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Parikh vectors
2IGH_1 1USW_1 2NOW_1 Letter Amino acid
0 6 9 R Arginine
3 14 15 N Asparagine
1 18 10 Q Glutamine
0 5 13 H Histidine
0 4 11 M Methionine
0 7 8 C Cysteine
4 22 58 G Glycine
3 15 45 L Leucine
1 10 10 P Proline
13 26 20 T Threonine
1 4 9 W Tryptophan
5 18 10 D Aspartic acid
1 13 34 I Isoleucine
7 4 9 K Lycine
2 8 23 F Phenylalanine
0 20 15 S Serine
3 16 11 Y Tyrosine
7 23 54 A Alanine
4 12 15 E Glutamic acid
6 15 45 V Valine 

2IGH_1|Chain A|PROTEIN G|Streptococcus sp. GX7805 (1325)
>1USW_1|Chain A|FERULOYL ESTERASE A|ASPERGILLUS NIGER (5061)
>2NOW_1|Chain A|Ammonia channel|Escherichia coli (562)
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 \(w=f(c)\)
2IGH , Knot 33 61 0.74 30 46 59 
LTPAVTTYKLVINGKTLKGETTTKAVDAETAEKAFKQYANDNGVDGVWTYDDATKTFTVTE
1USW , Knot 119 260 0.84 40 174 253 
ASTQGISEDLYNRLVEMATISQAAYADLCNIPSTIIKGEKIYNAQTDINGWILRDDTSKEIITVFRGTGSDTNLQLDTNYTLTPFDTLPQCNDCEVHGGYYIGWISVQDQVESLVKQQASQYPDYALTVTGHSLGASMAALTAAQLSATYDNVRLYTFGEPRSGNQAFASYMNDAFQVSSPETTQYFRVTHSNDGIPNLPPADEGYAHGGVEYWSVDPYSAQNTFVCTGDEVQCCEAQGGQGVNDAHTTYFGMTSGACTW
2NOW , Knot 165 424 0.78 40 196 380 
APAVADKADNAFMMICTALVLFMTIPGIALFYGGLIRGKNVLSMLTQVTVTFALVCILWVVYGYSLAFGEGNNFFGNINWLMLKNIELTAVMGSIYQYIHVAFQGSFACITVGLIVGALAERIRFSAVLIFVVVWLTLSYIPIAHMVWGGGLLASHGALDFAGGTVVEINAAIAGLVGAYLIGKRVGFGKEAFKPHNLPMVFTGTAILYIGWFGFNAGSAGTANEIAALAFVNTVVATAAAILGWIFGEWALRGKPSLLGACSGAIAGLVGVTPACGYIGVGGALIIGVVAGLAGLWGVTMLKRLLRVDDPCDVFGVHGVCGIVGCIMTGIFAASSLGGVGFAEGVTMGHQLLVQLESIAITIVWSGVVAFIGYKLADLTVGLRVPEEQEREGLDVNSHGENAYNADQAQQPAQADLEHHHHHH

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 \(c\).

\(|P_{f(2IGH_1)}(2) \setminus P_{f(1USW_1)}(2)|=16\), \(|P_{f(1USW_1)}(2) \setminus P_{f(2IGH_1)}(2)|=144\). 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 1:1011100001110100101000001101001001100010001101110000100010100
Pair \(Z_2\) Length of longest common subsequence
2IGH_1,1USW_1 160 3
2IGH_1,2NOW_1 184 3
1USW_1,2NOW_1 176 4

Newick tree

 
[
	2NOW_1:93.12,
	[
		2IGH_1:80,1USW_1:80
	]:13.12
]

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 \((0.85)(0.8)(\frac{321 }{\log_{20} 321}-\frac{61}{\log_{20}61})=83.0\)
Status Protein1 Protein2 d d1/2
Query variables 2IGH_1 1USW_1 107 64
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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