FIG. 2: Normalized unlooped lifetimes as a function of applied noise. The (square) data points were taken at a mean tension of h = 153 fN. The (diamond) and (circle) data points were taken at h = 123 and 175 fN respectively. Theoretical curves are shown for h = 123 , 153 , 175 fN. upon the formation of a loop: States S$_{1}$ and S$_{2}$ arise because there are multiple unlooped sub-states available to the protein- DNA system. S$_{1}$ represents a state with only one occupied operator, which may either loop at rate k$_{L}$ to form state L , or remain unlooped and convert at rate k$_{1}$ to state S$_{2}$ . State S$_{2}$ is an alternate configuration with both or neither operator occupied, which cannot form a loop, but may convert back to state S$_{1}$ at a rate k$_{2}$ . The first-order kinetics results in the following biexponential function for the cumulative probability distribution: where κ = k$_{2}$ + k$_{1}$ , α = [( κ + k$_{L}$ ) 2 − 4 k$_{2}$k$_{L}$ ] 1 / $^{2}$, c$_{±}$ = ( κ − k$_{L}$ ± α ) and the time constants are defined as τ$_{±}$ = 2 / ( κ + k$_{L}$ ± α ). From this fit equation we are able to extract the unlooped lifetime τ$_{u}$ = 1 /k$_{L}$ . In accord with the constant force results [7], the interconversion rates k$_{2}$ and k$_{1}$ were found to be essentially independent of the applied tension. Figure 2 displays the unlooped lifetimes as a function of the noise T$_{α}$/T normalized to the zero noise lifetime, τ$_{u}$ ( h ; T$_{α}$/T ) /τ$_{u}$ ( h ; 0), about a mean applied tension of 153 fN. The results demonstrate that fluctuations do indeed drive loop formation and coalesce nicely with previous observations that femtonewton forces can radically affect the rate at which 5 L ( t ) = 1 − 1 2 α [ c$_{+}$e − t/τ$_{−}$ − c$_{−}$e − t/τ + ] , (3) S$_{2}$ k 2 ⇌ k 1 S$_{1}$ k L −→ L. (2)