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Parikh vectors
4HCW_1 2GCT_1 1POZ_1 Letter Amino acid
21 0 11 D Aspartic acid
23 1 4 Q Glutamine
19 2 9 G Glycine
13 0 7 F Phenylalanine
14 2 10 P Proline
19 0 8 Y Tyrosine
19 1 12 A Alanine
6 1 6 C Cysteine
43 2 6 L Leucine
7 0 8 R Arginine
16 1 13 I Isoleucine
20 0 4 K Lycine
2 0 2 M Methionine
44 1 10 S Serine
17 0 16 T Threonine
1 0 0 W Tryptophan
21 0 12 N Asparagine
3 0 3 H Histidine
34 2 9 V Valine
15 0 9 E Glutamic acid 

4HCW_1|Chains A, B, C, D, E, F|thiaminase-I|Naegleria gruberi (5762)
>2GCT_1|Chain A|GAMMA-CHYMOTRYPSIN A|Bos taurus (9913)
>1POZ_1|Chain A|CD44 antigen|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 \(w=f(c)\)
4HCW , Knot 151 357 0.82 40 194 335 
HMSTQPKTLTVGLFPYLPSWNENGNEVKLINLIKDVLPTQVSGYNIEYTEFDCYSDASLQSLPDVFSTDSIFLPYLVSLGGVKSLDESLVRGVTGDLHSFVSSSASVNGSVYGFPQYLCSNFLLSSPNATQQASSLLELAQKVGYEQIVYPDVASSSSFTVFGLYQQLLQSSSSAAVDIKASDLPQSGDQVNKDITQKYRTILDSTVVASQREYINSVKQGKPISNYYVGYSESMCEIKDIIRDQQYNVQLIGTSDKPYVYTDVLALNSNLCDEKQKVAVEVIKNLLTNTLVLDLLGLGLTLPANKNGIAHLAKSSNFYAQLSQQFDAKESEVRVLRCVDFANKEVKNCAGVLRPFL
2GCT , Knot 11 13 0.72 18 12 11 
CGVPAIQPVLSGL
1POZ , Knot 76 159 0.80 38 122 155 
AQIDLNITCRFAGVFHVEKNGRYSISRTEAADLCKAFNSTLPTMAQMEKALSIGFETCRYGFIEGHVVIPRIHPNSICAANNTGVYILTSNTSQYDTYCFNASAPPEEDCTSVTDLPNAFDGPITITIVNRDGTRYVQKGEYRTNPEDIYPSNPTDDDV

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 \(c\).

\(|P_{f(4HCW_1)}(2) \setminus P_{f(2GCT_1)}(2)|=187\), \(|P_{f(2GCT_1)}(2) \setminus P_{f(4HCW_1)}(2)|=5\). 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 1:010001001011111011010001001011011001110010100100001000001010011011000011110110111100100011011010100110001010101011100100011100101000100110110011000110101100001011110001100000111010100110010010001000000110001110000010010010110000110000100100110000001011100001010001111000100000011101100110001110111111011100011101100001010100010100001011001011000100011110111
Pair \(Z_2\) Length of longest common subsequence
4HCW_1,2GCT_1 192 2
4HCW_1,1POZ_1 172 3
2GCT_1,1POZ_1 130 2

Newick tree

 
[
	4HCW_1:98.31,
	[
		1POZ_1:65,2GCT_1:65
	]:33.31
]

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 \((0.85)(0.8)(\frac{370 }{\log_{20} 370}-\frac{13}{\log_{20}13})=117.\)
Status Protein1 Protein2 d d1/2
Query variables 4HCW_1 2GCT_1 143 74
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]