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Parikh vectors
5NAW_1 8TLZ_1 4QWU_1 Letter Amino acid
27 8 18 A Alanine
16 13 14 D Aspartic acid
8 4 0 C Cysteine
8 11 6 Q Glutamine
22 10 23 G Glycine
22 9 16 V Valine
5 6 6 N Asparagine
6 3 15 I Isoleucine
6 8 18 K Lycine
8 13 19 T Threonine
9 14 17 E Glutamic acid
10 14 3 H Histidine
27 22 26 L Leucine
16 13 23 S Serine
18 16 9 R Arginine
2 5 4 M Methionine
0 5 9 F Phenylalanine
14 4 12 P Proline
4 5 2 W Tryptophan
4 5 10 Y Tyrosine 

5NAW_1|Chain A|Complement factor D|Homo sapiens (9606)
>8TLZ_1|Chains A, B|MHC class I polypeptide-related sequence A|Homo sapiens (9606)
>4QWU_1|Chains A, O|Proteasome subunit alpha type-2|Saccharomyces cerevisiae (559292)
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 \(w=f(c)\)
5NAW , Knot 103 232 0.80 38 145 218 
ILGGREAEAHARPYMASVQLNGAHLCGGVLVAEQWVLSAAHCLEDAADGKVQVLLGAHSLSQPEPSKRLYDVLRAVPHPDSQPDTIDHDLLLLQLSEKATLGPAVRPLPWQRVDRDVAPGTLCDVAGWGIVNHAGRRPDSLQHVLLPVLDRATCNRRTHHDGAITERLMCAESNRRDSCKGDSGGPLVCGGVLEGVVTSGSRVCGNRKKPGIYTRVASYAAWIDSVLASAAA
8TLZ , Knot 90 188 0.83 40 133 180 
EPHSLRYNLTVLSWDGSVQSGFLTEVHLDGQPFLRCDRQKCRAKPQGQWAEDVLGNKTWDRETRDLTGWGKDLRMTLAHIKDQKEGLHSLQEIRVCEIHEDNSTRSSQHFYYDGELFLSQNLETLEWTMPQSSRAQTLAMNVRNFLKEDAMETDTHYHAMRADCLSELRRYLKSGVVLRRTVHHHHHH
4QWU , Knot 111 250 0.81 38 159 239 
MTDRYSFSLTTFSPSGKLGQIDYALTAVKQGVTSLGIKATNGVVIATEKKSSSPLAMSETLSKVSLLTPDIGAVYSGMGPDYRVLVDKSRKVAHTSYKRIYGEYPPTKLLVSEVAKIMQEATQSGGVRPFGVSLLIAGHDEFNGFSLYQVDPSGSYFPWKATAIGKGSVAAKTFLEKRWNDELELEDAIHIALLTLKESVEGEFNGDTIELAIIGDENPDLLGYTGIPTDKGPRFRKLTSQEINDRLEAL

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 \(c\).

\(|P_{f(5NAW_1)}(2) \setminus P_{f(8TLZ_1)}(2)|=87\), \(|P_{f(8TLZ_1)}(2) \setminus P_{f(5NAW_1)}(2)|=75\). 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 1:1111001010101011010101101011111100111011001001101010111110010010100010011011101000100100011110100010111110111100100011110100111111100110010010011111100100000000011100011010000000001001111101111011100100101000011100011001111001110111
Pair \(Z_2\) Length of longest common subsequence
5NAW_1,8TLZ_1 162 3
5NAW_1,4QWU_1 168 4
8TLZ_1,4QWU_1 186 3

Newick tree

 
[
	4QWU_1:91.01,
	[
		5NAW_1:81,8TLZ_1:81
	]:10.01
]

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 \((0.85)(0.8)(\frac{420 }{\log_{20} 420}-\frac{188}{\log_{20}188})=68.5\)
Status Protein1 Protein2 d d1/2
Query variables 5NAW_1 8TLZ_1 84 76.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]