Quantifying the phenotypic information in mRNA abundance

Abstract

Quantifying the dependency between mRNA abundance and downstream cellular phenotypes is a fundamental open problem in biology. Advances in multimodal single-cell measurement technologies provide an opportunity to apply new computational frameworks to dissect the contribution of individual genes and gene combinations to a given phenotype. Using an information theory approach, we analyzed multimodal data of the expression of 83 genes in the Ca²⁺ signaling network and the dynamic Ca²⁺ response in the same cell. We found that the overall expression levels of these 83 genes explain approximately 60% of Ca²⁺ signal entropy. The average contribution of each single gene was 17%, revealing a large degree of redundancy between genes. Using different heuristics, we estimated the dependency between the size of a gene set and its information content, revealing that on average, a set of 53 genes contains 54% of the information about Ca²⁺ signaling. Our results provide the first direct quantification of information content about complex cellular phenotype that exists in mRNA abundance measurements.

Keywords: cellular heterogeneity; gene expression; information theory; mutual information; signaling dynamics.

Figure 1. Structure of gene and Ca²⁺ data

1. A

   Representative examples of Ca²⁺ dynamics of four cells in the dataset.

2. B

   A histogram of the pairwise gene correlation matrix (tri‐up) which highlights the relatively low correlations.

3. C

   Explained variance of mRNA transcript counts from PCA.

4. D

   Differential entropy of transcripts estimated by PCA.

5. E

   Plot of explained variance (panel C) vs differential entropy (panel D) with an increasing number of principal components.

6. F

   Dynamic Ca²⁺ signal periodogram (cropped to show only the lower wavelength, higher power frequencies). Ca²⁺ dynamic signals were found to contain a spectral entropy of 4.2 bits.

Data information: Collectively, panels (C–E) show that most of the entropy comes from components that do not explain much of the variance.

Figure 2. Mutual information between individual genes and Ca²⁺ signals

1. A

   I(G_i;Ca²⁺) sorted from least to greatest. Error bars show standard deviation of three technical replicates, that is three independent runs of the MINE algorithms on the same data.

2. B

   The blue line shows the cumulative sum of I(G_i;Ca²⁺) from (A) sorted from greatest to least, and individual genes appear to contain a lot of information (56 bits) about Ca²⁺ signals. The black dashed line shows mutual information between all 83 genes and Ca²⁺ dynamics estimated to be 2.5 bits.

3. C

   Histogram of (A) showing the mean I(G_i;Ca²⁺) is 0.7 bits.

4. D

   I(G_i;G_j) represents the pairwise mutual information between genes, the information that genes have about each other. This plot shows that genes that are more informative about other genes tend to be more informative about Ca²⁺ dynamics.

Figure 3. Synergy and redundancy of gene pairs with respect to Ca²⁺

1. A

   SRI(G_i,G_j¦C) sorted by average SRI.

2. B

   Histogram of SRI showing that most gene pairs are highly redundant with an average score of −0.43 bits.

3. C

   The mean rank of all synergistic pairs compared to the mutual information between that gene and Ca²⁺ signals, (spearman r = 0.5, P < 2e‐6), indicating that genes with more information about Ca²⁺ are also more redundant.

Figure 4. Mutual information between gene sets and Ca²⁺ signals

1. A

   Mean pairwise SRI from Fig 3 for sets based on annotation. MCF 10A differentiation and Ca²⁺‐dependent response are abbreviated.

2. B

   Gene sets of various sizes were constructed using three different strategies: an upper bound (green) that always adds the most informative gene to the set given the genes already included, random strategy (boxes, center band shows the median, box covers the 25–75% and whiskers show 95% confidence interval. Boxes are based on 12 random samples of genes) that samples random sets of genes, and a lower bound (purple) that always adds the least informative gene to the set given the genes already included.

3. C

   The blue line shows the fraction of redundant information using the expected value of I({G₀, …, G_n};Ca²⁺) from equation (3).

4. D

   A y‐y plot of gene differential entropy in blue (same as Fig 1D) and the mutual information between gene principal components and Ca²⁺ in orange. Both values are normalized by their respective max values.