Lac repressor hinge flexibility and DNA looping: single molecule kinetics by tethered particle motion

The tethered particle motion (TPM) allows the direct detection of activity of a variety of biomolecules at the single molecule level. First pioneered for RNA polymerase, it has recently been applied also to other enzymes. In this work we employ TPM for a systematic investigation of the kinetics of DNA looping by wild-type Lac repressor (wt-LacI) and by hinge mutants Q60G and Q60 + 1. We implement a novel method for TPM data analysis to reliably measure the kinetics of loop formation and disruption and to quantify the effects of the protein hinge flexibility and of DNA loop strain on such kinetics. We demonstrate that the flexibility of the protein hinge has a profound effect on the lifetime of the looped state. Our measurements also show that the DNA bending energy plays a minor role on loop disruption kinetics, while a strong effect is seen on the kinetics of loop formation. These observations substantiate the growing number of theoretical studies aimed at characterizing the effects of DNA flexibility, tension and torsion on the kinetics of protein binding and dissociation, strengthening the idea that these mechanical factors in vivo may play an important role in the modulation of gene expression regulation.

where T d is the filter's dead time and (5) where k 2 =J m k a ; k 1 and k -1 are the rates of transition between O-OR and the other prevalent biochemical state corresponding to a TPM unloop (i.e. RO-OR at high LacI concentrations and O-O at low concentrations). So that, at LacI concentrations of 100 pM and 20 pM k 1 corresponds to 2k d and k -1 to k a [LacI], while at a concentration of 4 pM k 1 corresponds to 2k a [LacI] and k -1 to k d . λ FL and λ FU (the frequencies of threshold-crossing for generating false loop and unloop, respectively) were calculated based on the definition provided by Colquhoun and Sigworth (2): where f c is the cutoff frequency of the Gaussian filter employed, φ is the signal amplitude (the distance between the two peaks in the <R(t)> histogram and σ n is the signal standard deviation in each of the two states (σ L is used for the calculation of λ FU and σ U is used for the calculation of λ FL ). The latter two parameters were measured experimentally on the TPM recordings based on the Gaussian parameters fitted to the <R(t)> distributions (see figure 2, right panels) on all the traces acquired in the presence of wt-LacI at a concentration of 100 pM. The values obtained, along with the resulting values of λ, are reported in the Table SI. Table SI. Signal characteristics of TPM measurements and derived frequencies of false thresholdcrossing. The standard deviations (σ) and separation of the peaks of TPM mobility (φ) were measured, as described in the text, on the <R(t)> distributions obtained with LacI at a concentration of 100 pM.
From these values the frequencies of threshold-crossing by residual noise (λ) and the average duration of the false events are calculated (see text for details). T r is the total recording time (in minutes); N r is the number of microspheres used for the statistics.
From these expressions, we describe the number of expected loop and unloop dwells (N Lc and N Uc ) and their average durations (τ Lc and τ Uc ), starting from their true (and unknown) values (N L , N U , τ L , and τ U ): τ FL and τ FU (the average durations of the false looped and unlooped states, respectively) were calculated with a numerical approach derived from Rice (3) and Watts (4), in any case effectively resulting in values close to T d . We then numerically obtain the values of J m and τ L that minimize, for each set of data, the following expression: The numerical minimization algorithm was implemented in Mathematica (v.5.0, Wolfram Research). From τ L , the value of α was calculated as (2k d τ L ) -1 .
To test the validity of this method, a series of numerical simulations were performed. In these simulations, the system was allowed to switch between the states shown in figure 6 with transition rates corresponding to the rate constants shown in the figure. The simulations were run at LacI concentrations of 100 pM, 20 pM, and 4 pM. In the simulations k a and k d were set to the same values used throughout this work. In each simulation the system starts from the O-O state; at each simulation step a doubleprecision floating-point random number between 0 and 1 is drawn and the value of this is compared with the probability of transition to another state accessible from the current state. If the latter probability exceeds the value of the random number, then the transition takes place, otherwise the system remains in the current state. The probability of transition to any state is calculated according to the scheme of figure  6 in the paper, using the values of k a and k d mentioned above, the LacI concentration selected and the values chosen for α and J m (see below) in the specific simulation running. The simulation samples the trajectory of the system on the biochemical scheme of figure 6 at the same frequency of the experimental recordings (50 Hz) and for a duration of one hour for each simulated trace. This procedure produces a trajectory of the system describing its occupancy of the states O-O, O-OR, RO-OR, and ORO as a function of time. For the purpose of simulating the TPM measurement, the O-O, O-OR and RO-OR states are mapped into the TPM unlooped state, while the ORO state is mapped into the TPM looped state. The R(t) trace is then generated from the simulated data adding to the simulated data a Gaussian noise with standard deviation which depends on whether the system is in the unlooped or looped state, so to closely reproduce the signal-to-noise ratio typical of the TPM experimental recordings. The R(t) traces are then filtered exactly as done for the experimental measurements. Figure S1 shows an example of a trace simulated choosing a concentration of LacI of 100 pM, α=1, and J m = 10 -10 M, and filtered with a Gaussian filter with cutoff frequency of 0.066 Hz. Different sets of simulations (each set is made of 30 simulated traces corresponding to one hour of TPM recording each) were obtained varying, independently, J m (between 10 -8 M and 10 -11 M) and α (between 0.1 and 10). Each set of simulated data was analyzed exactly as done for the experimental TPM recordings and the corrected values obtained for J m and α were compared with those that had been set to generate the simulated data. Comparisons between the corrected values obtained with our data analysis method and the true values of the parameters are shown in figure S2. From the figure it can be noticed that the system performs very well in a wide range of J m and α values, especially at the higher LacI concentrations (20 and 100 pM). For values of J m larger than about 10 -9 M, the estimation of α is evidently not reliable. This is an obvious limitation of the TPM measurements performed with microspheres as discussed above. In fact, those large values of J m would imply transitions O-OR →ORO with rates of the order of 1 s -1 or more; in these conditions, the duration of the unlooped state is largely dominated by the permanence of the system in the RO-OR state and the rate of looping will mainly depend on the rate of exit from this state, rendering the accurate measurement of J m not achievable. However, it should be noticed that the values of J m obtained with our method still give a reliable indication of whether this is the case or not. In fact, figure S2 shows that, at [LacI] of 20 pM and 100 pM, for any of the values of α chosen, there is a monotonic correlation of corrected J m versus true J m .
Obtaining corrected values of J m of the order of 10 -9 M from experimental measurements, thus, would indicate that the measurements are performed at (or below) the limits of the TPM technique. However, this is not the case for the measurements reported in this work, which are well within the range of optimal correlation between corrected and true values for both α and J m . At a LacI concentration of 4 pM, on the other hand, the reliability of the measurements is somewhat reduced, as also discussed in the paper on the basis of the limitations of the TPM technique itself. The simulations conducted at this lower LacI concentration (figure S2 bottom panels), in fact, show that a measured value of 10 -10 M for J m does not unambiguously correspond to a true value of 10 -10 M but could actually correspond also to larger true values. These simulations confirm that the estimation of α and J m from these measurements is not as reliable as at the higher concentrations. These results, combined with the considerations on the TPM method discussed in the paper, strongly indicate that great care needs to be taken when designing dynamic TPM measurements, so to keep the experimental system within the range of optimal validity of this single molecule experimental method and of the data filtering and correction methods that need to be applied to the experimental results. For the purpose of the present work, the LacI concentrations that are optimal for drawing quantitative conclusions from the experimental measurements are 20 pM and 100 pM, where the simulations indicate that no possible bias is taking place.
Estimation of the swelling force (F 0 ) due to the microsphere in TPM. The one-dimensional distributions (P x ) measured in the TPM experiment for the positions of the microsphere centroid (in the absence of LacI, so to avoid transitions between two levels of microsphere mobility) display a characteristic non-Gaussian shape, as noticed also by Pouget et al. (5). Theoretical models previously elaborated for the physical description of the TPM system (6) provided a good prediction of some aspects of the experiment (i.e. the dependence of the average range of microsphere mobility as a function of the DNA tether length) but did not attempt an analysis of P x . In fact, the description of the polymer, based, in those works, on a Gaussian model, would unavoidably lead to a Gaussian shape for P x . We have implemented Metropolis-Monte Carlo simulations of the TPM system based on a Worm-like chain (WLC) description of the polymer and an additional swelling force acting radially (F 0 ), describing the effects of the microsphere. Simulations were run for a 1300bp long DNA molecule with a persistence length of 50 nm (7), and varying the value of F 0 . At each value of F 0 the χ 2 between simulated and experimental data was calculated ( Figure S3). The figure shows that the best fit is obtained with F 0 = 127 fN. Confidence on the value of F 0 estimated with this method was evaluated following Press et al.    confidence interval on the measured parameter (see text for details). The continuous grey line represents a third-order polynomial fit to the χ 2 data, plotted to provide a means of interpolation between the points. The measured F 0 in the text is quoted as best fit ± ΔF 0 /2.