Fluctuation Analysis: Dissecting Transcriptional Kinetics with Signal Theory

Abstract

Recent live-cell microscopy techniques now allow the visualization in multiple colors of RNAs as they are transcribed on genes of interest. Following the number of nascent RNAs over time at a single locus reveals complex fluctuations originating from the underlying transcriptional kinetics. We present here a technique based on concepts from signal theory-called fluctuation analysis-to analyze and interpret multicolor transcriptional time traces and extract the temporal signatures of the underlying mechanisms. The principle is to generate, from the time traces, a set of functions called correlation functions. We explain how to compute these functions practically from a set of experimental traces and how to interpret them through different theoretical and computational means. We also present the major difficulties and pitfalls one might encounter with this technique. This approach is capable of extracting mechanistic information hidden in transcriptional fluctuations at multiple timescales and has broad applications for understanding transcriptional kinetics.

Keywords: Dynamics; Fluctuation analysis; Fluorescence; Imaging; RNA; Single cell; Single molecule; Splicing; Transcription.

Fig. 1

Transcriptional time traces and correlation function. (A) The MS2 and PP7 RNA-labeling technique consists in inserting, in one or two gene(s) of interest, two DNA cassettes (MS2 and PP7; here at two different locations in the same gene). They produce stem-loop structures in the nascent RNAs, which are bound by an MS2 or PP7 coat protein (MCP and PCP) fused to a fluorescent protein (eg, GFP and mCherry, respectively). (B) The transcription site (arrow) appears as a bright diffraction-limited spot in the nucleus in both fluorescence channels. (C) Recording its intensity fluctuations then yields a signal that is proportional to the number of nascent RNAs on the gene over time. (D) Using this signal as an example, the computation of a correlation function (here the covariance function) consists in shifting one signal relatively to the other and calculating the covariance between the values of the overlapping portions of the two signals as a function of the time-delay shift (Eqs. 1 and 2). (E) To analyze fluctuations at multiple timescales in a signal, computing the correlation function with the multipletau algorithm yields a somewhat uniform spacing of the time-delay points on a logarithmic scale (simulated data as in Fig. 2A). Panels (B) to (D): Data from Coulon, A., Ferguson, M. L., de Turris, V., Palangat, M., Chow, C. C., & Larson, D. R. (2014). Kinetic competition during the transcription cycle results in stochastic RNA processing. eLife, 3. http://doi.org/10.7554/eLife.03939.

Fig. 2

Biases due to the finiteness of time traces. (A) Shown is a portion of a signal used to illustrate the effect of inaccurate mean estimation. This 1000-min-long signal is partitioned into a set of 20-min-long signals. The true mean of the signal (ie, calculated on the long trace) is shown in gray and the inaccurate means of individual short traces are shown in black. (B) The autocovariance M(τ) of the entire signal shown in (A) is close to the expected curve (red circles vs gray curve). When averaging the autocovariances computed on each one of the 20-min-long traces, the resulting autocovariance deviates from the expected curve by a constant offset (green circles). (C) Performing the same calculation using a global estimation of the mean of the signals (ie, once over all the short signals) resolves the issue. (D) Another long signal is shown and partitioned into small sections to illustrate another artifact that may arise when averaging correlation functions G(τ). (E) The average (green circles) of the autocorrelation functions obtained on the 20-min-long sections of the signal shown in (D) deviates from the expected curve. This is due to an inaccurate weighting of the individual curves that occurs when averaging autocorrelation functions G(τ). As illustrated by the inset, the section that has a very low mean in (D) dominates the average. (F) As in (C), estimating the mean globally over all the signals solves the weighting problem. Both examples shown are simulated signals (A: Gaussian noise shaped in the Fourier domain, D: Monte Carlo simulation of transcription with Poisson initiation, distributed transcript dwell time and additive Gaussian noise). The “truth” curves in gray in (B), (C), (E), and (F) are the theoretical curves for both simulated situations.

Fig. 3

Decision chart for averaging method. To avoid introducing biases, the most appropriate method for averaging individual correlation functions depends on the experimenter’s knowledge of the fluorescence-to-RNA conversion factor, ie, the amount of fluorescence units that corresponds to a single, fully synthesized RNA.

Fig. 4

Flowchart for computing correlation functions with standard errors. (A) From a given set of time traces, one should first compute the average correlation function as appropriate (cf Fig. 3) and then perform the corrections described in Section 2.5 if needed. This yields a “corrected” correlation function. (B) To obtain the standard error by the bootstrap method, one should perform, at least 1000 times, the exact same computation as in (A), using each time a random sample of the time traces (same number of traces as the original set, and randomly drawn with replacement). This yields an estimate of the sample distribution, which standard deviation is the standard error on the correlation function calculated in (A). Intermediate results of the calculations are shown using a set of experimental time traces from Coulon et al. (2014) as an example. Shown are the average correlation function (C) before and (D) after baseline correction and renormalization, (E) multiple average correlation functions (as in D) resulting from the bootstrap loop, and (F) the average correlation function with standard errors.

Fig. 5

Principle of correlation function modeling. Although illustrated on a simplified single-color situation, the principle for modeling correlation functions presented here is general and holds for more complex descriptions. (A) Considering that the fluorescence time profile ${\widehat{a}}_p(t)$ recorded at the TS for a single nascent RNA is a rectangular function (rising when the MS2 cassette is transcribed and dropping when the RNA is release), then the covariance function ${\widehat{M}}_p(t)$ of this time profile is a triangle function. (B and C) When transcription initiation is considered homogeneous over time, the covariance function M(τ) can be understood as the average between individual correlation functions of single RNAs (Eq. 11). On the example of (A), if the dwell time X_p of individual RNAs is distributed, then the shape of M(τ) reveals information about initiation rate and dwell time distribution (mean, variability, etc.). (D) Several methods can be used to predict correlation functions from a given mechanistic scenario. As an alternative to a full Monte Carlo simulation approach, giving rather noisy results, and to analytical expressions, sometime difficult to derive, the hybrid approach described in Section 3.1.3 is both simple and precise. In this example, the hybrid method was performed over 100 single-RNA time profiles, and the simulation was performed over a signal that comprises 500 RNAs. The total computation time was similar in both cases. The analytical expression used is Eq. (12).

Fig. 6

Convergence when averaging correlation function. Single correlation functions are often misleading since their shape can show features that may look like regularities but are only due to the lack of data. Averaging multiple correlation functions together reduces this noise and allows one to calculate error bars. The more correlation functions are averaged together the more these spurious features disappear, leaving only the true regularities. The examples shown are simulated traces that are 100 data points eac