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Parikh vectors
4MVJ_1 2VHF_1 2VNS_1 Letter Amino acid
28 14 7 T Threonine
35 17 22 V Valine
26 22 10 D Aspartic acid
3 18 3 C Cysteine
6 16 8 Q Glutamine
32 19 12 G Glycine
27 29 9 K Lycine
16 26 15 E Glutamic acid
22 41 21 L Leucine
9 11 7 M Methionine
18 22 22 S Serine
36 15 23 A Alanine
12 17 11 R Arginine
20 14 5 N Asparagine
3 4 3 W Tryptophan
12 9 4 H Histidine
20 24 6 I Isoleucine
12 20 9 F Phenylalanine
9 19 16 P Proline
9 17 2 Y Tyrosine 

4MVJ_1|Chains A, B, C, K, N|Glyceraldehyde-3-phosphate dehydrogenase A|Escherichia coli (536056)
>2VHF_1|Chains A, B|UBIQUITIN CARBOXYL-TERMINAL HYDROLASE CYLD|HOMO SAPIENS (9606)
>2VNS_1|Chains A, B|METALLOREDUCTASE STEAP3|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 \(w=f(c)\)
4MVJ , Knot 153 355 0.84 40 201 335 
MHHHHHHSSGVDLGTENLYFQSNAMTIKVGINGFGRIGRIVFRAAQKRSDIEIVAINDLLDADYMAYMLKYDSTHGRFDGTVEVKDGHLIVNGKKIRVTAERDPANLKWDEVGVDVVAEATGLFLTDETARKHITAGAKKVVMTGPSKDNTPMFVKGANFDKYAGQDIVSNASCTTNCLAPLAKVINDNFGIIEGLMTTVHATTATQKTVDGPSHKDWRGGRGASQNIIPSSTGAAKAVGKVLPELNGKLTGMAFRVPTPNVSVVDLTVRLEKAATYEQIKAAVKAAAEGEMKGVLGYTEDDVVSTDFNGEVCTSVFDAKAGIALNDNFVKLVSWYDNETGYSNKVLDLIAHISK
2VHF , Knot 166 374 0.87 40 236 359 
GLEIMIGKKKGIQGHYNSCYLDSTLFCLFAFSSVLDTVLLRPKEKNDVEYYSETQELLRTEIVNPLRIYGYVCATKIMKLRKILEKVEAASGFTSEEKDPEEFLNILFHHILRVEPLLKIRSAGQKVQDCYFYQIFMEKNEKVGVPTIQQLLEWSFINSNLKFAEAPSCLIIQMPRFGKDFKLFKKIFPSLELNITDLLEDTPRQCRICGGLAMYECRECYDDPDISAGKIKQFCKTCNTQVHLHPKRLNHKYNPVSLPKDLPDWDWRHGCIPCQNMELFAVLCIETSHYVAFVKYGKDDSAWLFFDSMADRDGGQNGFNIPQVTPCPEVGEYLKMSLEDLHSLDSRRIQGCARRLLCDAYMCMYQSPTMSLYK
2VNS , Knot 97 215 0.80 40 136 202 
MPEEMDKPLISLHLVDSDSSLAKVPDEAPKVGILGSGDFARSLATRLVGSGFKVVVGSRNPKRTARLFPSAAQVTFQEEAVSSPEVIFVAVFREHYSSLCSLSDQLAGKILVDVSNPTEQEHLQHRESNAEYLASLFPTCTVVKAFNVISAWTLQAGPRDGNRQVPICGDQPEAKRAVSEMALAMGFMPVDMGSLASAWEVEAMPLRLLPAWKVP

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 \(c\).

\(|P_{f(4MVJ_1)}(2) \setminus P_{f(2VHF_1)}(2)|=65\), \(|P_{f(2VHF_1)}(2) \setminus P_{f(4MVJ_1)}(2)|=100\). 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 1:1000000001101100010100011010111011101101110110000010111100110100110110000001010101010010111010010101000110101001110111010111100001000101110011101100000111101101000110011001000000111110110001111011100101001000010110000101101100011100011101110111010101011110110101011010101001100001011101110101011110000011000101010001101011111000110110100000100001101110100
Pair \(Z_2\) Length of longest common subsequence
4MVJ_1,2VHF_1 165 3
4MVJ_1,2VNS_1 185 4
2VHF_1,2VNS_1 198 4

Newick tree

 
[
	2VNS_1:99.84,
	[
		4MVJ_1:82.5,2VHF_1:82.5
	]:17.34
]

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 \((0.85)(0.8)(\frac{729 }{\log_{20} 729}-\frac{355}{\log_{20}355})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 4MVJ_1 2VHF_1 133 127
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]