Shaped 3D singular spectrum analysis for quantifying gene expression, with application to the early zebrafish embryo

Abstract

Recent progress in microscopy technologies, biological markers, and automated processing methods is making possible the development of gene expression atlases at cellular-level resolution over whole embryos. Raw data on gene expression is usually very noisy. This noise comes from both experimental (technical/methodological) and true biological sources (from stochastic biochemical processes). In addition, the cells or nuclei being imaged are irregularly arranged in 3D space. This makes the processing, extraction, and study of expression signals and intrinsic biological noise a serious challenge for 3D data, requiring new computational approaches. Here, we present a new approach for studying gene expression in nuclei located in a thick layer around a spherical surface. The method includes depth equalization on the sphere, flattening, interpolation to a regular grid, pattern extraction by Shaped 3D singular spectrum analysis (SSA), and interpolation back to original nuclear positions. The approach is demonstrated on several examples of gene expression in the zebrafish egg (a model system in vertebrate development). The method is tested on several different data geometries (e.g., nuclear positions) and different forms of gene expression patterns. Fully 3D datasets for developmental gene expression are becoming increasingly available; we discuss the prospects of applying 3D-SSA to data processing and analysis in this growing field.

Figure 1

Zebrafish data [4] and the 3D-SSA approach: original nucleus position data for the spherical-cap gene expression region, with simulated intensity values (a schematic two-exponential pattern, with noise) represented by the color map; colors are given in the topographic scale, where the blue color corresponds to small values, when brown/red colors correspond to large values of intensity.

Figure 2

Processing steps in the 3D-SSA approach.

Figure 3

CAP-2EXP: the nuclear hulls, coloring based on original and pattern expression intensities.

Figure 4

CAP-2EXP: nuclei colored; an outside view.

Figure 5

CAP-2EXP: nuclei colored; an inside view.

Figure 6

CAP-BELL: the nuclear hulls, coloring based on original and pattern expression intensities.

Figure 7

CAP-BELL: nuclei colored; an inside view.

Figure 8

CAP-2EXP: 2D slices of 3D eigenarrays at d = L ₃/2.

Figure 9

CAP-2EXP: elementary component reconstructions, slices.

Figure 10

CAP-2EXP pattern: 2D slice with expression values, depth d = 82.5%; (a) the surface depicts the reconstructed pattern (in the regular grid points), the points show expression values in individual nuclei from a ±10% layer; (b) nuclear residuals from the reconstruction are scattered evenly around the zero plane.

Figure 11

CAP-2EXP pattern: 1D slice at 82.5% depth (d) and 50% width (φ). (a) Pattern on the grid (curve) and original values on the nuclei (points). (b) Residuals between the reconstruction and the nuclear data values.

Figure 12

CAP-2EXP pattern, normalized residuals. (a) Test for the additive ($\tilde{\alpha }=\mathrm{0}$) noise model. (b) Test for Poissonian noise ($\tilde{\alpha }=\mathrm{0.5}$). (c) Test for multiplicative noise ($\tilde{\alpha }=\mathrm{1}$). The homoscedasticity of the residuals on (c) indicates that the noise is multiplicative, confirming the noise model used in simulated generation of the data (3).

Figure 13

STRIP-2EXP pattern (3), nuclear hull, coloring based on the original simulated data; outside view, with shadow.

Figure 14

STRIP-2EXP: colored nuclear hull; outside views; without shadow.

Figure 15

STRIP-2EXP: colored nuclei; outside views; without shadow.

Figure 16

STRIP-2EXP: 1D slices for 50% depth (d) and width (ψ), signal reconstructed by 1 (a), 2 (b), 3 (c), and 4 (d) components.

Figure 17

The distribution of nuclei (red and green dots) in a shield stage early zebrafish embryo, with green dots corresponding to the nuclei expressing gene ntla. Higher magnification in the insert shows mixing of green and red (ntla off) nuclei along the boundary of the expression domain.