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Parikh vectors
3PIY_1 5OJV_1 2JXQ_1 Letter Amino acid
13 12 0 I Isoleucine
8 30 0 P Proline
18 27 0 V Valine
13 44 0 R Arginine
6 5 2 C Cysteine
10 14 0 Q Glutamine
18 31 5 G Glycine
6 19 0 W Tryptophan
15 15 0 Y Tyrosine
14 29 0 D Aspartic acid
25 26 0 E Glutamic acid
21 23 0 S Serine
11 23 0 T Threonine
12 18 0 F Phenylalanine
7 9 0 N Asparagine
6 10 0 H Histidine
19 7 0 K Lycine
13 9 0 M Methionine
12 52 3 A Alanine
27 45 0 L Leucine 

3PIY_1|Chain A|Tyrosine-protein kinase BTK|Homo sapiens (9606)
>5OJV_1|Chains A, B|Uncharacterized protein|Mycobacterium hassiacum (strain DSM 44199 / CIP 105218 / JCM 12690 / 3849) (1122247)
>2JXQ_1|Chain A|RNA (5'-R(*GP*CP*AP*GP*AP*GP*AP*GP*CP*G)-3')|
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 \(w=f(c)\)
3PIY , Knot 125 274 0.85 40 189 264 
GTAGLGYGSWEIDPKDLTFLKELGTGQFGVVKYGKWRGQYDVAIKMIKEGSMSEDEFIEEAKVMMNLSHEKLVQLYGVCTKQRPIFIITEYMANGCLLNYLREMRHRFQTQQLLEMCKDVCEAMEYLESKQFLHRDLAARNCLVNDQGVVKVSDFGLSRYVLDDEYTSSVGSKFPVRWSPPEVLMYSKFSSKSDIWAFGVLMWEIYSLGKMPYERFTNSETAEHIAQGLRLYRPHLASEKVYTIMYSCWHEKADERPTFKILLSNILDVMDEES
5OJV , Knot 185 448 0.84 40 232 420 
GAMPHDPSFTPTQLAARAAYLLRGNDLGTMTTAAPLLYPHMWSWDAAFVAIGLAPLSVERAVVELDTLLSAQWRNGMIPHIVFANGVDGYFPGPARWATATLADNAPRNRLTSGITQPPVHAIAVQRILEHARTRGRSTRAVAEAFLDRRWGDLMRWHRWLAECRDRNERGRITLYHGWESGMDNSPRWDSAYANVVPGKLPEYQRADNVIITDPSQRPSDGEYDRYLWLLEEMKAVRYDDERLPSVMSFQVEDVFFSAIFSVACQVLAEIGEDYKRPHADVKDLYLWAERFRAGVVETTDQRTGAARDFDVLAEKWLVTETAAQFAPLLCGGLPHDRERALLKLLEGPRFCGHPDLKYGLIPSTSPVSRDFRPREYWRGPVWPVLTWLFSWCFARRGWAERARLLRQEGLRQASDGSFAAYYEPFTGEPLGSMQQSWTAAAVLDWLG
2JXQ , Knot 6 10 0.46 6 5 6 
GCAGAGAGCG

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 \(c\).

\(|P_{f(3PIY_1)}(2) \setminus P_{f(5OJV_1)}(2)|=68\), \(|P_{f(5OJV_1)}(2) \setminus P_{f(3PIY_1)}(2)|=111\). 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 1:1011110101010100101100110101111001010100011101100101000011001011101000011010110000011111000110101100100100010000110100010011001000011000111000110001110100111000110000000110011101011011100010000011111111101001101100010000010011011010010110001001100010001000101011100110110000
Pair \(Z_2\) Length of longest common subsequence
3PIY_1,5OJV_1 179 4
3PIY_1,2JXQ_1 190 2
5OJV_1,2JXQ_1 231 2

Newick tree

 
[
	2JXQ_1:11.63,
	[
		3PIY_1:89.5,5OJV_1:89.5
	]:21.13
]

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 \((0.85)(0.8)(\frac{722 }{\log_{20} 722}-\frac{274}{\log_{20}274})=124.\)
Status Protein1 Protein2 d d1/2
Query variables 3PIY_1 5OJV_1 157 125.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]