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Parikh vectors
4XMD_1 1KFB_1 9LVY_1 Letter Amino acid
13 18 1 V Valine
12 6 1 T Threonine
4 10 2 Q Glutamine
10 20 1 G Glycine
3 5 0 H Histidine
7 18 2 I Isoleucine
27 8 0 K Lycine
12 40 0 A Alanine
13 13 0 D Aspartic acid
4 3 4 C Cysteine
14 28 2 L Leucine
10 12 0 F Phenylalanine
7 17 0 P Proline
13 16 2 S Serine
11 11 2 N Asparagine
10 16 2 E Glutamic acid
1 5 0 M Methionine
2 0 0 W Tryptophan
9 7 2 Y Tyrosine
2 15 0 R Arginine 

4XMD_1|Chain A|Nitrophorin-7|Rhodnius prolixus (13249)
>1KFB_1|Chain A|TRYPTOPHAN SYNTHASE ALPHA CHAIN|Salmonella typhimurium (90371)
>9LVY_1|Chains A, C|Insulin A chain|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 \(w=f(c)\)
4XMD , Knot 88 184 0.83 40 128 180 
PGECSVNVIPKKNLDKAKFFSGTWYETHYLDMDPQATEKFCFSFAPRESGGTVKEALYHFNVDSKVSFYNTGTGPLESNGAKYTAKFNTVDKKGKEIKPADEKYSYTVTVIEAAKQSALIHICLQEDGKDIGDLYSVLNRNKNALPNKKIKKALNKVSLVLTKFVVTKDLDCKYDDKFLSSWQK
1KFB , Knot 116 268 0.80 38 165 254 
MERYENLFAQLNDRREGAFVPFVTLGDPGIEQSLKIIDTLIDAGADALELGVPFSDPLADGPTIQNANLRAFAAGVTPAQCFEMLALIREKHPTIPIGLLMYANLVFNNGIDAFYARCEQVGVDSVLVADVPVEESAPFRQAALRHNIAPIFICPPNADDDLLRQVASYGRGYTYLLSRSGVVGAENRGALPLHHLIEKLKEYHAAPALQGFGISSPEQVSAAVRAGAAGAISGSAIVKIIEKNLASPKQMLAELRSFVSAMKAASRA
9LVY , Knot 15 21 0.72 22 20 19 
GIVEQCCTSICSLYQLENYCN

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 \(c\).

\(|P_{f(4XMD_1)}(2) \setminus P_{f(1KFB_1)}(2)|=63\), \(|P_{f(1KFB_1)}(2) \setminus P_{f(4XMD_1)}(2)|=100\). 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 1:1100010111000100101101010000010101010001010111000110100110010100010100010111000110001010010001001011000000010110110001110101000100110100110000011100010011001011100111000100000001100100
Pair \(Z_2\) Length of longest common subsequence
4XMD_1,1KFB_1 163 3
4XMD_1,9LVY_1 138 2
1KFB_1,9LVY_1 165 2

Newick tree

 
[
	1KFB_1:85.89,
	[
		4XMD_1:69,9LVY_1:69
	]:16.89
]

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 \((0.85)(0.8)(\frac{452 }{\log_{20} 452}-\frac{184}{\log_{20}184})=78.7\)
Status Protein1 Protein2 d d1/2
Query variables 4XMD_1 1KFB_1 100 84.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]