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SpatialDE is a method to identify genes which significantly depend on spatial coordinates in non-linear and non-parametric ways. The intended applications are spatially resolved RNA-sequencing from e.g. Spatial Transcriptomics, or in situ gene expression measurements from e.g. SeqFISH or MERFISH.

Additionally, SpatialDE provides automatic expression histology, a method that groups genes into common spatial patterns (and conversely reveal histological patterns based on gene coexpression).

This repository contains both the implementations of our methods, as well as case studies in applying it.

The key features of our method are

  • Unsupervised - No need to define spatial regions
  • Non-parametric and non-linear expression patterns
  • Automatic histology based on spatially coexpressed genes
  • Extremely fast - Transcriptome wide tests takes only a few minutes on normal computers

The primary implementation is as a Python 3 package, and can be installed from the command line by

$ pip install spatialde

(This should only take a minute or so on a typical system)

To see usage example of SpatialDE either keep reading, or look in the Analysis directory. The following examples are provided:

Transcriptome wide study on breast cancer tissue from Spatial Transcriptomics.
A time course of RNA-seq ("1-d space") of Xenopus development.
Expression from single cells in a region of an osteoblast culture using the MERFISH technology with 140 probes.
Spatial Transcriptomics assay of a slice of Mouse Olfactory Bulb. (Also see below)
Expression counts of single cells from mouse hippocampus using the SeqFISH technology with 249 probes.

If you wish to look at the data used or run the notebooks and scripts from start to finish, the data needs to be fetched using git lfs, a plugin to git for managing large files. Installation instructions are available on the projects website. Once git lfs is installed and you have cloned this repository, data can be downloaded by running git lfs pull from inside any repository directory.

If you are having problems using git lfs, all the data are available on figshare here:

Below follows a typical usage example in interactive form.

SpatialDE significance test example use

%pylab inline
import pandas as pd

rcParams['axes.spines.right'] = False
rcParams[''] = False

import NaiveDE
import SpatialDE
Populating the interactive namespace from numpy and matplotlib

As an example, let us look at spatially dependent gene expression in Mouse Olfactory Bulb using a data set published in Stahl et al 2016. With the authors method, hundreds of locations on a tissue slice can be sampled at once, and gene expression is measured by sequencing in an unbiased whole-transcriptome manner.

counts = pd.read_csv('Analysis/MouseOB/data/Rep11_MOB_0.csv', index_col=0)
counts = counts.T[counts.sum(0) >= 3].T  # Filter practically unobserved genes

counts.iloc[:5, :5]
(262, 14859)
  Nrf1 Zbtb5 Ccnl1 Lrrfip1 Bbs1
16.92x9.015 1 1 1 2 1
16.945x11.075 0 0 3 2 2
16.97x10.118 0 1 1 0 0
16.939x12.132 1 0 1 0 4
16.949x13.055 0 0 0 3 0
sample_info = pd.read_csv('Analysis/MouseOB/MOB_sample_info.csv', index_col=0)
counts = counts.loc[sample_info.index]  # Align count matrix with metadata table

.dataframe tbody tr th:only-of-type { vertical-align: middle; } .dataframe tbody tr th { vertical-align: top; } .dataframe thead th { text-align: right; }
x y total_counts
16.92x9.015 16.920 9.015 18790
16.945x11.075 16.945 11.075 36990
16.97x10.118 16.970 10.118 12471
16.939x12.132 16.939 12.132 22703
16.949x13.055 16.949 13.055 18641

We can plot the x and y coordinates in the sample info table to see which locations of the tissue slice has been sampled.

figsize(6, 4)
plt.scatter(sample_info['x'], sample_info['y'], c='k');


Our method assumes normally distributed noise, but the data we are using is from expression counts, and empirically seems to follow a negative binomial distribution. We use technique by Anscombe to approximately transform the data to normal distributed noise.

Secondly, library size or sequencing depth of the spatial samples will bias the expression of every gene. We use linear regression to account for this effect before performing the spatial test.

norm_expr = NaiveDE.stabilize(counts.T).T
resid_expr = NaiveDE.regress_out(sample_info, norm_expr.T, 'np.log(total_counts)').T

For the sake of this example, let's just run the test on 1000 random genes. This should just take a few seconds. With our very fast implementation, testing all 14,000 genes takes about 10 minutes.

sample_resid_expr = resid_expr.sample(n=1000, axis=1, random_state=1)

X = sample_info[['x', 'y']]
results =, sample_resid_expr)
INFO:root:Performing DE test
INFO:root:Pre-calculating USU^T = K's ...
INFO:root:Done: 0.11s
INFO:root:Fitting gene models
INFO:root:Model 1 of 10
INFO:root:Model 2 of 10
INFO:root:Model 3 of 10
INFO:root:Model 4 of 10
INFO:root:Model 5 of 10
INFO:root:Model 6 of 10
INFO:root:Model 7 of 10
INFO:root:Model 8 of 10
INFO:root:Model 9 of 10
INFO:root:Model 10 of 10

The result will be a DataFrame with P-values and other relevant values for each gene.

The most important columns are

  • g - The name of the gene
  • pval - The P-value for spatial differential expression
  • qval - Significance after correcting for multiple testing
  • l - A parameter indicating the distance scale a gene changes expression over
.dataframe tbody tr th:only-of-type { vertical-align: middle; } .dataframe tbody tr th { vertical-align: top; } .dataframe thead th { text-align: right; }
0 1 2 3 4
FSV 0.999955 2.0597e-09 2.0597e-09 2.0597e-09 2.0597e-09
M 4 4 4 4 4
g 2410016O06Rik Arpp19 Srsf7 Wbp7 Cpsf3l
l 0.402001 0.402001 0.402001 0.402001 0.402001
max_delta 4.53999e-05 4.85165e+08 4.85165e+08 4.85165e+08 4.85165e+08
max_ll -52.2589 -107.685 -114.477 -112.664 -49.1672
max_mu_hat -0.826851 -2.21845 -6.67811 -2.25044 0.146089
max_s2_t_hat 0.666985 1.04203e-08 9.22126e-08 1.07257e-08 2.20142e-10
model SE SE SE SE SE
n 260 260 260 260 260
s2_FSV 1.94342 0.253788 47.2945 0.363388 4.48293
s2_logdelta 6.81931e+08 4.3315e+16 8.07194e+18 6.20209e+16 7.65119e+17
time 0.00134182 0.00104499 0.000994921 0.000999928 0.00106692
BIC 126.761 237.613 251.196 247.571 120.577
max_ll_null -53.706 -107.686 -114.478 -112.665 -49.1681
LLR 1.44715 0.000964007 0.000964011 0.000964007 0.00096401
pval 0.228986 0.975231 0.975231 0.975231 0.975231
qval 0.975231 0.975231 0.975231 0.975231 0.975231
results.sort_values('qval').head(10)[['g', 'l', 'qval']]
.dataframe tbody tr th:only-of-type { vertical-align: middle; } .dataframe tbody tr th { vertical-align: top; } .dataframe thead th { text-align: right; }
g l qval
890 Kcnh3 1.907609 0.001512
772 Pcp4 1.135190 0.013843
736 Igfbp2 1.135190 0.013843
800 Gng13 1.907609 0.022632
646 Naaa 0.675535 0.051705
749 Map1b 1.135190 0.051705
826 Gng4 1.907609 0.051705
724 Fmo1 1.135190 0.096710
714 Slc38a3 1.135190 0.096710
712 Hpcal4 1.135190 0.107360

We detected a few spatially differentially expressed genes, Cck and Ptn for example.

A simple way to visualize these genes is by plotting the x and y coordinates as above, but letting the color correspond to expression level.

figsize(10, 3)
for i, g in enumerate(['Kcnh3', 'Pcp4', 'Igfbp2']):
    plt.subplot(1, 3, i + 1)
    plt.scatter(sample_info['x'], sample_info['y'], c=norm_expr[g]);



For reference, we can compare these to genes which are not spatially DE

results.sort_values('qval').tail(10)[['g', 'l', 'qval']]
.dataframe tbody tr th:only-of-type { vertical-align: middle; } .dataframe tbody tr th { vertical-align: top; } .dataframe thead th { text-align: right; }
g l qval
334 Tmem70 0.402001 0.975231
335 Rnf20 0.402001 0.975231
336 Zfp85-rs1 0.402001 0.975231
337 C1qtnf7 0.402001 0.975231
338 Ap4b1 0.402001 0.975231
339 Psma4 0.402001 0.975231
340 Aldh3b1 0.402001 0.975231
341 Hdx 0.402001 0.975231
328 Zfp113 0.402001 0.975231
999 Preb 9.052138 0.975231
figsize(10, 3)
for i, g in enumerate(['Myo9b', 'Sc4mol', 'Phf11b']):
    plt.subplot(1, 3, i + 1)
    plt.scatter(sample_info['x'], sample_info['y'], c=norm_expr[g]);



In regular differential expression analysis, we usually investigate the relation between significance and effect size by so called volcano plots. We don't have the concept of fold change in our case, but we can investigate the fraction of variance explained by spatial variation.

figsize(5, 4)

plt.scatter(results['FSV'], results['qval'], c='black')

plt.axhline(0.05, c='black', lw=1, ls='--');

plt.xlabel('Fraction spatial variance')
plt.ylabel('Adj. P-value');


Automatic expression histology

To perform automatic expression histology (AEH), the genes should be filtered by SpatialDE significance. For this example, let us use a very weak threshold. But in typical use, filter by qval < 0.05

sign_results = results.query('qval < 0.05')

AEH requires two parameters: the number of patterns, and the characteristic lengthscale for histological patterns.

For some guidance in picking the lengthscale l we can look at the optimal lengthscale for the signficant genes.

1.135190    11
1.907609     4
0.675535     4
3.205604     1
Name: l, dtype: int64

Here we see that the lengthscale on average is ~1.5, to use some extra spatial covariance, we put this paramater to l = 1.8.

For the number of patterns, we try C = 3.

histology_results, patterns = SpatialDE.aeh.spatial_patterns(X, resid_expr, sign_results, C=3, l=1.8, verbosity=1)
iter 0, ELBO: -9.48e+08
iter 1, ELBO: -4.20e+08, delta_ELBO: 5.28e+08
iter 2, ELBO: -4.20e+08, delta_ELBO: 7.63e+02
iter 3, ELBO: -4.20e+08, delta_ELBO: 2.07e+02
iter 4, ELBO: -4.20e+08, delta_ELBO: 8.03e+01
iter 5, ELBO: -4.20e+08, delta_ELBO: 3.40e+00
iter 6, ELBO: -4.20e+08, delta_ELBO: 6.62e-02
iter 7, ELBO: -4.20e+08, delta_ELBO: 2.75e-03
iter 8, ELBO: -4.20e+08, delta_ELBO: 3.96e-03
iter 9, ELBO: -4.20e+08, delta_ELBO: 7.49e-05
Converged on iter 9

After fitting the AEH model, the function returns two DataFrames, one with pattern membership information for each gene:

.dataframe tbody tr th:only-of-type { vertical-align: middle; } .dataframe tbody tr th { vertical-align: top; } .dataframe thead th { text-align: right; }
g membership pattern
564 AI593442 1.0 1
619 Arhgef9 1.0 1
632 6330403K07Rik 1.0 1
646 Naaa 1.0 0
712 Hpcal4 1.0 2

And one with realizations for the underlying expression for each histological pattern.

We can visualize this underlying expression in the tissue context as we would for any individual gene.

figsize(10, 3)
for i in range(3):
    plt.subplot(1, 3, i + 1)
    plt.scatter(sample_info['x'], sample_info['y'], c=patterns[i]);
    plt.title('Pattern {} - {} genes'.format(i, histology_results.query('pattern == @i').shape[0] ))


It is usually interesting to see what the coexpressed genes determining a histological pattern are:

for i in histology_results.sort_values('pattern').pattern.unique():
    print('Pattern {}'.format(i))
    print(', '.join(histology_results.query('pattern == @i').sort_values('membership')['g'].tolist()))
Pattern 0
Naaa, Aebp1, Mfap3l, Fmo1, 2810002D19Rik, Gng13

Pattern 1
Map2, Arhgef9, AI593442, 6330403K07Rik, Slc38a3, Igfbp2, Nmb, Map1b

Pattern 2
Hpcal4, Snap25, Pcp4, Gng4, Ppfia2, Kcnh3