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Parikh vectors
2ARQ_1 1QAZ_1 3SUF_1 Letter Amino acid
19 15 4 N Asparagine
7 8 4 H Histidine
16 9 4 M Methionine
22 21 19 V Valine
15 22 14 R Arginine
16 12 9 I Isoleucine
25 18 8 K Lycine
24 14 3 F Phenylalanine
14 20 11 P Proline
18 22 19 T Threonine
28 35 16 A Alanine
4 4 5 C Cysteine
12 16 10 Q Glutamine
25 23 23 G Glycine
33 19 20 S Serine
4 7 2 W Tryptophan
17 23 7 D Aspartic acid
39 27 13 L Leucine
12 18 5 Y Tyrosine
32 18 7 E Glutamic acid 

2ARQ_1|Chain A|Plasminogen activator inhibitor-2|Homo sapiens (9606)
>1QAZ_1|Chain A|PROTEIN (ALGINATE LYASE A1-III)|Sphingomonas sp. (28214)
>3SUF_1|Chains A, B, C, D|NS3 protease, NS4A protein|Hepatitis C virus (2847144)
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 \(w=f(c)\)
2ARQ , Knot 160 382 0.83 40 212 360 
MEDLCVANTLFALNLFKHLAKASPTQNLFLSPWSISSTMAMVYMGSRGSTEDQMASVLQFNEVGAAADKIHSSFRSLSSAINASTGNYLLESVNKLFGEKSASFREEYIRLCQKYYSSEPQAVDFLECAEEARKKINSWVKTQTKGKIPNLLPEGSVDGDTRMVLVNAVYFKGKWKTPFEKKLNGLFPFRVNSAQRTPVQMMYLREKLNIGYIEDLKAQILELPYAGDVSMFLLLPDEIADVSTGLELLESEITYDKLNKWTSKDKMAEDEVEVYIPQFKLEEHYELRSILRSMGMEDAFNKGRANFSGMSERNDLFLSEVFHQAMVDVNEEGTEAAAGTGGVMTGRTGHGGPQFVADHPFLFLIMHKITNCILFFGRFSSP
1QAZ , Knot 155 351 0.86 40 218 340 
GSHPFDQAVVKDPTASYVDVKARRTFLQSGQLDDRLKAALPKEYDCTTEATPNPQQGEMVIPRRYLSGNHGPVNPDYEPVVTLYRDFEKISATLGNLYVATGKPVYATCLLNMLDKWAKADALLNYDPKSQSWYQVEWSAATAAFALSTMMAEPNVDTAQRERVVKWLNRVARHQTSFPGGDTSCCNNHSYWRGQEATIIGVISKDDELFRWGLGRYVQAMGLINEDGSFVHEMTRHEQSLHYQNYAMLPLTMIAETASRQGIDLYAYKENGRDIHSARKFVFAAVKNPDLIKKYASEPQDTRAFKPGRGDLNWIEYQRARFGFADELGFMTVPIFDPRTGGSATLLAYKP
3SUF , Knot 96 203 0.83 40 143 197 
GSHMASMKKKGSVVIVGRINLSGDTAYAQQTRGEEGCQETSQTGRDKNQVEGEVQIVSTATQTFLATSINGVLWTVYHGAGTRTIASPKGPVTQMYTNVDKDLVGWQAPQGSRSLTPCTCGSSDLYLVTRHADVIPVRRRGDSRGSLLSPRPISYLKGSSGGPLLCPAGHAVGIFRAAVCTRGVAKAVAFIPVESLETTMRSP

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 \(c\).

\(|P_{f(2ARQ_1)}(2) \setminus P_{f(1QAZ_1)}(2)|=83\), \(|P_{f(1QAZ_1)}(2) \setminus P_{f(2ARQ_1)}(2)|=89\). 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 1:1001011001111011001101010001110110100011110110010000011011010011111001000100100110100100110010011100010100001010000000010110110010010001001100000101101110101010001111011010101001100010111110100100011011010001011010010101101101101011111100110100110110001000010010000011000101011010100000100110011100110010101011000001110011001110100010011110111101001011101110011111110010001111101001
Pair \(Z_2\) Length of longest common subsequence
2ARQ_1,1QAZ_1 172 4
2ARQ_1,3SUF_1 183 4
1QAZ_1,3SUF_1 179 3

Newick tree

 
[
	3SUF_1:91.95,
	[
		2ARQ_1:86,1QAZ_1:86
	]:5.95
]

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 \((0.85)(0.8)(\frac{733 }{\log_{20} 733}-\frac{351}{\log_{20}351})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 2ARQ_1 1QAZ_1 132 126.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]