Figure 1 — The Whorlmap

Figure 1 — The Whorlmap

From bootstrap distribution to brain-wide sex differences in gene expression

The whorlmap is a matrix visualisation where each cell encodes a full bootstrap distribution, not just a scalar mean. Within each 7×7-pixel cell, pixels are laid out in an outside-in spiral; their position encodes the quantile rank of the corresponding bootstrap sample, and their colour encodes the effect size (mean difference). Taken together, the shape, spread, and skewness of any distribution become readable directly from the mosaic pattern of a single cell.

This notebook generates all panels and assembles Figure 1. Data path: - CI / default: loads gtex_data/figure1_cache.pkl.gz (committed to repo, ~few MB) - Full reproducibility: set USE_CACHE = False to rebuild from gtex_data/gene_expr.pkl (requires running helpers/survey.ipynb and helpers/make_cache.ipynb first)

1. Setup

%matplotlib inline
%config InlineBackend.figure_format = 'retina'

import pathlib, pickle, gzip, warnings
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib.patches as mpatches
from matplotlib.colors import LinearSegmentedColormap, Normalize, TwoSlopeNorm
from matplotlib.colorbar import ColorbarBase
from matplotlib.cm import ScalarMappable
from matplotlib.patches import Rectangle as _Rect
from matplotlib.lines import Line2D as _L2D
from scipy.stats import gaussian_kde as _gkde
import seaborn as sns
import dabest
from dabest.multi import combine
from IPython.display import Image as IPImage, display

plt.rcParams.update({'font.size': 9, 'axes.titlesize': 10})
IMAGES = pathlib.Path('images')
IMAGES.mkdir(exist_ok=True)

Pre-compiling numba functions for DABEST...

Compiling numba functions:   0%|          | 0/11 [00:00<?, ?it/s]
Compiling numba functions:   9%|▉         | 1/11 [00:00<00:02,  3.57it/s]OMP: Info #276: omp_set_nested routine deprecated, please use omp_set_max_active_levels instead.

Compiling numba functions: 100%|██████████| 11/11 [00:00<00:00, 33.20it/s]

Numba compilation complete!

# ── Colour maps ───────────────────────────────────────────────────────────────
# CMAP_SWOR: used in Panels A and B (wider range, 4-stop)
CMAP_SWOR = LinearSegmentedColormap.from_list(
    'steelwhiteorangered',
    [(0.00, (70/255, 130/255, 180/255)),
     (0.50, (1.0,   1.0,   1.0  )),
     (0.75, (1.0,   0.55,  0.0  )),
     (1.00, (0.86,  0.08,  0.24 ))],
    N=256,
)
# CMAP: used in Panels C (steelblue–white–orangered, symmetric)
CMAP  = LinearSegmentedColormap.from_list(
    'swo', ['steelblue', 'white', 'orangered'], N=256
)
DNORM = TwoSlopeNorm(vcenter=0, vmin=-3.5, vmax=3.5)

# ── Spiral utilities (Panel A + B whorl cell construction) ───────────────────

def _spiralize(fill, m, n):
    """Fill an m×n grid via outside-in spiral traversal."""
    i = 0; j = 0; k = 0; array = np.zeros((m, n))
    while m > 0 and k < len(fill):
        jj = j; ii = i
        for j in range(j, n):
            if k >= len(fill): break
            array[i, j] = fill[k]; k += 1
        for i in range(ii + 1, m):
            if k >= len(fill): break
            array[i, j] = fill[k]; k += 1
        for j in range(n - 2, jj - 1, -1):
            if k >= len(fill): break
            array[i, j] = fill[k]; k += 1
        for i in range(m - 2, ii, -1):
            if k >= len(fill): break
            array[i, j] = fill[k]; k += 1
        m -= 1; n -= 1; j += 1
    return array

def _spiral_path(n):
    """Return list of (row, col) in outside-in spiral order for an n×n grid."""
    path = []
    top, bot, lft, rgt = 0, n - 1, 0, n - 1
    while top <= bot and lft <= rgt:
        for c in range(lft, rgt + 1): path.append((top, c))
        top += 1
        for r in range(top, bot + 1): path.append((r, rgt))
        rgt -= 1
        if top <= bot:
            for c in range(rgt, lft - 1, -1): path.append((bot, c))
            bot -= 1
        if lft <= rgt:
            for r in range(bot, top - 1, -1): path.append((r, lft))
            lft += 1
    return path

def _ring_structure(n):
    """Decompose an n×n grid into concentric rings."""
    rings, pos, k = [], 0, 0
    while True:
        sz = n - 2 * k
        if sz <= 0: break
        nc = 1 if sz == 1 else 4 * (sz - 1)
        rings.append({'k': k, 'start': pos, 'end': pos + nc, 'sz': sz})
        pos += nc; k += 1
        if sz == 1: break
    return rings

2. Panel A — From Bootstrap Distribution to Whorl Cell

We start with a simple synthetic two-group comparison (n = 20 per group) to show the encoding step-by-step:

• Ai (Gardner-Altman plot): the raw data and the floating bootstrap mean difference with 95% CI — the standard dabest output.
• Aii (whorl cell): the same bootstrap distribution, re-encoded as a 7×7 pixel grid filled via outside-in spiral. Each pixel represents one quantile rank; its colour is the effect size at that quantile.
• Aiii (stacked histogram): the bootstrap distribution segmented by the same quantile boundaries, with colours matching Aii. This makes the link between distribution shape and pixel pattern explicit.

# Synthetic two-group data (seed fixed for reproducibility)
_rng    = np.random.default_rng(42)
noise_a = _rng.normal(0, 3., 20)
noise_b = _rng.normal(0, 3., 20)
group_a = noise_a - noise_a.mean()
group_b = noise_b - noise_b.mean() + 0.5
_df_a   = pd.DataFrame({'group': ['A'] * 20 + ['B'] * 20,
                         'value': np.concatenate([group_a, group_b])})
_dobj_a = dabest.load(_df_a, x='group', y='value', idx=('A', 'B'))

2.1 Panel Ai — Gardner-Altman estimation plot

fig_ai = _dobj_a.mean_diff.plot(fig_size=(4.5, 5), float_contrast=True, show_pairs=False)
fig_ai.savefig(IMAGES / 'panel_ai.svg', bbox_inches='tight')
fig_ai.savefig(IMAGES / 'panel_ai.png', dpi=600, bbox_inches='tight')
plt.show()

2.2 Panels Aii & Aiii — Whorl cell + segmented histogram

The bootstrap distribution (5 000 resamples, 2.5% tails removed) is ranked and assigned to the 7² = 49 pixels via outside-in spiral ordering. The outermost ring (q0–q23) spans the most extreme values; the innermost pixel (q48) holds the median. The segmented histogram on the right uses the same quantile boundaries and colours.

_multi_a = combine([[_dobj_a]], ['gene'], row_labels=['region'], effect_size='mean_diff')
bs_raw   = sorted(_multi_a.bootstraps[0])
chop     = int(np.ceil(len(bs_raw) * 0.025))
bs_c     = bs_raw[chop:-chop]

n = 7; N = n * n
ranks = np.linspace(0, len(bs_c), N, dtype=int); ranks[0] = 1
fv    = np.array([bs_c[r - 1] for r in ranks])
if sum(v > 0 for v in bs_c) < len(bs_c) / 2:
    fv = fv[::-1]

XLIM       = (-3.5, 3.5)
rings      = _ring_structure(n)
fv_sorted  = np.clip(np.sort(fv), *XLIM)
cell_vals  = _spiralize(fv.tolist(), n, n)
bound      = max(2.0, np.ceil(max(abs(fv_sorted[0]), abs(fv_sorted[-1])) * 2) / 2)
VALUE_NORM = Normalize(vmin=-bound, vmax=bound)
spiral_pos = _spiral_path(n)

fig = plt.figure(figsize=(11, 5.4))
ax_cell = fig.add_axes([0.03, 0.18, 0.32, 0.68])
ax_cbar = fig.add_axes([0.03, 0.08, 0.32, 0.05])
ax_hist = fig.add_axes([0.43, 0.12, 0.54, 0.74])

ax_cell.imshow(cell_vals, cmap=CMAP_SWOR, norm=VALUE_NORM,
               origin='upper', interpolation='nearest', aspect='equal')

def _ring_arrows(ax, ring_k, sz, avg_val):
    mid = ring_k + sz // 2
    bg  = CMAP_SWOR(VALUE_NORM(avg_val))
    col = 'w' if (0.299*bg[0] + 0.587*bg[1] + 0.114*bg[2]) < 0.52 else 'k'
    ap  = dict(arrowstyle='->', color=col, lw=1.0, mutation_scale=9)
    d   = 0.25
    ax.annotate('', xy=(mid+0.6, ring_k-d),          xytext=(mid-0.6, ring_k-d),          arrowprops=ap, zorder=8, annotation_clip=False)
    ax.annotate('', xy=(ring_k+sz-1+d, mid+0.6),     xytext=(ring_k+sz-1+d, mid-0.6),     arrowprops=ap, zorder=8, annotation_clip=False)
    ax.annotate('', xy=(mid-0.6, ring_k+sz-1+d),     xytext=(mid+0.6, ring_k+sz-1+d),     arrowprops=ap, zorder=8, annotation_clip=False)
    ax.annotate('', xy=(ring_k-d, mid-0.6),           xytext=(ring_k-d, mid+0.6),           arrowprops=ap, zorder=8, annotation_clip=False)

for ring in rings[:-1]:
    avg_val = float(np.mean(fv_sorted[ring['start']:ring['end']]))
    _ring_arrows(ax_cell, ring['k'], ring['sz'], avg_val)

for k, (r, c) in enumerate(spiral_pos):
    val = float(fv[k])
    bg  = CMAP_SWOR(VALUE_NORM(val))
    lum = 0.299*bg[0] + 0.587*bg[1] + 0.114*bg[2]
    tc  = 'w' if lum < 0.52 else 'k'
    ax_cell.text(c, r, f'q{k}\n{val:.1f}',
                 ha='center', va='center', fontsize=4.5, color=tc,
                 zorder=10, linespacing=1.1, multialignment='center')

ax_cell.set_xlim(-0.5, n - 0.5); ax_cell.set_ylim(n - 0.5, -0.5)
ax_cell.axis('off')
ax_cell.set_title('7×7 whorl cell — outside-in spiral fill\n(pixel colour = effect size)', fontsize=9)

cb = ColorbarBase(ax_cbar, cmap=CMAP_SWOR, norm=VALUE_NORM, orientation='horizontal')
cb_ticks = np.arange(-bound, bound + 0.01, 0.5)
cb.set_ticks(cb_ticks); cb.set_ticklabels([f'{t:g}' for t in cb_ticks])
cb.ax.tick_params(labelsize=7.5)
cb.set_label('Effect size (bootstrap mean difference)', fontsize=8)

BIN_W = 0.20
bins  = np.arange(XLIM[0], XLIM[1] + BIN_W, BIN_W)
counts, _ = np.histogram(np.clip(bs_c, *XLIM), bins=bins, density=True)

seg_bounds = np.empty(N + 1)
seg_bounds[0] = XLIM[0]; seg_bounds[-1] = XLIM[1]
for k in range(1, N):
    seg_bounds[k] = (fv_sorted[k - 1] + fv_sorted[k]) / 2.0
seg_cols = [CMAP_SWOR(VALUE_NORM(float(v))) for v in fv_sorted]

best_piece = {}
for lo, hi, ht in zip(bins[:-1], bins[1:], counts):
    if ht == 0: continue
    y_bot = 0.0
    for k in range(N):
        overlap = max(0., min(hi, seg_bounds[k + 1]) - max(lo, seg_bounds[k]))
        if overlap <= 0: continue
        h = ht * overlap / (hi - lo)
        ax_hist.bar(lo, h, width=hi - lo, bottom=y_bot, align='edge',
                    color=seg_cols[k], edgecolor='none')
        if k not in best_piece or h > best_piece[k][2]:
            best_piece[k] = (lo + (hi - lo) / 2, y_bot, h)
        y_bot += h

for k, (xc, yb, h) in best_piece.items():
    if h < counts.max() * 0.005: continue
    val = float(fv_sorted[k])
    col = CMAP_SWOR(VALUE_NORM(val))
    lum = 0.299*col[0] + 0.587*col[1] + 0.114*col[2]
    tc  = 'w' if lum < 0.52 else 'k'
    ax_hist.text(xc, yb + h / 2, f'q{k}\n{val:.1f}',
                 ha='center', va='center', fontsize=3.5, color=tc,
                 zorder=11, linespacing=1.0, multialignment='center')

margin = (fv_sorted[-1] - fv_sorted[0]) * 0.08
ax_hist.set_xlim(fv_sorted[0] - margin, fv_sorted[-1] + margin)
ax_hist.set_ylim(0, counts.max() * 1.03)
for sp in ['left', 'right', 'top']:
    ax_hist.spines[sp].set_visible(False)
ax_hist.set_yticks([])
ax_hist.tick_params(labelsize=8)
ax_hist.set_xlabel('Bootstrap mean difference', fontsize=9)
ax_hist.set_title('Bootstrap distribution, segmented by quantile rank\n'
                  '(bar colours match spiral position on left)', fontsize=9)

fig.savefig(IMAGES / 'panel_a.svg', bbox_inches='tight')
fig.savefig(IMAGES / 'panel_a.png', dpi=600, bbox_inches='tight')
plt.show()

3. Panel B — Bootstrap Distribution Taxonomy

Five stylised distributions show how different statistical patterns map onto different whorl cell appearances:

Panel	Distribution	What to look for
Bi	Large positive, narrow CI	Solid warm (orange-red) outer ring; smooth gradient inward
Bii	Large negative, narrow CI	Solid cool (blue) outer ring
Biii	Near-zero effect	Pale cell with almost no colour — small n, no signal
Biv	Broad / uncertain	Mixed warm and cool in the outer ring; wide CI
Bv	Multi-modal(discrete, low sample size)	Two-colour checkerboard pattern

The bar charts below each cell show the bootstrap distribution on the same x-axis.

def _make_1x1_multi(ga, gb):
    df   = pd.DataFrame({'group': ['A'] * len(ga) + ['B'] * len(gb),
                         'value': np.concatenate([ga, gb])})
    dobj = dabest.load(df, x='group', y='value', idx=('A', 'B'))
    return combine([[dobj]], [''], row_labels=[''], effect_size='mean_diff')

_rng2 = np.random.default_rng(1234)
_ga1  = _rng2.normal(0, 1.5, 22);  _gb1 = _rng2.normal( 3.0, 1.5, 22)
_ga2  = _rng2.normal(0, 1.5, 22);  _gb2 = _rng2.normal(-3.0, 1.5, 22)
_ga3  = _rng2.normal(0, 0.4, 5);   _gb3 = _rng2.normal( 0.05, 0.4, 5)
_ga4  = _rng2.normal(0, 0.5, 15);  _gb4 = _rng2.normal( 0.3,  6.0, 15)
_ga5  = np.array([0, 0, 0, 0, 0])
_gb5  = np.array([2.1, 2, 2.05, -3, -3.05])

scenarios = [
    ('(i) Large positive,\nnarrow',       _make_1x1_multi(_ga1, _gb1)),
    ('(ii) Large negative,\nnarrow',      _make_1x1_multi(_ga2, _gb2)),
    ('(iii) Near-zero\neffect',           _make_1x1_multi(_ga3, _gb3)),
    ('(iv) Broad / uncertain\n(wide CI)', _make_1x1_multi(_ga4, _gb4)),
    ('(v) Multi-modal\n(discrete, low sample size)',      _make_1x1_multi(_ga5, _gb5)),
]

XLIM_B = (-5, 5);  BIN_W_B = 0.14
BINS_B = np.arange(XLIM_B[0], XLIM_B[1] + BIN_W_B, BIN_W_B)
BINCX  = 0.5 * (BINS_B[:-1] + BINS_B[1:])

hists_b = []
for _, multi_s in scenarios:
    bs = np.clip(multi_s.bootstraps[0], *XLIM_B)
    counts_s, _ = np.histogram(bs, bins=BINS_B, density=True)
    hists_b.append(counts_s)
ymax_b = max(h.max() for h in hists_b) * 1.05

fig_b, axes_b = plt.subplots(
    2, 5, figsize=(14, 5),
    gridspec_kw={'hspace': 0.08, 'wspace': 0.35, 'height_ratios': [1.8, 1]},
)

for j, ((label, multi_s), counts_s) in enumerate(zip(scenarios, hists_b)):
    with warnings.catch_warnings():
        warnings.simplefilter('ignore')
        multi_s.whorlmap(
            n=21, chop_tail=2.5, cmap=CMAP, vmin=-3.5, vmax=3.5,
            ax=axes_b[0, j],
            heatmap_kwargs={'cbar': False, 'xticklabels': [''], 'yticklabels': [''],
                            'linewidths': 0},
        )
    axes_b[0, j].set_aspect('equal')
    for _c in axes_b[0, j].collections:
        _c.set_rasterized(True)

    ax = axes_b[1, j]
    cols = CMAP(DNORM(BINCX))
    for lo, hi, ht, c in zip(BINS_B[:-1], BINS_B[1:], counts_s, cols):
        ax.bar(lo, ht, width=hi - lo, align='edge', color=c,
               edgecolor='lightgray', linewidth=.5)
    ax.set_xlim(*XLIM_B);  ax.set_ylim(0, ymax_b)
    ax.spines['left'].set_position('zero')
    ax.spines['right'].set_visible(False);  ax.spines['top'].set_visible(False)
    ax.set_yticks([]);  ax.set_xticks([-3, 0, 3]);  ax.tick_params(labelsize=7)
    ax.text(0.5, -0.42, label, transform=ax.transAxes,
            ha='center', fontsize=8.5, va='top', linespacing=1.3)

fig_b.subplots_adjust(top=0.92, right=0.88)
fig_b.text(0.5, 0.99, ' ', fontsize=14, va='top', color='white')
cbar_ax_b = fig_b.add_axes([0.90, 0.18, 0.015, 0.60])
sm_b = ScalarMappable(cmap=CMAP, norm=DNORM)
sm_b.set_array([])
cbar_b = fig_b.colorbar(sm_b, cax=cbar_ax_b)
cbar_b.set_label('Effect size', fontsize=8)
cbar_b.set_ticks([-3, 0, 3]);  cbar_b.ax.tick_params(labelsize=7)

fig_b.savefig(IMAGES / 'panel_b.svg', dpi=600, bbox_inches='tight')
fig_b.savefig(IMAGES / 'panel_b.png', dpi=600, bbox_inches='tight')
plt.show()

/Applications/anaconda3/lib/python3.10/site-packages/dabest/_effsize_objects.py:523: UserWarning: The lower limit of the BCa interval of baseline curve cannot be computed. It is set to the effect size itself. All bootstrap values were likely all the same.
  warnings.warn(err_temp.substitute(lim_type="lower"), stacklevel=0)
/Applications/anaconda3/lib/python3.10/site-packages/dabest/_effsize_objects.py:527: UserWarning: The upper limit of the BCa interval of baseline curve cannot be computed. It is set to the effect size itself. All bootstrap values were likely all the same.
  warnings.warn(err_temp.substitute(lim_type="upper"), stacklevel=0)

4. Panel C — GTEx Sex Differences in Brain Gene Expression

The question: Do any of these 20 neuroactive / glial genes show sex differences (Female − Male, log₂ TPM+1) in human brain, and if so, where?

The data: GTEx v8; 13 brain regions; male n ≈ 100–200, female n ≈ 50–100 per region. Effect size = bootstrap mean difference (5 000 resamples). Data source: GTEx Consortium. Science 369, 1318–1330 (2020). https://doi.org/10.1126/science.aaz1776

The highlighted pairs reveal something a scalar heatmap hides: - Pair A (blue): SST/Spinal Cord (a1) and FKBP5/Putamen (a2) have similar scalar means, but their bootstrap distributions differ dramatically in width. - Pair B (maroon): AIF1/Substantia Nigra (b1) and CYP19A1/Caudate (b2) — similar means, but different distributional shapes.

# ── Load expression data from small cache, then run dabest live ──────────────
# Cache path: expression values for the 20 display genes (~500 KB, committed).
# dabest.combine() runs here every time — bootstraps computed fresh, not loaded.
USE_CACHE = True   # set False to rebuild from full gene_expr.pkl instead

BRAIN_REGIONS = {
    'Hypothalamus':      'Brain - Hypothalamus',
    'Amygdala':          'Brain - Amygdala',
    'Hippocampus':       'Brain - Hippocampus',
    'Ant. Cing. Ctx':    'Brain - Anterior cingulate cortex (BA24)',
    'Frontal Cortex':    'Brain - Frontal Cortex (BA9)',
    'Cortex':            'Brain - Cortex',
    'Caudate':           'Brain - Caudate (basal ganglia)',
    'Putamen':           'Brain - Putamen (basal ganglia)',
    'Nucleus Accumbens': 'Brain - Nucleus accumbens (basal ganglia)',
    'Cerebellum':        'Brain - Cerebellum',
    'Cerebellar Hemi.':  'Brain - Cerebellar Hemisphere',
    'Substantia Nigra':  'Brain - Substantia nigra',
    'Spinal Cord':       'Brain - Spinal cord (cervical c-1)',
}
RIGHT_GENES  = [
    'TPH2', 'CHRNA7', 'ESR1',
    'TH', 'SLC6A3', 'DDC', 'AGRP',
    'SST', 'PENK', 'GAD1', 'GAD2', 'CRH',
    'DRD5', 'CHRM1', 'BDNF', 'CYP19A1',
    'AIF1', 'MAOA', 'FKBP5', 'GFAP',
]
region_order = list(BRAIN_REGIONS.keys())
DATA_DIR     = pathlib.Path('gtex_data')

expr_cache_path = DATA_DIR / 'figure1_expr_cache.pkl.gz'
gene_expr_path  = DATA_DIR / 'gene_expr.pkl'

if USE_CACHE and expr_cache_path.exists():
    print('Loading expression cache …')
    with gzip.open(expr_cache_path, 'rb') as f:
        _cache = pickle.load(f)
    expr_subset  = _cache['expr_subset']
    RIGHT_GENES  = _cache.get('gene_names',   RIGHT_GENES)
    region_order = _cache.get('region_order', region_order)
    print(f'  {len(expr_subset)} genes loaded from cache.')

elif gene_expr_path.exists():
    print('Loading full gene_expr.pkl and slicing to display genes …')
    with open(gene_expr_path, 'rb') as f:
        gene_expr = pickle.load(f)
    expr_subset = {gene: gene_expr[gene] for gene in RIGHT_GENES}

else:
    raise FileNotFoundError(
        'No expression data found.\n'
        '  Default path: gtex_data/figure1_expr_cache.pkl.gz\n'
        '  Run helpers/make_cache.ipynb (needs gene_expr.pkl from helpers/survey.ipynb).'
    )

sample_meta = pd.read_csv(DATA_DIR / 'sample_meta.csv', index_col=0)
print(f'sample_meta: {len(sample_meta)} samples ready.')

Loading expression cache …
  20 genes loaded from cache.
sample_meta: 3326 samples ready.

# ── Build dabest grid and run combine() ──────────────────────────────────────
# This is where the bootstrapping happens. Runtime: ~10–15 min.
# combine() runs 5 000 resamples for each of the 13 x 20 = 260 comparisons.
print('Building dabest grid (260 comparisons) …')
dabest_grid = []
for region in region_order:
    row = []
    for gene in RIGHT_GENES:
        gd   = expr_subset[gene]
        mask = sample_meta['region'] == region
        sids = sample_meta.index[mask]
        recs = [{'expr': gd[s], 'sex': sample_meta.loc[s, 'sex']}
                for s in sids if s in gd]
        df   = pd.DataFrame(recs)
        dobj = dabest.load(df, x='sex', y='expr', idx=('Male', 'Female'))
        row.append(dobj)
    dabest_grid.append(row)
    print(f'  {region}', flush=True)

print('Running combine() — bootstrapping all 260 distributions …')
with warnings.catch_warnings():
    warnings.simplefilter('ignore')
    multi_gtex = combine(dabest_grid, RIGHT_GENES,
                         row_labels=region_order, effect_size='mean_diff')
print('multi_gtex ready.')

# Extract mean_df via a throwaway whorlmap call
_fig_tmp, _ax_tmp = plt.subplots()
with warnings.catch_warnings():
    warnings.simplefilter('ignore')
    _, mean_df = multi_gtex.whorlmap(
        cmap=CMAP, chop_tail=2.5, ax=_ax_tmp,
        heatmap_kwargs={'cbar': False},
    )
plt.close(_fig_tmp)
print(f'mean_df: {mean_df.shape}')
mean_df

Building dabest grid (260 comparisons) …
  Hypothalamus
  Amygdala
  Hippocampus
  Ant. Cing. Ctx
  Frontal Cortex
  Cortex
  Caudate
  Putamen
  Nucleus Accumbens
  Cerebellum
  Cerebellar Hemi.
  Substantia Nigra
  Spinal Cord
Running combine() — bootstrapping all 260 distributions …
multi_gtex ready.
mean_df: (13, 20)

	TPH2	CHRNA7	ESR1	TH	SLC6A3	DDC	AGRP	SST	PENK	GAD1	GAD2	CRH	DRD5	CHRM1	BDNF	CYP19A1	AIF1	MAOA	FKBP5	GFAP
Hypothalamus	0.002471	-0.078216	-0.202824	-0.236558	-0.040501	-0.242780	-0.333785	-0.625742	-0.379131	-0.147960	-0.248164	-0.434383	-0.039303	-0.190621	-0.314026	-0.070229	0.066999	-0.049785	0.212946	0.091599
Amygdala	0.004224	0.021629	0.057536	-0.095235	0.050534	-0.010169	0.046908	-0.118074	0.030236	0.004639	-0.028158	-0.115287	-0.095850	-0.085870	0.174417	0.033436	0.080271	0.206954	0.107901	0.016422
Hippocampus	-0.013413	-0.012011	0.024586	-0.169880	0.019526	-0.023512	0.016705	-0.755915	-0.026221	-0.142294	-0.311973	-0.277202	-0.200212	-0.305467	-0.106461	0.081772	0.198744	0.051490	0.203935	0.014617
Ant. Cing. Ctx	-0.010697	-0.048152	0.038611	-0.065446	0.015582	0.002727	0.039664	0.012542	-0.121524	-0.116034	-0.183992	-0.261952	-0.046899	-0.074761	-0.009989	0.070013	0.246765	0.092146	0.139307	0.298177
Frontal Cortex	-0.005462	-0.060029	0.022910	0.012009	-0.004358	-0.005849	0.002491	-0.353670	-0.187935	-0.348132	-0.432378	-0.563705	-0.105192	-0.264026	-0.078788	0.083868	0.385709	0.128605	0.232678	0.549721
Cortex	-0.003155	-0.012599	-0.020425	-0.057507	0.013593	0.002610	0.014544	-0.516722	-0.129005	-0.294562	-0.349322	-0.428082	-0.106968	-0.167752	-0.094582	0.007727	0.124050	0.166903	0.352072	0.239641
Caudate	0.000111	-0.012347	-0.005830	-0.485342	0.000455	-0.107604	0.054822	-0.241599	-0.567500	-0.221010	-0.267398	-0.067347	-0.100244	-0.348139	0.029047	0.020339	-0.024881	0.131590	0.124888	0.109370
Putamen	-0.001948	0.004223	0.035270	-0.155316	0.014303	-0.172177	-0.029438	-0.333530	-0.537429	-0.238785	-0.275519	-0.260649	-0.089451	-0.261923	0.029819	0.010675	-0.003506	0.172450	0.271297	0.420710
Nucleus Accumbens	-0.002875	0.006152	0.076322	-0.114431	0.021339	-0.122800	0.062277	-0.254735	-0.429800	-0.200148	-0.232545	-0.037078	-0.080491	-0.200439	0.090668	0.031644	0.022592	0.138890	0.201405	0.129288
Cerebellum	-0.011175	-0.013842	-0.018375	0.004076	-0.001549	0.003004	-0.031730	0.028492	-0.008536	-0.206766	-0.231278	-0.006447	-0.006789	0.023700	0.026706	0.004489	0.058325	0.218870	0.184843	0.293985
Cerebellar Hemi.	-0.005442	-0.003028	-0.007613	-0.038241	-0.000041	-0.025308	-0.007630	0.069120	-0.294402	-0.405329	-0.330673	0.030845	0.021944	0.099452	-0.131114	0.000877	-0.059243	0.142170	0.111113	0.199126
Substantia Nigra	-0.019343	0.016725	0.010195	0.643285	0.669986	0.339492	0.012263	-0.383516	-0.322631	-0.309855	-0.355314	-0.140006	0.029543	-0.003036	0.165325	0.004940	0.018600	-0.016308	-0.108976	-0.009088
Spinal Cord	-0.017001	-0.023671	0.019967	-0.132838	0.007276	-0.017949	-0.073639	0.281962	0.073077	0.076268	0.015630	-0.160210	-0.018017	0.034281	0.028183	-0.060264	0.203038	0.036479	0.119745	0.070303

# ── Shared colour scale and helper ───────────────────────────────────────────
vabs        = 1.5
shared_norm = Normalize(vmin=-vabs, vmax=vabs)

def _add_shared_cbar(cax, orientation='horizontal'):
    cb = ColorbarBase(cax, cmap=CMAP, norm=shared_norm, orientation=orientation)
    cb.set_label('Mean diff F−M (log₂TPM+1)', fontsize=11)
    cb.set_ticks([-1.5, -0.5, 0, 0.5, 1.5])
    cb.ax.tick_params(labelsize=11)

# ── Highlighted pairs ────────────────────────────────────────────────────────
_N_COLS_G  = len(RIGHT_GENES)
_PAIR_COL  = ["#B90CC5", "#217690"]  # blue=A, maroon=B

# (row, col, pair_idx, within_pair_idx, label)  — display order: a1, a2, b1, b2
_ORDERED_CELLS = [
    (12,  7, 0, 0, 'a1'),  # SST / Spinal Cord
    ( 7, 18, 0, 1, 'a2'),  # FKBP5 / Putamen
    (11, 16, 1, 0, 'b1'),  # AIF1 / Substantia Nigra
    ( 6, 15, 1, 1, 'b2'),  # CYP19A1 / Caudate
]

_pair_bs = {
    0: [np.asarray(multi_gtex.bootstraps[12*_N_COLS_G +  7]),
        np.asarray(multi_gtex.bootstraps[ 7*_N_COLS_G + 18])],
    1: [np.asarray(multi_gtex.bootstraps[11*_N_COLS_G + 16]),
        np.asarray(multi_gtex.bootstraps[ 6*_N_COLS_G + 15])],
}
_pair_xlims = {}
_pair_ymax  = {}
for _pi, _blist in _pair_bs.items():
    _lo = min(_b.min() - 0.3*np.std(_b) for _b in _blist)
    _hi = max(_b.max() + 0.3*np.std(_b) for _b in _blist)
    _pair_xlims[_pi] = (_lo, _hi)
    _xr = np.linspace(_lo, _hi, 300)
    _pair_ymax[_pi] = max(_gkde(_b, bw_method='silverman')(_xr).max() for _b in _blist) * 1.1

print('mean_df (13 regions × 20 genes):')
mean_df

mean_df (13 regions × 20 genes):

	TPH2	CHRNA7	ESR1	TH	SLC6A3	DDC	AGRP	SST	PENK	GAD1	GAD2	CRH	DRD5	CHRM1	BDNF	CYP19A1	AIF1	MAOA	FKBP5	GFAP
Hypothalamus	0.002471	-0.078216	-0.202824	-0.236558	-0.040501	-0.242780	-0.333785	-0.625742	-0.379131	-0.147960	-0.248164	-0.434383	-0.039303	-0.190621	-0.314026	-0.070229	0.066999	-0.049785	0.212946	0.091599
Amygdala	0.004224	0.021629	0.057536	-0.095235	0.050534	-0.010169	0.046908	-0.118074	0.030236	0.004639	-0.028158	-0.115287	-0.095850	-0.085870	0.174417	0.033436	0.080271	0.206954	0.107901	0.016422
Hippocampus	-0.013413	-0.012011	0.024586	-0.169880	0.019526	-0.023512	0.016705	-0.755915	-0.026221	-0.142294	-0.311973	-0.277202	-0.200212	-0.305467	-0.106461	0.081772	0.198744	0.051490	0.203935	0.014617
Ant. Cing. Ctx	-0.010697	-0.048152	0.038611	-0.065446	0.015582	0.002727	0.039664	0.012542	-0.121524	-0.116034	-0.183992	-0.261952	-0.046899	-0.074761	-0.009989	0.070013	0.246765	0.092146	0.139307	0.298177
Frontal Cortex	-0.005462	-0.060029	0.022910	0.012009	-0.004358	-0.005849	0.002491	-0.353670	-0.187935	-0.348132	-0.432378	-0.563705	-0.105192	-0.264026	-0.078788	0.083868	0.385709	0.128605	0.232678	0.549721
Cortex	-0.003155	-0.012599	-0.020425	-0.057507	0.013593	0.002610	0.014544	-0.516722	-0.129005	-0.294562	-0.349322	-0.428082	-0.106968	-0.167752	-0.094582	0.007727	0.124050	0.166903	0.352072	0.239641
Caudate	0.000111	-0.012347	-0.005830	-0.485342	0.000455	-0.107604	0.054822	-0.241599	-0.567500	-0.221010	-0.267398	-0.067347	-0.100244	-0.348139	0.029047	0.020339	-0.024881	0.131590	0.124888	0.109370
Putamen	-0.001948	0.004223	0.035270	-0.155316	0.014303	-0.172177	-0.029438	-0.333530	-0.537429	-0.238785	-0.275519	-0.260649	-0.089451	-0.261923	0.029819	0.010675	-0.003506	0.172450	0.271297	0.420710
Nucleus Accumbens	-0.002875	0.006152	0.076322	-0.114431	0.021339	-0.122800	0.062277	-0.254735	-0.429800	-0.200148	-0.232545	-0.037078	-0.080491	-0.200439	0.090668	0.031644	0.022592	0.138890	0.201405	0.129288
Cerebellum	-0.011175	-0.013842	-0.018375	0.004076	-0.001549	0.003004	-0.031730	0.028492	-0.008536	-0.206766	-0.231278	-0.006447	-0.006789	0.023700	0.026706	0.004489	0.058325	0.218870	0.184843	0.293985
Cerebellar Hemi.	-0.005442	-0.003028	-0.007613	-0.038241	-0.000041	-0.025308	-0.007630	0.069120	-0.294402	-0.405329	-0.330673	0.030845	0.021944	0.099452	-0.131114	0.000877	-0.059243	0.142170	0.111113	0.199126
Substantia Nigra	-0.019343	0.016725	0.010195	0.643285	0.669986	0.339492	0.012263	-0.383516	-0.322631	-0.309855	-0.355314	-0.140006	0.029543	-0.003036	0.165325	0.004940	0.018600	-0.016308	-0.108976	-0.009088
Spinal Cord	-0.017001	-0.023671	0.019967	-0.132838	0.007276	-0.017949	-0.073639	0.281962	0.073077	0.076268	0.015630	-0.160210	-0.018017	0.034281	0.028183	-0.060264	0.203038	0.036479	0.119745	0.070303

4.1 Panel Ci — Whorlmap with bootstrap sparklines

Each cell is a 7×7 whorl cell. The four highlighted cells are connected to their full bootstrap distributions on the right. Within each pair (A or B), the x-axis is shared so width differences are directly visible.

# Shared geometry
FIGSIZE_BOTH = (16.5, 8)
HAX_RECT     = [0.09, 0.18, 0.55, 0.77]
CBAR_RECT    = [0.09, 0.035, 0.275, 0.038]

_H_SPARK = 0.17;  _SPARK_X = 0.685;  _SPARK_W = 0.26
_SG = 0.02;  _LG = 0.06;  _SB = HAX_RECT[1]
SPARK_RECTS = [
    [_SPARK_X, _SB + 3*_H_SPARK + 2*_SG + _LG, _SPARK_W, _H_SPARK],  # a1 (top)
    [_SPARK_X, _SB + 2*_H_SPARK + 1*_SG + _LG, _SPARK_W, _H_SPARK],  # a2
    [_SPARK_X, _SB + 1*_H_SPARK + 1*_SG,        _SPARK_W, _H_SPARK],  # b1
    [_SPARK_X, _SB,                              _SPARK_W, _H_SPARK],  # b2 (bottom)
]

fig_ci     = plt.figure(figsize=FIGSIZE_BOTH)
ax_ci      = fig_ci.add_axes(HAX_RECT)
cbar_ax_ci = fig_ci.add_axes(CBAR_RECT)

with warnings.catch_warnings():
    warnings.simplefilter('ignore')
    _, _ = multi_gtex.whorlmap(
        cmap=CMAP, chop_tail=2.5, vmin=-vabs, vmax=vabs,
        ax=ax_ci, heatmap_kwargs={'cbar': False, 'linewidths': 0},
    )
ax_ci.set_aspect('equal')
for _c in ax_ci.collections:
    _c.set_rasterized(True)
ax_ci.tick_params(axis='x', rotation=45, labelsize=11)
ax_ci.tick_params(axis='y', rotation=0,  labelsize=11)
_add_shared_cbar(cbar_ax_ci, orientation='horizontal')

# Extend limits by 3 units on all sides so edge indicator squares aren't clipped
_yl = ax_ci.get_ylim()
_xl = ax_ci.get_xlim()
ax_ci.set_ylim(_yl[0] + 3, _yl[1] - 3)
ax_ci.set_xlim(_xl[0] - 3, _xl[1] + 3)

_mn = mean_df.values

for _idx, (_ri, _ci, _pi, _wi, _lbl) in enumerate(_ORDERED_CELLS):
    _mean_val  = float(_mn[_ri, _ci])
    _bs        = _pair_bs[_pi][_wi]
    _hl_col    = _PAIR_COL[_pi]

    _cx0, _cy0 = _ci * 21, _ri * 21
    # Expand rect by 1 unit on each side so it clears the outermost ring
    ax_ci.add_patch(_Rect((_cx0 - 1, _cy0 - 1), 23, 23,
                          fill=False, edgecolor=_hl_col, linewidth=2.0, zorder=12))
    ax_ci.text(_cx0 + 2.5, _cy0 + 5, _lbl,
               fontsize=5.5, fontweight='bold', color=_hl_col, zorder=13, va='top', ha='left')

    _sr  = SPARK_RECTS[_idx]
    _sax = fig_ci.add_axes(_sr)
    _xlo, _xhi = _pair_xlims[_pi]
    _xr = np.linspace(_xlo, _xhi, 300)
    _yr = _gkde(_bs, bw_method='silverman')(_xr)

    # Fill KDE with cmap colours keyed to x-value, matching Panel B histogram style
    _dx = _xr[1] - _xr[0]
    _bar_cols = CMAP(shared_norm(_xr[:-1] + _dx / 2))
    for _x, _y, _bc in zip(_xr[:-1], _yr[:-1], _bar_cols):
        _sax.bar(_x, _y, width=_dx, align='edge', color=_bc, edgecolor='none')

    _sax.plot(_xr, _yr, color=_hl_col, lw=0.9)
    _sax.axvline(0,          color='#aaaaaa', lw=0.6, ls=(0, (3, 3)))
    _sax.axvline(_mean_val,  color=_hl_col,   lw=1.2)
    _sax.set_xlim(_xlo, _xhi);  _sax.set_ylim(0, _pair_ymax[_pi])
    _sax.set_yticks([]);  _sax.set_xticks([0])
    _sax.set_xticklabels(['0'], fontsize=5, color='#888')
    for _sp in ['top', 'right', 'left', 'bottom']:
        _sax.spines[_sp].set_visible(False)
    _sax.tick_params(length=2, width=0.5, pad=1)
    _sax.text(0.02, 0.97, _lbl, transform=_sax.transAxes,
              fontsize=7, fontweight='bold', ha='left', va='top', color=_hl_col)
    # Gene · region name in top-right corner, inside the axes
    _sax.text(0.98, 0.88, f'{RIGHT_GENES[_ci]} · {region_order[_ri]}',
              transform=_sax.transAxes, fontsize=7, ha='right', va='top',
              style='italic', color='#444')

# Orthogonal connection lines
fig_ci.canvas.draw()

def _d2f(_xd, _yd):
    _disp = ax_ci.transData.transform((_xd, _yd))
    return tuple(fig_ci.transFigure.inverted().transform(_disp))

_xlim_r = ax_ci.get_xlim()[1]
for _idx, (_ri, _ci, _pi, _wi, _lbl) in enumerate(_ORDERED_CELLS):
    _hl_col = _PAIR_COL[_pi]
    _sr = SPARK_RECTS[_idx]
    _p1 = _d2f((_ci + 1) * 21, _ri * 21 + 10.5)
    _p2 = _d2f(_xlim_r,         _ri * 21 + 10.5)
    _p3 = (_sr[0], _sr[1] + _sr[3] / 2)
    for _xs, _ys in [([_p1[0], _p2[0]], [_p1[1], _p2[1]]),
                     ([_p2[0], _p3[0]], [_p2[1], _p3[1]])]:
        fig_ci.add_artist(_L2D(_xs, _ys, transform=fig_ci.transFigure,
                               color=_hl_col, lw=0.85, zorder=15, solid_capstyle='round'))

fig_ci.savefig(IMAGES / 'panel_ci.svg', dpi=600, bbox_inches='tight')
fig_ci.savefig(IMAGES / 'panel_ci.png', dpi=600, bbox_inches='tight')
plt.show()

4.2 Panel Cii — Scalar heatmap

The same data as Ci, but each cell shows only the point estimate of the mean difference (annotated value). The highlighted pairs (same a1/a2/b1/b2 labels) have nearly identical scalar values — the distributional differences visible in Ci are invisible here.

fig_cii     = plt.figure(figsize=FIGSIZE_BOTH)
ax_cii      = fig_cii.add_axes(HAX_RECT)
cbar_ax_cii = fig_cii.add_axes(CBAR_RECT)

sns.heatmap(
    mean_df, ax=ax_cii,
    cmap=CMAP, vmin=-vabs, vmax=vabs, center=0,
    annot=True, fmt='.2f', annot_kws={'fontsize': 6},
    linewidths=0, linecolor='none',
    square=True, cbar=False,
)
for _c in ax_cii.collections:
    _c.set_rasterized(True)
ax_cii.set_title('Scalar mean (distribution information lost)', fontsize=10)
ax_cii.tick_params(axis='x', rotation=45, labelsize=11)
ax_cii.tick_params(axis='y', rotation=0,  labelsize=11)
_add_shared_cbar(cbar_ax_cii, orientation='horizontal')

# Phantom line at Ci's right edge (sparklines end at _SPARK_X + _SPARK_W ≈ 0.945)
# so bbox_inches='tight' captures the same width as panel_ci.png
fig_cii.add_artist(_L2D([_SPARK_X + _SPARK_W, _SPARK_X + _SPARK_W], [0.0, 1.0],
                        transform=fig_cii.transFigure, color='white', lw=0.5, alpha=0.01))

for _ri, _ci, _pi, _wi, _lbl in _ORDERED_CELLS:
    _hl_col = _PAIR_COL[_pi]
    # Expand box slightly beyond the 1×1 cell so it doesn't overlap cell content
    ax_cii.add_patch(_Rect((_ci - 0.08, _ri - 0.08), 1.16, 1.16,
                           fill=False, edgecolor=_hl_col, linewidth=2.0, zorder=12))
    ax_cii.text(_ci + 0.06, _ri + 0.18, _lbl,
                fontsize=6, fontweight='bold', color=_hl_col, zorder=13, va='top', ha='left')

fig_cii.savefig(IMAGES / 'panel_cii.svg', dpi=600, bbox_inches='tight')
fig_cii.savefig(IMAGES / 'panel_cii.png', dpi=600, bbox_inches='tight')
plt.show()

5. Assembly — Figure 1

Stitch all saved panel PNGs into a single 600 DPI figure using PIL. Sub-panel labels (Ai, Aii, etc.) are rendered on top with matplotlib.

from PIL import Image
import matplotlib.pyplot as _plt2

DPI      = 600
GAP_PX   = int(round(0.07 * DPI))
LABEL_PT = 30

def _load(name):
    return Image.open(IMAGES / name).convert('RGB')

def _scale_to_w(img, target_w):
    h = int(round(img.height * target_w / img.width))
    return img.resize((target_w, h), Image.LANCZOS)

p_ai = _load('panel_ai.png');  p_a = _load('panel_a.png')
_h_a       = p_a.height
_w_ai_s    = int(round(p_ai.width * _h_a / p_ai.height))
_p_ai_h    = p_ai.resize((_w_ai_s, _h_a), Image.LANCZOS)
_raw_w     = _w_ai_s + p_a.width
TARGET_W_PX = max(_raw_w, _load('panel_b.png').width,
                  _load('panel_ci.png').width, _load('panel_cii.png').width)

_f       = TARGET_W_PX / _raw_w
_row_a_h = int(round(_h_a * _f))
_p_ai_f  = _p_ai_h.resize((int(round(_w_ai_s * _f)), _row_a_h), Image.LANCZOS)
_p_a_f   = p_a.resize((int(round(p_a.width * _f)), _row_a_h), Image.LANCZOS)
row_a    = Image.new('RGB', (TARGET_W_PX, _row_a_h), (255, 255, 255))
row_a.paste(_p_ai_f, (0, 0));  row_a.paste(_p_a_f, (_p_ai_f.width, 0))

row_b   = _scale_to_w(_load('panel_b.png'),   TARGET_W_PX)
row_ci  = _scale_to_w(_load('panel_ci.png'),  TARGET_W_PX)
row_cii = _scale_to_w(_load('panel_cii.png'), TARGET_W_PX)

gap      = Image.new('RGB', (TARGET_W_PX, GAP_PX), (255, 255, 255))
_rows    = [row_a, gap, row_b, gap, row_ci, gap, row_cii]
TOTAL_H_PX = sum(r.height for r in _rows)
combined = Image.new('RGB', (TARGET_W_PX, TOTAL_H_PX), (255, 255, 255))
_y = 0
for _r in _rows:
    combined.paste(_r, (0, _y));  _y += _r.height

_y_a   = 0
_y_b   = row_a.height + GAP_PX
_y_ci  = _y_b + row_b.height + GAP_PX
_y_cii = _y_ci + row_ci.height + GAP_PX

_LOFF  = int(round(0.012 * DPI))
_YOFF  = int(round(0.03  * DPI))

# Locate the 5 B-panel column left edges by scanning row_b's histogram region (y≈60%)
_rb_arr     = np.asarray(row_b)
_rb_scan_y  = int(row_b.size[1] * 0.60)
_rb_content = np.any(_rb_arr[_rb_scan_y] < 220, axis=1)
_rb_trans   = np.diff(_rb_content.astype(int))
_rb_starts  = (np.where(_rb_trans == 1)[0] + 1).tolist()
_b_col_x    = _rb_starts[:5] if len(_rb_starts) >= 5 else [TARGET_W_PX * k // 5 for k in range(5)]
print(f'B column left-edges (TARGET_W_PX={TARGET_W_PX}): {_b_col_x}')

labels = {
    'Ai':   (_LOFF,                                                          _y_a   + _YOFF),
    # Aii: pull back from panel_a left edge to avoid overlap with Aiii region text
    'Aii':  (_p_ai_f.width - int(round(DPI * 0.2)) + _LOFF,                _y_a   + _YOFF),
    'Aiii': (_p_ai_f.width + int(_p_a_f.width * 0.41) + _LOFF,             _y_a   + _YOFF),
    'Bi':   (_b_col_x[0] + _LOFF, _y_b + _YOFF),
    'Bii':  (_b_col_x[1] + _LOFF, _y_b + _YOFF),
    'Biii': (_b_col_x[2] + _LOFF, _y_b + _YOFF),
    'Biv':  (_b_col_x[3] + _LOFF, _y_b + _YOFF),
    'Bv':   (_b_col_x[4] + _LOFF, _y_b + _YOFF),
    'Ci':   (_LOFF, _y_ci  + _YOFF),
    'Cii':  (_LOFF, _y_cii + _YOFF),
}

_fig_w = TARGET_W_PX / DPI;  _fig_h = TOTAL_H_PX / DPI
_fig, _ax = _plt2.subplots(figsize=(_fig_w, _fig_h))
_fig.subplots_adjust(0, 0, 1, 1)
_ax.imshow(np.asarray(combined), aspect='auto')
_ax.axis('off')
for _txt, (_px, _py) in labels.items():
    _ax.text(_px, _py, _txt, fontsize=LABEL_PT, fontweight='bold',
             fontfamily='Inter', color='black', va='top', ha='left',
             transform=_ax.transData)

out_png = IMAGES / 'figure1_assembly.png'
out_svg = IMAGES / 'figure1_assembly.svg'
_fig.savefig(out_png, dpi=DPI, bbox_inches='tight', pad_inches=0)
_fig.savefig(out_svg, bbox_inches='tight', pad_inches=0)
_plt2.close(_fig)
print(f'Saved {out_png}  ({out_png.stat().st_size / 1e6:.1f} MB)')
print(f'Saved {out_svg}  ({out_svg.stat().st_size / 1e6:.1f} MB)')

B column left-edges (TARGET_W_PX=9565): [754, 2583, 4411, 6240, 8057]
Saved images/figure1_assembly.png  (4.7 MB)
Saved images/figure1_assembly.svg  (0.9 MB)

Final Figure 1

display(IPImage(str(IMAGES / 'figure1_assembly.png'), width=1000))