the process 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 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 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 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 6 L ( t ) = 1 − 1 2 α [ ( κ − k$_{L}$ + α ) e − t/τ 1 − ( κ − k$_{L}$ − α ) e − t/τ 2 ] , (6) S$_{2}$ k + ⇌ k$_{−}$ S$_{1}$ k L −→ L, (5) S$_{1}$ ( t ) = 1 − e − k U $^{t}$, (4) L k U −→ S$_{1}$, (3)