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Parikh vectors
8TLY_1 7GOL_1 8PBW_1 Letter Amino acid
22 6 21 E Glutamic acid
7 7 3 Y Tyrosine
9 12 12 A Alanine
11 5 7 Q Glutamine
12 12 14 D Aspartic acid
12 13 6 I Isoleucine
24 13 14 L Leucine
10 6 6 M Methionine
17 10 10 F Phenylalanine
5 7 6 P Proline
7 17 7 T Threonine
10 10 5 R Arginine
18 18 12 G Glycine
13 6 6 H Histidine
14 7 8 K Lycine
20 4 11 S Serine
1 0 0 W Tryptophan
16 16 9 V Valine
13 10 10 N Asparagine
4 3 2 C Cysteine 

8TLY_1|Chains A, B|Activating signal cointegrator 1 complex subunit 1|Homo sapiens (9606)
>7GOL_1|Chains A, B|Protease 3C|enterovirus D68 (42789)
>8PBW_1|Chains A, B, C, D|HPt domain-containing protein|Thermochaetoides thermophila (209285)
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 \(w=f(c)\)
8TLY , Knot 109 245 0.81 40 167 233 
MGSSHHHHHHSSGLVPRGSHMTHFLAFFLNEVEVQEGFLRFQEEVLAKCSMDHGVDSSIFQNPKKLHLTIGMLVLLSEEEIQQTCEMLQQCKEEFINDISGGKPLEVEMAGIEYMNDDPGMVDVLYAKVHMKDGSNRLQELVDRVLERFQASGLIVKEWNSVKLHATVMNTLFRKDPNAEGRYNLYTAEGKYIFKERESFDGRNILKLFENFYFGSLKLNSIHISQRFTVDSFGNYASCGQIDFS
7GOL , Knot 86 182 0.82 38 139 179 
MGPGFDFAQAIMKKNTVIARTEKGEFTMLGVYDRVAVIPTHASVGEIIYINDVETRVLDACALRDLTDTNLEITIVKLDRNQKFRDIRHFLPRCEDDYNDAVLSVHTSKFPNMYIPVGQVTNYGFLNLGGTPTHRILMYNFPTRAGQCGGVVTTTGKVIGIHVGGNGAQGFAAMLLHSYFTD
8PBW , Knot 78 169 0.79 38 127 158 
GPGADRAQDHNNNNTPRNSNYVVEEEEVSEEEEEAIMMPDFGDHVDTSIFGQILEMDEGDDHDFSAPLVLNFFEQAEETFQKMETALNNKDLPELSKLGHFLKGSSATLGFTKIRDSCQLIQQYGHGLNVDGSSEPDEGVCLKKIAEALASARVDTVALHKMMREFFEY

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 \(c\).

\(|P_{f(8TLY_1)}(2) \setminus P_{f(7GOL_1)}(2)|=101\), \(|P_{f(7GOL_1)}(2) \setminus P_{f(8TLY_1)}(2)|=73\). 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 1:11000000000011110100100111111001010011101000111000100110001100100101011111110000100000110000001100101101101011110010001111011010101001000100110011001010111100100101010110011000101010001001010011000001010011011001011010100101000101001100100101010
Pair \(Z_2\) Length of longest common subsequence
8TLY_1,7GOL_1 174 3
8TLY_1,8PBW_1 156 4
7GOL_1,8PBW_1 158 3

Newick tree

 
[
	7GOL_1:84.72,
	[
		8TLY_1:78,8PBW_1:78
	]:6.72
]

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 \((0.85)(0.8)(\frac{427 }{\log_{20} 427}-\frac{182}{\log_{20}182})=72.3\)
Status Protein1 Protein2 d d1/2
Query variables 8TLY_1 7GOL_1 90 78.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]