First-principles calculation of DNA looping in tethered particle experiments

Abstract

We calculate the probability of DNA loop formation mediated by regulatory proteins such as Lac repressor (LacI), using a mathematical model of DNA elasticity. Our model is adapted to calculating quantities directly observable in tethered particle motion (TPM) experiments, and it accounts for all the entropic forces present in such experiments. Our model has no free parameters; it characterizes DNA elasticity using information obtained in other kinds of experiments. It assumes a harmonic elastic energy function (or wormlike chain type elasticity), but our Monte Carlo calculation scheme is flexible enough to accommodate arbitrary elastic energy functions. We show how to compute both the ‘looping J factor’ (or equivalently, the looping free energy) for various DNA construct geometries and LacI concentrations, as well as the detailed probability density function of bead excursions. We also show how to extract the same quantities from recent experimental data on TPM, and then compare to our model’s predictions. In particular, we present a new method to correct observed data for finite camera shutter time and other experimental effects. Although the currently available experimental data give large uncertainties, our first-principles predictions for the looping free energy change are confirmed to within about 1 kBT, for loops of length around 300 basepairs. More significantly, our model successfully reproduces the detailed distributions of bead excursion, including their surprising three-peak structure, without any fit parameters and without invoking any alternative conformation of the LacI tetramer. Indeed, the model qualitatively reproduces the observed dependence of these distributions on tether length (e.g., phasing) and on LacI concentration (titration). However, for short DNA loops (around 95 basepairs) the experiments show more looping than is predicted by the harmonic-elasticity model, echoing other recent experimental results. Because the experiments we study are done in vitro, this anomalously high looping cannot be rationalized as resulting from the presence of DNA-bending proteins or other cellular machinery. We also show that it is unlikely to be the result of a hypothetical ‘open’ conformation of the LacI tetramer.

Calibration of Tethered Particle Motion Experiments

“…Our results are preparatory to experimental [7] and theoretical [16] work on DNA looping in the lac operon system.…”

“…Physically, the point is that successive video frames are not independent draws from the distribution of particle positions, because the particle's motion is diffusive. The diffusion time τ diff of a particle in a trap increases with increasing trap radius and with increasing viscous drag constant for the particle, giving rise to the trends observed in the figure. (For a theoretical discussion see the Supplement to [16].) Similarly, the second row of graphs in Fig.…”

“…The excursion of the bead away from its attachment point on the microscope slide is affected by the length and stiffness of the DNA tether, the size of the bead, and the various interactions between the bead/wall, bead/tether, and wall/tether. To account for all these effects, we modified the Gaussian sampling Monte Carlo technique previously used in [13,9,4,8] (see [16] for details).…”

“…(For a procedure valid for any t, see the Supplement to [16].) We also applied a correction to this theoretical result, to account for the bead's motion during the rather long shutter time (see the following subsection).…”

Tetramer opening in LacI-mediated DNA looping

“…However, analysis of loop structures responsible for the observed TPM states is complicated by potential degeneracy of the TPM recording, where different loop structures may correspond to the same apparent tether length. Calculations by Towles et al (27), for example, show that wrap-away P2, A1, and A2 all result in similar values of ͗R͘, corresponding to a short-tether looped state, and antiparallel loops are energetically advantageous. The long tether, on the other hand, is attributed to P1, which, at 305 bp, features an energy intermediate between P2 and the antiparallel loops.…”