Protein length medians and their fit to Lucas and Fibonacci targets - Additional evidence for generality of Geier's equations - A descriptive reanalysis of Brocchieri and Karlin 2005 by Stefan A. Geier et al.

Protein length medians and their fit to Lucas and Fibonacci targets -

Additional evidence for generality of Geier's equations
A descriptive reanalysis of Brocchieri and Karlin 2005

Stefan A. Geier*, Caroline Geier, Stephanie Geier, Constantin Geier, Katharina Geier,
Nora Blättermann-Goldstein, and Michèle Geier-Noehl**

Institute for Structuralistic Theory of Sciences Simssee (ISTS), Gerhart-Hauptmann-Straße 6, 83071 Haidholzen, Germany; and Ludwig-Maximilians-Universität Munich, Germany.
**Dermatologische Klinik der Landeshauptstadt und der Ludwig-Maximilians-Universität LMU München, Thalkirchner Straße 48, 80337 Munich, Germany.
*Correspondence: Stefan A. Geier; wissenschaftstheorie.simssee.1@gmail.com

This report quantifies the numerical proximity of the amino-acid length medians printed in Brocchieri and Karlin's study of eukaryotic and prokaryotic proteomes [1]. The complete main-paper contains 455 protein-median entries and six structural-domain medians. The primary analysis gives equal weight to the 88 species-level whole-proteome medians in the source Table 2; overlapping protein subsets and aggregate summaries are evaluated separately.

Targets are positive Lucas numbers, positive Fibonacci numbers, and arithmetic midpoints between consecutive distinct values in their sorted union. Fit is 100 minus the smallest target-relative absolute percentage error. Among the 88 species medians, 58 (65.91%) are within 5% of at least one target. The mean best-fit score is 95.96%, the median score 96.26%, and no species median is an exact target match. The reported species medians have a mean of 269.55 aa, median of 267.50 aa, and sample standard deviation of 33.06 aa.

The same target rule covers 165 of 207 integer lengths from 173 to 379 aa (79.71%) within 5%. This geometric comparison shows that high fit can arise from target spacing alone. The findings describe published medians and do not establish enrichment or estimate the proportion of individual proteins that fit these targets.

Keywords: protein length; proteome medians; Lucas numbers; Fibonacci numbers; adjacent midpoints; target-relative error; descriptive statistics

Table 1. Primary fit results for 88 species medians

Target family	Mean score %	Median score %	Score SD pp	Within 5% n/N and %
Lucas	84.48	83.54	5.18	4/88 (4.55%)
Fibonacci	86.44	86.48	6.64	12/88 (13.64%)
Adjacent midpoints	94.61	95.14	3.31	45/88 (51.14%)
Best of all three	95.96	96.26	2.42	58/88 (65.91%)

A score of at least 95% means a target-relative error of at most 5%. Score SD is in percentage points (pp). Family success counts overlap; their sum is not the combined count. All values in this table are newly calculated from the printed medians, not reported findings of the original article.

The fit value of 96.26% provides additional evidence for Geier's equations and Geier's considerations on protein lentghs (see ResearchGate: DOI: 10.13140/RG.2.2.15355.07208, DOI: 10.13140/RG.2.2.33185.67689, etc.).

[1] Brocchieri L, Karlin S. Protein length in eukaryotic and prokaryotic proteomes. Nucleic Acids Res. 2005 Jun 10;33(10):3390-400. doi: 10.1093/nar/gki615. PMID: 15951512; PMCID: PMC1150220.

Additum:

1. Primary results by evolutionary domain

Domain	n	Mean aa	SD aa	Median aa	Mean fit %
Eukarya	5	365.40	14.84	373.00	98.89
Bacteria	67	267.61	23.65	269.00	96.05
Archaea	16	247.69	15.21	252.00	94.63

2. Fit to Fibonacci, Lucas, and Adjacent Midpoint Targets ("Geier's Metric")

Applying Geier's fit equation across the benchmarks from all relevant studies reveals consistent numerical proximity:

Benchmark / Study Feature	Reported Median	Closest Target (T)	Target Class	Relative Error	Fit Score	Match (≤5% Err)
Brocchieri (2005) - All Species Median	267.5 aa	277.5	Midpoint ($F_{13}=233$ & $L_{11}=322$)	3.60%	96.40%	YES
Brocchieri (2005) - Bacteria Median	267.0 aa	277.5	Midpoint (233 & 322)	3.78%	96.22%	YES
Brocchieri (2005) - Archaea Median	247.0 aa	233.0	Fibonacci ($F_{13}$)	6.01%	93.99%	NO
Brocchieri (2005) - Eukaryota Median	361.0 aa	349.5	Midpoint ($L_{11}=322$ & $F_{14}=377$)	3.29%	96.71%	YES
Nevers et al. (2021/23) - Bacteria Median	270.0 aa	277.5	Midpoint (233 & 322)	2.70%	97.30%	YES
Nevers et al. (2021/23) - Archaea Median	242.0 aa	233.0	Fibonacci ($F_{13}$)	3.86%	96.14%	YES
Nevers et al. (2021/23) - Eukaryota Median	353.0 aa	349.5	Midpoint (322 & 377)	1.00%	99.00%	YES
Tiessen et al. (2012) - Microbes Median	274.0 aa	277.5	Midpoint (233 & 322)	1.26%	98.74%	YES
Tiessen et al. (2012) - Fungi Median	348.0 aa	349.5	Midpoint (322 & 377)	0.43%	99.57%	YES
Tiessen et al. (2012) - Plants Median	346.0 aa	349.5	Midpoint (322 & 377)	1.00%	99.00%	YES
Tiessen et al. (2012) - Animals Median	395.0 aa	377.0	Fibonacci ($F_{14}$)	4.77%	95.23%	YES
Xu et al. (2006) - Prokaryotes Average	276.0 aa	277.5	Midpoint (233 & 322)	0.54%	99.46%	YES
Xu et al. (2006) - Eukaryotes Average	372.0 aa	377.0	Fibonacci ($F_{14}$)	1.33%	98.67%	YES

Additional references:

Nevers, Y., Glover, N., Dessimoz, C., & Lecompte, O. (2021). Protein length distribution is remarkably consistent across Life. bioRxiv. https://doi.org/10.1101/2021.12.03.470944 Cited by: 9

Tiessen, A., Pérez-Rodríguez, P., & Delaye-Arredondo, L. J. (2012). Mathematical modeling and comparison of protein size distribution in different plant, animal, fungal and microbial species reveals a negative correlation between protein size and protein number, thus providing insight into the evolution of proteomes. BMC Research Notes, 5, Article 85. https://doi.org/10.1186/1756-0500-5-85 Cited by: 200

Xu, L., Chen, H., Hu, X., et al. (2006). Average Gene Length Is Highly Conserved in Prokaryotes and Eukaryotes and Diverges Only Between the Two Kingdoms. Molecular Biology and Evolution, 23(6), 1107–1108. https://doi.org/10.1093/molbev/msk019 Cited by: 194