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Parikh vectors
2PVB_1 7XCM_1 1TVK_1 Letter Amino acid
1 15 18 Y Tyrosine
20 50 36 A Alanine
2 16 10 M Methionine
6 28 31 E Glutamic acid
1 17 13 H Histidine
12 36 19 K Lycine
10 18 20 F Phenylalanine
8 26 22 S Serine
14 31 27 D Aspartic acid
2 18 15 Q Glutamine
7 41 35 G Glycine
4 25 26 I Isoleucine
1 22 20 P Proline
2 28 30 T Threonine
0 5 4 W Tryptophan
1 11 21 R Arginine
1 7 12 C Cysteine
5 46 34 V Valine
1 21 16 N Asparagine
9 42 31 L Leucine 

2PVB_1|Chain A|PROTEIN (PARVALBUMIN)|Esox lucius (8010)
>7XCM_1|Chains A, B, C, D, E, F|Trimethylamine methyltransferase|Methanosarcina barkeri MS (1434108)
>1TVK_1|Chain A|Tubulin alpha chain|Bos taurus (9913)
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 \(w=f(c)\)
2PVB , Knot 54 108 0.78 40 77 104 
XSFAGLKDADVAAALAACSAADSFKHKEFFAKVGLASKSLDDVKKAFYVIDQDKSGFIEEDELKLFLQNFSPSARALTDAETKAFLADGDKDGDGMIGVDEFAAMIKA
7XCM , Knot 203 503 0.83 40 244 463 
MAKNNAVAGFNALNGVELNLFTTDELKAIHYATMEVLMDPGIQVSDPEARQIFKENGCEVNEKTNVVKIPEYLVRKALQLAPSRFVLWGRDKKFNTVQECGGKVHWTCFGTGVKVCKYQDGKYVTVDSVEKDIADIAKLCDWAENIDYFSLPVSARDIAGQGAQDVHETLTPLANTAKHFHHIDPVGENVEYYRDIVKAYYGGDEEEARKKPIFSMLLCPTSPLELSVNACQVIIKGARFGIPVNVLSMAMSGGSSPVYLAGTLVTHNAEVLSGIVLAQLTVPGAKVWYGSSTTTFDLKKGTAPVGSPELGLISAAVAKLAQFYGLPSYVAGSKSDAKVPDDQAGHEKTMTTLLPALAGANTIYGAGMLELGMTFSMEQLVIDNDIFSMVKKAMQGIPVSEETLAVESIQKVGIGNNFLALKQTRQLVDYPSNPMLLDRHMFGDWAAAGSKDLATVAHEKVEDVLKNHQVTPIDADIFKDMQAIVDKADKAFRGMGGHHHHHH
1TVK , Knot 189 440 0.87 40 253 420 
MRECISIHVGQAGVQIGNACWELYCLEHGIQPDGQMPSDKTIGGGDDSFNTFFSETGAGKHVPRAVFVDLEPTVIDEVRTGTYRQLFHPEQLITGKEDAANNYARGHYTIGKEIIDLVLDRIRKLADQCTGLQGFSVFHSFGGGTGSGFTSLLMERLSVDYGKKSKLEFSIYPAPQVSTAVVEPYNSILTTHTTLEHSDCAFMVDNEAIYDICRRNLDIERPTYTNLNRLIGQIVSSITASLRFDGALNVDLTEFQTNLVPYPRGHFPLATYAPVISAEKAYHEQLSVAEITNACFEPANQMVKCDPRHGKYMACCLLYRGDVVPKDVNAAIATIKTKRTIQFVDWCPTGFKVGINYEPPTVVPGGDLAKVQRAVCMLSNTTAIAEAWARLDHKFDLMYAKRAFVHWYVGEGMEEGEFSEAREDMAALEKDYEEVGVDSV

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 \(c\).

\(|P_{f(2PVB_1)}(2) \setminus P_{f(7XCM_1)}(2)|=14\), \(|P_{f(7XCM_1)}(2) \setminus P_{f(2PVB_1)}(2)|=181\). 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 1:001111001011111110011001000011101111000100100110110000011100001011100101010110010001111010001011111001111101
Pair \(Z_2\) Length of longest common subsequence
2PVB_1,7XCM_1 195 4
2PVB_1,1TVK_1 204 3
7XCM_1,1TVK_1 145 4

Newick tree

 
[
	2PVB_1:10.33,
	[
		7XCM_1:72.5,1TVK_1:72.5
	]:34.83
]

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 \((0.85)(0.8)(\frac{611 }{\log_{20} 611}-\frac{108}{\log_{20}108})=147.\)
Status Protein1 Protein2 d d1/2
Query variables 2PVB_1 7XCM_1 183 110
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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