The application of load selectively modulates rates within the biochemical cycle involving motion. A simple Boltzmann relation describes the resulting force–velocity relationship for many mechanoenzymes3,43,44
is the velocity at large assisting load, F½
is the force at which velocity reaches half its maximal value, kB
is Boltzmann’s constant, T
is the temperature, and δ is a parameter that represents the effective distance over which force acts. The physical interpretation of δ depends on the underlying model43
. In power stroke models, δ typically represents the distance from the pre-translocated position to the transition state. In such a model for RNAP inspired by ref. 29
, this transition state is located between the PPi
-bound and PPi
-released states, where PPi
is pyrophosphate (), and δ corresponds to some fraction of a single-base-pair separation. In brownian ratchet models, δ typically represents the characteristic distance associated with fluctuations between pre- and post-translocated states. For RNAP, this distance generally corresponds to one base pair. We considered two instances of a brownian ratchet: a model inspired by ref. 32
, where translocation precedes NTP binding (), and a model where translocation can either precede or follow NTP binding (). Because the enzyme active site is occluded by the 3′ end of RNA in the pre-translocated state, the latter model requires an incoming NTP to occupy a secondary binding site before being loaded into the active site. Such a secondary binding site might represent, for example, the ‘E site’ observed in crystal structures of polymerase II45,46
, or the templated binding sites proposed in biochemical studies27,28,47
. These binding sites are structurally distinct, but the binding of an NTP to either type of site may be modelled by the simplified kinetic pathway shown in (see also Supplementary Information).
Alternative kinetic models for RNAP translocation
To distinguish between the models of on the basis of force–velocity behaviour, it was critical to remove load-dependent, off-pathway events from records. In particular, we observed occasional pauses associated with both backtracking19
and backstepping (). These off-pathway events decrease overall elongation rates by differing amounts depending on load, obscuring the on-pathway load dependence. With the improved resolution obtained, it was possible to identify and remove all rearward motions greater than 1 nm from traces. The run velocity for each molecule was computed by dividing the total distance of advance by the total elapsed time minus any time spent in backtracked states. We note that off-pathway pauses that are insensitive to load (and therefore have no associated motion) do not affect the force dependence of nucleotide addition, because they occur with a fixed probability per unit time.
RNAP backstepping and backtracking resolved at high resolution
Elongation velocities were measured over a wide range of assisting and hindering loads (−18 to 28 pN) at four NTP concentrations (1, 10, 100 and 250 × [NTP]eq
), with an average of 32 molecules per data point (). With backtracking included, the velocity interpolated to zero load at [NTP]eq
was in excellent agreement with an independent estimate obtained through gel-based measurements. Once backtracking pauses were removed, each force–velocity curve was fit individually to equation (1)
. These unconstrained fits returned a characteristic distance parameter δ = 3.4 ± 0.5 Å (Supplementary Fig. S4).
Force–velocity and Michaelis–Menten relationships along with model fits
We generated global fits of all three models to our entire data set of velocities (N
= 40; ). Note that more force is required to hinder elongation with increasing [NTP] (that is, F½
decreases; ), a trend opposite to that predicted for a power stroke model coupled to PPi
release. The global fit to the power stroke model generated a poor fit (χv2
= 6.03; v
= 35; p
) = 5.3 × 10−27
; five parameters; v
is the number of degrees of freedom). These findings, together with a computed distance parameter corresponding to a full base-pair displacement (rather than some fraction of a base pair expected for a power stroke model) and previous results showing that elongation velocity is insensitive to PPi
, all argue against the mechanism of .
A global fit of our data to the simple brownian ratchet model of returned better results (χv2
= 2.67; v
= 37; p
) = 1.6 × 10−7
; three parameters), and qualitatively predicted the behaviour of F½
as a function of [NTP]. An even better fit was obtained for the ratchet model of , which invokes a secondary NTP binding site. This model predicted all unconstrained F½
values to within error, and was statistically consistent with the complete data set (χv2
= 0.64; v
= 36; p
) = 0.956; four parameters). The fit parameters suggest the presence of a small energetic penalty (~1 kT) associated with nucleotide binding to the secondary site when the molecule is pre-translocated, compared with the post-translocated binding energy (under standard conditions). Saturating NTP concentrations tend to ensure that the secondary binding site remains occupied and thereby bias the enzyme towards the post-translocated state. However, in contrast to the ratchet mechanism of ref. 32
, hindering loads can overcome this bias by forcing the NTP-bound form into a pre-translocated state, increasing the force sensitivity at higher NTP levels. This model seems attractive for its simplicity (four free parameters), its ability to fit all available force–velocity data, and the close correspondence to recent structural and biochemical studies supplying evidence for a secondary site27,28,45,46
. Clearly, alternative kinetic schemes may be formulated to fit the data presented here and elsewhere.
The marked improvement in resolution obtained in this optical trapping study has led to direct measurements of base-pair stepping by an individual enzyme and supplied insights into the molecular mechanism of transcription by RNAP. Our data argue directly against any power stroke mechanism that is tightly coupled to PPi
release. Furthermore, although other recent publications have supplied independent biochemical and biophysical evidence in support of various forms of a brownian ratchet mechanism30,31,48
, we propose a specific model incorporating a secondary NTP binding site that is consistent with our data and others27,45,46
. The techniques presented here are broadly applicable. In particular, it may be possible to use our approach to relate the behaviour of a nucleic acid-based enzyme directly to the underlying DNA sequence to which it is bound, facilitating studies of sequence-dependent effects in replication, transcription and translation, and possible use in single-molecule DNA sequencing. The ability to detect motions at the ångström scale in single enzymes opens new avenues for the study of biomolecules.