Inference of the Xenopus tropicalis embryonic regulatory network and spatial gene expression patterns

Abstract

Background: During embryogenesis, signaling molecules produced by one cell population direct gene regulatory changes in neighboring cells and influence their developmental fates and spatial organization. One of the earliest events in the development of the vertebrate embryo is the establishment of three germ layers, consisting of the ectoderm, mesoderm and endoderm. Attempts to measure gene expression in vivo in different germ layers and cell types are typically complicated by the heterogeneity of cell types within biological samples (i.e., embryos), as the responses of individual cell types are intermingled into an aggregate observation of heterogeneous cell types. Here, we propose a novel method to elucidate gene regulatory circuits from these aggregate measurements in embryos of the frog Xenopus tropicalis using gene network inference algorithms and then test the ability of the inferred networks to predict spatial gene expression patterns.

Results: We use two inference models with different underlying assumptions that incorporate existing network information, an ODE model for steady-state data and a Markov model for time series data, and contrast the performance of the two models. We apply our method to both control and knockdown embryos at multiple time points to reconstruct the core mesoderm and endoderm regulatory circuits. Those inferred networks are then used in combination with known dorsal-ventral spatial expression patterns of a subset of genes to predict spatial expression patterns for other genes. Both models are able to predict spatial expression patterns for some of the core mesoderm and endoderm genes, but interestingly of different gene subsets, suggesting that neither model is sufficient to recapitulate all of the spatial patterns, yet they are complementary for the patterns that they do capture.

Conclusion: The presented methodology of gene network inference combined with spatial pattern prediction provides an additional layer of validation to elucidate the regulatory circuits controlling the spatial-temporal dynamics in embryonic development.

Figure 1

Inference error versus number of observations. The proportional error (i.e., inference error) denotes the minimal cross-validation error divided by the minimal least-squares error of the linear regression without any regularization terms and averaged over five random networks. The proportional errors decrease with more observations and stabilize when there are enough observations.

Figure 2

Inference error versus number of random valid edges provided. The proportional error (i.e., inference error) denotes the minimal cross-validation error divided by the minimal least-squares error of the linear regression without any regularization terms and averaged over five random networks and five random sets of valid edges. Zero to 25 (i.e., the number of all edges in each random network) prior edges are provided respectively. (A), (B) and (C) are the results of 4, 5 and 6 observations, respectively. The proportional errors decrease when more valid edges are provided. Prior connections appear to be more important when only a few observations are available.

Figure 3

Inference error versus number of random invalid edges (without valid edges) provided. The proportional error (i.e., inference error) denotes the minimal cross-validation error divided by the minimal least-squares error of the linear regression without any regularization terms and averaged over five random networks. The red, blue and green curves represent 4, 6 and 10 observations, respectively. Providing invalid network edges only has little effect on the errors, especially when many observations are available. The errors become smaller as the number of observations increases, which is consistent with the results in Figure 1.

Figure 4

Inference error versus number of random invalid edges. The proportional error (i.e., inference error) denotes the minimal cross-validation error divided by the minimal least-squares error of the linear regression without any regularization terms and averaged over five random networks. The blue, green and red curves represent 0, 5 and 20 random valid edges, respectively. 4 and 2 observations are considered in (A) and (B), respectively. When there are more valid edges (e.g., valid = 20), the errors are generally smaller as a whole. When only a few observations are available, the valid edges appear to be even more important. The cross-validation errors are in a large scale (i.e., 10⁵) in (B), because they are divided by the least-squares errors, while fewer observations are easier to be over fitted with small least-squares errors (e.g., 10^-6 ~ 10^-4).

Figure 5

Inferred network from the linear ODE model. Positive interactions are shown in blue and negative interactions are shown in red. The strength of connections is indicated via the thickness of lines connecting the genes. The interactions vary from −1.2921 to 1.4132 and a threshold 0.25 is applied, i.e., all the edges with the weights smaller than 0.25 are discarded.

Figure 6

Inferred network from the linear Markov model. Positive interactions are shown in blue and negative interactions are shown in red. The strength of connections is indicated via the thickness of lines connecting the genes. The interactions vary from −1.1317 to 2.5607 and a threshold 0.2 is applied, i.e., all the edges with the weights smaller than 0.2 are discarded.

Figure 7

Spatial gene expression patterns in Xenopus embryos. (A) The ventx gene is expressed ventrally. (B) The gsc gene is expressed dorsally. (C) The bix1 gene is expressed in both the ventral and dorsal regions. (D) The gata4 gene is expressed in the vegetal region. The rectangular bar across the embryo indicates the portion of the image viewed for classification purposes, with the bar under the image showing the 1-dimensional representation of gene expression.

Figure 8

Percentage of correctly predicted patterns in the ODE spatial prediction model. Given a fixed number (from 1 to 26) of pre-defined patterns, the percentages are averaged over 20 sets (Figure (A)) and 100 sets (Figure (B)) of randomly chosen genes with pre-defined patterns. As expected, the percentages of correctly predicted patterns are more stable when they are averaged over more sets of randomly chosen genes. Also, the prediction percentages show an increasing trend as more pre-defined patterns are provided.

Figure 9

Estimated probability density distribution of the spatial patterns prediction. 1000 random networks derived from rearranging the inferred network were used to calculate the fractions of the correctly predicted genes in the 28 genes with prior patterns. (A) 1000 random networks were derived from rearranging the inferred 28-gene ODE network. They are mostly about 12% correct (the p-value is 0.007, i.e., there are only 0.7% of the 1000 random networks obtaining not less prior patterns than the inferred ODE network), while the inferred ODE network is 39.3% correct. (B) 1000 random networks were derived from rearranging the inferred 28-gene Markov network. They are mostly about 14% correct (the p-value is 0.002, i.e., there are 0.2% of the 1000 random networks obtaining not less prior patterns than the inferred Markov network), while the inferred Markov network is 39.3% correct.