Fig 1.
Imputation of spatial transcriptomes by graph-regularized tensor completion.
(A) The input sptRNA-seq data is modeled by a 3-way sparse tensor in genes (p-mode) and the (x, y) spatial coordinates (x-mode and y-mode) of the observed gene expressions. H&E image is also shown to visualize the cell morphologies aligned to the spots. (B) A protein-protein interaction network and a spatial graph are integrated as a product graph for tensor completion. The spatial graph is also a product graph of two chain graphs for columns (x-mode) and rows (y-mode) in the grid. (C) After the imputation, the CPD form of the complete tensor can be used to impute any missing gene expressions, e.g. the entry (k, j, i) can be reconstructed as the sum of the element-wise multiplications of the three components ,
and
.
Table 1.
Notations.
Table 2.
10x Genomics spatial transcriptome data from 10 tissue sections.
Fig 2.
Spot-wise cross-validation on 10x Genomics data.
The performances of the four compared methods on the 10 tissue sections are measured by 5-fold cross-validation. Each bar shows the mean of the imputation performance of one method on all the spatial spots. The result on each of the 10 datasets is shown in one vertical column separated by dashed lines. The means are also compared between FIST and each of the baseline methods in S1 Table by paired-sample t-tests.
Fig 3.
Gene-wise cross-validation on 10x Genomics data.
The performances of the five compared methods on the 10 tissue sections are measured by 5-fold cross-validation. Each error bar shows the mean and variance of the imputation performance for one method on all the genes. The result on each of the 10 datasets is shown in one vertical column separated by dashed lines.
Fig 4.
Analysis of Cartesian product graph regularization with varying network hyper-parameter in spot-wise evaluation.
The plots show the imputation performance of FIST on the ten 10x Genomics datasets with varying network hyper-parameters in {0, 0.1, 1} by MAE, MAPE and R2.
Fig 5.
Analysis of Cartesian product graph regularization on gene-wise evaluation.
(A) The percentages of genes benefit from the CPG are plotted by their densities in four different ranges. Each colored line represents one of the 10 datasets. (B) The total reduction of MAE using the original and permuted graphs are compared across the 10 tissue sections.
Fig 6.
Enrichment analysis on the sparse and imputed sptRNA-seq data.
The total number of significantly enriched clusters (with at least one enriched GO term with FDR adjusted p-value < 0.05) in the 10 tissue sections are shown.
Fig 7.
FIST recovers spatial gene expression patterns on Mouse Kidney Section.
The H&E image of the mouse kidnesy section is shown in the middle with circles roughly separating the tissue area of Cortex, the outer stripe of the outer medulla (OSOM) and the inner stripe of the outer medulla (ISOM) from outer to inner regions. The gene expression patterns of the clusters in each of the three regions are grouped in the same box labeled by the region.
Table 3.
Functional terms enriched by spatial gene clusters (most significantly relevant functions).
Fig 8.
Spot-wise imputation performance on mouse tissue replicates.
The performances of the four compared methods on the 3 replicates are measured by 5-fold cross-validation. The performance on each spatial spot is denoted by one dot in the box plots. The performances of different methods are shown in different colors.
Fig 9.
Imputation performance of FIST on mouse tissue replicates by varying network hyper-parameters.