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Parikh vectors
2XUP_1 1XUD_1 5MAP_1 Letter Amino acid
10 9 10 H Histidine
8 10 10 K Lycine
27 10 16 T Threonine
13 3 6 W Tryptophan
27 17 30 D Aspartic acid
6 0 1 C Cysteine
27 6 19 E Glutamic acid
27 13 17 F Phenylalanine
45 13 23 P Proline
45 7 22 V Valine
50 9 47 A Alanine
17 7 9 N Asparagine
58 13 31 L Leucine
15 7 9 I Isoleucine
8 5 10 M Methionine
32 12 21 S Serine
18 9 3 Y Tyrosine
37 3 34 R Arginine
22 2 14 Q Glutamine
51 16 44 G Glycine 

2XUP_1|Chains A, B|ACETYLCHOLINESTERASE|MUS MUSCULUS (10090)
>1XUD_1|Chains A, B|Collagenase 3|Homo sapiens (9606)
>5MAP_1|Chains A, B|DtpA|Streptomyces lividans TK24 (457428)
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 \(w=f(c)\)
2XUP , Knot 217 543 0.84 40 249 505 
EGREDPQLLVRVRGGQLRGIRLKAPGGPVSAFLGIPFAEPPVGSRRFMPPEPKRPWSGVLDATTFQNVCYQYVDTLYPGFEGTEMWNPNRELSEDCLYLNVWTPYPRPASPTPVLIWIYGGGFYSGAASLDVYDGRFLAQVEGAVLVSMNYRVGTFGFLALPGSREAPGNVGLLDQRLALQWVQENIAAFGGDPMSVTLFGESAGAASVGMHILSLPSRSLFHRAVLQSGTPNGPWATVSAGEARRRATLLARLVGCPPGGAGGNDTELIACLRTRPAQDLVDHEWHVLPQESIFRFSFVPVVDGDFLSDTPEALINTGDFQDLQVLVGVVKDEGSAFLVYGVPGFSKDNESLISRAQFLAGVRIGVPQASDLAAEAVVLHYTDWLHPEDPTHLRDAMSAVVGDHNVVCPVAQLAGRLAAQGARVYAYIFEHRASTLTWPLWMGVPHGYEIEFIFGLPLDPSLNYTTEERIFAQRLMKYWTNFARTGDPNDPRDSKSPQWPPYTTAAQQYVSLNLKPLEVRRGLRAQTCAFWNRFLPKLLSAT
1XUD , Knot 81 171 0.81 38 126 166 
YNVFPRTLKWSKMNLTYRIVNYTPDMTHSEVEKAFKKAFKVWSDVTPLNFTRLHDGIADIMISFGIKEHGDFYPFDGPSGLLAHAFPPGPNYGGDAHFDDDETWTSSSKGYNLFLVAAHEFGHSLGLDHSKDPGALMFPIYTYTGKSHFMLPDDDVQGIQSLYGPGDEDPN
5MAP , Knot 154 376 0.81 40 192 354 
ATPLTSLGSEQAMFHGKHQPGITTPMQARGHLVAFDLAAGAGRKEAAALLRRWSDTARRLMAGEPAGSRDTDVARDAGPSSLTVTFGFGHSFFGRTGLEKQRPVALDPLPDFSSDHLDKNRSNGDLWVQIGADDALVAFHALRAIQRDAGAAARVRWQMNGFNRSPGATAHPMTARNLMGQVDGTRNPKPGEADFDRRIFVPEEPEAGKGGPAWMANGSYVVVRRIRMLLDDWEELSLKAQEDVIGRRKSDGAPLSGGSGATESTEMDLEKTDGSGELVVPINAHARITRPDQNGGAAMVRRPFSYHDGFDADGVPDAGLLFVCWQADPLRGFVPVQRKLDRGDALSQFIRHEASGLFAVPGGAAEGEYVGQRLLE

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 \(c\).

\(|P_{f(2XUP_1)}(2) \setminus P_{f(1XUD_1)}(2)|=163\), \(|P_{f(1XUD_1)}(2) \setminus P_{f(2XUP_1)}(2)|=40\). 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 1:010001011101011010110101111110111111110111100011110100110111010010010000100101110100110100010000101011010101101011111101111001110101001011101011111010001101111111100011101111000111011000111111011010111001111011101101100011001110010101111010110100010111011101111111000011101000110011000101110001101011111010110001011100101001011111100010111101111100000011001011111011110100111011110000110100100100110111100011011101110111011010101100010010111111110100101111111010100000001110011001001100101001000001011100011000101010110100110100011100111011010
Pair \(Z_2\) Length of longest common subsequence
2XUP_1,1XUD_1 203 3
2XUP_1,5MAP_1 169 4
1XUD_1,5MAP_1 174 4

Newick tree

 
[
	1XUD_1:97.64,
	[
		2XUP_1:84.5,5MAP_1:84.5
	]:13.14
]

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 \((0.85)(0.8)(\frac{714 }{\log_{20} 714}-\frac{171}{\log_{20}171})=153.\)
Status Protein1 Protein2 d d1/2
Query variables 2XUP_1 1XUD_1 201 130.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]