Visualization and Characterization of Gene Group Distribution Patterns Across Genomes

Introduction

Gene order along eukaryotic chromosomes is not random [1]. Co-expressed and housekeeping genes cluster into chromosomal domains in yeast, fly and human [2, 3, 4, 5], and many gene families have grown by tandem duplication¹ and still sit together where they arose [7, 8, 9]. Other families are dispersed, having been spread by segmental or whole-genome duplication² [10, 11, 12], by transposition, or by chromosomal rearrangement [13]. A family can therefore be clustered, dispersed, or both. Yet how far apart family members sit, and whether their spacing follows any pattern, has rarely been measured consistently across genomes.

A simple way to see where genes lie is to draw them on an ideogram³ [14]. We built an interactive web application on which the members of any of the 1,396 FlyBase gene groups [15] can be displayed on the banded ideogram of the fruit fly Drosophila melanogaster (§ 3). Browsing it, two impressions recur (§ 4). Some groups contain sets of members that appear evenly spaced, with a middle gene almost equidistant from its two neighbours. And some members sit in tight clusters, within a few kilobases of each other, while others are megabases apart, often on different arms.

This paper turns those impressions into measurements. For every gene family we take each pair of consecutive members on the same chromosome or arm, a pair of neighbour genes, and measure the distance between them. In fly we first ask how common evenly spaced, symmetric sets are (§ 4.1), and what shape the neighbour-distance distribution has (§ 4.2). To test whether these patterns exceed chance, we used a random control that replaces each group with the same number of random genes from the same chromosome arms (§ 5). This control produced about 85% as many symmetric three-gene sets when no minimum distance between members is imposed, so even spacing arises largely by chance (§ 5.1). The control also reproduced the long-range population but not the short-range one. The short-range population is therefore a genuine clustering signal, most likely from tandem duplication (§ 5.2).

We then extend the analysis to human and six other model organisms (§ 6), the same eight genomes whose chromosomes we compared for functional element composition in a companion study [16], and ask whether the mixture of the two populations is the same on every chromosome of a genome (§ 7), and what sets the short-range distance scale: genome size, gene size, or the spacing of genes (§ 8). The short-range distance runs from 1.8 kb in yeast and fly to 69 kb in maize, and it follows the spacing of genes rather than genome size or gene length (§ 8). Finally, we show that differences in the weights of the two populations between genomes come mostly from group size, and then from clade (§ 9).

Data and methods

Genomes, genes and gene families

Eight assemblies were used, the same as in the companion study of functional element composition [16], each in its own coordinates with no liftover: D. melanogaster dm6 (release 6 of the Berkeley Drosophila Genome Project, BDGP6) [17, 18], human (Homo sapiens) GRCh38 (Genome Reference Consortium human build 38) [19, 20], mouse (Mus musculus) GRCm39 [21], zebrafish (Danio rerio) GRCz11 [22], the budding yeast Saccharomyces cerevisiae R64-1-1 [23], the nematode Caenorhabditis elegans WBcel235 [24], Arabidopsis thaliana TAIR10 (The Arabidopsis Information Resource release 10) [25, 26] and maize (Zea mays) Zm-B73-REFERENCE-NAM-5.0, the B73 assembly of the maize nested association mapping (NAM) founder lines [27, 28]. Chromosome names follow FlyBase/Ensembl style without a chr prefix. All coordinates were converted to 0-based half-open intervals at load; GFF3 (General Feature Format version 3), FlyBase and BioMart coordinates are 1-based inclusive and were shifted accordingly. Table 1 lists the sources.

Species	Assembly	Size (Mb)	Group source	Groups	Unit
S. cerevisiae	R64-1-1	12.1	InterPro Family	3,475	16 chromosomes
C. elegans	WBcel235	100.3	InterPro Family	4,656	6 chromosomes
Arabidopsis	TAIR10	135.0	InterPro Family	4,999	5 chromosomes
D. melanogaster	dm6	143.7	FlyBase gene groups	1,396	6 arms
zebrafish	GRCz11	1,373	InterPro Family	7,513	25 chromosomes
maize	Zm-B73 NAM-5.0	2,178	InterPro Family	4,623	10 chromosomes
mouse	GRCm39	2,723	InterPro Family	8,648	21 chromosomes
human	GRCh38	3,100	HGNC gene groups	2,153	48 p/q arms

Gene coordinates. Fly gene coordinates are the FlyBase dm6 gene list [15]. Human, mouse, zebrafish, yeast and C. elegans coordinates come from Ensembl BioMart [29]; Arabidopsis and maize coordinates were parsed from Ensembl Plants GFF3 files [30], because the Ensembl Plants BioMart service failed every query at the time of download.

Gene families. A gene group is any set of genes that a resource assigns to one family. Fly uses the 1,396 curated FlyBase gene groups (release FB2026_01) that have at least one placed member [15], and human the 2,153 curated gene groups of the HUGO (Human Genome Organisation) Gene Nomenclature Committee (HGNC) [31]. No downloadable curated table of the same kind exists for the other six genomes, so for them a group is an InterPro entry of type Family [32], assigned to genes through Ensembl BioMart or, for Arabidopsis and C. elegans, through the UniProt reference proteomes [33]. To test whether results depend on the group source we also built (i) InterPro Family groups for human, on the same coordinates and arms as the HGNC groups; (ii) groups from WormBase gene-class names (nhr, srh, col, . . .) for C. elegans [34]; and (iii) a looser definition, InterPro Family + Domain, for every InterPro genome. HGNC groups include a few large biotype collections (antisense RNAs, long non-coding RNAs, microRNAs, pseudogene classes) that are not families; they are kept in the primary analysis and removed in a robustness check (§ 6.3).

Units. Neighbour genes are paired within a unit. In fly the unit is the chromosome arm (X, 2L, 2R, 3L, 3R, 4); in human it is the p or q arm, read from the leading letter of each gene’s cytogenetic band. Splitting at the centromere keeps a pair from spanning megabases of centromeric heterochromatin. In the other six genomes the unit is the whole chromosome: mouse chromosomes are acrocentric, with no annotated genes on the short arm [21]; yeast centromeres are point centromeres of about 125 bp [35]; C. elegans chromosomes are holocentric [36]; and the zebrafish, Arabidopsis and maize annotations used here carry no centromere positions. A pair in these genomes may therefore span a centromere.

Ideograms and the interactive web application

The fly ideogram is drawn from the dm6 cytogenetic (Bridges) band map [37] distributed with ideogram.js [14], with rounded telomeres and hourglass centromeres; arms 2L/2R and 3L/3R are joined at their centromeres. No band map was used for the other seven genomes, whose chromosomes are drawn as plain bars scaled to assembled length. In the whole-genome views every located gene is drawn at its position and coloured by gene class, using the gene classification and colour palette of [16].

The web application is a Panel [38] application rendered with matplotlib [39], served from notebook ideograms/03 of the project repository. Its controls are listed in § 3. All of its group-level computations — members, neighbour pairs and symmetric sets — call the same library functions (fly_genome.io) that the analyses of this paper use, so a set seen in the application is a set counted in the statistics.

Symmetric gene sets

Symmetry is judged on midpoint-to-midpoint spacings, so that a gene’s own length does not enter. A symmetric three-gene set is three consecutive members of one group on one arm — left, middle and right — whose two flanking midpoint spacings $\delta_\ell$ and $\delta_r$ satisfy

for a symmetry tolerance $\tau$. A symmetric four-gene set is four consecutive members $a$, $b$, $c$, $d$ on one chromosome, either all on one arm or $a$, $b$ on one arm and $c$, $d$ on the other, whose two outer spacings ($a$–$b$ and $c$–$d$) satisfy the same rule. Optionally every compared spacing must be at least a minimum distance $d_{\min}$. The left-hand side is the set’s asymmetry; 0 is perfect symmetry. The application’s defaults are $\tau= 15\%$ and $d_{\min} = 10$ kb.

Neighbour genes and neighbour distances

For each group, the members on each unit are ordered by midpoint. Two members are neighbour genes if they lie on the same unit and no other member of the same group lies between them; each such consecutive pair is a neighbour pair. A group with $n$ members on a unit therefore has $n - 1$ neighbour pairs there, and members alone on their unit have none. The neighbour distance is the edge-to-edge distance

where $e_a$ is the end of the first gene and $s_b$ the start of the second. A distance is negative when the two genes overlap. As a check on unit assignment, no distance may exceed the span of its unit. Distances run over five or more orders of magnitude, so they are displayed as $\operatorname{sign}(d) \log_{10}(\lvert d\rvert+ 1)$, which shows overlaps and megabase distances on one axis. (Axis labels and some panel titles in the figures read “gap” or “intergenic gap”; it is the same quantity.)

Distribution fits

Four positive-support families were fitted to the positive neighbour distances by maximum likelihood with the location parameter fixed at zero: lognormal, gamma, Weibull and exponential. The exponential is the null of interest, since it is the distance distribution of points placed at random (a Poisson process) along a line. Models were ranked by the Akaike information criterion (AIC) [40], and the Kolmogorov–Smirnov (KS) statistic $D$, the largest difference between the fitted and empirical cumulative distribution functions, was used as a measure of shape mismatch.

Because the distances turned out to be bimodal, we also fitted Gaussian mixtures to $\log_{10} d$ with $k = 1,\ldots,4$ components, compared by the Bayesian information criterion (BIC) [41]. A Gaussian on $\log_{10} d$ is a lognormal in base pairs, so $k = 2$ is a mixture of a short-range and a long-range lognormal. We report the median (the back-transformed mean) and weight of each component, and denote the weight of the short-range component $w_{\mathrm{short}}$. BIC usually kept improving up to $k = 4$, but almost all of the gain came from the step $k = 1 \to2$ (Table 3); we report $k = 2$ throughout because it is interpretable and comparable between genomes. Fits used scipy [42] and scikit-learn [43] (five initialisations, fixed seed).

Species	Pairs	Overlap	$\Delta$BIC	Exp. $D$	Short mode	$w_{\mathrm{short}}$	Long mode	Unit
yeast	487	1.8%	266	0.18	1.77 kb	0.26	230 kb	chrom.
C. elegans	7,115	1.2%	3,150	0.37	1.92 kb	0.40	1.23 Mb	chrom.
Arabidopsis	11,355	0.6%	7,050	0.25	1.90 kb	0.26	2.49 Mb	chrom.
fly	6,393	1.5%	4,751	0.24	1.77 kb	0.28	2.42 Mb	arm
zebrafish	7,909	6.9%	3,979	0.45	9.38 kb	0.49	6.39 Mb	chrom.
maize	9,805	0.2%	6,027	0.21	69.2 kb	0.28	33.3 Mb	chrom.
mouse	9,499	3.6%	4,149	0.65	20.7 kb	0.71	14.0 Mb	chrom.
human	20,554	3.4%	2,157	0.38	22.9 kb	0.37	2.96 Mb	arm

Arm-matched random control

To separate the effect of gene-family biology from that of chromosome geometry, each fly gene group was replaced by a control set of the same size and arm composition: for every arm, the same number of genes the group has on that arm were drawn uniformly at random, without replacement, from all dm6 genes on that arm. The control therefore has the same number of sets, set sizes and neighbour pairs as the real groups, but no positional structure. The full analysis was repeated on the control (seed 0), and the two neighbour-distance distributions were compared by a two-sample KS test. Symmetric sets were compared over 20 independent control draws.

Per-unit fits and genome types

The same $k = 2$ mixture was fitted separately on every unit with at least 50 positive distances (25 for yeast Family groups). A forced two-component fit always returns two components, so a unit was called bimodal only if the component means were at least 0.5 dex (orders of magnitude, i.e. $\log_{10}$ units) apart and neither weight was below 0.10. Each component median was then related to the gene-bearing span of its unit by Spearman’s rank correlation $\rho$ and by a log–log regression. To summarise how consistent the per-unit mixtures of a genome are, we computed, over its bimodal units, the standard deviation (SD) of $\log_{10}$ of the short-range median, the SD of $\log_{10}$ of the long-range median, and the SD of $w_{\mathrm{short}}$. A genome is called Type I (consistent) when all three are small — at most 0.25 dex, 0.15 dex and 0.085 respectively — and Type II (variable) otherwise. The thresholds are descriptive; they sit in the clear break between the two groups of genomes (Table 4).

Type	Species	Units	Short-range median Range	Short-range median SD	Long-range median Range	Long-range median SD	$w_{\mathrm{short}}$ Range	$w_{\mathrm{short}}$ SD	Unit span (Mb)
I	fly	5	1.3–3.7 kb	0.16	2.25–2.67 Mb	0.03	0.22–0.30	0.027	23.5–32.1
I	Arabidopsis	5	1.6–2.3 kb	0.06	1.81–3.29 Mb	0.10	0.23–0.31	0.026	18.6–30.4
I	maize	10	47–186 kb	0.21	27.1–40.4 Mb	0.06	0.23–0.39	0.055	152–308
I	C. elegans	6	1.2–3.0 kb	0.13	0.72–1.91 Mb	0.14	0.23–0.45	0.081	13.8–20.9
II	human	41	4.1–178 kb	0.41	0.57–9.46 Mb	0.24	0.21–0.62	0.092	14.4–147
II	mouse	20	11–124 kb	0.22	5.15–35.1 Mb	0.20	0.48–0.83	0.097	57.9–192
II	zebrafish	25	5.1–23.9 kb	0.16	1.14–12.3 Mb	0.22	0.29–0.71	0.123	37.4–78.1
II	yeast	8	1.0–5.7 kb	0.29	0.18–0.35 Mb	0.08	0.13–0.41	0.107	0.75–1.53

Gene size and gene spacing

Two genome-wide quantities were measured from the whole gene pool of each genome, not from the group members alone: the median gene span (start to end, introns included) and the median distance between consecutive genes, the next-gene distance, taken over all placed genes sorted by position on each unit. Overlapping or nested gene pairs give non-positive distances and were excluded. In C. elegans the 15,363 21U-RNA (PIWI-interacting RNA, piRNA) loci, 21 bp each and packed into two clusters on chromosome IV, were excluded from the pool; no grouped gene can be a neighbour to them.

Size-matched component weights

A group with $n$ members on one unit contributes $n - 1$ neighbour distances, of which only a few can be short-range, so large groups lower $w_{\mathrm{short}}$ by construction. Group tables differ widely in their size distributions (HGNC groups have on average 16.3 placed members, InterPro families 4.9), so raw weights are not comparable between genomes. For a fair comparison, every genome was restricted to groups of 2–10 placed members (and, as checks, 2–5, 2–20 and 3–10), and the mixture was refitted.

Visualizing gene groups on the fly ideogram: an interactive web application

Figure 1 shows the fly genome on its banded ideogram, the frame on which the rest of the paper builds. On the left, one gene group, the aquaporins, is marked: bib on 2L and the adjacent pair Drip, Prip on 2R, about 7 kb apart. On the right, every located fly gene is drawn at its position and coloured by class. Genes cover the euchromatic arms densely and thin out towards the pericentromeric heterochromatin.

Gene content is broadly similar across the five large fly arms (Appendix Fig. A1): they are 23.5–32.1 Mb long, carry 2,639–4,147 genes at 112–147 genes/Mb, and are 63–72% genic by base pairs. Chromosome 4 has the fewest genes (110) but the highest genic coverage (79%), because its genes are long, and Y is gene-poor (31 genes/Mb, 30% genic). Gene length is close to lognormal on every arm (median 1.5–2 kb); a minor peak near 100 bp is formed by the short ncRNA classes (tRNA, miRNA, snoRNA, rRNA, snRNA; Appendix Fig. A2).

To see where the members of a gene family lie, and how they are spread along the genome, we built an interactive web application on the banded fly ideogram (Figure 2). Any of the 1,396 FlyBase gene groups can be chosen from a searchable box, and its members are marked on the ideogram by arrows with gene names; labels are spaced automatically so that they never overlap, and groups with more than 30 members are shown by markers alone. The group can also be chosen by its number, with a slider, with previous/next buttons, or by an automatic display that steps through the groups at a chosen interval (3–20 s). Two toggle buttons list the group’s members arm by arm, in ideogram order, and its neighbour pairs — the consecutive members on each arm (§2.4). The application runs locally with a single command and needs only a web browser.

A switch above the search box changes the application to Symmetric mode (Figure 3). The search list then narrows to groups that contain at least one symmetric three- or four-gene set (§2.3); a slider sets the symmetry tolerance (0–15%), a box sets the minimum distance (10–100 kb), and a live counter shows how many groups qualify (170 at the defaults). The ideogram draws dashed connectors for every symmetric set of the selected group — for a three-gene set, from the middle gene to its two flankers, with the middle gene ringed; for a four-gene set, across each of the two matched pairs — and the neighbour button becomes Show symmetric genes, which lists each set with its two spacings and their percentage difference. Group numbers stay those of the full list, so a group can be found again in either mode.

Two patterns seen by browsing

Browsing the application group by group gives two recurring impressions: some members look evenly spaced, and members tend to be either very close together or megabases apart. We describe each in turn, and test both against chance in § 5.

Symmetric gene sets

Browsing the application in Symmetric mode shows how common evenly spaced members are. At the defaults ($\tau= 15\%$, $d_{\min} = 10$ kb), 170 of the 1,396 fly groups contain at least one symmetric set, 396 sets in all; without the minimum distance, 240 groups contain 696 sets. Some are very tight. Table 2 lists the six most symmetric sets in the fly genome at the application’s defaults, each from a different group. The tightest, the RAB set, is shown in Figure 3; Figure 4 shows two more, the beta importins and the NEDD8 transferases, whose members are far enough apart on the whole-genome ideogram for the two equal spacings to be seen. The other three are not drawn, because at this scale their highlighted members lie too close together for the spacings to be judged by eye: the sets of the spliceosome complex P (SLP-C-P) and the cationic amino acid transporters (CAT) span 4.8 and 5.1 Mb, and bcd–Dfd–Scr of the Hox-like homeobox transcription factors (HOXL) only 79 kb, against 13.5 and 15.7 Mb for the two sets shown, so their rings and labels run into one another and, for SLP-C-P, into those of the group’s other 82 genes. In the RAB GTPases, Rab30–Rab6 and Rab14–Rab9 on 2L are 5,076,619 and 5,077,253 bp apart, a difference of 634 bp, or 0.012%. The beta importins Karybeta3, Ipo9 and cdm sit on 3R with Ipo9 within 2.3 kb of the midpoint of a 13.5 Mb interval.

The six most symmetric sets in the fly genome, one per group, at the application’s defaults ($\tau= 15\%$, $d_{\min} = 10$ kb). Spacings are midpoint to midpoint. Asymmetry: $|\delta_1 - \delta_2|/\max(\delta_1, \delta_2)$. FlyBase group abbreviations: RAB, RAB GTPases; SLP-C-P, spliceosome complex P; IMB, beta importins; NEDD8T, NEDD8 transferases; HOXL, Hox-like homeobox transcription factors; CAT, cationic amino acid transporters.

Group (No.)	Arm	Kind	Set	$\delta_1$ (bp)	$\delta_2$ (bp)	Asym.
RAB (962)	2L	four	Rab30–Rab6 · Rab14–Rab9	5,076,619	5,077,253	0.012%
SLP-C-P (1145)	2L	four	snRNA:U5:23D–CG15432 · Phf5a–SmE	1,393,887	1,394,691	0.058%
IMB (551)	3R	three	Karybeta3–Ipo9–cdm	6,762,432	6,767,045	0.068%
NEDD8T (756)	3L	three	Roc1b–UbcE2M–Diap1	7,838,684	7,847,805	0.116%
HOXL (523)	3R	three	bcd–Dfd–Scr	39,478	39,419	0.151%
CAT (192)	3L	three	CG7255–CG5535–CG13248	2,546,984	2,554,126	0.280%

Five of the six tightest sets span megabases. The exception is bcd–Dfd–Scr, in the Antennapedia complex on 3R: three Hox-like genes about 39 kb apart in a cluster that arose by tandem duplication. The same group holds a second set, Ubx–abd-A–Abd-B, in the bithorax complex (6.1% asymmetry).

Neighbour distances form two populations

The second impression, tight clusters next to members spread over megabases, concerns the distance between neighbour genes (§2.4). The 1,396 fly gene groups yield 6,393 neighbour pairs. Of these, 99 (1.5%) overlap. The positive neighbour distances are clearly bimodal (Figure 5a): a minor population of about 1–2 kb and a dominant one in the megabase range. No single distribution fits. Gamma is the best by AIC, but with $D = 0.096$, and lognormal ($D = 0.19$) and exponential ($D = 0.24$) are worse; every unimodal curve misses the short-range peak. A two-component lognormal mixture lowers the BIC by 4,751 relative to one component and separates a short-range population with median 1.77 kb and weight 0.28 from a long-range population with median 2.42 Mb and weight 0.72 (Figure 5b).

A random control separates biology from geometry

Both patterns raise the same question: how much of each is due to gene families, and how much to the simple fact of choosing a few genes on a fly arm? To answer it, each fly gene group was replaced by the same number of genes drawn at random from the same chromosome arms (§ 2.6), and the analysis was repeated on these controls.

Symmetric sets are mostly chance

Are symmetric sets more common than chance would produce (Figure 6)? Counting three-gene sets only, at $\tau= 15\%$ with no minimum distance, 186 fly groups contain at least one symmetric set, 414 sets in all (the other 282 of the 696 sets counted in § 4.1 are four-gene sets). The control yields 353 $\pm$ 22 sets (mean $\pm$ SD, 20 draws), so chance accounts for about 85% of them. The real excess is consistent but small: $z = +2.5$ to $+3.4$ at every tolerance from 1% to 15%.

With a minimum distance of 10 kb, the default of the application, the sign reverses: 234 real three-gene sets against 353 $\pm$ 22 in the control ($z = -5.4$; Figure 6b). The reason is mechanical. The gate removes 43% of the real sets, those with a neighbour closer than 10 kb, and none of the control sets, since random genes are almost never that close. For the same reason the symmetric sets that pass a 10 kb gate and a 5% tolerance are dominated by widely spaced members: of 131 such sets (71 three-gene and 60 four-gene sets), 74% have a first spacing of 1–10 Mb (Appendix Fig. A3). A tight relative tolerance is easy to meet when genes are megabases apart and hard when they are a few kilobases apart.

A single symmetric set, however tight, is therefore not evidence of a special arrangement; it is the expected yield of choosing a few genes on a fly arm. The only symmetric sets that a random control cannot produce are the tandem-scale ones, such as bcd–Dfd–Scr — the very ones a minimum-distance gate removes. This points to the real structure in the data: the distance between neighbour genes, not its symmetry.

The short-range population is the clustering signal

For neighbour distances, the control shows how much of Figure 5 is due to gene families, and how much to the simple fact of choosing a few genes on a 25 Mb arm (Figure 7). Its distance distribution has one mode, at 1–5 Mb, that coincides with the long-range mode of the real groups. The short-range population is absent: only 3 of 6,393 control pairs overlap (against 99 real pairs), and the control has essentially no distances below 10 kb. A single gamma distribution fits the control almost exactly ($D = 0.015$), with shape 0.74–0.81 on the large arms. A shape near 1 is close to exponential, the spacing law of random points. When a two-component mixture is forced on the control, its ‘short’ component sits at about 450 kb, the left shoulder of the single mode, three orders of magnitude above the 1.8 kb short-range mode of the real groups. Real and control distributions differ strongly (KS $D = 0.228$, $p \approx10^{-144}$), and the difference lies entirely in the short-range population, which holds about a quarter of the real pairs.

The control also reproduces two features of the per-arm fits below: the near-identity of the five large arms (2L–3L, 2L–3R and 3L–3R are statistically indistinguishable in the control as in the real data) and the outlier status of 4 (KS $D = 0.75$–0.80 against every arm). These features follow from arm length and gene density, not from family biology. Two findings survive the control: the short-range population, and the overlapping pairs.

Two populations in eight genomes

Gene positions on the ideograms of seven further genomes

The whole-genome view of Figure 1b was drawn for human and the six other model species (Figure 8–Figure 10). It makes the differences in chromosome architecture that matter later in the paper visible at a glance: the 48 human arms span more than a tenfold range of length, from the gene-poor acrocentric short arms to 1p and 2q; mouse chromosomes are all acrocentric; the five Arabidopsis chromosomes, the six C. elegans chromosomes and the ten maize chromosomes each vary in length by about a factor of two; and the sixteen yeast chromosomes are all shorter than 1.6 Mb. In C. elegans, 96% of the 15,363 piRNA loci lie in two clusters on IV, visible as two dense bands.

The two populations recur in every genome

The analysis was repeated on human, mouse, zebrafish, yeast, Arabidopsis, maize and C. elegans (Figure 11; Table 3). The two-population shape is found in all of them. The exponential null is rejected in every genome, most strongly in mouse ($D = 0.65$) and least strongly in yeast ($D = 0.18$). Among single distributions the Weibull is best in human, the lognormal in mouse and the gamma in yeast, fly, Arabidopsis, maize and C. elegans; in zebrafish gamma wins by AIC and lognormal by KS. As in fly, no single distribution fits well, and in every genome the step from one to two mixture components gives by far the largest BIC gain.

The genomes differ in three ways that are visible in Table 3.

Overlap. The fraction of overlapping pairs ranges from 0.2% (maize) and 0.6% (Arabidopsis) to 3.4–3.6% in the mammals and 6.9% in zebrafish. Overlapping pairs are typically genes nested within a neighbour’s intron (median depth 7.8 kb in mouse, 14 kb in zebrafish), not coincident loci. Zebrafish, with large tandem-expanded families such as rhodopsin-like G-protein-coupled receptors (GPCRs; 731 genes), NOD-like (nucleotide-binding oligomerization domain-like) receptors (345) and olfactory receptors (119), has the most.

Short-range mode. It is 1.8–1.9 kb in yeast, fly, Arabidopsis and C. elegans (yeast and fly differ by less than 1 bp: 1,768 and 1,769 bp), 9.4 kb in zebrafish, 21–23 kb in the two mammals and 69 kb in maize. That fly and Arabidopsis, genomes of almost the same size (144 and 135 Mb) separated by more than a billion years [44], agree to within 8% in the short-range mode (1.77 and 1.90 kb) and 3% in the long-range mode (2.42 and 2.49 Mb) was the first hint of what sets the scale (§ 8).

Weights. Raw $w_{\mathrm{short}}$ ranges from 0.26 to 0.71. § 9 shows that most of this range comes from differences in group size.

Robustness to the group definition

Loosening the InterPro definition from Family to Family + Domain roughly doubles the number of groups and multiplies the pairs by 2.5–4, but leaves the two-population shape intact in all InterPro genomes. The short-range mode moves by 7% in C. elegans, by about 20% in mouse, by 40% in zebrafish and Arabidopsis, and by a factor of two in maize (69 to 140 kb); the maize mode is therefore best quoted as of order $10^{5}$ bp. In C. elegans, WormBase gene classes give a short-range mode of 1.98 kb and a long-range mode of 1.19 Mb, within 3–4% of the InterPro values. In human, removing the 29 biotype collections from the HGNC groups removes about 45% of the pairs and moves both modes (the short-range peak sharpens near 10 kb and the long-range one shifts towards 10 Mb), but the distribution stays bimodal.

Two types of genome: consistent and variable per-chromosome mixtures

The pooled fits hide how the two populations behave chromosome by chromosome. Fitting the mixture separately on every unit divides the eight genomes into two types (Table 4; Figure 12 and Figure 13). In Type I genomes every chromosome or arm gives nearly the same mixture: the short-range and long-range peaks sit in the same places and carry the same weights, so the per-unit curves lie on top of one another. In Type II genomes the per-unit mixtures fan out, in the position of the long-range peak, in the weight of the short-range one, or in both.

Type I: fly, maize, Arabidopsis and C. elegans

Fly is the clearest Type I genome (Figure 12, top row). The five large arms give short-range medians of 1.3–3.7 kb with weights 0.22–0.30, and long-range medians of 2.25–2.67 Mb (SD 0.03 dex), and their fitted curves almost coincide. Pairwise KS distances between the large arms are small ($D = 0.04$–0.10), and several pairs, such as 2L–3L and 3L–3R, are statistically indistinguishable. Chromosome 4 (15 pairs) is the only outlier, fitted with a single component, because a 1.3 Mb chromosome cannot hold megabase distances.

Maize, a genome fifteen times larger, behaves the same way (Figure 12, second row). All ten chromosomes are bimodal, the long-range medians lie between 27 and 40 Mb (SD 0.06 dex), and the short-range weights between 0.23 and 0.39. The short-range medians spread more (47–186 kb), driven by chromosomes 6 and 8, but the curves still share one shape. The long-range median rises gently with chromosome span ($\rho= 0.70$, $p = 0.025$; slope 0.49), as random placement predicts.

Arabidopsis is as consistent as fly (Figure 12, third row): all five chromosomes are bimodal, with short-range medians of 1.6–2.3 kb (SD 0.06 dex, the tightest of all eight genomes), long-range medians of 1.8–3.3 Mb that track chromosome span ($\rho= 0.90$, $p = 0.04$), and weights of 0.23–0.31.

C. elegans is Type I at the edge of the definition (Figure 12, bottom row): the short-range medians (1.2–3.0 kb) and long-range medians (0.7–1.9 Mb) are consistent, but the short-range weight varies from 0.23 on X to 0.45 on II and V, and chromosome V, which holds 38% of all pairs, has the lowest long-range median (0.72 Mb). Both deviations follow from the tandem-arrayed chemoreceptor families that crowd the arms of the autosomes, above all V [45] (§ 8).

What the four Type I genomes share is that their units are of similar size. The five large fly arms span 23.5–32.1 Mb, the Arabidopsis chromosomes 18.6–30.4 Mb, the C. elegans chromosomes 13.8–20.9 Mb and the maize chromosomes 152–308 Mb: no more than a twofold range in any of them. Since the long-range population is random placement (§ 5.2), units of similar length give long-range peaks in similar places.

Type II: human, mouse, zebrafish and yeast

Human is the clearest Type II genome (Figure 13, top row). Of the 42 arms with at least 50 positive distances, 41 are bimodal (5p is the exception); the four acrocentric short arms (13p, 14p, 15p, 22p) carry no group members at all. The per-arm curves fan out across two orders of magnitude: long-range medians run from 0.57 Mb (Yq) to 9.5 Mb (2q), short-range medians from 4 kb (14q, 15q) to 178 kb (3q), and short-range weights from 0.21 (16q) to 0.62 (19q). Part of the spread is geometry: human arms differ more than tenfold in length, and the long-range median climbs with arm span (Spearman $\rho= 0.80$, $p = 2 \times10^{-10}$; log–log slope 0.77, $R^2 = 0.62$), whereas the short-range median does not ($\rho= 0.29$, $p = 0.06$; $R^2 = 0.03$; Figure 13, top row, b). Part is family biology: the highest short-range weights are on 19p and 19q (0.53, 0.62), home to the largest clusters of KRAB (Krüppel-associated box) zinc-finger genes [46].

Mouse (Figure 13, second row) shows the same scaling more cleanly: 20 of 21 chromosomes are

bimodal (all but Y), the long-range median scales with chromosome span ($\rho= 0.85$, $p = 10^{-6}$; 5.2–35 Mb), and the short-range median is flat ($R^2 = 0.00$; autosomal medians 11–42 kb across chromosomes of 58–192 Mb). The weights vary more than in any Type I genome (0.48–0.83), and X stands out with a short-range median of 124 kb.

Zebrafish (Figure 13, third row) shows that Type II is not only a matter of unit length. Its 25 chromosomes vary in span by only a factor of two (37–78 Mb), less than maize’s, and the long-range median does not scale with span ($\rho= 0.37$, $p = 0.07$). Yet the short-range weight ranges from 0.29 to 0.71 — the largest SD of all eight genomes — and chromosome 4, whose long arm carries large tandem gene clusters [22], has a long-range median of 1.1 Mb, two to ten times below the others. The variation here comes from where the tandem-expanded families lie, not from chromosome geometry.

Yeast (Figure 13, bottom row) is Type II for a different reason. The long-range medians are consistent (0.18–0.35 Mb), but each chromosome holds only 26–103 pairs, and the short-range component is fitted on a handful of them: its median ranges from 1.0 to 5.7 kb and its weight from 0.13 to 0.41, and chromosome XIII is not bimodal. With the larger Family + Domain set (12 chromosomes; Appendix Figure A4) the weights become more consistent but the short-range medians spread further (0.5–14 kb), and the long-range median now scales with chromosome length ($\rho= 0.65$, $p = 0.02$). With the Family set alone only nine chromosomes can be fitted and no trend is detectable. Yeast’s variability is thus partly a sampling limit of a small genome.

What the two types share

Across both types, the long-range population behaves exactly as random placement predicts: the distance between randomly chosen genes grows with the room available. The long-range median rises significantly with unit span in human and mouse, whose units span a more than threefold range, and also in maize, Arabidopsis and yeast with Family + Domain, where the range is only about twofold and the trend is shallow (slope 0.49 in maize). It does not in C. elegans ($\rho= -0.54$, $p = 0.3$), whose six chromosomes span only a 1.5-fold range, or in zebrafish ($\rho= 0.37$, $p = 0.07$), where chromosome 4 is an outlier (§ 7). Within a twofold range the long-range peak moves little, which is why such genomes can still be Type I. In yeast, randomly placed members of a family with 2–3 members on a 0.9 Mb chromosome would be about 200–300 kb apart, which matches the fitted 230 kb, and it explains why the exponential null is only barely rejected there. Long-range modes can therefore be compared only between genomes with the same unit. Mouse’s 14 Mb and maize’s 33 Mb reflect whole-chromosome units of 58–308 Mb, not a property of families, and the agreement between fly (2.4 Mb) and human (3.0 Mb) is a coincidence of arm lengths.

The short-range median, in contrast, does not follow unit span in any genome. It is nearly constant across the units of every Type I genome, and in Type II genomes it varies without regard to span. Its scale is a property of the genome as a whole, which the next section explains.

The short-range scale is set by gene spacing

The short-range population is the genuine clustering signal, so its scale is the quantity worth explaining. Three hypotheses were tested (Figure 14; Table 5).

Species	Genome (Mb)	Gene span (bp)	Next-gene dist. (bp)	Overlapping	Short mode (bp)	Mode / dist.
yeast	12.1	1,023	349	0.10	1,768	5.07
C. elegans	100.3	1,259	855	0.21	1,919	2.25
Arabidopsis	135.0	2,078	891	0.13	1,902	2.14
fly	143.7	1,709	622	0.21	1,769	2.84
zebrafish	1,373	10,433	6,926	0.15	9,383	1.35
maize	2,178	2,410	23,468	0.10	69,196	2.95
mouse	2,723	3,961	14,128	0.29	20,748	1.47
human	3,100	4,727	14,603	0.34	22,938	1.57
Spearman $\rho$ with the short mode: genome size 0.83; gene span 0.71; next-gene distance 0.98.

H1: genome size. The short-range mode increases with genome size ($\rho= 0.83$, $p = 0.010$; log–log $R^2 = 0.77$), but slowly (slope 0.63), and with clear exceptions: maize, at 70% of the size of the human genome, has a mode three times larger, and zebrafish lies well below the trend. H2: gene size. If a tandem copy must clear the span of the gene it copies, the mode should follow gene length. It does not ($\rho= 0.71$, $p = 0.047$; $R^2 = 0.33$). The two exceptions come from opposite directions. Maize genes are as compact as Arabidopsis genes (median span 2.4 vs 2.1 kb), but its mode is 29 times its median gene; zebrafish has the longest genes in the set (median 10.4 kb, more than twice human’s), but a mode below both mammals’.

H3: gene spacing. What maize has that Arabidopsis lacks, and what zebrafish lacks relative to the mammals, is DNA between adjacent genes. The median next-gene distance explains the mode across all eight genomes ($\rho= 0.98$, $p < 0.001$; $R^2 = 0.94$, slope 0.84; Figure 14c). Expressed as a multiple of the next-gene distance, the mode varies only from 1.35 to 5.07 (0.57 dex), compared with a 39-fold (1.59 dex) range in the raw modes; gene spacing thus accounts for about two thirds of the variation between genomes. Every genome lies above the 1:1 line: a short-range pair is typically one to a few genes apart, not strictly adjacent.

Two genomes were not used to formulate H3 and serve as out-of-sample checks. Zebrafish, with a next-gene distance of 6.9 kb, is the only genome between the small-genome cluster (0.35–0.9 kb) and the large-genome cluster (14–23 kb), and it falls on the relationship. C. elegans was predicted, before fitting, to have a mode of a few kilobases from its 855 bp next-gene distance; the fitted mode is 1.92 kb (ratio 2.25). It falls inside an already populated cluster, so it is a weaker test.

A within-genome test in C. elegans. C. elegans chromosomes have gene-dense centres with low recombination, flanked by arms with high recombination [47]. Pairs on the arms are far more often short-range ($w_{\mathrm{short}} = 0.50$, against 0.32 in the centres; Figure 15), consistent with the concentration of rapidly duplicating chromosome repeat families on the arms [45]. H3 predicts slightly longer short-range distances on the arms, where genes are 25% further apart (next-gene distance 954 vs 762 bp). The raw short-range distances point that way (median 1,540 vs 1,212 bp, Mann–Whitney $p = 0.009$, a ratio of 1.27 against a spacing ratio of 1.25), but the fitted modes (1,857 vs 1,936 bp) cannot resolve a contrast this small. The test is consistent with H3 but could not have refuted it.

Component weights: group size, then clade

Raw short-range weights range from 0.26 (yeast, Arabidopsis) to 0.71 (mouse), and human sits at 0.37. Three explanations are confounded: the group source, the size distribution of groups, and clade. Human separates the first two, because it has two group tables on the same coordinates and arms (Table 6a). HGNC groups give $w_{\mathrm{short}} = 0.37$ and InterPro families 0.62, but restricted to groups of 2–10 members they give 0.635 and 0.643. The group source makes no difference; the size distribution does. The short-range mode is barely affected (22.9, 23.8, 18.8 and 23.3 kb), so the conclusions of §8 stand.

(a) Human
Group table	Mean size	Pairs	$w_{\mathrm{short}}$	Short mode
HGNC groups, all	16.3	20,554	0.372	22.9 kb
HGNC groups, 2–10 members	4.4	1,141	0.635	18.8 kb
InterPro Family, all	4.9	6,038	0.618	23.8 kb
InterPro Family, 2–10 members	3.2	1,723	0.643	23.3 kb
(b) Groups of 2–10 members
Species	Clade	Mean size	Pairs	$w_{\mathrm{short}}$	Range over windows
yeast	non-vertebrate	2.8	279	0.221	0.18–0.23
Arabidopsis	non-vertebrate	3.8	3,348	0.231	0.23–0.27
C. elegans	non-vertebrate	3.4	1,618	0.274	0.27–0.31
fly	non-vertebrate	4.3	1,705	0.289	0.26–0.29
maize	non-vertebrate	4.0	2,142	0.307	0.29–0.36
zebrafish	vertebrate	3.5	1,655	0.444	0.42–0.44
mouse	vertebrate	3.3	1,916	0.629	0.61–0.63
human (InterPro)	vertebrate	3.2	1,723	0.643	0.62–0.66

Matched for size, the genomes separate cleanly by clade (Table 6b): the five non-vertebrates have $w_{\mathrm{short}} = 0.22$–0.31 and the three vertebrates 0.44–0.64, with no overlap under any of the four size windows (closest: maize 0.360 at 2–5 members against zebrafish 0.416). The direction of the size bias depends on the genome. In human, large groups dilute the tandem signal (0.37 raw, 0.64 matched); in C. elegans, the largest groups are the tandem-arrayed chemoreceptor families, so matching lowers the weight (0.40 raw, 0.27 matched). The three vertebrates are also three of the four Type II genomes: a larger share of tandem pairs, concentrated in particular families on particular chromosomes, is one way per-chromosome mixtures come to differ.

Discussion

The study began with a picture. Drawing every gene of eight genomes on ideograms, and the members of any fly gene group on a banded ideogram in an interactive application, made the spatial distribution of gene families something one can browse, group by group. Two impressions from browsing — members that look evenly spaced, and tight clusters next to megabase-scale spreads — were then measured.

Evenly spaced members turned out to be the weaker of the two. Sets can be strikingly symmetric — in the RAB GTPases two spacings of about 5 Mb differ by only 634 bp, about 0.01% — but an arm-matched random control produces about 85% as many, and a minimum-distance filter that hides tandem clusters turns a modest enrichment into an apparent depletion. The few symmetric sets that chance does not produce are tandem-scale ones such as bcd–Dfd–Scr in the Antennapedia complex, and these are better described by their short distances than by their symmetry. Visual patterns on an ideogram need the same random control as any other statistic.

The distances between neighbour genes have the same structure across eight eukaryotic genomes, three kingdoms (animals, plants and fungi) and a 250-fold range of genome size: a short-range population and a long-range population. The random control in fly and the per-unit fits in the other genomes assign them different meanings. The long-range population is what randomly placed genes produce: it is reproduced by the control in fly, nearly exponential where it dominates (yeast), and its scale follows the length of the chromosome or arm. The short-range population is what random placement cannot produce, and it is the signature of family members that lie close together, most plausibly as products of tandem duplication [6, 8, 9].

Fitted chromosome by chromosome, the genomes fall into two types. In Type I genomes (fly, maize, Arabidopsis, C. elegans) the per-unit mixtures nearly coincide, because their units are of similar length and, apart from the chemoreceptor-rich arms of C. elegans, their tandem families are spread fairly evenly across them. In Type II genomes (human, mouse, zebrafish, yeast) they vary, for three separable reasons: a wide range of unit lengths moves the long-range peak (human, mouse); tandem-expanded families concentrated on particular chromosomes move the short-range weight (human 19, zebrafish 4); and too few pairs per chromosome make the short-range fit noisy (yeast). The typology is therefore descriptive, not a single biological property, and its interest lies in which of these reasons applies. The division nonetheless echoes the companion comparison of the same eight genomes [16], in which gene-family and protein-function composition was more uniform along the chromosomes of the plants and the fly than along those of the vertebrates; there, C. elegans, the borderline Type I genome here, was the least uniform of all.

The scale of the short-range population follows the spacing of genes in each genome. This is not obvious in advance. One might expect it to follow genome size, since larger genomes have more DNA between everything, or gene size, since a tandem copy must clear the gene it copies. Maize and zebrafish separate these alternatives because they break the usual correlation between genome size, gene size and gene spacing in opposite directions. Maize packs plant-sized genes into mammal-sized spacing, its large genome consisting mostly of retrotransposons between genes [27]. Zebrafish has the longest genes in the set and less intergenic DNA than the mammals. In both, the short-range mode follows the spacing. A natural reading is that a tandem copy, once it has drifted away from its original by the insertion of other genes, other sequences or rearrangements, ends up a couple of gene spacings away, so the clustering scale is measured in genes, not in base pairs.

The weights of the two populations are a different matter. Raw weights mostly reflect how large the groups in each table are, a point that matters for any comparison of clustering between genomes annotated by different resources. Once group size is matched, the three vertebrates carry about twice the share of short-range pairs found in the five non-vertebrates. Why is open. The three vertebrates share ancestry, so this is one contrast, not three independent observations, and it could reflect lineage-specific expansions of large tandem families (olfactory receptors, zinc-finger genes) [48, 46].

Caveats

• Group definitions differ. Fly and human use curated tables; the other six use InterPro protein families, which miss non-coding families and can split curated groups. Two-source comparisons in human and C. elegans, and the Family + Domain check, show that the modes are stable to within 3–40% in all genomes except maize (a factor of two).

• Units differ. Fly and human pairs are confined to arms; elsewhere a pair may span a centromere, which inflates long-range distances. Long-range modes are compared only within a unit convention. Splitting Arabidopsis and maize chromosomes at their centromeres is an obvious refinement.

• The Type I/Type II division is descriptive. Its thresholds were set at the break between the observed genomes, C. elegans lies close to it, and the per-unit SDs depend on the number of pairs per unit (yeast). It summarises the per-unit fits; it is not a test.

• The application covers fly only. Symmetric sets were counted and controlled in fly; the same functions run on the other genomes, but no interactive view or random control was built for them.

• Annotation density differs. Mammalian annotations contain many nested and antisense genes (29–34% of adjacent pairs overlap). Dropping them biases the mammalian next-gene distance downward; correcting it would widen, not narrow, the spread in Table 5.

• Few genomes. H3 rests on eight points, with no genomes at 0.9–6.9 kb and 6.9–14 kb spacing. A rank correlation of $\rho= 0.98$ at eight genomes (one adjacent swap) has an exact two-sided permutation $p$ of about $4 \times10^{-4}$, and the yeast and fly short-range modes (1,768 and 1,769 bp) are effectively tied; it is a pattern worth testing on more genomes, not a scaling law.

• The random control was run only in fly. The reading of the long-range component as random placement in other genomes rests on its scaling with unit length and, in yeast, on a back-of-envelope estimate. Running the control in each genome is the most direct next step.

• Fly gene biotypes in the arm statistics of the Appendix were inferred from FlyBase symbol prefixes, not from the GFF3 biotype field.

• Human coordinates were re-downloaded after an initial BioMart download silently dropped chromosomes 1 and 2; all numbers here are from the corrected data.

Data and code availability

All analyses use public releases: FlyBase FB2026_01 gene list and gene-group report (dm6) [15]; HGNC complete set [31]; Ensembl BioMart (GRCh38, GRCm39, GRCz11, R64-1-1, WBcel235) [29]; Ensembl Plants GFF3 (TAIR10, Zm-B73-REFERENCE-NAM-5.0) [30]; UniProt reference proteomes UP000006548 (Arabidopsis) and UP000001940 (C. elegans) [33]; the InterPro entry list [32]; WormBase gene classes [34]; and the dm6 cytogenetic band map distributed with ideogram.js [14]. Data were downloaded between May and September 2026. No new sequence data were generated. All figures are produced by the Jupyter notebooks 01_ideogram, 02_gene_markers, 03_ideogram_app, 04_chromosome_arm_stats, 05_random_control_gene_distances, 06_fly_neighbour_gene_distances, 07–12_<species>_neighbour_gene_distances (human, mouse, zebrafish, yeast, Arabidopsis, maize), 13_gene_class1_ideogram_all_species and 14_celegans_neighbour_gene_distances, using the fly_genome.io, fly_genome.viz and fly_genome.neighbour modules. The interactive application is notebook 03_ideogram_app, served with panel serve; its screenshots (Figure 2 and Figure 3) were taken at its default settings, and the ideograms of Figure 3 and Figure 4 were redrawn with the same plotting functions so that gene names are shown for every group. The source code is available from the author upon reasonable request.

Acknowledgements

This work used no funding and no computing resources beyond a laptop. Analyses were carried out in Python with numpy [49], pandas [50], scipy [42], scikit-learn [43], matplotlib [39] and panel [38].

The author would like to sincerely thank Anthropic’s Claude Opus 5.5 for its incredibly intelligent aid in the baking of this manuscript.

Appendix: supplementary figures

References