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Parikh vectors
3FQN_1 2CFV_1 9FRK_1 Letter Amino acid
17 23 2 L Leucine
14 19 3 S Serine
23 19 0 T Threonine
24 14 2 A Alanine
23 15 5 R Arginine
17 17 4 Q Glutamine
13 13 0 H Histidine
2 16 0 N Asparagine
4 5 4 C Cysteine
14 19 6 Y Tyrosine
16 24 3 V Valine
22 19 3 E Glutamic acid
4 20 2 I Isoleucine
8 12 2 F Phenylalanine
10 4 1 W Tryptophan
11 14 2 P Proline
18 22 1 D Aspartic acid
20 14 6 G Glycine
10 18 7 K Lycine
5 9 1 M Methionine 

3FQN_1|Chain A|HLA class I histocompatibility antigen, A-2 alpha chain|Homo sapiens (9606)
>2CFV_1|Chain A|HUMAN PROTEIN TYROSINE PHOSPHATASE RECEPTOR TYPE J|HOMO SAPIENS (9606)
>9FRK_17|Chain Q[auth P]|Small ribosomal subunit protein uS14|Saccharolobus solfataricus P2 (273057)
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 \(w=f(c)\)
3FQN , Knot 122 275 0.83 40 182 263 
GSHSMRYFFTSVSRPGRGEPRFIAVGYVDDTQFVRFDSDAASQRMEPRAPWIEQEGPEYWDGETRKVKAHSQTHRVDLGTLRGYYNQSEAGSHTVQRMYGCDVGSDWRFLRGYHQYAYDGKDYIALKEDLRSWTAADMAAQTTKHKWEAAHVAEQLRAYLEGTCVEWLRRYLENGKETLQRTDAPKTHMTHHAVSDHEATLRCWALSFYPAEITLTWQRDGEDQTQDTELVETRPAGDGTFQKWAAVVVPSGQEQRYTCHVQHEGLPKPLTLRWE
2CFV , Knot 143 316 0.86 40 203 301 
MHHHHHHSSGVDLGTENLYFQSMKLIRVENFEAYFKKQQADSNCGFAEEYEDLKLVGISQPKYAAELAENRGKNRYNNVLPYDISRVKLSVQTHSTDDYINANYMPGYHSKKDFIATQGPLPNTLKDFWRMVWEKNVYAIIMLTKCVEQGRTKCEEYWPSKQAQDYGDITVAMTSEIVLPEWTIRDFTVKNIQTSESHPLRQFHFTSAPDHGVPDTTDLLINFRYLVRDYMKQSPPESPILVHCSAGVGRTGTFIAIDRLIYQIENENTVDVYGIVYDLRMHRPLMVQTEDQYVFLNQCVLDIVRSQKDSKVDLIY
9FRK , Knot 32 54 0.78 34 46 50 
MGKYKPPAERKYGKGVQSCQRCGSKDSVIQKYGIYLCRQCFREVAYELGFRKYW

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 \(c\).

\(|P_{f(3FQN_1)}(2) \setminus P_{f(2CFV_1)}(2)|=73\), \(|P_{f(2CFV_1)}(2) \setminus P_{f(3FQN_1)}(2)|=94\). 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 1:10001001100100110101011111010000110100011000101011110001100101000010100000010110101000000110001001010011001011010000100100011100010010110111000000101101100101010100101100010010001000011000100011000010100111010110101010001000000001100011101010011111110100000000100011101101010
Pair \(Z_2\) Length of longest common subsequence
3FQN_1,2CFV_1 167 5
3FQN_1,9FRK_1 186 3
2CFV_1,9FRK_1 193 3

Newick tree

 
[
	9FRK_1:98.23,
	[
		3FQN_1:83.5,2CFV_1:83.5
	]:14.73
]

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 \((0.85)(0.8)(\frac{591 }{\log_{20} 591}-\frac{275}{\log_{20}275})=88.9\)
Status Protein1 Protein2 d d1/2
Query variables 3FQN_1 2CFV_1 116 105.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]