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Parikh vectors
7DWN_1 7HOI_1 9GTR_1 Letter Amino acid
14 9 38 R Arginine
3 6 0 C Cysteine
16 2 35 K Lycine
14 4 34 P Proline
16 9 46 D Aspartic acid
18 5 23 Q Glutamine
2 6 9 H Histidine
35 12 47 L Leucine
10 8 13 Y Tyrosine
27 10 58 A Alanine
21 7 15 E Glutamic acid
6 2 10 M Methionine
8 3 20 F Phenylalanine
13 6 51 T Threonine
19 14 61 V Valine
7 5 22 N Asparagine
13 14 75 G Glycine
24 5 32 I Isoleucine
23 14 46 S Serine
4 3 8 W Tryptophan 

7DWN_1|Chains A, B, C, D|Predicted DNA-binding transcriptional regulator|Aliivibrio fischeri ES114 (312309)
>7HOI_1|Chain A|Protease 2A|Coxsackievirus A16 (31704)
>9GTR_1|Chains A[auth q], B[auth r], C[auth s]|Gp5/Type VI secretion system Vgr protein OB-fold domain-containing protein|Streptomyces coelicolor A3(2) (100226)
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 \(w=f(c)\)
7DWN , Knot 127 293 0.82 40 177 275 
MNVESKWLEDFLVLAKVKNFSQAAELRNVTQPAFSRRIRLLEDTVGAELVDRKSKPIELTPSGKLFRITARTLVNQIEAGISQISDLSQLGGNVVQVAAAHSLATSLIPKMQQAFDEGDYKPILSVEAIDVDEATKELREGACDILLAFDDDILRLPPYQSQLIAKTELLPVSACDEMGKPIYDFISQGAVPWLTYSSTSYMGRQVEIIREQVALTPIFSSSMTDMLKILVLNKQGIAWLPAYSIQEELAQKKVAIIGEQSLRLPIEYYAYRYQARLHPAGEKVWSILCNLDT
7HOI , Knot 70 144 0.80 40 112 139 
SGAIYVGNYRVVNRHLATHNDWANLVWEDSSRDLLVSSTTAQGCDTIARCDCQTGVYYCSSRRKHYPVSFSKPSLIFVEASEYYPARYQSHLMLAVGHSEPGDCGGILRCQHGVVGIVSTGGNGLVGFADVRDLLWLDEEAMEQ
9GTR , Knot 249 643 0.83 38 241 577 
MVRPSFSSIVEVKIGGAKLPDTIAPMLTDGWVDQGVNVPAAFRLTFRDPYHRLLGDLNVQFGTKVVITPIADGQGKNNPLLTGEVTGLETDYDGTGSFTVIRGYDYGHRLMRQRRVAAYRNQKASDIARKLVQMDGVSVGRIQPTKGTYAFIAQPNVTDWDFLARLADENKMIMYLDSKGKFRFVTPKPSAGAPSPNTDGDQSAFVLQAGHDILRLRAAVTAADQIGKVESRGWNVTTKKKITEIAPATTDPGISIKWTPGTAAGKFKPGKLVETANPYDKQDEVQNAAKALASDVTASFSELEVAANGHPDLRPGVPVALSDVGTPFEGKYTVTSVRHHFGDGVAYESWITVSGRQWRSLYGLASGGGGSDPASATRLPSVANAIVTDVQDPLKQGRVKLQFPWLDDTYVSDWARSVQMGGVAGGGIFPMDVGDEVLVAFDRGALDHPFVIGGLYNGRDVPTKSDVPLHDGLKKKAVRHTLSDRQGNRVDLLSQRTGGRKQGVRIASGNDKLTINLDRTKTEITVDSKGSVSITGSRSVSVDAGTDLSLSARRSLTIKSGGPLSIQGGSTINMRSGGLVGVNAGGALNLGAGGAATLSAGAATTISGTVNVQINSVMTRVIGATFEVKTPIFKVATIPVPGL

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 \(c\).

\(|P_{f(7DWN_1)}(2) \setminus P_{f(7HOI_1)}(2)|=117\), \(|P_{f(7HOI_1)}(2) \setminus P_{f(7DWN_1)}(2)|=52\). 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 1:10100011001111101001001101001001110001011000111011000001101010101101010011001011100100100111011011110011001110100110010001110101101001000100110011111000110111000011100011110100011011001100111111000000011001011000111011100010011011110001111111001000110001111100010111000100001010111001101100100
Pair \(Z_2\) Length of longest common subsequence
7DWN_1,7HOI_1 169 3
7DWN_1,9GTR_1 150 5
7HOI_1,9GTR_1 207 4

Newick tree

 
[
	7HOI_1:10.13,
	[
		7DWN_1:75,9GTR_1:75
	]:25.13
]

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 \((0.85)(0.8)(\frac{437 }{\log_{20} 437}-\frac{144}{\log_{20}144})=87.3\)
Status Protein1 Protein2 d d1/2
Query variables 7DWN_1 7HOI_1 112 81
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]