«Abstract. We calculate the probability of DNA loop formation mediated by regulatory proteins such as Lac repressor (LacI), using a mathematical model ...»
First-principles calculation of DNA looping in tethered
Kevin B Towles1, John F Beausang1, Hernan G Garcia2, Rob Phillips3,
and Philip C Nelson1
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia PA 19104
Department of Physics, California Institute of Technology, Pasadena CA 91125
Division of Engineering and Applied Science, California Institute of Technology, Pasadena
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 tethered particle motion, 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 kB T, for loops of length around 300 basepairs. More signiﬁcantly, our model successfully reproduces the detailed distributions of bead excursion, including their surprising three-peak First-principles calculation of DNA looping in tethered particle experiments 2 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
PACS numbers: 87.14.gk,87.80.Nj, 82.37.Rs, 82.35.Pq, 36.20.Ey Submitted to: Phys. Biol.
Keywords: Tethered particle, DNA looping, Brownian motion, single molecule, lac repressor, Monte Carlo
1. Introduction and summary
1.1. Background Living cells must orchestrate a multitude of biochemical processes. Bacteria, for example, must rigorously suppress any unnecessary activities to maximize their growth rate, while maintaining the potential to carry out those activities should conditions change. For example, in a glucose-rich medium E. coli turn off the deployment of the machinery needed to metabolize lactose; when starved of glucose, but supplied with lactose, they switch this machinery on. This switch mechanism—the “lac operon”—was historically the ﬁrst genetic regulatory system to be discovered. Physically, the mechanism involves the binding of a regulatory protein, called LacI, to a speciﬁc sequence of DNA (the “operator”) situated near the beginning of the set of genes coding for the lactose metabolism enzymes. Some recent First-principles calculation of DNA looping in tethered particle experiments 3 reviews of the lac system include Refs. [1–4]; see also Ref.  for looping in the lambda system.
Long after the discovery of genetic switching, it was found that some regulatory proteins, including LacI, exist in multimeric forms with two binding heads for DNA, and that their normal operation involves binding both sites to distant operators, forming a loop [6–11].
The looping mechanism seems to confer advantages in terms of function . From the biophysical perspective, it is remarkable that in some cases loop formation, and its associated gene repression, proceed in vivo even when the distance between operators is much less than a persistence length of DNA . For this and other reasons, a number of experimental methods have been brought to bear on reproducing DNA looping in vitro, to minimize the effects of unknown factors and focus on the one process of interest. Reconstituting DNA looping behavior in this way is an important step in clarifying the mechanism of gene regulation.
Tethered particle motion (TPM) is an attractive technique for this purpose . In this method, a long DNA construct is prepared with two (or more) operator sequences at a desired spacing near the middle. One end is anchored to a wall, and the other to an otherwise free, optically visible bead. The bead motion is passively monitored, typically by tracking microscopy, and used as an indirect reporter of conformational changes in the DNA, including loop formation and breakdown (Fig. 1).
1.2. Goals of this paper The recent surge of interest in DNA looping motivated us to ask: Can we understand TPM data quantitatively, starting from simple models of DNA elasticity? What is the simplest model that captures the main trends? How well can we predict data from TPM experiments, using no ﬁtting parameters?
To answer such questions, we had to combine and improve a number of existing calculation tools. This paper explains how to obtain a simple elastic-rod model for DNA, First-principles calculation of DNA looping in tethered particle experiments 4
freely pivoting attachments (not to scale). The motion of the bead’s center is observed and tracked, for example as described in Ref. . In each video frame, the position vector, usually projected to the xy plane, is found. After drift subtraction, the mean of this position vector deﬁnes the anchoring point. The projected distance from this anchoring point to the instantaneous bead center is the bead excursion ρ. A regulatory protein, for example a LacI tetramer, is shown bound to a speciﬁc “operator” site
an actual representative looped conﬁguration from the simulations described in this paper, drawn to scale. Figs. 2 and 6 explain the graphical representations of DNA and
and a geometric characterization of the repressor–DNA complex, from existing (non-TPM) experiments. From this starting point, with no additional ﬁtting parameters, we show how to calculate experimentally observable quantities of TPM experiments (such as the fraction of time spent in various looped states and the distribution of bead excursions), as functions of experimentally controlled parameters (operator separation and repressor concentration), and compare to recent experiments.
Although our main interest is TPM experiments, our method is more generally applicable. Thus as a secondary project, we also compute looping J factors for a DNA construct with no bead or wall (“pure looping”). This situation is closer to the one that prevails in vivo; although in that case many other uncertainties enter, it is nevertheless interesting to First-principles calculation of DNA looping in tethered particle experiments 5 compare our results to the experimental data.
1.3. Assumptions, methods, and results of this paper Supplementary Information Sect. S1 gives a summary of the notation used in this paper. Some readers may wish to skip to Sect. 1.3.3, where we summarize our results. Sect. 1.3.4 gives an outline of the main text and the supplement; in addition, the other subsections of this introduction give forward references showing where certain key material can be found.
1.3.1. Outline of assumptions First we summarize key assumptions and simpliﬁcations made in our analysis. Some will be justiﬁed in the main text, whereas others are taken in the spirit of seeking the model that is “as simple as possible, but not more so.” All our results are obtained using equilibrium statistical mechanics; we make no attempt to obtain rate constants, although these are experimentally available from TPM data [14, 16–18]. Our model treats DNA as a homogeneous, helical, elastic body, described by a 3 × 3 elastic compliance matrix (discussed in Sect. 3). Thus we neglect, for now, the effect of DNA sequence information , so our results may be compared only to experiments done with random-sequence DNA constructs. Despite this reduction, our model is more realistic than ones that have previously been used for TPM theory; for example, we include the substantial bend anisotropy, and twist–bend coupling, of DNA elasticity. We also neglect long-range electrostatic interactions (as is appropriate at the high salt conditions in the experiments we study), assuming that electrostatic effects can be summarized in effective values of the elastic compliances.
The presence of a large reporter bead at one end of the DNA construct, and a wall at the other end, signiﬁcantly perturb looping in TPM experiments. We treat the bead as a sphere, the wall as a plane, and the steric exclusion between them as a hard-wall interaction. We neglect nonspeciﬁc DNA–protein interactions (“wrapping” ).
First-principles calculation of DNA looping in tethered particle experiments 6 1.3.2. Outline of methods Our method builds on prior work [15, 21]. Sect. 7 discusses other theoretical approaches in the literature.
Our calculations must include the effects of chain entropy on loop formation, because we consider loop lengths as large as 510 basepairs. We must also account for entropic-force effects created by the large bead at one end of the DNA and the wall at the other end, in addition to the speciﬁc orientation constraints imposed on the two operators by the repressor protein complex. To our knowledge such a complete, ﬁrst-principles approach to calculating DNA looping for tethered particle motion has not previously been attempted. In part because of these complications, we chose to calculate using a Monte Carlo method called “Gaussian sampling” (discussed in Sect. 4 and Sects. S6–S7). Gaussian sampling is distinguished from Markov-chain methods (e.g., Metropolis Monte Carlo) in that successive sampled chains are independent of their predecessors.
We must also address a number of points before we can compare our results to experiments. For example, DNA simulations report a quantity called the “looping J factor.” But TPM experiments instead report the time spent in looped versus unlooped states, which depends on both J and a binding constant Kd. We present a method to extract both J and Kd separately from TPM data (discussed in Sect. S5). We also describe two new data-analysis tools: (1) A correction to our theoretical results on bead excursion, needed to account for the effect of ﬁnite camera shutter time on the experimental results (discussed in Sect. S2), and (2) another correction needed to make contact with a widely used statistic, the ﬁnite-sample RMS bead excursion (discussed in Sect. S3). (To be precise, the latter two corrections do both involve phenomenological parameters, but we obtain these from TPM data that are different from the ones we are seeking to explain. Each correction could in any case be avoided by taking the experimental data differently, as described in the Supplementary Information.) 1.3.3. Outline of results Some of our results were ﬁrst outlined in Refs. [22, 23]. The assumptions sketched above amount to a highly reductionist approach to looping. Moreover, First-principles calculation of DNA looping in tethered particle experiments 7 we have given ourselves no freedom to tweak the model with adjustable parameters, other than the few obtained from non-TPM experiments (four elastic constants and the geometry of the repressor tetramer); all other parameters we used had known values (e.g., bead size and details of the DNA construct). So it is not surprising that some of our results are only in
qualitative agreement with experiment. Nevertheless, we ﬁnd that:
• Our physical model quantitatively predicts basic aspects of the TPM experiments, such as the effects of varying tether length and bead size (see Fig. 3).
• The model can roughly explain the overall value of the looping J factor obtained in experiments for a range of loop lengths near 300 basepairs (discussed in Sect. 5).
• Perhaps most surprising, the same simple model predicts rather well the observed, detailed structure of the distribution of bead excursions, including its dependence on loop lengths near 300 basepairs (see Fig. 12). The distinctive three-peaks structure of this distribution [24–26] has sometimes been taken as prima facie evidence for a hypothetical alternate “open” conformation of the repressor protein. But we show that it can also arise without that hypothesis, as a consequence of the contributions of loops with different topologies.
• Notwithstanding those successes, our simple model does not successfully extrapolate to predict the magnitude of the J factor for loop lengths near 100 basepairs, at least according to the limited, preliminary experimental data now available. Instead, there it underestimates J, pointing to a breakdown of some of its hypotheses in this high-strain situation. Perhaps the needed modiﬁcation is a nonlinear elastic theory of DNA [27, 28], signiﬁcant ﬂexibility in the tetramer, additional nonspeciﬁc binding of DNA to the repressor protein, or some combination of these.
• However, our model does give a reasonable account of the structure of the bead excursion distribution even for loop lengths near 100 bp (see Fig. 13).
• Because previous authors have proposed the speciﬁc hypothesis that one of the excursiondistribution peaks reﬂects an “open” conformation of LacI, we simulated that situation as well.