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Parikh vectors
2FYC_1 1HTB_1 7FVY_1 Letter Amino acid
2 7 2 H Histidine
8 16 2 C Cysteine
4 6 2 Q Glutamine
4 2 2 W Tryptophan
3 12 4 N Asparagine
10 22 8 I Isoleucine
10 19 9 E Glutamic acid
7 38 11 G Glycine
10 29 5 L Leucine
12 32 14 K Lycine
1 7 5 M Methionine
5 24 12 T Threonine
7 31 6 A Alanine
2 10 6 R Arginine
5 6 2 Y Tyrosine
3 20 1 P Proline
10 21 10 S Serine
4 39 17 V Valine
12 17 11 D Aspartic acid
4 16 6 F Phenylalanine 

2FYC_1|Chains A, C|Alpha-lactalbumin|Mus musculus (10090)
>1HTB_1|Chains A, B|BETA3 ALCOHOL DEHYDROGENASE|Homo sapiens (9606)
>7FVY_1|Chain A|Fatty acid-binding protein, adipocyte|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 \(w=f(c)\)
2FYC , Knot 63 123 0.82 40 98 118 
TELTKCKVSHAIKDIDGYQGISLLEWACVLFHTSGYDTQAVVNDNGSTEYGLFQISDRFWCKSSEFPESENICGISCDKLLDDELDDDIACAKKILAIKGIDYWKAYKPMCSEKLEQWRCEKP
1HTB , Knot 159 374 0.84 40 204 359 
STAGKVIKCKAAVLWEVKKPFSIEDVEVAPPKAYEVRIKMVAVGICRTDDHVVSGNLVTPLPVILGHEAAGIVESVGEGVTTVKPGDKVIPLFTPQCGKCRVCKNPESNYCLKNDLGNPRGTLQDGTRRFTCRGKPIHHFLGTSTFSQYTVVDENAVAKIDAASPLEKVCLIGCGFSTGYGSAVNVAKVTPGSTCAVFGLGGVGLSAVMGCKAAGAARIIAVDINKDKFAKAKELGATECINPQDYKKPIQEVLKEMTDGGVDFSFEVIGRLDTMMASLLCCHEACGTSVIVGVPPASQNLSINPMLLLTGRTWKGAVYGGFKSKEGIPKLVADFMAKKFSLDALITHVLPFEKINEGFDLLHSGKSICTVLTF
7FVY , Knot 69 135 0.83 40 104 129 
GSHMCDAFVGTWKLVSSENFDDYMKEVGVGFATRKVAGMAKPNMIISVNGDVITIKSESTFKNTEISFILGQEFDEVTADDRKVKSTITLDGGVLVHVQKWDGKSTTIKRKREDDKLVVECVMKGVTSTRVYERA

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 \(c\).

\(|P_{f(2FYC_1)}(2) \setminus P_{f(1HTB_1)}(2)|=40\), \(|P_{f(1HTB_1)}(2) \setminus P_{f(2FYC_1)}(2)|=146\). 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 1:001000010011001010011011011011100010000111000100001110100011000001100001011000011000100011010011110110010100110000100100001
Pair \(Z_2\) Length of longest common subsequence
2FYC_1,1HTB_1 186 3
2FYC_1,7FVY_1 132 3
1HTB_1,7FVY_1 170 4

Newick tree

 
[
	1HTB_1:95.55,
	[
		2FYC_1:66,7FVY_1:66
	]:29.55
]

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 \((0.85)(0.8)(\frac{497 }{\log_{20} 497}-\frac{123}{\log_{20}123})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 2FYC_1 1HTB_1 141 93.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]