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Parikh vectors
2GKW_1 9GQJ_1 5AMD_1 Letter Amino acid
15 7 19 D Aspartic acid
2 0 1 C Cysteine
9 9 9 I Isoleucine
20 9 26 L Leucine
11 6 12 T Threonine
8 8 13 A Alanine
10 3 7 R Arginine
6 4 10 N Asparagine
10 6 12 F Phenylalanine
14 8 18 S Serine
4 3 12 H Histidine
9 2 1 M Methionine
9 6 17 P Proline
9 4 8 Y Tyrosine
12 9 17 V Valine
8 2 11 Q Glutamine
6 10 13 E Glutamic acid
15 15 22 G Glycine
13 16 24 K Lycine
2 1 7 W Tryptophan 

2GKW_1|Chain A|TNF receptor-associated factor 3|Homo sapiens (9606)
>9GQJ_1|Chains A, B|Peptidyl-prolyl cis-trans isomerase FKBP5|Homo sapiens (9606)
>5AMD_1|Chain A|CARBONIC ANHYDRASE 2|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 \(w=f(c)\)
2GKW , Knot 92 192 0.84 40 141 186 
NTGLLESQLSRHDQMLSVHDIRLADMDLRFQVLETASYNGVLIWKIRDYKRRKQEAVMGKTLSLYSQPFYTGYFGYKMCARVYLNGDGMGKGTHLSLFFVIMRGEYDALLPWPFKQKVTLMLMDQGSSRRHLGDAFKPDPNSSSFKKPTGEMNIASGCPVFVAQTVLENGTYIKDDTIFIKVIVDTSDLPDP
9GQJ , Knot 64 128 0.80 38 103 125 
GAPATVTEQGEDITSKKDRGVLKIVKRVGNGEETPMIGDKVYVHYKGKLSNGKKFDSSHDRNEPFVFSLGKGQVIKAWDIGVATMKKGEIAHLLIKPEYAYGSAGSLPKIPSNATLFFEIELLDFKGE
5AMD , Knot 111 259 0.79 40 175 248 
SHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK

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 \(c\).

\(|P_{f(2GKW_1)}(2) \setminus P_{f(9GQJ_1)}(2)|=93\), \(|P_{f(9GQJ_1)}(2) \setminus P_{f(2GKW_1)}(2)|=55\). 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 1:001110001000001101001011010101011001000111110100000000011110010100011001011001010101010111010010111111010001111111000101111001000001101101010000100101010110101111100110010010000111011100001101
Pair \(Z_2\) Length of longest common subsequence
2GKW_1,9GQJ_1 148 3
2GKW_1,5AMD_1 176 3
9GQJ_1,5AMD_1 168 3

Newick tree

 
[
	5AMD_1:89.67,
	[
		2GKW_1:74,9GQJ_1:74
	]:15.67
]

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 \((0.85)(0.8)(\frac{320 }{\log_{20} 320}-\frac{128}{\log_{20}128})=59.2\)
Status Protein1 Protein2 d d1/2
Query variables 2GKW_1 9GQJ_1 75 61.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]