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Parikh vectors
1BBI_1 7ROZ_1 2QMT_1 Letter Amino acid
3 38 2 Q Glutamine
4 91 5 E Glutamic acid
2 61 2 F Phenylalanine
6 44 0 P Proline
9 96 0 S Serine
1 78 4 V Valine
3 34 3 N Asparagine
8 69 5 D Aspartic acid
1 25 0 H Histidine
1 34 1 M Methionine
0 16 1 W Tryptophan
2 40 3 Y Tyrosine
14 23 0 C Cysteine
2 65 1 I Isoleucine
2 112 3 L Leucine
2 57 10 T Threonine
4 75 6 A Alanine
2 67 0 R Arginine
0 64 4 G Glycine
5 83 6 K Lycine 

1BBI_1|Chain A|TRYPSIN/CHYMOTRYPSIN BOWMAN-BIRK INHIBITOR|Glycine max (3847)
>7ROZ_1|Chain A|RNA-dependent RNA polymerase 2|Arabidopsis thaliana (3702)
>2QMT_1|Chain A|Immunoglobulin G-binding protein G|Staphylococcus aureus (1280)
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 \(w=f(c)\)
1BBI , Knot 35 71 0.70 36 59 69 
DDESSKPCCDQCACTKSNPPQCRCSDMRLNSCHSACKSCICALSYPAQCFCVDITDFCYEPCKPSEDDKEN
7ROZ , Knot 421 1172 0.84 40 339 1027 
MVSETTTNRSTVKISNVPQTIVADELLRFLELHLGEDTVFALEIPTTRDNWKPRDFARVQFTTLEVKSRAQLLSSQSKLLFKTHNLRLSEAYDDIIPRPVDPRKRLDDIVLTVGFPESDEKRFCALEKWDGVRCWILTEKRRVEFWVWESGDCYKIEVRFEDIIETLSCCVNGDASEIDAFLLKLKYGPKVFKRVTVHIATKFKSDRYRFCKEDFDFMWIRTTDFSGSKSIGTSTCFCLEVHNGSTMLDIFSGLPYYREDTLSLTYVDGKTFASAAQIVPLLNAAILGLEFPYEILFQLNALVHAQKISLFAASDMELIKILRGMSLETALVILKKLHQQSSICYDPVFFVKTQMQSVVKKMKHSPASAYKRLTEQNIMSCQRAYVTPSKIYLLGPELETANYVVKNFAEHVSDFMRVTFVEEDWSKLPANALSVNSKEGYFVKPSRTNIYNRVLSILGEGITVGPKRFEFLAFSASQLRGNSVWMFASNEKVKAEDIREWMGCFRKIRSISKCAARMGQLFSASRQTLIVRAQDVEQIPDIEVTTDGADYCFSDGIGKISLAFAKQVAQKCGLSHVPSAFQIRYGGYKGVIAVDRSSFRKLSLRDSMLKFDSNNRMLNVTRWTESMPCFLNREIICLLSTLGIEDAMFEAMQAVHLSMLGNMLEDRDAALNVLQKLSGENSKNLLVKMLLQGYAPSSEPYLSMMLRVHHESQLSELKSRCRILVPKGRILIGCMDEMGILEYGQVYVRVTLTKAELKSRDQSYFRKIDEETSVVIGKVVVTKNPCLHPGDIRVLDAIYEVHFEEKGYLDCIIFPQKGERPHPNECSGGDLDGDQFFVSWDEKIIPSEMDPPMDYAGSRPRLMDHDVTLEEIHKFFVDYMISDTLGVISTAHLVHADRDPEKARSQKCLELANLHSRAVDFAKTGAPAEMPYALKPREFPDFLERFEKPTYISESVFGKLYRAVKSSLAQRKPEAESEDTVAYDVTLEEAGFESFIETAKAHRDMYGEKLTSLMIYYGAANEEEILTGILKTKEMYLARDNRRYGDMKDRITLSVKDLHKEAMGWFEKSCEDEQQKKKLASAWYYVTYNPNHRDEKLTFLSFPWIVGDVLLDIKAENAQRQSVEEKTSGLVSIKLENLYFQGSAWSHPQFEKGGGSGGGSGGSAWSHPQFEK
2QMT , Knot 32 56 0.76 30 46 54 
MQYKLILNGKTLKGETTTEAVDAATAEKVFKQYANDNGVDGEWTYDDATKTFTVTE

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 \(c\).

\(|P_{f(1BBI_1)}(2) \setminus P_{f(7ROZ_1)}(2)|=7\), \(|P_{f(7ROZ_1)}(2) \setminus P_{f(1BBI_1)}(2)|=287\). 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 1:00000010000010000011000000101000001000010110011001010100100010010000000
Pair \(Z_2\) Length of longest common subsequence
1BBI_1,7ROZ_1 294 3
1BBI_1,2QMT_1 97 2
7ROZ_1,2QMT_1 297 4

Newick tree

 
[
	7ROZ_1:16.29,
	[
		1BBI_1:48.5,2QMT_1:48.5
	]:11.79
]

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 \((0.85)(0.8)(\frac{1243 }{\log_{20} 1243}-\frac{71}{\log_{20}71})=321.\)
Status Protein1 Protein2 d d1/2
Query variables 1BBI_1 7ROZ_1 403 213
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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