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Parikh vectors
4XNY_1 6BRD_1 1WBD_1 Letter Amino acid
14 45 64 R Arginine
19 55 100 L Leucine
6 7 26 Y Tyrosine
11 26 42 D Aspartic acid
15 1 6 C Cysteine
24 43 64 E Glutamic acid
7 12 24 M Methionine
16 19 35 P Proline
33 28 48 T Threonine
22 48 40 V Valine
35 5 25 N Asparagine
27 10 45 I Isoleucine
21 7 27 K Lycine
12 14 22 F Phenylalanine
18 15 36 S Serine
8 7 5 W Tryptophan
18 46 85 A Alanine
13 21 40 Q Glutamine
24 51 51 G Glycine
11 16 15 H Histidine 

4XNY_1|Chain A[auth G]|Envelope glycoprotein gp160,Envelope glycoprotein gp160,Envelope glycoprotein gp160|Human immunodeficiency virus 1 (11676)
>6BRD_1|Chains A, B, C|Rifampin monooxygenase|Streptomyces venezuelae (strain ATCC 10712 / CBS 650.69 / DSM 40230 / JCM 4526 / NBRC 13096 / PD 04745) (953739)
>1WBD_1|Chains A, B|DNA MISMATCH REPAIR PROTEIN MUTS|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 \(w=f(c)\)
4XNY , Knot 149 354 0.82 40 208 331 
VWKEAETTLFCASDAKAYETEKHNVWATHACVPTDPNPQEIHLENVTEEFNMWKNNMVEQMHTDIISLWDQSLKPCVKLTGGSAITQACPKVTFEPIPIHYCAPAGFAILKCKDEEFNGIGPCKNVSTVQCTHGIKPVVSTQLLLNGSLAEKEVKIRCENITNNAKTIIVQLVNPVKINCTRPNNGGSGSGGDIRQAHCNVNRTEWNNTLHQVVEQLRKHFNKTINFANSTGGDLEITTHSFNCGGEFFYCNTTNLFNSTWNHTASMNSTESNDTIILPCRIKQIINMWQRVGQAMYAPPIRGVIRCESNITGLILTRDGGNTNSTRETFRPGGGDMRDNWRSELYKYKVVKIE
6BRD , Knot 187 476 0.80 40 202 434 
MFDVIVVGGGPTGLMLAGELRLHGVRVLVLEKETEPTRQSRAQGLHVRSIEVMAQRGLLERFLERGHTVAVGGFFAGLATSWPERLDTAHSYVLAVPQVITEQLLAEHATALGAEIRRGRALVGLRQDEDGVTVDLADGEQLRARYVVGCDGGRSTVRKLLGVAFPGEPSRVETLLGEMEMTASQEELTSVMTEVRKTQQRFGAMPLGDGVFRVVVPAEGVAEDRTASPTLDEFKQQLRAHAGTDFGVHSPRWLSRFGDATRQAERYRVDRVFLAGDAAHIHPPTGGQGLNLGIQDAFNLGWKLAAEVDGWAPEGLLDTYHAERHPVATEVLDNTRAQIQLMSTEPGPQAVRRLMAELVEFENVNRYLIEKITAISVRYDVGEGHELLGRRMRDLALKHGRLYERMHEGRGLLLDQTGRLSVAGWEDRVDHVVEVSEELDVPAVLLRPDGHVVWAGEDQQELLTRMPAWFGAATAG
1WBD , Knot 306 800 0.85 40 291 717 
MSAIENFDAHTPMMQQYLRLKAQHPEILLFYRMGDFYQLFYDDAKRASQLLDISLTKRGASAGEPIPMAGIPYHAVENYLAKLVNQGESVAICEQIGDPATSKGPVERKVVRIVTPGTISDEALLQERQDNLLAAIWQDSKGFGYATLDISSGRFRLSEPADRETMAAELQRTNPAELLYAEDFAEMSLIEGRRGLRRRPLWEFEIDTARQQLNLQFGTRDLVGFGVENAPRGLCAAGCLLQYAKDTQRTTLPHIRSITMEREQDSIIMDAATRRNLEITQNLAGGAENTLASVLDCTVTPMGSRMLKRWLHMPVRDTRVLLERQQTIGALQDFTAGLQPVLRQVGDLERILARLALRTARPRDLARMRHAFQQLPELRAQLETVDSAPVQALREKMGEFAELRDLLERAIIDTPPVLVRDGGVIASGYNEELDEWRALADGATDYLERLEVRERERTGLDTLKVGFNAVHGYYIQISRGQSHLAPINYMRRQTLKNAERYIIPELKEYEDKVLTSKGKALALEKQLYEELFDLLLPHLEALQQSASALAELDVLVNLAERAYTLNYTCPTFIDKPGIRITEGRHPVVEQVLNEPFIANPLNLSPQRRMLIITGPNMGGKSTYMRQTALIALMAYIGSYVPAQKVEIGPIDRIFTRVGAADDLASGRSTFMVEMTETANILHNATEYSLVLMDEIGRGTSTYDGLSLAWACAENLANKIKALTLFATHYFELTQLPEKMEGVANVHLDALEHGDTIAFMHSVQDGAASKSYGLAVAALAGVPKEVIKRARQKLRELESIS

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 \(c\).

\(|P_{f(4XNY_1)}(2) \setminus P_{f(6BRD_1)}(2)|=93\), \(|P_{f(6BRD_1)}(2) \setminus P_{f(4XNY_1)}(2)|=87\). 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 1:110010001101001010000000111001011001010010100100010110001100100011011000101010101101100101010101111000111111110000001011110001001000011011100011101011000101000010001001110110110100001001101011010010001000010001001100100010001011000110101000010011011000000110001000101000000001111001001101100110110111101110000010111100011000000001011110100010001000011010
Pair \(Z_2\) Length of longest common subsequence
4XNY_1,6BRD_1 180 4
4XNY_1,1WBD_1 175 4
6BRD_1,1WBD_1 145 4

Newick tree

 
[
	4XNY_1:93.55,
	[
		1WBD_1:72.5,6BRD_1:72.5
	]:21.05
]

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 \((0.85)(0.8)(\frac{830 }{\log_{20} 830}-\frac{354}{\log_{20}354})=128.\)
Status Protein1 Protein2 d d1/2
Query variables 4XNY_1 6BRD_1 162 143.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]