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Parikh vectors
7ZQO_1 5OAZ_1 1PVD_1 Letter Amino acid
0 2 37 I Isoleucine
0 3 26 P Proline
0 3 50 A Alanine
3 1 4 C Cysteine
3 6 42 G Glycine
0 5 40 V Valine
0 2 23 F Phenylalanine
0 4 32 S Serine
0 2 7 W Tryptophan
0 5 54 L Leucine
0 3 35 K Lycine
0 1 12 M Methionine
0 4 42 T Threonine
0 3 17 Y Tyrosine
0 3 28 D Aspartic acid
0 2 21 Q Glutamine
0 4 29 E Glutamic acid
0 1 15 R Arginine
0 8 29 N Asparagine
0 1 12 H Histidine 

7ZQO_1|Chains A[auth AAA], B[auth BBB]|DNA (5'-D(*CP*GP*CP*GP*CP*G)-3')|synthetic construct (32630)
>5OAZ_1|Chains A, B|Tyrosine-protein kinase ABL1|Homo sapiens (9606)
>1PVD_1|Chains A, B|PYRUVATE DECARBOXYLASE|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
7ZQO , Knot 3 6 0.29 4 2 2 
CGCGCG
5OAZ , Knot 38 63 0.83 40 56 60 
MENDPNLFVALYDFVASGDNTLSITKGEKLRVLGYNHNGEWCEAQTKNGQGWVPSNYITPVNS
1PVD , Knot 219 555 0.83 40 249 512 
SEITLGKYLFERLKQVNVNTVFGLPGDFNLSLLDKIYEVEGMRWAGNANELNAAYAADGYARIKGMSCIITTFGVGELSALNGIAGSYAEHVGVLHVVGVPSISSQAKQLLLHHTLGNGDFTVFHRMSANISETTAMITDIATAPAEIDRCIRTTYVTQRPVYLGLPANLVDLNVPAKLLQTPIDMSLKPNDAESEKEVIDTILALVKDAKNPVILADACCSRHDVKAETKKLIDLTQFPAFVTPMGKGSISEQHPRYGGVYVGTLSKPEVKEAVESADLILSVGALLSDFNTGSFSYSYKTKNIVEFHSDHMKIRNATFPGVQMKFVLQKLLTNIADAAKGYKPVAVPARTPANAAVPASTPLKQEWMWNQLGNFLQEGDVVIAETGTSAFGINQTTFPNNTYGISQVLWGSIGFTTGATLGAAFAAEEIDPKKRVILFIGDGSLQLTVQEISTMIRWGLKPYLFVLNNDGYTIEKLIHGPKAQYNEIQGWDHLSLLPTFGAKDYETHRVATTGEWDKLTQDKSFNDNSKIRMIEIMLPVFDAPQNLVKQAKLT

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 \(c\).

\(|P_{f(7ZQO_1)}(2) \setminus P_{f(5OAZ_1)}(2)|=2\), \(|P_{f(5OAZ_1)}(2) \setminus P_{f(7ZQO_1)}(2)|=56\). 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 1:010101
Pair \(Z_2\) Length of longest common subsequence
7ZQO_1,5OAZ_1 58 1
7ZQO_1,1PVD_1 251 1
5OAZ_1,1PVD_1 223 3

Newick tree

 
[
	1PVD_1:13.04,
	[
		7ZQO_1:29,5OAZ_1:29
	]:10.04
]

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 \((0.85)(0.8)(\frac{69 }{\log_{20} 69}-\frac{6}{\log_{20}6})=26.3\)
Status Protein1 Protein2 d d1/2
Query variables 7ZQO_1 5OAZ_1 35 18.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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