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Parikh vectors
7ZCM_1 8IHT_1 4YLY_1 Letter Amino acid
11 4 10 F Phenylalanine
6 0 0 W Tryptophan
1 2 10 H Histidine
32 12 14 L Leucine
1 8 6 Q Glutamine
7 7 16 E Glutamic acid
7 2 12 M Methionine
27 6 16 V Valine
11 18 11 R Arginine
0 0 1 C Cysteine
15 7 14 I Isoleucine
27 20 7 A Alanine
2 1 13 N Asparagine
2 13 14 K Lycine
9 6 5 P Proline
8 6 6 S Serine
15 10 8 T Threonine
11 3 6 Y Tyrosine
6 4 12 D Aspartic acid
27 6 17 G Glycine 

7ZCM_1|Chain A|Sensory rhodopsin-2|Natronomonas pharaonis (2257)
>8IHT_1|Chains A, E|Histone H3|Xenopus laevis (8355)
>4YLY_1|Chains A, B|Peptidyl-tRNA hydrolase|Staphylococcus aureus (1280)
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 \(w=f(c)\)
7ZCM , Knot 97 225 0.77 38 124 211 
MVGLTTLFWLGAIGMLVGTLAFAWAGRDAGSGERRYYVTLVGISGIAAVAYVVMALGVGWVPVAERTVFAPRYIDWILTTPLIVYFLGLLAGLDSREFGIVITLNTVVMLAGFAGAMVPGIERYALFGMGAVAFLGLVYYLVGPMTESASQRSSGIKSLYVRLRNLTVILWAIYPFIWLLGPPGVALLTPTVDVALIVYLDLVTKVGFGFIALDAAATLRAEHGE
8IHT , Knot 65 135 0.78 36 100 126 
ARTKQTARKSTGGKAPRKQLATKAARKSAPATGGVKKPHRYRPGTVALREIRRYQKSTELLIRKLPFQRLVREIAQDFKTDLRFQSSAVMALQEASEAYLVALFEDTNLAAIHAKRVTIMPKDIQLARRIRGERA
4YLY , Knot 93 198 0.82 38 148 191 
MKCIVGLGNIGKRFELTRHNIGFEVVDYILEKNNFSLDKQKFKGAYTIERMNGDKVLFIEPMTMMNLSGEAVAPIMDYYNVNPEDLIVLYDDLDLEQGQVRLRQKGSAGGHNGMKSIIKMLGTDQFKRIRIGVGRPTNGMTVPDYVLQRFSNDEMVTMEKVIEHAARAIEKFVETSRFDHVMNEFNGEVKLEHHHHHH

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 \(c\).

\(|P_{f(7ZCM_1)}(2) \setminus P_{f(8IHT_1)}(2)|=81\), \(|P_{f(8IHT_1)}(2) \setminus P_{f(7ZCM_1)}(2)|=57\). 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 1:111100111111111111011111110011010000010111101111110111111111111100011110010111001111011111111000011111010011111111111111100011111111111111001111100010000011001010100101111110111111111111110101011111010110011111111011101010010
Pair \(Z_2\) Length of longest common subsequence
7ZCM_1,8IHT_1 138 4
7ZCM_1,4YLY_1 152 4
8IHT_1,4YLY_1 154 3

Newick tree

 
[
	4YLY_1:78.84,
	[
		7ZCM_1:69,8IHT_1:69
	]:9.84
]

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 \((0.85)(0.8)(\frac{360 }{\log_{20} 360}-\frac{135}{\log_{20}135})=68.5\)
Status Protein1 Protein2 d d1/2
Query variables 7ZCM_1 8IHT_1 82 66
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]