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Parikh vectors
4IBM_1 5ZPB_1 9GQE_1 Letter Amino acid
10 34 9 I Isoleucine
15 30 6 F Phenylalanine
9 19 4 Y Tyrosine
9 15 2 Q Glutamine
26 46 10 E Glutamic acid
7 14 3 H Histidine
28 45 9 L Leucine
15 13 16 K Lycine
17 51 8 A Alanine
18 48 3 R Arginine
17 43 7 D Aspartic acid
18 36 8 S Serine
14 37 6 T Threonine
27 53 9 V Valine
5 3 0 C Cysteine
17 37 6 P Proline
5 12 1 W Tryptophan
13 23 4 N Asparagine
21 55 15 G Glycine
15 6 2 M Methionine 

4IBM_1|Chains A, B|Insulin receptor|Homo sapiens (9606)
>5ZPB_1|Chains A, B|Phenylethylamine oxidase|Arthrobacter globiformis (1665)
>9GQE_1|Chains A, B|Peptidyl-prolyl cis-trans isomerase FKBP5|Homo sapiens (9606)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
4IBM , Knot 135 306 0.84 40 186 293 
VFPCSVYVPDEWEVSREKITLLRELGQGSFGMVYEGNARDIIKGEAETRVAVKTVNESASLRERIEFLNEASVMKGFTCHHVVRLLGVVSKGQPTLVVMELMAHGDLKSYLRSLRPEAENNPGRPPPTLQEMIQMAAEIADGMAYLNAKKFVHRDLAARNCMVAHDFTVKIGDFGMTRDIYETDYYRKGGKGLLPVRWMAPESLKDGVFTTSSDMWSFGVVLWEITSLAEQPYQGLSNEQVLKFVMDGGYLDQPDNCPERVTDLMRMCWQFNPNMRPTFLEIVNLLKDDLHPSFPEVSFFHSEENK
5ZPB , Knot 240 620 0.83 40 262 572 
ASPFRLASAGEISEVQGILRTAGLLGPEKRIAYLGVLDPARGAGSEAEDRRFRVFIHDVSGARPQEVTVSVTNGTVISAVELDTAATGELPVLEEEFEVVEQLLATDERWLKALAARNLDVSKVRVAPLSAGVFEYAEERGRRILRGLAFVQDFPEDSAWAHPVDGLVAYVDVVSKEVTRVIDTGVFPVPAEHGNYTDPELTGPLRTTQKPISITQPEGPSFTVTGGNHIEWEKWSLDVGFDVREGVVLHNIAFRDGDRLRPIINRASIAEMVVPYGDPSPIRSWQNYFDTGEYLVGQYANSLELGCDCLGDITYLSPVISDAFGNPREIRNGICMHEEDWGILAKHSDLWSGINYTRRNRRMVISFFTTIGNYDYGFYWYLYLDGTIEFEAKATGVVFTSAFPEGGSDNISQLAPGLGAPFHQHIFSARLDMAIDGFTNRVEEEDVVRQTMGPGNERGNAFSRKRTVLTRESEAVREADARTGRTWIISNPESKNRLNEPVGYKLHAHNQPTLLADPGSSIARRAAFATKDLWVTRYADDERYPTGDFVNQHSGGAGLPSYIAQDRDIDGQDIVVWHTFGLTHFPRVEDWPIMPVDTVGFKLRPEGFFDRSPVLDVPAN
9GQE , Knot 64 128 0.80 38 103 125 
GAPATVTEQGEDITSKKDRGVLKIVKRVGNGEETPMIGDKVYVHYKGKLSNGKKFDSSHDRNEPFVFSLGKGQVIKAWDIGVATMKKGEIAHLLIKPEYAYGSAGSLPKIPSNATLFFEIELLDFKGE

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(4IBM_1)}(2) \setminus P_{f(5ZPB_1)}(2)|=54\), \(|P_{f(5ZPB_1)}(2) \setminus P_{f(4IBM_1)}(2)|=130\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:111001011001010000101100110101111001010011010100011100100010100010110010110110000110111110010101111011101010001001010100011011101001101110110111010100110001110001110010101101110001000000001101111101111001001110000011011111101001100100110000110111011010010001001001101010101010101101101100010101101011000000
Pair \(Z_2\) Length of longest common subsequence
4IBM_1,5ZPB_1 184 5
4IBM_1,9GQE_1 163 3
5ZPB_1,9GQE_1 209 3

Newick tree

 
[
	5ZPB_1:10.48,
	[
		4IBM_1:81.5,9GQE_1:81.5
	]:21.98
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{926 }{\log_{20} 926}-\frac{306}{\log_{20}306})=167.\)
Status Protein1 Protein2 d d1/2
Query variables 4IBM_1 5ZPB_1 213 156.5
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]