke mechanism, while the broken
line shows the predicted behavior of a random power
stroke mechanism.
Mechanochemistry of T7 DNA Helicase
paper show that (i) at saturating dTTP concentrations, up to four dTTPs are hydrolyzed in a single
exponential phase. (ii) At saturating dTTP concentrations, the rate-limiting step in a single dTTPase
cycle is the release of the product Pi. dTTP
hydrolysis is faster at the hexamer subunits, with
the result that more than two subunits of the
hexamer lag behind in Pi release. (iii) We have
shown that DNA binds to the hexamer in the dTTPbound state,7 and experiments described here
indicate that DNA releases immediately after
dTTP is hydrolyzed. (iv) The mutant poisoning
experiments rule out a random power stroke
mechanism and support a sequential power stroke
mechanism.
Using the available data, we develop below a
kinetic model that globally fits all the data, and
construct a numerical optimization method to
deduce the rate constants of the individual steps.
We also develop a computational approach to
identify the main pathways by which a multi-site
enzyme system such as T7 helicase coordinates the
activity at each subunit to drive translocation along
DNA.
The DNA translocation power stroke occurs
during dTTP binding
Previous studies have shown that the affinity of
DNA for helicase is high in the dTTP-bound states,
but low in both the dTDP-bound state and the
empty state.7 Here, we show that the dTDP-Pibound state also has low affinity for DNA. In order
to translocate DNA, the subunit must be in the highaffinity state for DNA. Thus, the DNA translocation
step can only take place in the dTTP-bound state.
In order to describe the translocation of DNA as
the hydrolysis cycle proceeds in each subunit we
require the following minimal kinetic steps: (1)
weak dTTP binding (or “docking”); (2) strong dTTP
binding; (3) dTTP hydrolysis; (4) Pi release; and (5)
dTDP release. In addition, (6) DNA binding and (7)
unbinding must also be considered. Here, we
assume that, like many other motors (e.g. F1ATPase, kinesin, and myosin) Pi is released before
dTDP. This is because Pi has many fewer hydrogen
bonds than dTDP, and additionally dTDP has
hydrophobic interactions with the catalytic site, so
that the energy required for Pi release is smaller
than that for dTDP release.
Thus, the kinetic paths in each subunit can be
written in terms of the states of its catalytic site:
(1)
Here, E is the empty state, T* is the weakly bound
dTTP state, T is the tightly bound dTTP state, DP is
the dTDP-Pi state (after hydrolysis), and D is the
dTDP state (after Pi release). The lower row (with N
prefix) corresponds to DNA-bound states. Since
461
Mechanochemistry of T7 DNA Helicase
N$DP is a short-lived state, the energy well for this
state is not very deep and so we omit this state.
Mathematically, it is always possible to lump two
states together into one state and use rate constants
equivalent to the collective jump probabilities of the
original states.
Thus, we can write the principal pathway for a
subunit as:
E4 T*4 N,T*4 N,T4 DP4 D4 E
Power
Stroke
(2)
Structural considerations suggest two possible
mechanisms for the translocation power stroke.
First, the crystal structure suggests that the relative
rotational angles of adjacent subunits of the
hexamer are different for the empty state and the
dTTP-bound state. In this “subunit rotation” model,
when dTTP binds to the catalytic site, it triggers the
alignment of the positive charge groups constituting the “arginine finger” with the negatively
charged g-phosphate of dTTP.29 This alignment
drives the relative rotation between neighboring
subunits, and so the DNA contact residues rotate
with it to translocate the DNA. A second possibility
is that dTTP binding directly deforms the local bsheet structure, as in F1-ATPase.30,31 In this “direct
drive” model, the deformation induced in the bsheet propagates to the DNA binding loop that also
emanates from the b-sheet, and thence to DNA
contact residues to translocate the DNA strand.
Recent crystal structures of the f12 P4 hexameric
packaging motor show its RNA contact residues
undergo a large conformational change from an
ATP-bound state to an ADP-bound state.14 The
translocation stroke is coupled to a different kinetic
step from the helicase studied here, presumably
due to the difference in the high-affinity state to the
substrate RNA or DNA. However, the idea being
that subunit deformation induced by nucleotide
binding and release is the same. These two
mechanisms differ in how stress radiates outward
from the catalytic site to drive the translocation
power stroke. Both mechanisms begin with the
dTTP binding site undergoing the “binding zipper”
transition: the formation of the hydrogen bond
network during dTTP binding.32,33 In both cases,
the power stroke of each subunit is driven by the
binding zipper transition of dTTP from its weakly
bound to the tightly bound state, N$T*/N$T. The
conformational changes accompanying NTP binding
have been shown for F1-ATPase32,33 and myosin.34
The translocation power strokes take place
sequentially around the ring
One of the most important questions that relate to
the mechanism of the ring helicases is the sequence
of the power strokes that drive the movement of the
ring along the DNA strand. Knowing the order of
the power strokes helps clarify how the hexamer
subunits communicate with each other. We address
the following questions: are the translocation power
strokes in the hexamer subunits simultaneous,
random, or sequential? Four power stroke
sequences must be considered.
Simultaneous
Experiments indicate that the helicase translocates an average of 3 nt/dTTP hydrolyzed.8
Since nucleotides in DNA are separated by
w0.3 nm, the distance translocated per dTTP is
w0.9 nm. If all six subunits work simultaneously
(i.e. the power strokes act in parallel), then the
translocation distance for each subunit would have
to be 6!0.9Z5.4 nm. Since the channel height of the
T7 hexamer is only w5 nm, this is very unlikely.
Furthermore, we have mentioned that the reset
transition requires the subunits have a low affinity
for DNA, and this synchronous low affinity for
DNA corresponds to reducing the duty ratio of the
motor, which would allow DNA to diffuse away.
Therefore, we discard this possibility.
Paired sequential
Based on the “dimer of trimers” arrangement in the
crystal structure, Singleton et al. suggested that the
power strokes progress sequentially around the ring,
but with diametrically opposing subunits in the same
state.5 For example, subunit 4 is opposite subunit 1
and so the two pass through the hydrolysis cycle
synchronously. In this scheme, two dTTPs are
consumed during each translocation step, and so
the translocation step should be w1.8 nm. This is
inconsistent with the 0.6–0.7 nm distance of subunit
rotation in the possible DNA contact regions determined from the crystal structure.5 Thus, we consider
the paired power stroke sequence unlikely; indeed,
Singleton et al.5 pointed out that, in the presence of
DNA, asymmetry may occur and the paired sequential mechanism could be invalid.
Random
There are two possible random power stroke
mechanisms. (1) Random in time: the power stroke
of each subunit starts and finishes at random times
independent of other subunits. This mechanism is
not consistent with the very high processivity
observed in experiments because there is an
appreciable probability that none of the subunits
bind DNA at the same time. Thus, we can discard
this mechanism. (2) Random in sequence: the power
strokes are sequential in time (i.e. each subunit can
only commence after another subunit finishes), but
the order of the power strokes around the ring is
random. That is, the sequence of the power strokes
is not 1-2-3-4-5-6, but random, as in rolling a die.
After subunit 1 binds and moves the DNA strand,
subunits 2–6 have equal probabilities of binding
DNA and moving it. In terms of binding DNA, it is
still sequential (one after the other), but in terms of
the power stroke order, it is random. The mutant
poisoning experiment does not support the random
mechanisms.
462
Sequential
This mechanism carries out power strokes in
strict sequential order: 1-2-3-4-5-6. After one subunit finishes a power stroke, the adjacent subunit
binds DNA and executes the next power stroke.
This scheme requires cooperativity between adjacent subunits to coordinate the sequence. We
consider this mechanism most likely, for it is
consistent with the structural features that permit
adjacent subunit coupling discussed below.
It is also possible that the power strokes are
partially sequential; that is, sequential but with
some random variation. For example, after translocation by subunit 1, there is a probability of
continuing the next translocation in subunit 2 and
another probability to continue in subunit 3.
Currently, there is no evidence that can unequivocally demonstrate that the order of translocation
power strokes is strictly sequential or mixed. The
only thing known is that power strokes cannot take
place simultaneously. Here, we use the purely
sequential mechanism to illustrate how “one after
the other” power strokes can work. This sequential
mechanism is consistent with the electron
microscopy observations and the mechanism
shown for the P4 packaging motor.13,14 We will see
that this fits all of the experimental observations
satisfactorily.
Mechanical stress coordinates the hydrolysis
cycles between subunits
Based on experimental data and structural
considerations we develop a set of rules that govern
sequential hydrolysis of dTTP and processive DNA
translocation. If the power strokes are sequential,
the translocation step N$T*/N$T can take place
only when the DNA is bound to only one subunit.
Otherwise, each translocation step would be
impeded by the DNA being tethered to an adjacent
subunit. After completing one power stroke, the
next subunit must bind DNA to continue translocation. Using the kinetic notation in equation (2), the
DNA binding step in the next subunit, T*/N$T*,
commences when the previous subunit in the
sequence has completed its power stroke and is in
the N$T state. Geometrically, this is possible if the
power stroke of the previous subunit brings the
DNA strand into a position where it can quickly
fluctuate to the next subunit.
Since hydrolysis enables release of the DNA
strand, in order to ensure high processivity the
unbinding of DNA in one subunit must take place
after the binding of DNA to the next subunit. Thus,
the transition N$T/DP in one catalytic site must
follow the binding of nucleotide to the next site, i.e.
state T*/N$T*. (Recall: N$DP is omitted from the
kinetics in equation (2) because it is a short-lived
state.) The most likely explanation for this is that
efficient hydrolysis is greatly enhanced when the
two subunits that are bound to DNA form a
mechanical stress loop through the arginine finger
Mechanochemistry of T7 DNA Helicase
(see Figure A1 in Appendix A).5,29 The effect of this
transmitted stress is to ensure the catalytic residues
and the catalytic water molecules are constrained in
the right configurations to form the transition state
for hydrolysis.
To summarize, three cooperative steps are
required for sequential DNA bind–release cycles
that also assure processivity of translocation (they
are shown in Figure 8):
(1) The power stroke, N$T*/N$T, takes place
when the DNA is bound to one subunit.
(2) The transition T*/N$T* requires the previous
subunit to be in the N$T state.
(3) The transition N$T/DP requires the next
subunit to be in the N$T* state.
We also examined the necessity of each of these
three steps, and the simulations showed that these
three steps are essential (see Appendix A).
In order to model the three cooperative steps, we
applied a rate enhancement factor to each of the
above steps. For all other transitions the rate
constant is independent of the states of other
subunits. To ensure a detailed balance, which is
necessary for the kinetic scheme to obey the Second
Law of Thermodynamics, if a forward rate is
enhanced then the same enhancement must be
applied to the reverse rate constant. The standard
free energy of 12.5 kBT is used for the hydrolysis
cycle, so one of the reverse rate constants is
predefined by this energy relationship. This simple
model with cooperativity only in the above three
transitions reduces the total number of fitting
parameters to six forward rate constants, five
reverse constants, and three enhancement factors,
for a total of 14 unknown parameters. The forward
rate constants are kE/T*, kT*/NT*, kNT*/NT, kNT/DP,
kDP/D*, and kD/E. The backward rate constants,
including the one obtained from the free energy
constraint, are kT*/E, kNT*/T*, kNT/NT*, kDP/NT,
kD/DP, and kE/D. The three enhancement factors
are fNT*/NT, fT*/NT*, fNT/DP*, corresponding to the
three enhancement rules stated above. With these
parameters, we can construct the rate equations and
solve them.
Kinetic pathways
As shown above in equation (2), there are six
nucleotide ligation states in each subunit of the
hexamer, so that the total number of states is 66Z
46,656. Each hexamer state can be denoted by
EEEEEE (all sites empty), EEDDDD (four sites
occupied by dTDP), etc. The rate equation for all
states can be written as pseudo-first-order
equations:
dc
Z K,c
dt
(3)
where c(t) is a concentration vector of all the
possible kinetic states. Thus, c(t) is a concentration
463
Mechanochemistry of T7 DNA Helicase
vector with 46,656 elements, and K is the 46,656!
46,656 rate matrix. For example, k12 is the rate
constant for the change of state [EEEEEE] (i.e. the
concentration of state EEEEEE) as a function of
[EEEEET*]:kEEEEET*/EEEEEE. Equation (3) corresponds to the experimental conditions where the
concentrations of dTTP, dTDP, and Pi do not vary
too much over the course of an experiment, so that
their concentrations can be incorporated into the
rate matrix, K. (However, the algorithm we use is
not confined by the linear approximation.)
Notice that the complexity of this large system of
equations is purely computational: the underlying
chemistry includes only the three cooperative steps,
similar to the classical treatment of subunit interactions.35–37 These three cooperative steps developed from structural and mechanochemical
considerations impose constraints on the system
such that many of the 46,656 states are either totally
inaccessible or have very low probability. The
computational results will report these inaccessible
states as well as the time-dependent concentrations
of those accessible states. The purpose of the
algorithm is to find the main pathway(s) by fitting
this equation to the data. The difficulty arises from
the large number of coupled differential equations.
However, the rate matrix, K, while large, is sparse,
so that an efficient numerical method can be used to
integrate the equations.38
The initial conditions can be inferred from the
chase-time experiments. The optimization algorithm is used to obtain 14 parameters needed for the
computation. The details of obtaining initial conditions and defining the necessary parameters are
given in Appendix A.
The results of the calculations are shown in
Figure 4, and the computed rate constants are given
in Table 1. The fit to the data is quite good in both
the transient and steady-state phases. The free
energy diagram calculated from these best-fit rate
constants is given in Appendix A.
A sensitivity test was conducted to find the valid
ranges of the rate constants obtained from the fitting
algorithm. To determine the range of one rate
constant, we varied that constant while keeping
all other constants fixed, and then computed the
time trajectories and the corresponding residual
error of the new parameters. The residual error is
Scase i St DtðCðtÞi;compute K CðtÞi;measure Þ2 , summing
the concentration differences of all cases and all
time steps. When that constant is different from the
best-fit rate constant, the residual error is greater
than the best-fit residual error. We chose a tolerance
residual error roughly equivalent to 10% prediction
error on average. The upper and lower bounds of
each rate constant reported in Table 1 are identified
when the corresponding residual error equals the
tolerance residual error. The Table shows that some
rate constants are sensitive while others are not.
Thus, our experiments can determine some rate
constants with good accuracy, leaving other constants underdetermined. The bold-face numbers in
Table 1 are those whose best-fit rate constants are of
Box 1. Sketch of the computational algorithm
The solution to equation (3) determines the rate
constants for the pre-steady-state kinetic data shown
in Figure 4. The flow chart for the computational
algorithm is shown in Figure 10 in Appendix A,
where more details can be found. The underlying
principle is to first solve the rate equations with a
trial set of rate constants, and then to use the
optimization algorithm to find a set of best-fit rate
constants. The best fit is defined as the least-squares
error of the prediction compared with the experimental data. First, a set of 14 estimated parameters
(11 rate constants and three enhancement factors) are
assigned to the corresponding kinetic steps as
elements of the rate matrix K. For example, the
element in the matrix K is the rate constant, since it
corresponds to the transition T*/E in the last
subunit. Another example: the element is assigned
to, since it corresponds to the mechanical coupling
enhancement for the hydrolysis step, with the next
subunit ready to take over the next power stroke.
Once the matrix is constructed, the rate equation
dc/dtZK$c can be solved using an iterative method
(the
quasi-minimal
residual
method
in
MATLABe).38 Simply guessing the set of initial
parameters gives a poor fit, so we used the simplex
optimization method.39 An initial 14 dimensional
simplex is constructed consisting of 14C1 vertices.
Each vertex contains a set of 14 fitting parameters.
For each vertex (i.e. each set of rate constants), a rate
matrix K is constructed and the rate equation is
solved. To compare the predictions with the
measurements, the computed concentrations must
be converted to the corresponding experimental
concentrations. For each vertex, we obtain the sum of
the square of the difference between predictions and
measurements:, where cp is the predicted concentrations and cm is the measured concentrations. The
prediction error, E, is to be minimized. After
prediction errors are obtained for all vertices, the
vertices of the simplex are moved according to the
Nelder–Mead method so that the size of the simplex
gradually shrinks corresponding to convergence to a
minimum.39 The best-fit parameters are obtained
when the size of the simplex is smaller than a
predetermined cutoff.
the same order of magnitude. For example, our
experiments can determine the dTTP docking rate,
the Pi release rate, and the dTTP synthesis rate.
Some rates have well-defined lower bounds while
others have well-defined upper bounds. A sensitivity test of the Pi and dTDP binding rates is not
possible because dTDP and Pi concentrationdependent experiments are not available.
With the best-fit rate constants and their corresponding ranges being considered, Table 1 shows
that when the initial concentration, [dTTP]0, is low,
the E/T* step is rate-limiting. When [dTTP]0 is
high, the E/T* step and the DP/D steps are of the
same order, and a combined effect must be
considered. Changing [dTTP]0 shifts the rate limiting step. The slow DP/D step is consistent with
the phase delay between hydrolysis and Pi release.
The mechanical power stroke step, N$T*/N$T, is
464
fast compared to other kinetic steps. A mechanical
step is generally not equivalent to a kinetic step. If a
mechanical step is rate limiting, the dynamics of the
system is dominated by spatial motion. A kinetic
process is an instantaneous jump transition with a
distribution of waiting times in the source state;
therefore, it is generally not a good model for a
mechanical step. On the other hand, if the mechanical step is fast compared to other kinetic steps, the
dynamics of the system are dominated by the
kinetic process. In this case, it is appropriate to use a
kinetic step to approximate the mechanical step,
with equivalent mean passage time chosen for the
kinetic approximation.
We used the same approach for simulations of the
DNA binding-deficient mutant, as illustrated in
Figure 6. First, we assume the mutant species have
the same property as the wild-type species in
forming hexamers. Thus, different proportions of
wild-type versus mutant species give different
distributions of various mixed hexamers, such as
WWWWWW, WWWWWM, etc., where W represents the wild-type subunit while M represents
the mutant subunit. There are a total of 64 species of
mixed hexamers and their initial distributions can
be computed from the [R487C] versus [wild-type]
ratio. For each species, the kinetic equations are
formulated and the steady-state hydrolysis rate can
be obtained. Different species have a different rate
matrix. For a mutant subunit, because it is unable to
bind DNA, the states N$T* and N$T are not
reachable. This subunit, however, is still able to
hydrolyze dTTP as observed in the all mutant
hexamers. Thus, we assign a detour state for mutant
subunits: E4T*4T4DP4D4E. We also make a
preliminary assumption that all rate constants,
except the one of the hydrolysis step, are the same
as those for the wild-type subunits. The power
stroke here T*/T has the same rate constants as
NT*/NT. The hydrolysis step T/DP, however,
becomes the rate-limiting step because of the lack of
the subunit–subunit coordination as described
above. This slow hydrolysis rate can be easily
found by fitting to the data of all mutant species
(MMMMMM) shown on the right in Figure 6. With
this rate constant and all others known, we
construct the rate matrix for each species and obtain
its corresponding steady-state hydrolysis rate.
Knowing the distributions of all species under a
certain [R487C] versus [wild-type] ratio, we then
weight them accordingly and obtain the average
hydrolysis rate for each case. The simulated results
for this sequential power stroke mechanism are
shown as the continuous line in Figure 6. The trend
of the simulation is similar to the trend of the
experimental data, while in the middle range of
mixing the simulation underestimates the rate. One
possible explanation for this discrepancy is that the
mixing of the mutant and the wild-type species may
not be uniform, so that it may have a higher
probability to form homogeneous wild-type rings
than calculated. Another possibility is that the
sequential power stroke is not as strict as we use
Mechanochemistry of T7 DNA Helicase
to compute, but follows the partially sequential
mechanism we describe above.
What is the “main” kinetic pathway?
One goal of the calculation is to find the “main
pathway(s)” amongst the very large number of
possible pathways. One quantitative definition of a
main pathway is that, at steady-state, it carries the
maximum kinetic flux. Using the best fit rate
constants, the steady-state concentrations of all
states can be determined, and the fluxes of all
kinetic steps computed (see, e.g. Hill40). Many
pathways transmit negligible flux and can be
ignored. Those kinetic pathways with a flux
threshold above 3% of the total flux were identified.
Figure 7 shows the steady-state kinetic flux network
under two different concentration conditions. The
thickness of the connections between any two states
is proportional to the kinetic flux between them.
Figure 7 shows that the distribution of pathways
depends on the initial conditions. For physiological
conditions (assumed to be [dTTP]Z100 mM,
[dTDP]Z10 mM, and [Pi]Z1 mM), the flux passes
mostly through one or two pathways with low
occupancy, while under experimental conditions of
[dTTP]0Z500 mM, the kinetics spread over many
pathways. In the latter case, it is not possible to
define a single principal pathway. With a 3% flux
threshold, the net flux in the partial network shown
in Figure 7(a) is about 91% of the total flux, while in
Figure 7(b) the net flux is only about two-thirds of
the total flux. Figure 7(b) also shows the shifting of
kinetic fluxes toward high-occupancy states compared to Figure 7(a), and this is consistent with the
experiments where increasing [dTTP]0 corresponds
to increasing average occupancy. The calculation of
the kinetic flux network shown here can be applied
to any cooperative multiple-protein kinetics.
Translocation along single-stranded DNA
involves multiple kinetic pathways
Here, we summarize how the subunits of the
hexamer coordinate their actions to translocate
along DNA. The three required cooperative steps
in the reaction network are shown schematically in
Figure 8(a). Previous results have shown that T7
helicase translocates on an average 3 nt per dTTP
hydrolyzed; therefore, we can assume that after one
dTTP hydrolysis cycle, the ssDNA is translocated
by 3 nt. Figure 8(b) shows the pathways that were
extracted from the high-flux pathways shown in
Figure 7(b). Within this subset, the translocation can
take place by several pathways. For example, the
first step in Figure 8(a) is the hydrolysis step; this
step occurs when DNA is bound to two subunits
and, as shown in Figure 8(b), one in which the third
subunit is E (in step 1a) and the other where the
third subunit is DP (in step 1b). After dTTP
hydrolysis the DNA dissociates from that subunit
and the N$T* state goes to N$T, which is the power
stroke step that translocates the DNA. Depending
465
Mechanochemistry of T7 DNA Helicase
Figure 7. The steady-state kinetic flux networks for hexameric T7 helicase. The mechanism behind part of these flux
networks is illustrated in Figure 8(b). Here, each hexagon represents a kinetic state of the hexamer. Each subunit of the
hexamer is colored with the corresponding occupancy state shown in the key: white, empty state (E); cyan, dTTP
docking (weakly bound) state (T*); pink, dTTP docking with DNA bound (N$T*); red, dTTP tightly bound to DNA
(N$T); yellow, the dTDP-Pi state (DP); and green, dTDP bound (D). The reaction cycle proceeds across the columns with
the possible reaction pathways linked by lines whose thickness reflect the fractions of the total flux carried through that
reaction branch. Thus, the magnitude of the flux (line thickness) illustrates the relative probabilities of each pathway. The
rightmost column is the same state as the first column but shifted one subunit to form a repeated kinetic cycle. Lines
representing the power stroke driving DNA translocation are colored red; all other transitions are colored blue. The
lower rows have higher occupancy (increasing ligand concentration). The pattern of flux pathways depends on the
solution concentrations. For physiological conditions ([dTTP]0Z100 mM, [dTDP]0Z10 mM, [Pi]0Z1 mM), there is a
dominant kinetic pathway with low occupancy, while for [dTTP]0Z500 mM, the kinetics spread over many pathways. In
the latter case, the concept of a “principal pathway” is not well-defined. Along different pathways the order of the
reaction steps is different. The states shown here are only the most populous of the 66Z46,656 possible states, i.e. those
above a threshold flux of 3% of the total flux.
on what is bound to the third subunit, the power
stroke step result in (DP, N$T, E) or (DP, N$T, DP)
states, shown as states after step 2a and step 2b,
respectively. In order to pass the DNA to the third
subunit for the next power stroke, the third subunit
must be in the T* state. The (DP, N$T, E) state can
reach this state via two pathways: through steps 3a,
4a, and 5a to (E, N$T, T*) state, or through steps 4b
and 5b to (DP, N$T, T*). The cycle is then reset after
the subunit in the T* state binds DNA to become the
N$T* state through steps 6a or 6b, where two
subunits once again bind DNA. In this simplified
network, (DP, N$T, E) is the bottleneck that every
pathway must pass through. If the network
includes all six subunits there are many more
pathways and the bottleneck may not exist for
there are multiple kinetic pathways for T7 helicase
forming a reaction network to accomplish the
sequential translocation task processively.
Discussion
Based on a combination of experimental observations and computational methods, we propose a
consistent mechanochemical model for ssDNA
translocation by the hexameric T7 helicase. We
analyzed data from (i) DNA and nucleotide binding
affinities, (ii) pre-steady-state dTTP hydrolysis
kinetics in the presence and absence of DNA, (iii)
medium oxygen exchange, and (iv) a mutant
poisoning experiment. We combined these observations with the structural arrangement of the
catalytic sites to deduce a cooperative mechanism
466
Mechanochemistry of T7 DNA Helicase
Figure 8. Cooperative steps and multiple pathways of the T7 helicase kinetic network. (a) Schematic plot of the
sequential steps in a typical translocation cycle. The vertical bars represent ssDNA with the red intervals representing the
binding phosphate groups three nucleotides apart. The orientations of DNA contacting loops are illustrated on both
sides of DNA by the “lever arms” (see Figure 1). The contacts between these loops emanating from subunits 1, 2, and 3
(number in the upper left corner) are shown on both sides of the ssDNA. DNA is translocated downward in this
reference frame by power strokes driven by dTTP binding to the catalytic site. Only three steps are necessary to assure a
highly processive sequential mechanism. First, hydrolysis takes place when the next subunit binds to DNA ensuring that
DNA will not diffuse away. When only one subunit binds to DNA, the power stroke can proceed without constraint.
After each power stroke, the next subunit must proceed to the N$T* state (weak binding to DNA) in order for the whole
cycle to repeat. (b) Multiple pathways extracted from the high-flux pathways shown in Figure 7(b) with matching colors
of states. The helicase is shown “unwrapped” to form a band of subunits, and the position of each nucleotide binding site
is shown. The diagonal orange line represents phosphate groups of ssDNA which are translocated downward. The blue
dots represent the positions of ssDNA binding residues. Only three out of six subunits adjacent to the DNA binding
subunit are shown; the other three subunits can be in several possible states (E, T*, DP, or D). The network shown here is
meant only to represent a typical sequence amongst multiple pathways. Some kinetic steps are projected onto the
corresponding steps shown in (a).
that translocates ssDNA sequentially. In the absence
of DNA, we observed several sites that turned over
dTTP slowly. The studies and analyses in the
presence of DNA presented here show different
behavior and support the view that all subunits of
the T7 helicase hexamer are catalytic. Although we
observed that less than six sites are hydrolytic at
any one time, this is likely due to the negative
Mechanochemistry of T7 DNA Helicase
cooperativity in NTP binding that has been
observed in other hexameric helicases.18–21 The
experimental data show that the rate-limiting step
changes from dTTP binding to Pi release as the
dTTP concentration increases. The hydrolysis of
dTTP constitutes an important step in inter-subunit
communication and in modulating the DNA binding affinity. The dTTP bound state has a high affinity
for DNA, whereas the dTDP-Pi, dTDP, and empty
states have low affinity for DNA. This information
allowed us to isolate the mechanochemical coupling
step that transduces dTTP binding energy to DNA
translocation. DNA binds to a subunit when that
subunit binds dTTP; the power stroke that drives
DNA translocation must occur as the dTTP anneals
into a tight binding configuration in the catalytic
site. Fast dTTP hydrolysis dissociates the DNA
from the driving loop, but allows the next subunit
in the dTTP bound state to bind DNA and continue
the cycle of DNA translocation.
Sequential mechanisms in which subunits of the
protein bind and release DNA successively have
been proposed for ring-shaped5,15,20,21 and nonring-shaped helicases3 as well as proteins such as
the f12 P4 hexameric packaging motor.14 These
models explain processive translocation of these
motor proteins along nucleic acids. The models
based on crystal structure data for ring-shaped
helicases are based on structures in the absence of
the nucleic acid bound in the central channel.5,41,42
Only two ring helicases, Rho protein and T7
helicase, have been characterized by detailed
biochemical and kinetic studies in the presence of
nucleic acid. Sequential hydrolysis of ATP and RNA
binding and release around the hexamer ring has
been proposed for the Rho protein where three sites
have been invoked in the mechanism,20,43 which is
different from the six-site mechanism that we
propose here for T7 helicase in the presence of
DNA.
Recently, Gai et al.44 have suggested that the
hexameric helicase of simian virus 40 (SV40) large T
antigen works by a concerted mechanism. This
model is also based on structural data obtained in
the absence of DNA. The structures of SV40 large T
antigen in the absence of nucleic acid demonstrate
many important aspects of the protein function.
They show conformational changes inside a subunit
and between adjacent subunits when in different
nucleotide binding states and these conformational
changes are proposed to move DNA. It is possible
that when DNA is present the resulting asymmetric
forces will induce only part of all subunits to change
conformations and to move DNA. The shapes and
the charge distributions are highly asymmetric, not
surprising considering the major and minor
grooves of DNA, as well as the base and the
phosphate backbone in any cross-section of ssDNA
and dsDNA. These asymmetric properties of the
DNA when bound in the central channel introduce
asymmetric forces on the surrounding subunits.
Without the imposition of these asymmetric forces,
the ring will show symmetric arrangements, as
467
observed in the crystal structures. Although, the
concerted mechanism cannot be ruled out, it is not
easy to see how all subunits can reset in concert
without losing contact with DNA. If the motor does
not hold DNA most of the time (high duty ratio),
DNA will diffuse away or at least slip several basepairs.
We developed a computational algorithm to
determine the rate constants for the NTP hydrolysis
cycle and to determine the kinetic fluxes through
the reaction network, which is a priori quite large.
This allowed us to deduce a reduced set of states to
characterize the helicase. The results reveal that
dTTP hydrolysis can occur by multiple pathways,
particularly at saturating dTTP concentrations.
Multiple pathways provide more degrees of freedom for the system to find the “low impedance”
sequence, as shown in Figure 8(a). Parallel kinetic
pathways also exist in other multi-subunit motors
such as myosin V.45 In multimeric proteins the
existence of multiple pathways reduces the necessity for strict cooperativity among subunits at each
kinetic step. For some required sequential steps the
cooperativity must be strong enough to ensure that
the motor proteins can work processively, but for
other kinetic steps many pathways are possible. The
presence of both high and low cooperative interactions ensure robustness of these processive motor
proteins.
The elucidation of the kinetic pathways described
here has been influenced by reasoning about
mechanical stresses to identify and quantify possible cooperative interactions and to eliminate
unacceptable ones. We still do not have a complete
knowledge of the detailed interactions between the
subunits and how these interactions change as the
helicase moves along the DNA. However, because
the only possible mechanism for cooperative
interactions is via mechanical stress, the number of
possible coupling mechanisms is limited. We tried
several possible mechanisms and the one shown
here is the only one we found that could explain the
data. There may be other complicated mechanical
interactions that we did not consider, and our
methods cannot rule out those possibilities. The
detailed distributions of the kinetic pathways may
require revision as future studies reveal more
detailed information about conformational changes
in the protein.
The large matrix analysis we have employed here
to model the data has several advantages over the
usual Michaelis–Menten analysis. Michaelis–Menten analysis may provide insights into the individual steps, including the binding events, inhibition
effects, and a lumped rate for steps other than the
binding step. However, it is based on the assumptions that the system is in the steady-state and there
is only a single kinetic pathway, which is almost
never the case. Departure from these assumptions
produces non-hyperbolic enzyme kinetics and
requires more complicated kinetic equations,
especially when considering the effects of subunit
interactions. The method we have employed to
468
analyze our experiments is a modification of
classical models for subunit interactions, including
the Monod–Wyman–Changeaux (MWC) and the
Koshland–Nemethy–Filmer (KNF)-type models.35,46
Since these methods were proposed, computational
power has advanced to the point that we could
include all six subunits in the simulation. This large
matrix simulation allows us to identify the distribution of pathways for the T7 helicase (see Figure
7), and enables fitting to both transient and steadystate phases of the data. Because the matrix is large,
it might appear that the results are not reliable.
However, the situation is simpler than the dimensionality might suggest, since all six subunits have
exactly the same kinetic cycle, and there are only
three cooperative steps. Our method is similar to
the free energy of activation treatment proposed by
Ricard et al.36,37 The complexity lies only in
computation, not in the underlying physics; that
is, our method provides a way, based on simple
physics, to attack multimeric time-dependent kinetics problems.
The algorithm developed here for T7 helicase can
be applied to any multimeric, multi-state enzyme,
including ring-shape enzymes such as the virus
portal protein, helicases, GroEL, and F1 ATP
synthase. It is also easy to implement for walking
motors such as kinesins or double-headed myosins,
although the total number of states for these motors
is within the purview of the analytical method of
Fisher & Kolomeisky.47–49 Our method is not
restricted to steady-state kinetics; indeed, presteady-state kinetics and mechanical measurements
(e.g. force–velocity curves) can be incorporated, so
long as the power stroke step can be modeled as a
thermally activated event. In a subsequent publication this restriction will be lifted so that the power
stroke can be described by an arbitrary mechanochemical potential. Within its limitations, this
algorithm provides a systematic procedure to
study the multiple pathways of cooperative, allosteric enzymes.
Mechanochemistry of T7 DNA Helicase
Pre-steady-state chase-time kinetics of dTTP
hydrolysis
The experiments were conducted using a rapid quenchflow instrument at 25 8C. T7 helicase was mixed with
dTTPC[a-32P]dTTP, M13 ssDNA, and EDTA in the
helicase buffer and immediately loaded into one syringe
of the quench-flow instrument. After three minutes, 24 ml
of the enzyme complex was rapidly mixed with an equal
volume of unlabeled dTTP and MgCl2 from a second
syringe of the instrument. The final concentrations after
mixing were as follows: T7 helicase (1 mM hexamer),
radiolabeled dTTP (60 mM–500 mM), chase dTTP (5 mM),
MgCl2 (20 mM) and M13 ssDNA (15 nM). After various
chase-times, the reactions were quenched with 4 M
formic acid and products were analyzed as described.12
The chase-time kinetics were fit to equation (5) and the
linear phase of dTTP hydrolysis due to inefficient chase
was subtracted. The resulting chase-time kinetics was fit
to a single exponential:
DðtÞ Z Að1 K eKk1 t Þ
(4)
where D(t) is dTDP at time t, A is the burst amplitude, and
k1 is the exponential rate constant.
Pre-steady-state acid-quench kinetics of dTTP
hydrolysis
The experiments were conducted at 25 8C using a rapid
chemical quench-flow instrument (KinTek RQF3 software, State College, PA). T7 helicase was mixed with
dTTPC[a-32P]dTTP, M13 ssDNA, and EDTA in the
helicase buffer and immediately loaded into one syringe
of the quench-flow instrument. After three minutes, 24 ml
of the enzyme complex was rapidly mixed with an equal
volume of MgCl2 from a second syringe of the instrument.
The final concentrations after mixing were as follows: T7
helicase (1 mM hexamer), dTTP (60–500 mM), MgCl2
(20 mM) and M13 ssDNA (15 nM). The mixed reactions
were quenched after millisecond to second intervals with
4 M formic acid and products analyzed as described.12
The acid-quench kinetics were fit to the burst equation:
DðtÞ Z Að1 K eKk1 t Þ C mt
(5)
where D(t) is dTDP at time t, A is the burst amplitude, k1
is the exponential rate constant, and m is the slope of the
linear phase.
Experimental Procedures
dTTPase activity of mixed wild-type and mutant T7
helicase hexamers
Protein, nucleotides, and buffer
R487C and wild-type T7 helicase proteins in different
ratios (final concentration of protein mixture, 100 nM
hexamer) were mixed in helicase buffer for 30 minutes
and the dTTPase reaction was initiated by adding Mg2C,
2 mM dTTP spiked with [a-32P]dTTP, and 2 nM ssM13
DNA. The reactions were stopped after 15, 30, 45, 60, 120,
180, 240 seconds with 4 M formic acid, and dTTP and
dTDP were quantified as described above. The steadystate dTTPase rate was calculated from the time-course
(slope) and plotted as a function of [R487C].
T7 gp4A 0 protein (referred to as the T7 helicase) is a
M64L mutant of T7 helicase–primase protein and
R487C were over-expressed and purified as
described.18,50 The protein concentration was determined both by absorbance measurements at 280 nm in
8 M urea (the extinction coefficient is 76,100 MK1 cmK1) and
by the Bradford assay using bovine serum albumin as a
standard. Both methods provided similar concentrations.
The M13 ssDNA (M13mp18) was purified as described.51
dTTP and dTDP were purchased from Sigma Chemicals,
and [g-32P]dTTP was obtained from Amersham Pharmacia
Biotech. Helicase buffer (50 mM Tris–HCl (pH 7.6), 40 mM
NaCl, and 10% (v/v) glycerol) was used throughout the
experiments unless specified otherwise.
Stopped-flow kinetics of inorganic phosphate release
The A197C phosphate-binding protein (PBP) was
purified and labeled with MDCC as described.23
Inorganic phosphate release reactions were performed
469
Mechanochemistry of T7 DNA Helicase
at 25 8C using a stopped-flow instrument manufactured
by KinTek Corp. (State College, PA). The instrument
syringes were treated with the phosphate mop (0.5 unit/
ml of PNPase and 300 mM 7-methylguanosine) to remove
contaminating Pi.
A phosphate calibration curve was created as follows.
To obtain a baseline value of fluorescence intensity, C0,
PBP-MDCC (10 mM) from one syringe of the stopped-flow
instrument was mixed with the helicase buffer from a
second syringe. The fluorescence intensity, which
remained unchanged with time for at least 300 ms, was
measured using a 450 nm long pass filter (Corion LL-450
F) after excitation at 425 nm to obtain the C0 value. This
baseline fluorescence intensity value was determined
before each Pi standard reaction. Immediately after C0
determination, PBP-MDCC (10 mM) from one syringe was
mixed with a known concentration of Pi (0.1–0.8 mM) from
a second syringe. The fluorescence intensity increased
with time due to Pi binding to the PBP-MDCC protein,
and the maximum fluorescence intensity provided Ci. The
value Ci–C0 was plotted against the final Pi concentration
to generate the phosphate calibration curve.
The Pi-release kinetics were measured in the stoppedflow instrument by reacting 40 ml of solution A with
solution B. Solution A consisted of T7 helicase (0.2 mM,
hexamer), EDTA (5 mM), Pi -mop (0.5 unit/ml of PNPase
with 300 mM 7-MEG), M13 ssDNA (3 nM), and dTTP
(200–3200 mM), and solution B consisted of MgCl2 (45.2–
48.2 mM), Pi-mop, and 10 mM PBP-MDCC. A total of four
or five kinetic traces were averaged and the fluorescence
intensity time-courses were converted to the kinetics of Pi
release using the phosphate calibration curve. We ensured
that the phosphate mop did not compete with PBP-MDCC
for phosphate (kcat/Km for the PNPase reaction with Pi is
3.2!106 MK1 sK1).52 A control experiment was carried
out for each dTTP concentration by mixing solution A
with solution B lacking Mg2C. The control experiment
was subtracted from the experiment with Mg2C.
Phosphate–water oxygen exchange experiments
[18O]Pi was prepared by reacting PCl5 (Sigma) with
99% H18
2 O (Aldrich Chemical) followed by ion-exchange
chromatography (Bio-Rad; AG1-X4).53 The resulting
inorganic phosphate had an enrichment of 96% 18O.
The concentration of [18O]Pi was determined by the
molybdate method.54 T7 helicase (1 mM hexamer) was
incubated with 10 mM MgCl2, 0.5 mM EDTA, 6 mM
dTDP, 20 mM [18O]Pi, M13 ssDNA (15 nM), and 60%
(v/v) 2H2O in 0.5 ml of helicase buffer at 22 8C for 2–12
hours. The reactions were quenched by vortexing with
chloroform to denature the enzyme, and aqueous phase
was extracted for 31P NMR analysis. The distribution of
P18O16
j O4Kj species, where 0%j%4, was determined by
31
P NMR.55 The spectra were obtained with JEOL GX-400
instrument at a frequency of 162 MHz for phosphorus.
Five hundred scans were accumulated and then Fourier
transformed. The P18O16
j O4Kj peaks were each separated by
shifts of 0.02 ppm.55 Fractional contribution of each [18O]Pi
species to the spectrum was evaluated directly from the
peak area.
Acknowledgements
S.S.P. and Y.-J. J. were supported by NIH grant
GM55310 and J.-C.L. and G.O. were supported by
NIH grant GM59875-02. The authors thank Dale
Wigley, Oleg Igoshin, Jianhua Xing, Michael Grabe,
and Joshua Adelman for valuable discussions.
Appendix A
Determination of the optimal concentration of
M13 ssDNA
T7 helicase (2 mM, hexamer) was incubated with
M13 ssDNA (0–50 nM, molecules) in the helicase
buffer containing 5 mM EDTA and 1 mM dTTP.
After 15 minutes, radiolabeled 76-mer ssDNA (final
6 mM) was added and the samples were filtered
through a NC-DEAE membrane assembly. The
membranes were washed before and after filtration
of samples with the membrane wash buffer (50 mM
Tris–HCl (pH 7.5), 5 mM NaCl). After the samples
were filtered, radioactivity on both NC and DEAE
filters was quantified using a Phosphoimager
(Molecular Dynamics). The molar amount of T7
helicase–76-mer complex was calculated from the
ratio of radioactivity on NC over the sum of
radioactivity on both membranes, as described.7
In the absence of M13 ssDNA, a large amount of
radiolabeled 76-mer–gp4A 0 complex was observed.
As the concentration of M13 ssDNA molecules was
increased in the reaction, the amount of helicase–76mer complex decreased, and beyond 20 nM M13
ssDNA, no helicase–76-mer complex was observed,
indicating that no free helicase was present in the
reaction to bind radiolabeled 76-mer. These experiments indicate that for each mmol of T7 helicase
hexamer, we need at least 10 nmol of M13 ssDNA
molecule to saturate the binding of protein to DNA.
We therefore used a ratio of 15 nM M13 ssDNA per
mM T7 helicase hexamer in all our experiments. In a
separate experiment, we varied the time of incubation of the helicase with DNA and dTTP. No free
protein was detectable even after a short (two
minutes) period of incubation (data not shown).
Based on this experiment, we chose a three minute
pre-incubation period for all experiments, which
also gives us enough time to load the pre-incubated
sample in the quench-flow instrument.
The effect of adding Mg2C in the kinetic steps
In quench, chase, or Pi release experiments, Mg2C
is added after dTTP and the helicase is allowed to
reach equilibrium. That is, an extra state should be
included to distinguish the states with and without
Mg2C in the catalytic site. Experiments show that
the hydrolysis rate is negligible before Mg2C is
added to the solution.17 Therefore, before Mg2C is
added, the subunits can only be in E, T**, NE, NT**
states, where T** represents the dTTP docking state
without Mg2C. After adding Mg2C, Mg2C can
diffuse into the catalytic site to become the T* or
NT* state, or Mg–dTTP can be formed in solution
and directly dock to the catalytic site. In the latter
470
case, the E (NE) state can directly go to the T* (NT*)
state without passing through the T** (NT**) state.
Since Mg2C is much smaller than dTTP, its
diffusion to the catalytic site is very fast. Therefore,
T**/T* is very fast and so is NT**/NT*. Under
this assumption, the T** and NT** states can be
omitted. The assumption may break down if, after
Mg2C diffuses into the catalytic site, Mg–dTTP
takes a long time to rearrange its coordination so
that the correct configuration of T* state is achieved.
In this case, the rate of E to T** is not equivalent to E
to T*, and states T** and NT** must be retained in
the model. In the main text, we lump states T** and
T* as one state, T*.
Alternative explanations for oxygen exchange
experiments
Considering multi-subunit interactions provides
alternate explanations for the oxygen exchange
observations. dTMPPCP is a non-hydrolyzable
version of dTTP. When dTTP is replaced by
dTMPPCP at one site, it may delay, or block, the
Mechanochemistry of T7 DNA Helicase
kinetic steps following dTDP-Pi (DP) or dTDP (D) at
the adjoining site, increasing the probability of the
reverse step dTDP-Pi/dTTP. For example, if dTTP
hydrolysis at one site affects Pi release at an adjacent
site, then non-hydrolyzable dTMPPCP will prevent
release of Pi, giving time for dTDP–Pi/dTTP.
Mechanically, hydrolysis in one subunit can induce
stress in adjacent subunits facilitating Pi release.
DNA provides coordination between these subunits through an additional stress loop shown in
Figure A1(c), so in Figure 5(d) when DNA is absent,
synthesis events are not observed. Under this
hypothesis, in the presence of DNA, Pi is stuck in
the catalytic site unless dTTP hydrolyzes in another
site.
Another possibility is that DNA binding, T*/
NT*, in one site triggers the release of Pi in the
adjacent site. Notice that the binding of DNA
requires the subunit to be in its weakly bound
dTTP state. When dTMPPCP is trapped in a
catalytic site, that subunit is in its tightly bound
state and the corresponding DNA binding residues
are not in position to bind DNA. Therefore, Pi is
Figure A1. Stress paths between subunits in the T7 helicase. (a) Stereo view of the T7 hexamer showing (in red) the
main-chain connections between adjacent catalytic sites. (b) Detail showing the main stress pathways between adjacent
catalytic sites, assuming motif III is the DNA contact residues. (c) Cartoon illustrating the circumferential stress path and
the stress loop passing through the nucleotide in the (N$T, N$T*) state. The springs represent the b-sheet and the
continuous red lines represent the loops emanating from them.
471
Mechanochemistry of T7 DNA Helicase
trapped in its site waiting for dTMPPCP to
dissociate. This waiting time can be sufficient for
some synthesis events.
These hypotheses rest on cooperativity between
the hexamer subunits for dTTP hydrolysis. The
absence of medium [ 18O]Pi exchange with
dTMPPCP without DNA indicates that regulation
or cooperativity among subunits occurs via DNA,
but not because dTTP and dTMPPCP have different
binding affinities.
The complete simulation
We have presented alternate explanations for the
results of the oxygen exchange experiments. In
order to distinguish one explanation from the other,
a complete kinetic cycle must be considered
explicitly, including both D/DP and N$D/
N$DP steps. Also, the effect of non-hydrolyzable
dTMPPCP binding must be included. The
minimal kinetics required to describe each subunit
is:
T^
h
4
^
T
h
4
E
h
4
T
h
4
T
h
dTTP binding events probably occur in almost all
subunits. Therefore, to obtain the initial concentrations of all states we modeled a random dTTP
binding process with binding probability given by
the chase occupancy.
Alternative optimization methods
We used the downhill simplex method for the
optimization to obtain rate constants that fit the
experimental data well. Several other optimization
methods have been devised, including those
specifically designed for least-squares fitting,56
maximum likelihood methods,57,58 and genetic
algorithms.59,60 However, because the function to
be minimized is highly non-linear, none of these
methods guarantee finding a global minimum. We
have compared the simplex method with several
other non-linear solvers and none has provided
any compelling advantage over the simplest
approach.
4
DP
4
h
D
h
4
E
h
(A1)
^
N,T^ 4 N,T
4 N,E 4 N,T 4 N,T 4 N,DP 4 N,D 4 N,E
where T^ represents the dTMPPCP-bound state.
Because dTMPPCP is not hydrolyzable, the states
in the left side are terminated before the hydrolysis
events. The complete kinetic network has 146Z
7,529,536 states. Although our method applies in
principle, this system is too large to be addressed
computationally within current computer power
and memory limits.
Initial conditions
The concentrations of all states at the initial time
are required to solve the kinetic equations. Chase
experiments provide information to deduce the
initial conditions. For this, a simulation was carried
out by considering only three possible states before
Mg2C is added (i.e. hydrolysis has not yet
occurred): E, T*, and NT*. Using these states, the
initial occupancy can be inferred from the chase
experiments. In order for sequential translocation to
proceed, only one subunit can bind tightly to DNA
at a time. That is, at most one subunit can be in state
NT*; all other subunits are in states E or T*. The acid
quench data show the transient phase takes about
four or five hydrolysis cycles to reach the steadystate slope. The hydrolysis speed (slope) in this
transient phase is greater than the steady-state
speed. This is because some subunits are already in
the T* state, ready to hydrolyze. The [dTTP]0
dependence of steady-state slopes indicates dTTP
binding is the rate-limiting step. Because dTTP
binding is rate limiting, the subunits, which already
have dTTP bound, bypass this slow step. The four
or five startup hydrolysis cycles imply that initial
Flux methods
The flux diagram method developed by T. Hill
has been successful in computing states and fluxes
for small kinetic networks.40 However, the network
for the T7 helicase is too large for the diagram
method, and so a numerical scheme such as we
developed here is required. Recently, Fisher &
Kolomeisky developed kinetic models with
sequences and branches applied to motor proteins.47–49 In these models, analytical solutions for
any number of sequential states with branches and
jumps were obtained and used to fit data for myosin
and kinesin dimers with two catalytic sites. Rajendran et al. used the analytical solution of a
sequential kinetic model for another ring helicase
DnaB.19 However, it is difficult to extend these
models to the large kinetic network that characterizes the T7 helicase, so the problem is better
approached by numerical schemes rather than
analytical solutions. The mechanical considerations
we used to prune the possible kinetic pathways are
also an important element in distinguishing our
method from others.
Three cooperative steps are necessary
The fitting is based on the three required
cooperative steps involving multi-subunit interactions. To examine the necessity of each of these
three steps, we also carried out simulations relaxing
one required step at a time. In each case either the
power strokes are not sequential, or DNA loses
contact to all subunits of the helicase, thus
472
Mechanochemistry of T7 DNA Helicase
Figure A2. Flow chart illustrating
the computational scheme used to
obtain the best-fit rate constants to
the kinetic experiments.
losing tight coupling between hydrolysis and
translocation.
To further test the model, extra requirements
were also imposed in addition to the three required
cooperative steps. For example, in F1 ATP synthase
ADP release is enhanced by ATP binding on
another subunit.31 Therefore, an additional requirement was imposed that dTDP release is triggered by
dTTP binding at adjacent sites. Simulation based on
this additional cooperative requirement shows that
dTDP is released too slowly so that the occupancy
does not correspond to the measurements.18 Since
the data also show Pi release has a phase delay after
hydrolysis, we also simulated the situation that Pi
release is triggered by dTTP binding or the power
stroke. In both cases Pi release becomes the ratelimiting step under all initial [dTTP]0, and fitting for
different slopes under different [dTTP]0 is impossible. We conclude that the phase delay of Pi release
results simply because the rate is intrinsically slow,
not because this step is affected by multi-subunit
interactions.
Predictions for single molecule force–velocity
experiments
Based on the computed rate constants, we can
estimate how this protein might function under a
mechanical load. Assume a single molecule force–
velocity measurement can be accomplished for T7
helicase, analogous to the DNA pulling experiment
carried out for the f29 portal protein.61 As the force
opposing translocation increases, the translocation
power stroke, NT*/NT, becomes the rate-limiting
step as the hydrolysis cycle slows. Because the
hydrolysis cycle is tightly coupled to translocation,
as the stall force is approached, the rate of the
hydrolysis cycle becomes too small to be measured.
The prediction of force dependence is shown in
Figure A3 based on the assumption that the power
stroke can be modeled as a thermally activated
event, with the forward rate modified by a
Boltzmann factor, exp(KFd/2kBT), and the backward rate modified by exp(Fd/2kBT). Here, F is the
external load force and d is the power stroke
displacement, dw0.9 nm. The initial conditions are
the same as those shown in Figure 4. Physiological
conditions are assumed to be [dTTP]Z100 mM,
[dTDP]Z10 mM, and [Pi]Z1 mM, as for Figure 7.
For low initial [dTTP]0, a force less than 5 pN does
not have a large effect on the hydrolysis rate. This is
because dTTP binding, rather than the power
stroke, is the rate-limiting step at low load. As the
load force increases, the hydrolysis rates for all
473
Mechanochemistry of T7 DNA Helicase
Figure A3. Force and free energies of translocation. (a) Predicted force dependence for the steady-state dTTP
hydrolysis rate under different initial conditions assuming a Boltzmann force dependence; i.e. the power stroke is a
thermally activated step. Since N$T*/N$T is the power stroke, this step is affected by the load force. When the load force
is large, this step becomes rate limiting and the hydrolysis cycle slows. The values at FZ0 correspond to the steady-state
slopes in Figure 4. The prediction for physiological conditions is also shown ([dTTP]Z100 mM, [dTDP]Z10 mM, [Pi]Z
1 mM). The stall force is estimated to be about 30 pN. (b) The free energy levels for physiological conditions under
different load forces. The corresponding hydrolysis rates under these four different forces are marked in (a). The load
force only changes the free energy of the power stroke step, N$T*/N$T. As the load force increases, this step eventually
becomes endothermic, and the corresponding hydrolysis rate decreases.
curves drop until, around 30 pN, the hydrolysis rate
is very small, and can be considered an estimate of
the stall force. An actual mechanical potential
model, rather than the exponential Boltzmann
model used here, can be implemented to obtain a
different force–velocity relationship, especially near
the stall force.31,33
As shown in Figure A3(a), under physiological
conditions the hydrolysis rate at zero force is about
13.5 sK1. The hydrolysis rate also decreases with
increasing load force; values of four different load
forces are shown: 0 pN, 10 pN, 20 pN, and 30 pN.
For each of these forces, the computed free energy
level of each chemical state is plotted in Figure
A3(b), illustrating the free energy drop in each
kinetic step. Of course, there is an activation free
energy barrier in each kinetic step; however, since
the barrier heights are not determined (i.e. the
frequency factor of each step is not known), we link
the free energy levels by broken lines. Note that
only the power stroke step is affected by the load
force, the free energy differences of all other steps
remain unaltered. At physiological conditions, the
concentration of Pi is high, and so the Pi release step
is endothermic. This is consistent with the experimental observation that Pi release is slow. Brownian fluctuations drive the system through this
unfavorable kinetic step, which will be rectified by
the subsequent steps (D/E, E/T*, etc) with large
free energy drops. As the load force increases, the
power stroke step becomes endothermic, and this
step becomes rate limiting.
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Edited by R. Ebright
(Received 24 January 2005; received in revised form 22 April 2005; accepted 22 April 2005)