content_image/373270.jpg content_image/373269.jpg content_image/373268.jpg content_image/373267.jpg
the process 

<img src='content_image/373270.jpg'>

where L is the looped state, S$_{1}$ is the state of the DNA with only one operator bound to a protein, and k$_{U}$ is the unlooping rate. Therefore, the time dependent probability of unlooping is 

<img src='content_image/373269.jpg'>

which corresponds to the fit function (Eq. 1) for the lifetimes of the looped state. The kinetics of loop formation, however, are more complicated because there are different unlooped sub- states that cannot be distinguished within our experiment. We start by collecting all time intervals that begin at an unlooping event and end with the formation of a loop. The kinetics may be represented as 

<img src='content_image/373268.jpg'>

where S$_{1}$ is the state of one vacant and one occupied operator and may directly convert to the looped state L at a rate k$_{L}$ , or remain unlooped and convert to state S$_{2}$ at a rate k$_{−}$ . State S$_{2}$ , however, is an alternate configuration with both or neither operator occupied by a protein, which is not able to directly form a loop, but may convert to state S$_{1}$ at a rate k$_{+}$ . With the initial condition S$_{1}$ (0) = 1, the first-order kinetics above may be solved for the time-dependent probability of forming a loop 

<img src='content_image/373267.jpg'>

where κ = k$_{+}$ + k$_{−}$ , α = [( κ + k$_{L}$ ) 2 − 4 k$_{+}$k$_{L}$ ] 1 / $^{2}$, and the time constants are defined as τ$_{1}$ = 2 / ( κ + k$_{L}$ − α ) and τ$_{2}$ = 2 / ( κ + k$_{L}$ + α ). Equation 6 is again a biexponential distribution and corresponds to the fit function of Eq. 2. Therefore, we can unambiguously extract the four rate constants in our kinetic model. The results are shown in Table 2 and plotted in Fig. 3. Our main observation is that, within the uncertainties of our measurements, k$_{+}$ , k$_{−}$ , and k$_{U}$ are independent of force whereas k$_{L}$ is acutely force-sensitive on the hundred- femtonewton scale. This is consistent with the conventional expectation that the rate of conversion between the unlooped states S$_{1}$ and S$_{2}$ does not vary significantly as a function of the stretching force on this scale. The insensitivity to applied force of the unlooping rate k$_{U}$ can be explained by considering the binding energy of the LacI protein to the DNA, whose disassociation from the lac binding site is necessary to break a loop. With a binding energy 

