DNA Looping Kinetics Analyzed Using Diffusive Hidden Markov Model

Abstract

Tethered particle experiments use light microscopy to measure the position of a micrometer-sized bead tethered to a microscope slide via an approximately micrometer-length polymer, to infer the behavior of the invisible polymer. Currently, this method is used to measure rate constants of DNA loop formation and breakdown mediated by repressor protein that binds to the DNA. We report a new technique for measuring these rates using a modified hidden Markov analysis that directly incorporates the diffusive motion of the bead, which is an inherent complication of tethered particle motion because it occurs on a timescale between the sampling frequency and the looping time. We compare looping lifetimes found with our method, which are consistent over a range of sampling frequencies, to those obtained via the traditional threshold-crossing analysis, which vary depending on how the raw data are filtered in the time domain. Our method does not involve such filtering, and so can detect short-lived looping events and sudden changes in looping behavior.

One mechanism for regulating DNA transcription is for a protein to bind to specific operator sites in the DNA sequence, thereby enhancing or diminishing the expression of adjacent genes. In an elaboration of this idea, multiple operators recruit copies of the repressor protein, bind to each other, and bend the DNA into a loop, for example in the λ-system (1). One goal of in vitro DNA looping experiments is to determine the rate constants for DNA loop breakdown/formation and gain insight into how the physical process of looping influences the biochemistry of transcription. The purpose of this letter is to present a new diffusive hidden Markov method (DHMM) for determining the looping kinetics from data measured in tethered particle experiments. The main advantage of our method is that by directly incorporating the dynamics of particle diffusion we do not need to filter the raw data. Consequently, DHMM has better time resolution and more consistent results than the traditional threshold-crossing analyses.

The tethered particle method (TPM, Fig. 1) consists of measuring the Brownian motion of a small bead attached to a microscope slide via a short polymer tether, to learn about the tether's behavior (2–6). Our setup uses DIC imaging of a 480-nm diameter polystyrene bead tethered to the slide via a 3477-bp DNA construct containing two sets of three wild-type λ-operator sites separated by 2317 bp, as described previously (7). The (x,y) coordinates of up to six well-spaced beads are recorded simultaneously with 20-ms time resolution using custom particle tracking software and a CCD camera with a 1-ms shutter to reduce blurring. Bead positions are first recorded for ∼10 min to ensure uniform behavior. Upon addition of 200 nM cI repressor protein, dynamic exchanges between unlooped and looped tether lengths—consistent with the known construct length and operator spacing—are observed. After recording for 30–60 min, the data are corrected for microscope drift and screened for anomalous sticking events using methods described previously (8).

FIGURE 1.

Schematic showing DNA loop formation and breakdown in a tethered bead. The character ρ denotes the plane-projected distance from the attachment point to the bead center.

After drift correction, transitions are clearly visible when the data are plotted as the radial distance from the anchor point where t is an index indicating which video frame (Fig. 2). The equilibrium distributions of ρ are well understood (8,9), but the large overlap between unlooped and looped distributions at small values of ρ prevents us from unambiguously determining the state of the DNA at particular times—loop formation is not directly observable. Typically, this ambiguity is reduced by filtering ρ (we find the variance over windows of time width W); however, the time resolution is then degraded by at least the same amount (6,7). Filtering helps remove false events at very short times introduced by natural Brownian motion of the bead, but actual events are missed due to the reduced time resolution. Unfortunately, we show below that in our system, looping lifetimes determined by this technique depend strongly on the chosen value of W.

FIGURE 2.

Dynamic looping seen in tethered particle time-series data. The region from time 200–1200 s shows dynamic looping behavior and was used in the subsequent analysis. The ▾ indicates a brief 4.5 s sticking event that was omitted from the analysis by concatenating the drift-corrected data. The inset shows the corresponding filtered time series with window size W = 4 s. Red lines are inferred transition sequences for DHMM and threshold-crossing analyses, respectively. Dashed line = 270 nm threshold.

Hidden Markov methods (10,11), however, do not require such smoothing. These methods allow for analysis of the unfiltered data, once we overcome one obstacle: In traditional HMM applications, the uninteresting part of the observed signal (or “noise”) has no correlations apart from those introduced by the underlying hidden process. Unfortunately, the tethered Brownian motion of our bead has an intrinsic timescale that is slow compared with our 20-ms sampling frequency, but faster than the looping lifetime. The basic physics is easily reproduced by a particle diffusing in a harmonic potential well (a problem similar to TPM motion), with diffusion constant D = 480,000 nm²/s and spring constant κ = 0.65 × 10⁻³ pN/nm obtained from fits to the autocorrelation of the measured position, the characteristic decay time τ_D = k_BT/Dκ is ∼140 ms (12). This diffusive motion not only prevents efficient filtering, but also the direct application of traditional hidden Markov methods to TPM.

More precisely, standard HMM supposes that an observed signal reflects two processes (10): A hidden process that generates a time series {q_t} according to an autonomous Markov process with some time-step distribution D(q_t+1|q_t), and an observed signal {r_t} that, at each instant t, is drawn from a probability distribution P(r_t|q_t), which depends only on the current value of q_t. This framework is appropriate for the case where q_t is the internal state of an ion channel and r_t is the instantaneous current through the channel. We might be tempted to apply it to our case as well, letting q_t denote the looping state of our DNA tether and r_t = (x_t, y_t). But the ability to form a loop depends on the location of the bead: For example, if the bead is too far from the attachment point, then loop formation is impossible until the bead has wandered closer, invalidating the assumption made in standard HMM. Moreover, the next bead location r_t+1 depends not only on the present looping state, but also on the present bead location (if the chosen time step is not much longer than the bead diffusion time). For both of these reasons, we must modify the usual formulation of HMM.

To find the required modification, we first note when no cI protein is present, the bead executes tethered Brownian motion, and this motion is itself a Markov process: The bead's displacement r_t+1 depends only on r_t, not on earlier positions. We extracted the “unlooped” probability distribution for the next position, D_un(r_t+1|r_t), from observed time series in a control experiment. Then we found the analogous distribution D_loop(r_t+1|r_t) for permanently looped tethers. Our two distributions D_un and D_loop were thus determined phenomenologically, with no attempt to model the dynamical details of tethered Brownian motion near a wall. As functions of r_t+1, the distributions D_un and D_loop are both roughly two-dimensional Gaussians centered about a point that depends on r_t, with widths that reflect the random excursions of Brownian motion in one time step. We checked that simulating Markov processes with these distributions gave good agreement with the control data, both for the probability distributions of the radial distance ρ, and for the autocorrelation functions of x and y, two nontrivial consistency checks on our data and theory.

To incorporate the hidden state dependence, we constructed a heuristic joint distribution function D_DHMM(q_t+1, r_t+1|q_t, r_t), the probability of observing q_t+1, r_t+1 given q_t, r_t, as follows. If the DNA is initially unlooped (q_t = 1) and ρ_t is too large to permit loop formation, then the DNA must remain unlooped in the final state: D_DHMM = D_un(r_t+1|r_t) for q_t+1 = 1 and D_DHMM = 0 for q_t+1 = 2.

However, if q_t =1 and ρ_t is less than the maximum excursion observed for beads with a permanently looped tether (observed in a separate experiment and verified via Monte Carlo simulation (9)), then both final states are allowed, and we take D_DHMM = (1−Δt/τ_LF)D_un(r_t+1|r_t) for q_t+1 = 1 and D_DHMM = (Δt/τ_LF)D_loop(r_t+1|r_t) for q_t+1 = 2. The rate constant 1/τ_LF is a parameter of the model, the probability per time to form a loop when permitted. A similar construction gives the case when the DNA is initially looped (q_t = 2), in terms of a second unknown rate constant 1/τ_LB for loop breakdown.

We repeated the above calculation for all pairs of data points and summed over all possible sets of the hidden variables q_t (10), resulting in a likelihood function

with two unknown fit parameters, τ_LF and τ_LB, which are the quantities of interest to us. We evaluated the likelihood for various values of the parameters, expressing it as a two-dimensional surface on a logarithmically spaced grid. The resulting surface is smooth, so the peak likelihood can be determined by fitting a two-dimensional quadratic in the neighborhood of the optimum lifetimes, including error bars corresponding to a range of lifetimes enclosing the maximum with 97% confidence. A computer code implementing our algorithm is available as Supplementary Material.

We tested our DHMM by analyzing multiple data subsets obtained by thinning the data, by either a factor of two (Δt = 40 ms) or four (Δt = 80 ms). All computed lifetimes and cross validation of the likelihoods between independent data subsets agreed within uncertainty. To test the algorithm further, we generated a Monte Carlo simulation of the 40-ms looping data, assuming the values of τ_LB and τ_LF determined from the experimental data. Then we applied our DHMM method to the simulated data, and checked that it again found the known values and that the event detection corresponded to the time series of the hidden looping transitions (which were known in the simulated data). In contrast to these consistent results, we found that the threshold-crossing method resulted in a lifetime that depends on the filter window size W; see Fig. 3 where, for simplicity, only τ_LF is shown. (Additional tests and further mathematical details of the method will be discussed elsewhere.)

FIGURE 3.

Lifetime results for the threshold method for DNA loop formation (points) of the measured data shown in Fig. 2, as a function of filter window W. Dwell times between threshold crossings in the unlooped/looped state longer than twice the filter dead time, i.e., 2W, were fit to a histogram to determine τ_LF and τ_LB (13). Our DHMM method uses no window; its result is shown as the dashed line. The shaded region corresponds to the error estimate discussed in the text.

We have developed a new method for assessing DNA looping rates from data obtained by the tethered particle method. We tested it on actual and simulated data and determined lifetimes that were independent of sampling frequency. DHMM should improve TPM as a quantitative tool, providing results that are more consistent with improved time resolution compared to the threshold-crossing method.

SUPPLEMENTARY MATERIAL

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