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Parikh vectors
6SBG_1 3OCS_1 3GDN_1 Letter Amino acid
13 12 26 F Phenylalanine
13 15 25 Y Tyrosine
17 8 28 N Asparagine
21 24 23 E Glutamic acid
14 13 24 I Isoleucine
9 19 19 K Lycine
23 10 36 T Threonine
15 6 4 W Tryptophan
18 18 44 V Valine
5 6 5 C Cysteine
20 11 12 Q Glutamine
24 16 48 G Glycine
10 6 14 H Histidine
37 27 48 L Leucine
17 8 32 P Proline
17 13 17 R Arginine
15 14 30 D Aspartic acid
22 20 53 S Serine
54 11 30 A Alanine
3 14 3 M Methionine 

6SBG_1|Chain A|MstE|Scytonema sp. PCC 10023 (1680591)
>3OCS_1|Chain A|Tyrosine-protein kinase BTK|Homo sapiens (9606)
>3GDN_1|Chains A, B|R-oxynitrile lyase isoenzyme 1|Prunus dulcis (3755)
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 \(w=f(c)\)
6SBG , Knot 153 367 0.82 40 209 339 
GSTLQPLENSTRQEKLLYPKLNQLSNSINAAVAFLLEARNLEGWWQDFNFPQAASIGDEWVTAYVGTMLATLPYAHVHEALMQAWELLKIRDHRPTGEWGYNYILCGDADTTGWALQLAAAVGASDSERAQQARAALATHLQPNGGIATFAEESIRAYIKVPDLANVSFQGWCGAHTCVSAAVAALPEFRSRLHDYLRVTQTSQGNWEGYWWSDHEYTTALTAEALAAGGQAADQPSIEQAVAWGLKRLCPQGFVATSKHPNGSTFATAWCLRLLLLNTVDAEVKAARAAAIGWLLEQQRPNGSWVSSAYLRIPYPFDRNPNQFPHWRYYDEIEGDKAFEGSIIFDHNSIFTTATVVNSLVKAAPML
3OCS , Knot 126 271 0.86 40 189 260 
MGSWEIDPKDLTFLKELGTGQFGVVKYGKWRGQYDVAIKMIKEGSMSEDEFIEEAKVMMNLSHEKLVQLYGVCTKQRPIFIITEYMANGCLLNYLREMRHRFQTQQLLEMCKDVCEAMEYLESKQFLHRDLAARNCLVNDQGVVKVSDFGLSRYVLDDEYTSSVGSKFPVRWSPPEVLMYSKFSSKSDIWAFGVLMWEIYSLGKMPYERFTNSETAEHIAQGLRLYRPHLASEKVYTIMYSCWHEKADERPTFKILLSNILDVMDENLYGQ
3GDN , Knot 205 521 0.82 40 239 478 
LATTSDHDFSYLSFAYDATDLELEGSYDYVIVGGGTSGCPLAATLSEKYKVLVLERGSLPTAYPNVLTADGFVYNLQQEDDGKTPVERFVSEDGIDNVRGRVLGGTSIINAGVYARANTSIYSASGVDWDMDLVNQTYEWVEDTIVYKPNSQSWQSVTKTAFLEAGVHPNHGFSLDHEEGTRITGSTFDNKGTRHAADELLNKGNSNNLRVGVHASVEKIIFSNAPGLTATGVIYRDSNGTPHQAFVRSKGEVIVSAGTIGTPQLLLLSGVGPESYLSSLNIPVVLSHPYVGQFLHDNPRNFINILPPNPIEPTIVTVLGISNDFYQCSFSSLPFTTPPFGFFPSSSYPLPNSTFAHFASKVAGPLSYGSLTLKSSSNVRVSPNVKFNYYSNLTDLSHCVSGMKKIGELLSTDALKPYKVEDLPGVEGFNILGIPLPKDQTDDAAFETFCRESVASYWHYHGGCLVGKVLDGDFRVTGINALRVVDGSTFPYTPASHPQGFYLMLGRYVGIKILQERSASD

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 \(c\).

\(|P_{f(6SBG_1)}(2) \setminus P_{f(3OCS_1)}(2)|=103\), \(|P_{f(3OCS_1)}(2) \setminus P_{f(6SBG_1)}(2)|=83\). 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 1:1001011000000001101010010001011111110100101110010110110110011010110111011010100111011011010000101011000110101000111101111111000001001011110010101111011000101010110110101011011000101111111010001000101000001010101100000001101011111101100101001111110010101111000010100110110101111001010101101111111100001010110010101101100010011010000010100110101110000110010110011011111
Pair \(Z_2\) Length of longest common subsequence
6SBG_1,3OCS_1 186 3
6SBG_1,3GDN_1 142 4
3OCS_1,3GDN_1 180 4

Newick tree

 
[
	3OCS_1:97.39,
	[
		6SBG_1:71,3GDN_1:71
	]:26.39
]

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 \((0.85)(0.8)(\frac{638 }{\log_{20} 638}-\frac{271}{\log_{20}271})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 6SBG_1 3OCS_1 129 113
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]