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Parikh vectors
1FML_1 1TAM_1 1ONO_1 Letter Amino acid
19 4 15 N Asparagine
42 14 43 L Leucine
26 13 15 K Lycine
14 6 21 T Threonine
5 2 3 W Tryptophan
14 6 31 V Valine
19 1 11 F Phenylalanine
20 10 49 A Alanine
19 9 21 R Arginine
3 2 8 C Cysteine
9 10 24 Q Glutamine
15 10 21 G Glycine
8 3 8 H Histidine
15 1 19 M Methionine
22 3 18 P Proline
24 4 22 D Aspartic acid
12 6 16 I Isoleucine
23 4 6 Y Tyrosine
30 13 21 E Glutamic acid
12 11 26 S Serine 

1FML_1|Chains A, B|RETINOL DEHYDRATASE|Spodoptera frugiperda (7108)
>1TAM_1|Chain A|HIV-1 MATRIX PROTEIN|HIV-1 M:B_HXB2R (11706)
>1ONO_1|Chains A, B|1-deoxy-D-xylulose 5-phosphate reductoisomerase|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 \(w=f(c)\)
1FML , Knot 152 351 0.84 40 198 334 
MEKQQDLPFPYEFRELNPEEDKLVKANLGAFPTTYVKLGPKGYMVYRPYLKDAANIYNMPLRPTDVFVASYQRSGTTMTQELVWLIENDLNFEAAKTYMSLRYIYLDGFMIYDPEKQEEYNDILPNPENLDMERYLGLLEYSSRPGSSLLAAVPPTEKRFVKTHLPLSLMPPNMLDTVKMVYLARDPRDVAVSSFHHARLLYLLNKQSNFKDFWEMFHRGLYTLTPYFEHVKEAWAKRHDPNMLFLFYEDYLKDLPGCIARIADFLGKKLSEEQIQRLCEHLNFEKFKNNGAVNMEDYREIGILADGEHFIRKGKAGCWRDYFDEEMTKQAEKWIKDNLKDTDLRYPNMEL
1TAM , Knot 65 132 0.80 40 107 129 
MGARASVLSGGELDRWEKIRLRPGGKKKYKLKHIVWASRELERFAVNPGLLETSEGCRQILGQLQPSLQTGSEELRSLYNTVATLYCVHQRIEIKDTKEALDKIEEEQNKSKKKAQQAAADTGHSSQVSQNY
1ONO , Knot 167 398 0.83 40 216 375 
MKQLTILGSTGSIGCSTLDVVRHNPEHFRVVALVAGKNVTRMVEQCLEFSPRYAVMDDEASAKLLKTMLQQQGSRTEVLSGQQAACDMAALEDVDQVMAAIVGAAGLLPTLAAIRAGKTILLANKESLVTCGRLFMDAVKQSKAQLLPVDSEHNAIFQSLPQPIQHNLGYADLEQNGVVSILLTGSGGPFRETPLRDLATMTPDQACRHPNWSMGRKISVDSATMMNKGLEYIEARWLFNASASQMEVLIHPQSVIHSMVRYQDGSVLAQLGEPDMRTPIAHTMAWPNRVNSGVKPLDFCKLSALTFAAPDYDRYPCLKLAMEAFEQGQAATTALNAANEITVAAFLAQQIRFTDIAALNLSVLEKMDMREPQCVDDVLSVDANAREVARKEVMRLAS

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 \(c\).

\(|P_{f(1FML_1)}(2) \setminus P_{f(1TAM_1)}(2)|=133\), \(|P_{f(1TAM_1)}(2) \setminus P_{f(1FML_1)}(2)|=42\). 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 1:100000111100100101000011010111110001011101011001010011010011101001111000001001000111110001010110001010010101111001000000001110100101000111100000110011111110000110001110111101100101101100100111001001011011000001001101100110010101001001110000101111100001001110110110111001000010010001010010001110100000111110100110010110100010001000100110001000010010101
Pair \(Z_2\) Length of longest common subsequence
1FML_1,1TAM_1 175 4
1FML_1,1ONO_1 186 3
1TAM_1,1ONO_1 181 4

Newick tree

 
[
	1ONO_1:93.13,
	[
		1FML_1:87.5,1TAM_1:87.5
	]:5.63
]

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 \((0.85)(0.8)(\frac{483 }{\log_{20} 483}-\frac{132}{\log_{20}132})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 1FML_1 1TAM_1 128 87.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]