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volume exclusion effects arising from entropic interactions between the microsphere and the coverslip [9]. The looped and unlooped states of the DNA molecule, which correspond to different axial positions of the microsphere, were measured by analyzing defocused images acquired on a CCD camera at 100 fps in the presence of 100 pM of LacI protein. This method provides excellent temporal resolution for detecting loop formation and breakdown events with time windows as short as 300 ms. 

Fluctuating forces were applied to the DNA by modulating the intensity of the trapping laser with an AOM connected to a data acquisition board and controlled by a custom LabVIEW program. The program generated Gaussian white noise, at a sampling rate of 1 /δt , which was superimposed upon a set average optical force. The modulation was performed such that the force applied to the trapped microsphere was randomly chosen from a normalized Gaussian distribution of standard deviation σ . 

It is the tension along the DNA that should display white noise statistics; however, by optically shaking the microsphere the resulting time correlation of the induced tension will effectively be low-pass filtered: 

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where η ( t ) is the DNA tension and α = σ $^{2}$δt is the noise strength. The characteristic time may be approximated by τ$_{c}$ ≈ γ/κ , where γ = 6 πηr is the hydrodynamic friction coefficent of the microsphere, η is the viscosity of the medium and r is the radius of the microsphere. If we only consider events that happen on timescales greater than τ$_{c}$ we may approximate the colored spectrum of Eq. (1) by purely white noise such that Eq. (1) reduces to lim$_{τ}$ c → 0 〈 η ( t ) η ( t $^{′}$) 〉 ≈ 2 αδ ( t − t $^{′}$). From the worm-like chain model for DNA, τ$_{c}$ ranges from 5-8ms at tensions from 180-120 fN. Therefore, if we set δt below the cutoff imposed by τ$_{c}$ (for our experiments, we fix δt at 2 ms (1 /δt = 500 Hz)) then the applied fluctuations are essentially white up to frequencies 1 /τ$_{c}$ , and we may adjust the strength of the noise by tuning the width of the distribution σ . It is more intuitive, perhaps, to consider the applied fluctuations as generating an additional energy contribution to the thermal modes of the DNA of magnitude k$_{B}$T$_{α}$ = σ $^{2}$δt/ 2 γ , and to parametrize the fluctuations relative to the ambient temperature of thermal fluctuations T$_{α}$/T . 

Figure 1 shows the distribution of lifetimes for the looped and unlooped states at a mean tension of 153 fN and with fluctuations of T$_{α}$/T = 0 , 0 . 01 , 0 . 05 , and 0 . 12. As indicated 

