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Parikh vectors
3PEY_1 9CQI_1 3HLA_1 Letter Amino acid
17 11 2 N Asparagine
8 10 14 Y Tyrosine
23 13 17 Q Glutamine
20 26 21 E Glutamic acid
32 31 10 K Lycine
19 31 14 S Serine
23 25 20 G Glycine
5 14 13 H Histidine
28 12 4 I Isoleucine
16 13 11 P Proline
12 12 5 M Methionine
16 15 8 F Phenylalanine
26 19 22 T Threonine
29 34 24 A Alanine
20 20 22 R Arginine
27 22 18 D Aspartic acid
3 1 4 C Cysteine
41 47 16 L Leucine
1 5 9 W Tryptophan
29 26 16 V Valine 

3PEY_1|Chain A|ATP-dependent RNA helicase DBP5|Saccharomyces cerevisiae (4932)
>9CQI_1|Chain A|Serpin H1|Canis lupus familiaris (9615)
>3HLA_1|Chain A|CLASS I HISTOCOMPATIBILITY ANTIGEN (HLA-A2.1) (ALPHA 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 \(w=f(c)\)
3PEY , Knot 167 395 0.84 40 225 378 
GAMAKSFDELGLAPELLKGIYAMKFQKPSKIQERALPLLLHNPPRNMIAQSQSGTGKTAAFSLTMLTRVNPEDASPQAICLAPSRELARQTLEVVQEMGKFTKITSQLIVPDSFEKNKQINAQVIVGTPGTVLDLMRRKLMQLQKIKIFVLDEADNMLDQQGLGDQCIRVKRFLPKDTQLVLFSATFADAVRQYAKKIVPNANTLELQTNEVNVDAIKQLYMDCKNEADKFDVLTELYGLMTIGSSIIFVATKKTANVLYGKLKSEGHEVSILHGDLQTQERDRLIDDFREGRSKVLITTNVLARGIDIPTVSMVVNYDLPTLANGQADPATYIHRIGRTGRFGRKGVAISFVHDKNSFNILSAIQKYFGDIEMTRVPTDDWDEVEKIVKKVLKD
9CQI , Knot 165 387 0.84 40 213 365 
GAGALSPKAATLAERSAGLAFSLYQAMAKDQAVENILLSPVVVASSLGLVSLGGKATTASQAKAVLSAEQLRDEEVHAGLGELLRSLSNSTARNVTWKLGSRLYGPSSVSFAEDFVRSSKQHYNCEHSKINFRDKRSALQSINEWAAQTTDGKLPEVTKDVERTDGALLVNAMFFKPHWDEKFHHKMVDNRGFMVTRSYTVGVTMMHRTGLYNYYDDEKEKLQIVEMPLAHKLSSLIILMPHHVEPLERLEKLLTKEQLKIWMGKMQKKAVAISLPKGVVEVTHDLQKHLAGLGLTEAIDKNKADLSRMSGKKDLYLASVFHATAFEWDTEGNPFDQDIYGREELRSPKLFYADHPFIFLVRDTQSGSLLFIGRLVRPKGDKMRDEL
3HLA , Knot 120 270 0.83 40 181 259 
GSHSMRYFFTSVSRPGRGEPRFIAVGYVDDTQFVRFDSDAASQRMEPRAPWIEQEGPEYWDGETRKVKAHSQTHRVDLGTLRGYYNQSEAGSHTVQRMYGCDVGSDWRFLRGYHQYAYDGKDYIALKEDLRSWTAADMAAQTTKHKWEAAHVAEQLRAYLEGTCVEWLRRYLENGKETLQRTDAPKTHMTHHAVSDHEATLRCWALSFYPAEITLTWQRDGEDQTQDTELVETRPAGDGTFQKWAAVVVPSGQEQRYTCHVQHEGLPKPL

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 \(c\).

\(|P_{f(3PEY_1)}(2) \setminus P_{f(9CQI_1)}(2)|=81\), \(|P_{f(9CQI_1)}(2) \setminus P_{f(3PEY_1)}(2)|=69\). 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 1:11110010011111011011011010010010001111110011001110000101001110101100101001010110111000110001011001101001000111100100000101011110110110110001101001011110010011000111000101001110000111101011011000100111010010100001010110010100000100101100101110110011111000010110101000100101101010000000110010010001110001110110110101110001101101010110010011001011001111011000001011011000110101001100010010011001100
Pair \(Z_2\) Length of longest common subsequence
3PEY_1,9CQI_1 150 4
3PEY_1,3HLA_1 190 3
9CQI_1,3HLA_1 178 5

Newick tree

 
[
	3HLA_1:97.06,
	[
		3PEY_1:75,9CQI_1:75
	]:22.06
]

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 \((0.85)(0.8)(\frac{782 }{\log_{20} 782}-\frac{387}{\log_{20}387})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 3PEY_1 9CQI_1 138 135
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]