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Parikh vectors
2JYQ_1 8UZW_1 1RWF_1 Letter Amino acid
5 54 107 A Alanine
8 13 25 K Lycine
5 7 22 F Phenylalanine
10 27 51 S Serine
4 22 17 Q Glutamine
7 32 31 E Glutamic acid
7 31 80 G Glycine
0 11 11 M Methionine
1 4 19 W Tryptophan
5 11 27 N Asparagine
4 23 42 D Aspartic acid
3 18 27 I Isoleucine
11 68 69 L Leucine
3 20 30 P Proline
6 9 19 Y Tyrosine
5 33 53 V Valine
8 36 39 R Arginine
3 6 7 C Cysteine
3 11 14 H Histidine
8 27 67 T Threonine 

2JYQ_1|Chain A|Tyrosine-protein kinase transforming protein Src|Rous sarcoma virus (11886)
>8UZW_1|Chains A, B, C, D, E, F, G, H, I, J|L-seryl-tRNA(Sec) selenium transferase|Escherichia coli (83333)
>1RWF_1|Chain A|chondroitin AC lyase|Arthrobacter aurescens (43663)
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 \(w=f(c)\)
2JYQ , Knot 57 106 0.83 38 88 102 
QAEEWYFGKITRRESERLLLNPENPRGTFLVRESETTKGAYCLSVSDFDNAKGLNVKHYKIRKLDSGGFYITSRTQFSSLQQLVAYYSKHADGLCHRLTNVCPTSK
8UZW , Knot 185 463 0.81 40 214 422 
MTTETRSLYSQLPAIDRLLRDSSFLSLRDTYGHTRVVELLRQMLDEAREVIRGSQTLPAWCENWAQEVDARLTKEAQSALRPVINLTGTVLHTNLGRALQAEAAVEAVAQAMRSPVTLEYDLDDAGRGHRDRALAQLLCRITGAEDACIVNNNAAAVLLMLAATASGKEVVVSRGELVEIGGAFRIPDVMRQAGCTLHEVGTTNRTHANDYRQAVNENTALLMKVHTSNYSIQGFTKAIDEAELVALGKELDVPVVTDLGSGSLVDLSQYGLPKEPMPQELIAAGVSLVSFSGDKLLGGPQAGIIVGKKEMIARLQSHPLKRALRADKMTLAALEATLRLYLHPEALSEKLPTLRLLTRSAEVIQIQAQRLQAPLAAHYGAEFAVQVMPCLSQIGSGSLPVDRLPSAALTFTPHDGRGSHLESLAARWRELPVPVIGRIYDGRLWLDLRCLEDEQRFLEMLLK
1RWF , Knot 283 757 0.82 40 282 656 
EAEPGAAEFAALRNRWVDQITGRNVIQAGDPDFAKAITALNNKAADSLAKLDAAAGRTSVFTDLSLAKDAEMVTTYTRLSQLATAWATPTAAVFGDAAVLAAIKAGLADANTLCYNDRKEEVGNWWSWEIGVPRALADAMVLLHAELSAAERTAYCAAIDHFVPDPWLQFPPKRGKITSVGANRVDLCQGIIIRSLAGEDPTKLNHAVAGLSQVWQYVTSGDGIFRDGSFIQHSTTPYTGSYGVVLLTGLSKLFSLLGGTAFEVSDPTRSIFFDAVEGSFAPVMINGAMADAVRGRSISREANTGYDLGASAIEAILLLARAMDPATAARWRGLCAGWIARDTYRPILNSASVPRTALVKQLEATGVAPVAEATGHKLFPAMDRTMHRGPGWALSLALSSNRIAWYECGNGENNRGYHTGSGMTYFYTSDLGQYDDAFWATANYNRLPGITVDTTPLPDKVEGQWGAAVPADEWSGATALGEVAAVGQHLVGPGRTGLTARKSWFVSGDVTVCLGADISTASGAKVETIVDHRNLHQGSNTLTTAAGTIAGTAGTVEVLGDGRWVHLEGFGGYAMLDDSPLHVLRETRSGSWSGVNINGSATVQQRNFATLYVNHGVGPVAGSYAYMVAPGASVDLTRKLLEGNKYSVIRNDATAQSVEFKTAKTTAATFWKPGMAGDLGASGPACVVFSRHGNELSLAVSEPTQKAAGLTLTLPEGTWSSVLEGAGTLGTDADGRSTLTLDTTGLSGKTKLIKLKR

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 \(c\).

\(|P_{f(2JYQ_1)}(2) \setminus P_{f(8UZW_1)}(2)|=36\), \(|P_{f(8UZW_1)}(2) \setminus P_{f(2JYQ_1)}(2)|=162\). 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 1:0100101101000000011101001010111000000011001010010010110100001001001110100000100100111000001011000100101000
Pair \(Z_2\) Length of longest common subsequence
2JYQ_1,8UZW_1 198 3
2JYQ_1,1RWF_1 224 4
8UZW_1,1RWF_1 154 5

Newick tree

 
[
	2JYQ_1:11.66,
	[
		8UZW_1:77,1RWF_1:77
	]:36.66
]

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 \((0.85)(0.8)(\frac{569 }{\log_{20} 569}-\frac{106}{\log_{20}106})=136.\)
Status Protein1 Protein2 d d1/2
Query variables 2JYQ_1 8UZW_1 170 103.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]