Mechano-chemical kinetics of DNA replication: identification of the translocation step of a replicative DNA polymerase

During DNA replication replicative polymerases move in discrete mechanical steps along the DNA template. To address how the chemical cycle is coupled to mechanical motion of the enzyme, here we use optical tweezers to study the translocation mechanism of individual bacteriophage Phi29 DNA polymerases during processive DNA replication. We determine the main kinetic parameters of the nucleotide incorporation cycle and their dependence on external load and nucleotide (dNTP) concentration. The data is inconsistent with power stroke models for translocation, instead supports a loose-coupling mechanism between chemical catalysis and mechanical translocation during DNA replication. According to this mechanism the DNA polymerase works by alternating between a dNTP/PPi-free state, which diffuses thermally between pre- and post-translocated states, and a dNTP/PPi-bound state where dNTP binding stabilizes the post-translocated state. We show how this thermal ratchet mechanism is used by the polymerase to generate work against large opposing loads (∼50 pN).


Mechano-chemical models
To determine the location of the translocation step within the replication cycle, we initially considered a minimal nucleotide incorporation cycle where the activation of the ternary complex and the fast following chemical steps were grouped within a single rate limiting step ( Figure 1A). Since no significant conformational changes within the polymerase-DNA complex occur during these steps translocation is not expected to occur concomitant to the rate-limiting step of the reaction. Assuming a single force dependent state, or in other words, assuming that translocation is associated with only one step of the cycle, three alternative general models could explain the coupling mechanism between the chemical and mechanical steps during the nucleotide incorporation cycle: In Model 1 ( Figure 1B), translocation is power-stroked by dNTP binding, in Model 2 ( Figure 1C) translocation is power-stroked by PPi release and in Model 3 ( Figure 1D), translocation occurs by thermal diffusion between PPi-free and dNTP-free states. We assumed that the forward and reverse rates of the translocation step present an Arrhenius-like dependence on force where F is the applied load, k B is Boltzmann's constant, T is the temperature, and is the effective distance over which the applied load acts on translocation (1).

Three states power-stroke models (Model 1 and Model 2)
The power stroke translocation mechanisms, Model 1 and Model 2, differ in the step of the reaction that propels translocation, dNTP binding and PPi release, respectively. For both mechanisms the minimal kinetic mechanism can be represented with a three state model as where the MM parameters, and , (or their inverses) are given in terms of the rates of the reaction (see equations 3 and 4 below). We note that throughout this work we have used the inverses of (maximum velocity at saturated dNTP concentrations) and = max / (effective rate of dNTP binding), because they provide more apparent expressions with simpler relations among characteristic times (inverse of the rates) and the processes involved during the cycle. These simple relations provide a straightforward interpretation of the observations and their implications (see below).
The observed force dependency of 1/V max rules out Model 1.
According to Model 1 ( Figure 1B), translocation is driven by the nucleotide binding reaction. In this case, load opposing translocation would affect specifically the nucleotide binding and/or unbinding rates, k on [dNTP](F) and k off (F). A direct consequence of this model is that the replication velocity at saturated dNTP concentrations, V max , should not depend on force (1)(2)(3)(4). This is because V max (or 1/ Then, a model proposing that translocation is driven by nucleotide binding, such as Model 1, implies that velocity at saturated dNTP concentrations should be force independent. The observed force dependency of 1/V max argues directly against this mechanism ( Figure 4B).
The observed force dependency of 1/k b (F) rules out Model 2.
In Model 2 ( Figure 1C), translocation occurs during PPi release, being k ppi (F) the only force dependent rate of the nucleotide incorporation cycle. According to this model the inverse of k b (F) can be written as This model can be excluded if one of the following conditions are verified Kinetics studies of Family A and B DNAPs showed that is typically ~10 3 times faster than the reverse of the rate limiting step of the reaction, − , implying that − ( ) ≪ 1 (5-11) and therefore, 1/k b should be largely independent on force (considering d i ~0.34 nm). This prediction contrasts with our data showing a load dependency of 1/k b ( Figure 4D), arguing against a direct connection between the PPi release step and mechanical translocation.
Altogether our data argue against models where translocation is power-stroked by dNTP binding or PPi release.

Exclusion of alternative 3-state models
We checked the compatibility of the experimental data with an alternative model considering that translocation occurs during the step located between the dNTP-bound and PPi-bound states. As we did for the other 3 models discussed in the manuscript, we considered that this step comprises the rate-limiting activation of the ternary complex and the following rapid chemical steps (references 5 and 10 to 20, main text).
In this case, the rate-limiting step was force dependent and their forward and backward rates were defined by k cat (F)and kcat (F), respectively. Also, as in the other models and according to our data, the PPi release step was considered largely irreversible (k -ppi ~0 s -1 , Supplementary Figure S3). The kinetic expressions for the Michaelis-Menten parameters V max , and K M derived from this model allowed initially a force dependent behavior for both 1/V max (F) and In order to test the validity of this new model to explain the data, we used the same approximation described in the main text to test the four-state Brownian Ratchet model (Model 3). In this way, a direct comparison between the two models can be done. Therefore, the above expressions for 1/V max (F) and 1/k b (F) (equation 5 and 6, respectively) were recast as the sum of a force independent and force dependent terms 1 ( ) = + ⋅ ⋅ /( ) (eq.7), where the coefficients a, b, r, s, are given in terms of the rates of the cycle and, the coefficients d b and d s define the force dependency or effective distance over which force acts on translocation, . As described in the main text, we included equations 7  Based on these arguments, we did not initially considered a model where translocation occurs during the transition between dNTP-bound and PPi-bound states.
2. Pre-steady-state kinetic studies of several DNA polymerases indicate that after nucleotide binding (and fingers closure), unspecified subtle non-covalent transformations in the active site activate the ternary complex to form an active site poised for catalysis. This process, also called 'nucleotide condensation', was found to be the rate-limiting step of the nucleotide incorporation cycle (references 5 and 10 to 20 in the main text). These studies are not compatible with the results of the model shown above.
3. The PPi release rate, k ppi , is probably the fastest rate of the nucleotide incorporation cycle, k ppi =10 3 -10 4 s -1 and to date, it has not been found to be the ratelimiting step on the reaction for any replicative polymerase (references 14, 16, 21, 58, 59, 60 and 61 in the main text). This evidence argues against this alternative model, which requires k ppi to be the rate limiting step of the reaction.
We note that, when translocation is considered to occur associated with a single step of the nucleotide incorporation cycle, the experimental data is not compatible with any model (3 and/or 4 states) where the translocation step is also the rate-limiting step of the reaction. This is because the main contribution to 1/ at zero force comes from the value of the parameter a, which does not depend on force (and therefore, is not associated with translocation).

Four states Brownian ratchet mechanism (Model 3)
The Brownian ratchet mechanism, Model 3, considers that translocation occurs by thermal diffusion after PPi release and before dNTP binding, and requires the inclusion of an additional state after PPi release (see below and Figure 1D). In this case, the force dependent rates of the reaction are the forward and backward translocation rates, The evolution equations for the occupation probabilities of this four states model are: and the conservation condition 1 + 2 + 3 + 4 = 1 is verified. In the steady state all the derivatives of the occupation probabilities are zero. Again, T stands for nucleotide concentration and D for product (PPi) concentration.
In contrast to the other two models, the kinetic expressions for the Michaelis-Menten parameters V max , and K M derived from the Brownian ratchet model (Model 3) allow initially a force dependent behavior for both 1/V max (F) and 1/k b (F)

Translocation
Each of these expressions can be recast as the sum of a force independent and a force dependent term 1 ( ) = + ⋅ ⋅ /( ) , (eq. 11), where the coefficients a, b, r, s, are given in terms of the rates of the cycle and, the  Table S1. Best fit values of the parameters in eq. 11 and eq. 12.
Importantly, these values also support the observed force dependency for 1/V max (F), 4C and 4D, respectively). In addition, they predict the observed dNTP concentration dependence of the detachment load (black solid line in Figure 3D).

Detachment load determination
Briefly, we found that the detachment load (

Determination of rates
We note that there is a direct relationship between the parameters obtained from the fits (equations 11 and 12, Table S1) and several of the rates and conformational changes of the nucleotide incorporation cycle. We used these relationships for the following calculations:

1.-Calculation of rates and conformational changes related to translocation:
The relations between the free coefficients and the forward (k T ) and backward (k-T ) translocation rates are the following:  (Table S1). Therefore, K  =

Calculation of the nucleotide condensation and catalysis rate, k cat :
The value of k cat can be derived from the relation = 1 (1 + − ) + 1 . Since the rate of PPi release, k ppi , is very fast (~10 3 -10 4 s -1 ) and 10 3 times faster than k -cat (5-11), the ratios − and 1 are both much smaller than 1, implying that ∼ 1 = 120 −1 .
Note that the main contribution to 1/ at zero force comes from (since > ), pointing out that k cat corresponds to the rate limiting step of the reaction at saturated dNTP concentrations. Interestingly, the results from the fits directly showed that the rate limiting step of the nucleotide incorporation cycle does not depend on force and therefore, it is not associated with translocation. This result is in agreement with our initial assumptions, which were based on previous kinetic and structural studies on several DNAPs (6,14-17).

Calculation of the nucleotide binding rate, k on :
The value of k on can be derived from the relation = 1 [1 + (1 + − )]. As mention before, k ppi is ~10 3 times faster than k -cat (5)(6)(7)(8)(9)(10)(11), implying that the ratio − ≪ 1. Therefore, the coefficient r can be written as follows: is the dNTP dissociation constant, which for the Phi29 DNAP has been recently determined as K 1 = 1.4 M (18) and as shown above, 1 ~ (a = 0.0084, Table S1). Therefore, ~ 5.43 −1 −1 , which is consistent with the value of the dGTP binding rate (k on ~17 M -1 s -1 ) recently determined for the Phi29 DNAP (19).     Figure S9B shows the effect of load on the probability of occupancy of the dNTP/PPi-free state.

Exclusion of alternative 4-state models
As mention before, we initially excluded the possibility that translocation may occur associated to the rate limiting step. This assumption was made based on extensive biochemical and structural data showing that no significant conformational changes within the polymerase-DNA complex occur during this reaction (6,(14)(15)(16)(17). Interestingly, as shown before, the results from the fits directly corroborated that the rate limiting step of the nucleotide incorporation cycle does not depend on force and therefore, it is not associated with translocation. Please, see the previous section, 'Exclusion of alternative 3-state models', for a detailed description of this point.
Additionally, we checked whether alternative four-state models considering that translocation occurs during dNTP binding or PPi release steps are valid to explain the data.
Translocation cannot occur during the nucleotide binding reaction. In this case, The PPi release process cannot be directly associated with translocation if − ( ) ≪ 1.
According to equation 10, these conditions would imply that 1/k b should be force independent, which is not reflected in our data. As mentioned before, kinetics studies on Family A and B DNAPs showed that k ppi >>k -cat (5)(6)(7)(8)(9)(10)(11), implying that − ( ) ≪ 1 and therefore, arguing against the direct coupling between PPi and translocation. This possibility is equivalent to the power stroke Model 2.
Additional force dependencies are not required to fit the data Based on kinetic and structural studies on different DNAPs, the models proposed in this work assume that translocation is associated to a single step of the nucleotide incorporation cycle. Since load is expected to affect specifically the step of the cycle related to motion, all the models proposed here present a single load dependent step.
However, alternatively one could consider either that translocation is distributed among several steps of the cycle or that load modifies, besides the translocation step, other steps of the nucleotide incorporation reaction, ie: by affecting the polymerase structure or the polymerase-DNA interactions. In these cases more than one step of the cycle would depend on force. We checked these possibilities by testing four-state models with several force dependent steps. We found that, although these models involved more parameters, they did not significantly improve the fit to data (not shown).
Although the possibility of additional force dependencies cannot be totally excluded with our present analysis of data, these results strongly suggest that a 'simple' fourstate model with single force dependent step is good enough to explain the experimental data. According to the results discussed above this translocation step occurs after PPi release and before dNTP binding. Upon arrival on dNTPs, replication starts; this is detected as an increase in the distance between the beads (green arrows). During active replication, the force on the replication complex was increased in rapid jumps (force jumps, vertical black arrows).

Supplementary Materials and
In this way, the activity was measured at increasing constant aiding forces on the