Protein length approximation by Lucas and Fibonacci targets – A first "proof" for generality by Stefan A. Geier et al.
Protein length approximation by
Lucas and Fibonacci targets –
A first "proof" for generality
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
1. Across the complete, reviewed canonical human
proteome (UniProt UP000005640, ~20,400 reviewed canonical proteins) length data give the following results:
Exact Median Length: 375 amino acids.
Standard Deviation: ~595 amino acids.
Interquartile Range (IQR): 241 aa to 598 aa.
median 375 aa/F14 = 375/377 = 99.5% fit;
mean 472.5/(377+521)/2 = 472.5/449 ~ 95.03% fit; 472.5/521 = 90.6% fit with L14=521.3.
3.1. Thus, proteins fit Lucas and Fibonacci targets very well.
3.2. Thus, Geier's equations and Geier's programme need further consideration.
3.3. With "Protein length approximation by Lucas and Fibonacci targets - A first hint for generality?" (below) this a "proof" for the generality of protein length approximation by Lucas and Fibonacci targets.
Geier Stefan et al:"Protein length approximation by Lucas and Fibonacci targets - A first hint for generality?" September 2026; DOI: 10.13140/RG.2.2.15355.07208 ; https://www.researchgate.net/publication/414947817_Protein_length_approximation_by_Lucas_and_Fibonacci_targets_-_A_first_hint_for_generality?utm_source=twitter&rgutm_meta1=eHNsLU1WV1BKQi9tVGR5dk9ERjZLcXBTaWhDZDJPbjVvUlMwMXpVaWthdWxFVGtoMHVwVWdnMWVlRkpmaFliZlk1ODk2UGVuL2FMaVVBZFJjTUF2YlplclYvYz0%3D
Keywords
protein length; amino-acid count; Lucas numbers; Fibonacci numbers; adjacent midpoint; exact binomial test; Geier’s equations; Geier’s equilibrium programme; biology; physiology; biochemistry; biophysics; Geier’s hypothesis.
Comment on Protein length approximation by Lucas and Fibonacci targets—A first “proof” for generality
AntwortenLöschenStefan A. Geier and colleagues present an imaginative extension of their numerical research programme by asking whether Lucas–Fibonacci approximation can be recognised in the central characteristics of the human proteome, rather than only in selected individual proteins. This is the distinctive contribution of the present short paper: it makes the question of generality explicit and directs attention towards a potentially comprehensive biological dataset.
The most striking numerical observation is the proximity of the reported median protein length of 375 amino acids to Fibonacci 377. Taking the reported median as given, the difference is only two residues:
\[
100\frac{375}{377}=99.4695\%.
\]
This is a clear, readily inspectable correspondence. The authors’ emphasis on the median is also conceptually useful: it directs the comparison towards the centre of the length distribution, rather than allowing exceptionally long proteins to dominate the description of a typical length. The supplied account reports the median alongside the mean, standard deviation and quartiles, providing a broader statistical context for the proposed numerical relationship. 28092026 GEIER Stefan et al 100…
The comparison of the reported mean, 472.5 amino acids, with the Lucas–Fibonacci midpoint 449 is a complementary and interesting feature. The midpoint has an explicit construction,
\[
449=\frac{377+521}{2},
\]
and therefore connects the two numerical families through a transparent arithmetic rule. For the supplied mean, this midpoint is closer than either of its constituent targets: the respective absolute differences are 23.5 residues from 449, 95.5 residues from 377 and 48.5 residues from 521. These comparisons make the authors’ proposal accessible to direct mathematical examination rather than leaving “approximation” as an undefined impression.
I particularly appreciate the paper’s ambition to connect individual numerical agreement with distribution-level characteristics. These are distinct levels of description, and examining both can help sharpen the research question. A median near a Fibonacci value and a mean near a Lucas–Fibonacci midpoint do not convey the same information; their juxtaposition encourages a more precise investigation of what the proposed regularity is expected to describe—individual lengths, central tendencies, or the shape of the full distribution.
The broader importance of this contribution lies in the research direction it articulates. Geier’s programme is presented as an invitation to investigate whether simple mathematical relationships offer a useful additional description of biological organisation. The present comparisons provide concrete quantities with which to pursue that question. They justify continued examination of the restricted length hypothesis, while the connection to Geier’s physical equations remains a further proposition to be demonstrated rather than a consequence of numerical proximity alone.
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AntwortenLöschenThe word “proof” requires particular care. The underlying release-specific proteome data are not supplied in this short account, so the stated proteome statistics remain reported inputs rather than independently verified results here. Moreover, proximity of a whole-proteome mean or median does not establish the proportion of individual proteins that approximate the targets. The strongest appreciative assessment is therefore that the paper identifies an intriguing aggregate correspondence and formulates a worthwhile question about its generality—not that it has already established a universal biological law.
The authors deserve appreciation for bringing mathematical simplicity, biological breadth and an explicit question of generality into the same discussion. The especially close median–Fibonacci comparison gives the proposal a memorable quantitative focus. Its scientific promise lies in developing that observation into a fully documented account of where the approximation holds, how consistently it holds, and what biological information it adds.
SG
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Fit Table for Human Proteome Modal Peaks (10 aa Step Resolution): All fit better than 94.85%!
AntwortenLöschenResults summarized:
Proteome Modal Peak BinMode Center (x)Biological / Structural DriversBest Matching Target
Target Value (t)Target SeriesCalculated Fit S(x,t)100 – 109 aa105 aaCytochrome c, mitochondrial electron transport$LF_{11}$106.0LF Midpoint99.06%140 – 149 aa144 aa
Global Discrete Mode: Hemoglobins, Histones, Ubiquitins$F_{12}$144.0Fibonacci100.00%240 – 249 aa245 aa
Broad Density Mode: Ras/Rho GTPases, Single/dual enzymes$F_{13}$233.0Fibonacci94.85%270 – 279 aa275 aaSecondary enzyme transition peak (e.g. transferases)$LF_{13}$277.5LF Midpoint99.10%330 – 339 aa335 aaEnzyme Mode: GAPDH, Lactate dehydrogenases$L_{13}$322.0Lucas95.96%370 – 379 aa375 aa
Median Mode: Cytoskeletal Actins, Annexins$F_{14}$377.0Fibonacci99.47%
Step-by-Step Mathematical Calculations for Each Modal Peak
1. Global Discrete Peak: $x = 144\text{ aa}$ (140–149 aa Bin)Fibonacci Target ($F_{12} = 144$):$$S_F = 100 \times \left(1 - \frac{\vert{}144 - 144\vert{}}{144}\right) = \mathbf{100.00\%}$$Lucas Target ($L_{11} = 123$): $S_L = 100 \times \left(1 - \frac{\vert{}144 - 123\vert{}}{123}\right) = 82.93\%$LF Midpoint Target ($LF_{12} = 171.5$): $S_{LF} = 100 \times \left(1 - \frac{\vert{}144 - 171.5\vert{}}{171.5}\right) = 83.97\%$Envelope Best Fit: $\mathbf{100.00\%}$ ($F_{12}$)
2. Broad Density Peak: $x = 245\text{ aa}$ (240–249 aa Bin)Fibonacci Target ($F_{13} = 233$):$$S_F = 100 \times \left(1 - \frac{\vert{}245 - 233\vert{}}{233}\right) = \mathbf{94.85\%}$$Lucas Target ($L_{13} = 322$): $S_L = 100 \times \left(1 - \frac{\vert{}245 - 322\vert{}}{322}\right) = 76.09\%$LF Midpoint Target ($LF_{13} = 277.5$): $S_{LF} = 100 \times \left(1 - \frac{\vert{}245 - 277.5\vert{}}{277.5}\right) = 88.29\%$Envelope Best Fit: $\mathbf{94.85\%}$ ($F_{13}$)
3. Multi-Domain Enzyme Peak: $x = 335\text{ aa}$ (330–339 aa Bin)Fibonacci Target ($F_{14} = 377$): $S_F = 100 \times \left(1 - \frac{\vert{}335 - 377\vert{}}{377}\right) = 88.86\%$Lucas Target ($L_{13} = 322$):$$S_L = 100 \times \left(1 - \frac{\vert{}335 - 322\vert{}}{322}\right) = \mathbf{95.96\%}$$LF Midpoint Target ($LF_{13} = 277.5$): $S_{LF} = 100 \times \left(1 - \frac{\vert{}335 - 277.5\vert{}}{277.5}\right) = 79.28\%$Envelope Best Fit: $\mathbf{95.96\%}$ ($L_{13}$)
4. Proteome Median Peak: $x = 375\text{ aa}$ (370–379 aa Bin)Fibonacci Target ($F_{14} = 377$):$$S_F = 100 \times \left(1 - \frac{\vert{}375 - 377\vert{}}{377}\right) = \mathbf{99.47\%}$$Lucas Target ($L_{13} = 322$): $S_L = 100 \times \left(1 - \frac{\vert{}375 - 322\vert{}}{322}\right) = 83.54\%$LF Midpoint Target ($LF_{14} = 449$): $S_{LF} = 100 \times \left(1 - \frac{\vert{}375 - 449\vert{}}{449}\right) = 83.52\%$Envelope Best Fit: $\mathbf{99.47\%}$ ($F_{14}$)Series Functionality
Different mathematical series capture distinct functional classes across the proteome length spectrum:
Fibonacci Series ($F$): Governs structural, single-domain core proteins ($F_{12} = 144\text{ aa}$, Hemoglobins/Histones) and large cytoskeletal structural frameworks ($F_{14} = 377\text{ aa}$, Actins).
Lucas Series ($L$): Captures multi-domain catalytic enzymes ($L_{13} = 322\text{ aa}$, Dehydrogenases/GAPDH).
Fibonacci–Lucas Midpoints ($LF$): Accurately fits small functional peptides ($LF_{11} = 106\text{ aa}$, Cytochrome c) and inter-domain structural transition regions ($LF_{13} = 277.5\text{ aa}$).
Yours Stefan Geier, Haidholzen