Single-cell Bayesian deconvolution

Abstract

Individual cells exhibit substantial heterogeneity in protein abundance and activity, which is frequently reflected in broad distributions of fluorescently labeled reporters. Since all cellular components are intrinsically fluorescent to some extent, the observed distributions contain background noise that masks the natural heterogeneity of cellular populations. This limits our ability to characterize cell-fate decision processes that are key for development, immune response, tissue homeostasis, and many other biological functions. It is therefore important to separate the contributions from signal and noise in single-cell measurements. Addressing this issue rigorously requires deconvolving the noise distribution from the signal, but approaches in that direction are still limited. Here, we present a non-parametric Bayesian formalism that performs such a deconvolution efficiently on multidimensional measurements, providing unbiased estimates of the resulting confidence intervals. We use this approach to study the expression of the mesodermal transcription factor Brachyury in mouse embryonic stem cells undergoing differentiation.

Keywords: Biocomputational method; Complex system biology; Optical Signal Processing; Technical aspects of cell biology.

Graphical abstract

Figure 1

Scheme of the deconvolution process The signal and noise mixture distributions, together with the observed data (top row), define the posterior distribution over the parameter space of the mixture, Equation 6 (middle row). This distribution can present multiple peaks, sometimes degenerate with respect to basis label exchange, each corresponding to a different mixture description of the observed and target distributions. The red arrow in the Gibbs Markov Chain Monte Carlo sampling plot (bottom left) represents an unlikely jump between two peaks separated by a relatively wide probability valley.

Figure 2

Deconvolution instance for a target bimodal distribution (light blue in B and C) corrupted by a normally distributed noise with a signal-to-noise ratio $\text{SNR}=2$ (magenta distribution in the inset of A), applying the Bayesian Deconvolution method (B) and the FFT method (C) (A and B) The total distribution from which the noise is deconvolved is shown in green in panel (A). The red lines in panel (B) depict samples of the Bayesian fitting process. In this case, the noise distribution is a single normal function with mean ${\mu }^{\xi }=0$ and standard deviation ${\sigma }^{\xi }=0.5$, and the target is a mixture of two normal functions with means ${\mu }_1^T=-0.43$ and ${\mu }_2^T=1.67$, standard deviations ${\sigma }_1^{\xi }={\sigma }_2^{\xi }=0.6$, and weights ${\omega }_1^T=0.8$ and ${\omega }_2^T=0.2$.

Figure 3

Similarity between the deconvolved and ground-truth target distributions as expressed by the Mean Integrated Overlap (MIO) for the two deconvolution methods (x and y axis), at different sampling sizes for the synthetic datasets shown in Figure S1, with $\text{SNR}=1$ (circles) and $\text{SNR}=10$ (squares) For each value of N and SNR, all nine noise and target combinations of Figure S1 are shown.

Figure 4

Deconvolution of the CMFDA dye in condition c4 (N2B27+CHIR99+CMFDA) (A–C) Overlaying the distributions we show realizations of the Bayesian sampling process (red lines) for the three distributions (convolution (A), noise (B) and target (C) obtained during the fitting. The convolved and target distributions (green and light blue, respectively) come from the real data. The distribution of the dye (in magenta (B) results from sampling the inferred dye distribution, as described in the text.

Figure 5

Two-dimensional scatterplot showing a sample of the results of the autofluorescence in a multichannel system Light green circles and orange dots are samples of the real data from the convolved and autofluorescence distributions, respectively. Light blue circles are samples from the results of the deconvolution for the two channels. Light gray circles are the result of subtracting the mean autofluorescence from the convolved distribution, the most traditional approach to correct for autofluorescence. The Gaussian mixture model has two main components in the convolved data (accounting together for more than 95% of the probability density, as seen by the weight contributions of each component shown in the inset), which are shown in red and blue for the convolved and deconvolved distributions.