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Parikh vectors
8SMK_1 5QOW_1 1WBD_1 Letter Amino acid
27 11 64 R Arginine
47 12 51 G Glycine
22 2 15 H Histidine
45 13 27 K Lycine
42 8 48 T Threonine
34 8 85 A Alanine
19 3 6 C Cysteine
35 10 22 F Phenylalanine
50 5 40 V Valine
40 13 42 D Aspartic acid
67 13 100 L Leucine
47 12 35 P Proline
11 3 5 W Tryptophan
14 5 26 Y Tyrosine
25 8 25 N Asparagine
31 2 40 Q Glutamine
43 11 64 E Glutamic acid
31 14 45 I Isoleucine
17 3 24 M Methionine
48 11 36 S Serine 

8SMK_1|Chains A, D|Protein-arginine deiminase type-4|Homo sapiens (9606)
>5QOW_1|Chain A|DCP2 (NUDT20)|Homo sapiens (9606)
>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 \(w=f(c)\)
8SMK , Knot 271 695 0.85 40 304 631 
HHHHHHASGGLNDIFEAQKIEWHEENLYFQGTSAQGTLIRVTPEQPTHAVCVLGTLTQLDICSSAPEDCTSFSINASPGVVVDIAHGPPAKKKSTGSSTWPLDPGVEVTLTMKAASGSTGDQKVQISYYGPKTPPVKALLYLTGVEISLCADITRTGKVKPTRAVKDQRTWTWGPCGQGAILLVNCDRDNLESSAMDCEDDEVLDSEDLQDMSLMTLSTKTPKDFFTNHTLVLHVARSEMDKVRVFQATRGKLSSKCSVVLGPKWPSHYLMVPGGKHNMDFYVEALAFPDTDFPGLITLTISLLDTSNLELPEAVVFQDSVVFRVAPWIMTPNTQPPQEVYACSIFENEDFLKSVTTLAMKAKCKLTICPEEENMDDQWMQDEMEIGYIQAPHKTLPVVFDSPRNRGLKEFPIKRVMGPDFGYVTRGPQTGGISGLDSFGNLEVSPPVTVRGKEYPLGRILFGDSCYPSNDSRQMHQALQDFLSAQQVQAPVKLYSDWLSVGHVDEFLSFVPAPDRKGFRLLLASPRSCYKLFQEQQNEGHGEALLFEGIKKKKQQKIKNILSNKTLREHNSFVERCIDWNRELLKRELGLAESDIIDIPQLFKLKEFSKAEAFFPNMVNMLVLGKHLGIPKPFGPVINGRCCLEEKVCSLLEPLGLQCTFINDFFTYHIRHGEVHCGTNVRRKPFSFKWWNMVP
5QOW , Knot 81 167 0.82 40 127 164 
SMGVPTYGAIILDETLENVLLVQGYLAKSGWGFPKGKVNKEEAPHDCAAREVFEETGFDIKDYICKDDYIELRINDQLARLYIIPGIPKDTKFNPKTRREIRNIEWFSIEKLPCHRNDMTPKSKLGLAPNKFFMAIPFIRPLRDWLSRRFGDSSDSDNGFSSTGSTP
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 \(c\).

\(|P_{f(8SMK_1)}(2) \setminus P_{f(5QOW_1)}(2)|=192\), \(|P_{f(5QOW_1)}(2) \setminus P_{f(8SMK_1)}(2)|=15\). 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 1:00000010111001101001010000101010010101101010010011011101001010001100000101010111110110111100000100011101110101010110100100010100011001110111010110101010100010101001100000101110101111110000001000110000001100001001011010000100110000111011000100101101001010000011111011000111111000101010111110001111101010110000101101111000111011111101000110010100110000110010011101000101010000100011000101101011000111110010001100111001111011010011001110110011010101110101000111011110000100000010011001101001011101000110110100110111110001101111010000011000000101011110110000000100110000100000110001010001100011110001101101101001001011110110111110011110111111010001000100110111100011001100010010100100100011010110111
Pair \(Z_2\) Length of longest common subsequence
8SMK_1,5QOW_1 207 4
8SMK_1,1WBD_1 121 5
5QOW_1,1WBD_1 206 4

Newick tree

 
[
	5QOW_1:11.99,
	[
		8SMK_1:60.5,1WBD_1:60.5
	]:53.49
]

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 \((0.85)(0.8)(\frac{862 }{\log_{20} 862}-\frac{167}{\log_{20}167})=193.\)
Status Protein1 Protein2 d d1/2
Query variables 8SMK_1 5QOW_1 248 151.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]