  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. [...] 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 tethered particle motion, and then compare to our model's
predictions. [...] 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.
