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Parikh vectors
6FMV_1 9FSA_1 4NNB_1 Letter Amino acid
28 15 44 G Glycine
13 6 14 H Histidine
25 6 29 I Isoleucine
11 5 33 P Proline
6 0 10 Y Tyrosine
23 20 42 V Valine
10 1 49 R Arginine
9 12 34 E Glutamic acid
7 10 34 D Aspartic acid
5 0 18 Q Glutamine
25 32 68 A Alanine
2 7 11 N Asparagine
3 0 9 W Tryptophan
3 0 2 C Cysteine
15 5 7 M Methionine
23 3 16 F Phenylalanine
17 6 26 S Serine
12 7 23 T Threonine
37 12 58 L Leucine
8 10 15 K Lycine 

6FMV_1|Chain A|Undecaprenyl-diphosphatase|Escherichia coli K-12 (83333)
>9FSA_1|Chain A|S-layer protein|Methanococcus voltae (2188)
>4NNB_1|Chain A|OBCA, Oxalate Biosynthetic Component A|Burkholderia glumae (626418)
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 \(w=f(c)\)
6FMV , Knot 120 282 0.80 40 162 267 
MHHHHHHGSMSDMHSLLIAAILGVVEGLTEFLPVSSTGHMIIVGHLLGFEGDTAKTFEVVIQLGSILAVVVMFWRRLFGLIGIHFGRPLQHEGESKGRLTLIHILLGMIPAVVLGLLFHDTIKSLFNPINVMYALVVGGLLLIAAECLKPKEPRAPGLDDMTYRQAFMIGCFQCLALWPGFSRSGATISGGMLMGVSRYAASEFSFLLAVPMMMGATALDLYKSWGFLTSGDIPMFAVGFITAFVVALIAIKTFLQLIKRISFIPFAIYRFIVAAAVYVVFF
9FSA , Knot 69 157 0.74 32 88 139 
MVEKIGDVEGFKVIDNGEPTADIVVGSTAAAADVVSAANVAAKVGSMMFKEGEGGSDAKAPVAFKAPLAVLDTEVSLDAANKKLILVGGPVANALTKELADAGKIEMTVESPATLAVVAGAANGNDVLVVAGGDRAATAEAANALIEMLLEHHHHHH
4NNB , Knot 214 542 0.82 40 236 494 
GHMTSLYITAAPIGAVPKFLDPFEATFIPSFLLEGFFDADRCASIAADLKTDGWEVVPAGGRLLQVGHAQPIDERLLAGNAQAATIRQALEAARWTRRDGAWHPPRLAAPNAAHFPKPWLAALSNKLARRIVLQLTTYGWIVSEQGDLLWEHERQHHYLPPALIEAIEKESPALLKNMEEAGWIACAAGYWQAGKARSPYLPITPEAITEETIRSMRAGAAVVHLHTRDLSDRRRIEIPGLGVVTVGSQRNQIVLDDYDAIVPMVKKREPAAILNLSTSVRGDRHGARSKLRRAHLKFYDDVGSAPEVASLSPAAVVFQGGGGYDNAPDFLDAQFDHFERVGTRPEVEVFNHAIVDNATSLYRDRLLRTGKPVLFMLVAGVDQYRRDPITGEVEDDSLIARVVREEISSLLADESADSHRRAVELAIGQLRPVVERLRASFPVSKISILLPGPMQNLLVDVALGLGLDGIRVGLEDGLTVNDARVPGGVRKARGTWEQVSLVREELLGRGATILTAAQVRDMFGLGIKPAARRERDPQTAAG

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 \(c\).

\(|P_{f(6FMV_1)}(2) \setminus P_{f(9FSA_1)}(2)|=108\), \(|P_{f(9FSA_1)}(2) \setminus P_{f(6FMV_1)}(2)|=34\). 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 1:100000010100100111111111101100111100010111110111101001001011101101111111110011111110110110001000101011011111111111111100010011011011011111111111100101001011110010000111110100111111100011010111111100011001011111111111011010001111001011111111101111111110011011001011111100111111101111
Pair \(Z_2\) Length of longest common subsequence
6FMV_1,9FSA_1 142 6
6FMV_1,4NNB_1 178 4
9FSA_1,4NNB_1 182 4

Newick tree

 
[
	4NNB_1:95.50,
	[
		6FMV_1:71,9FSA_1:71
	]:24.50
]

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 \((0.85)(0.8)(\frac{439 }{\log_{20} 439}-\frac{157}{\log_{20}157})=83.7\)
Status Protein1 Protein2 d d1/2
Query variables 6FMV_1 9FSA_1 104 77.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]