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Parikh vectors
8YSZ_1 6ZMT_1 2ENT_1 Letter Amino acid
10 24 2 T Threonine
20 15 5 R Arginine
7 15 1 D Aspartic acid
19 21 1 L Leucine
7 11 2 K Lycine
10 10 2 F Phenylalanine
12 14 12 S Serine
10 25 2 E Glutamic acid
3 4 2 H Histidine
4 15 0 I Isoleucine
5 7 0 Y Tyrosine
17 37 1 A Alanine
4 7 0 M Methionine
1 10 2 W Tryptophan
2 9 0 N Asparagine
8 2 2 C Cysteine
5 15 0 Q Glutamine
12 15 9 G Glycine
15 20 4 P Proline
9 19 1 V Valine 

8YSZ_1|Chains A[auth C], D, E|Insulin-like growth factor II|Homo sapiens (9606)
>6ZMT_1|Chain A[auth B]|40S ribosomal protein SA|Homo sapiens (9606)
>2ENT_1|Chain A|Krueppel-like factor 15|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 \(w=f(c)\)
8YSZ , Knot 83 180 0.79 40 125 176 
MGIPMGKSMLVLLTFLAFASCCIAAYRPSETLCGGELVDTLQFVCGDRGFYFSRPASRVSRRSRGIVEECCFRSCDLALLETYCATPAKSERDVSTPPTVLPDNFPRYPVGKFFQYDTWKQSTQRLRRGLPALLRARRGHVLAKELEAFREAKRHRPLIALPTQDPAHGGAPPEMASNRK
6ZMT , Knot 127 295 0.81 40 176 277 
MSGALDVLQMKEEDVLKFLAAGTHLGGTNLDFQMEQYIYKRKSDGIYIINLKRTWEKLLLAARAIVAIENPADVSVISSRNTGQRAVLKFAAATGATPIAGRFTPGTFTNQIQAAFREPRLLVVTDPRADHQPLTEASYVNLPTIALCNTDSPLRYVDIAIPCNNKGAHSVGLMWWMLAREVLRMRGTISREHPWEVMPDLYFYRDPEEIEKEEQAAAEKAVTKEEFQGEWTAPAPEFTATQPEVADWSEGVQVPSVPIQQFPTEDWSAQPATEDWSAAPTAQATEWVGATTDWS
2ENT , Knot 25 48 0.67 30 36 43 
GSSGSSGTGEKPFACTWPGCGWRFSRSDELSRHRRSHSGVKPSGPSSG

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 \(c\).

\(|P_{f(8YSZ_1)}(2) \setminus P_{f(6ZMT_1)}(2)|=56\), \(|P_{f(6ZMT_1)}(2) \setminus P_{f(8YSZ_1)}(2)|=107\). 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 1:111111001111101111100011100100010110110010110100110100110010000011100001000011110000101100000100110111001100111011000010000001001111110100101110010110010000111111000110111110110000
Pair \(Z_2\) Length of longest common subsequence
8YSZ_1,6ZMT_1 163 4
8YSZ_1,2ENT_1 137 3
6ZMT_1,2ENT_1 182 2

Newick tree

 
[
	6ZMT_1:91.56,
	[
		8YSZ_1:68.5,2ENT_1:68.5
	]:23.06
]

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 \((0.85)(0.8)(\frac{475 }{\log_{20} 475}-\frac{180}{\log_{20}180})=86.3\)
Status Protein1 Protein2 d d1/2
Query variables 8YSZ_1 6ZMT_1 108 86.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]