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Parikh vectors
7TXX_1 3MWP_1 4AKM_1 Letter Amino acid
6 27 8 N Asparagine
11 40 11 G Glycine
14 30 7 I Isoleucine
11 47 15 S Serine
0 6 0 W Tryptophan
9 39 8 A Alanine
3 31 4 R Arginine
5 23 9 E Glutamic acid
11 41 8 K Lycine
5 20 2 M Methionine
6 19 3 P Proline
7 45 10 D Aspartic acid
13 28 12 Q Glutamine
6 8 10 H Histidine
5 13 10 F Phenylalanine
11 34 11 T Threonine
0 9 4 C Cysteine
11 68 12 L Leucine
8 14 7 Y Tyrosine
5 35 15 V Valine 

7TXX_1|Chain A|BA5|Bartonella henselae str. Houston-1 (283166)
>3MWP_1|Chains A, B, C|Nucleoprotein|Lassa virus Josiah (11622)
>4AKM_1|Chains A, B|LYSOSOME-ASSOCIATED MEMBRANE GLYCOPROTEIN 3|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 \(w=f(c)\)
7TXX , Knot 71 147 0.80 36 107 137 
MAHHHHHHMGTLEAQTQGPGSMSSGSSQIYKEGYILDKNALDSISIGSSQEQVILALGTPSLKTKYDNEVFYYISQTRYRGMQFMKTKIIDRKVLAIYFNKNDQVTKIANYGLQDGQVFDFIAQTTPTATKEQPLLIQIIKGPANLP
3MWP , Knot 227 577 0.83 40 254 536 
GAMDHVEFMSASKEIKSFLWTQSLRRELSGYCSNIKLQVVKDAQALLHGLDFSEVSNVQRLMRKERRDDNDLKRLRDLNQAVNNLVELKSTQQKSILRVGTLTSDDLLILAADLEKLKSKVIRTERPLSAGVYMGNLSSQQLDQRRALLNMIGMSGGNQGARAGRDGVVRVWDVKNAELLNNQFGTMPSLTLACLTKQGQVDLNDAVQALTDLGLIYTAKYPNTSDLDRLTQSHPILNMIDTKKSSLNISGYNFSLGAAVKAGACMLDGGNMLETIKVSPQTMDGILKSILKVKKALGMFISDTPGERNPYENILYKICLSGDGWPYIASRTSITGRAWENTVVDLESDGKPQKADSNNSSKSLQSAGFTAGLTYSQLMTLKDAMLQLDPNAKTWMDIEGRPEDPVEIALYQPSSGCYIHFFREPTDLKQFKQDAKYSHGIDVTDLFATQPGLTSAVIDALPRNMVITCQGSDDIRKLLESQGRKDIKLIDIALSKTDSRKYENAVWDQYKDLCHMHTGVVVEKKKRGGKEEITPHCALMDCIMFDAAVSGGLNTSVLRAVLPRDMVFRTSTPRVVL
4AKM , Knot 81 166 0.83 38 131 160 
SVKTGIYQVLNGSRLCIKAEMGIQLIVQDKESVFSPRRYFNIDPNATQASGNCGTRKSNLLLNFQGGFVNLTFTKDEESYYISEVGAYLTVSDPETVYQGIKHAVVMFQTAVGHSFKCVSEQSLQLSAHLQVKTTDVQLQAFDFEDDHFGNVDECSSDYTHHHHHH

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 \(c\).

\(|P_{f(7TXX_1)}(2) \setminus P_{f(3MWP_1)}(2)|=31\), \(|P_{f(3MWP_1)}(2) \setminus P_{f(7TXX_1)}(2)|=178\). 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 1:110000001101010001110100100010001011000110010110000011111101010000000110010000001101100011000111101000001001100110010110111000101000011110110111011
Pair \(Z_2\) Length of longest common subsequence
7TXX_1,3MWP_1 209 3
7TXX_1,4AKM_1 152 6
3MWP_1,4AKM_1 195 4

Newick tree

 
[
	3MWP_1:10.13,
	[
		7TXX_1:76,4AKM_1:76
	]:32.13
]

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 \((0.85)(0.8)(\frac{724 }{\log_{20} 724}-\frac{147}{\log_{20}147})=163.\)
Status Protein1 Protein2 d d1/2
Query variables 7TXX_1 3MWP_1 207 127.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]