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Parikh vectors
3REJ_1 5MKV_1 3QFW_1 Letter Amino acid
19 40 66 A Alanine
1 17 10 N Asparagine
1 8 3 C Cysteine
7 36 22 I Isoleucine
6 35 17 S Serine
8 18 15 Q Glutamine
7 31 18 E Glutamic acid
6 41 37 G Glycine
13 28 7 K Lycine
2 11 7 M Methionine
4 17 12 F Phenylalanine
10 35 13 T Threonine
0 7 0 W Tryptophan
6 43 26 V Valine
4 34 16 D Aspartic acid
2 12 12 H Histidine
12 32 43 L Leucine
6 25 20 P Proline
18 21 26 R Arginine
3 13 8 Y Tyrosine 

3REJ_1|Chains A, E|Histone H3.2|Xenopus laevis (8355)
>5MKV_1|Chains A, B, C, D|Dihydropyrimidinase-related protein 2|Homo sapiens (9606)
>3QFW_1|Chains A, B|Ribulose-1,5-bisphosphate carboxylase/oxygenase large subunit|Rhodopseudomonas palustris (316056)
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 \(w=f(c)\)
3REJ , Knot 65 135 0.78 38 102 126 
ARTKQTARKSTGGKAPRKQLATKAARKSAPATGGVKKPHRYRPGTVALREIRRYQKSTELLIRKLPFQRLVREIAQDFKTDLRFQSSAVMALQEASEAYLVALFEDTNLCAIHAKRVTIMPKDIQLARRIRGERA
5MKV , Knot 207 504 0.85 40 252 476 
TSDRLLIKGGKIVNDDQSFYADIYMEDGLIKQIGENLIVPGGVKTIEAHSRMVIPGGIDVHTRFQMPDQGMTSADDFFQGTKAALAGGTTMIIDHVVPEPGTSLLAAFDQWREWADSKSCCDYSLHVDISEWHKGIQEEMEALVKDHGVNSFLVYMAFKDRFQLTDCQIYEVLSVIRDIGAIAQVHAENGDIIAEEQQRILDLGITGPEGHVLSRPEEVEAEAVNRAITIANQTNCPLYITKVMSKSSAEVIAQARKKGTVVYGEPITASLGTDGSHYWSKNWAKAAAFVTSPPLSPDPTTPDFLNSLLSCGDLQVTGSAHCTFNTAQKAVGKDNFTLIPEGTNGTEERMSVIWDKAVVTGKMDENQFVAVTSTNAAKVFNLYPRKGRIAVGSDADLVIWDPDSVKTISAKTHNSSLEYNIFEGMECRGSPLVVISQGKIVLEDGTLHVTEGSGRYIPRKPFPDFVYKRIKARSRLAELRGVPRGLYDGPVCEVSVTPKTVTPA
3QFW , Knot 153 378 0.80 38 182 345 
MSLTEPRIVATYHIASDAERIEQRALALAIEQSVECPLEAIGDPAIVANIVGRVEDVAELQPGRYAVRIGLAAATAPAEPGQLLNMLFGNSSIQPDIALADVELPAHYLTAFGGPRVGLAGIRTLTGAQSRALTASALKPQGLSPAALASIAHQLALGGVDLIKDDHGLADQAFSPFAERAAAVGKAVREANAARGGRTLYAPNISGTLDDMRRQLGVIRDEGIGAVLVAPMIVGVSNFHAIVKEAAGLVVVAHPAMAGAAKIAAPLLLGRLFRLFGADATVFPNYGGRFAYSTASCLALAQAARDPFGKLNACIPTPAGGIMLQRVNELLRFYGQDVMLLIGGSLLASRERLTEQASRFVNKVADYGQREGHHHHHH

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 \(c\).

\(|P_{f(3REJ_1)}(2) \setminus P_{f(5MKV_1)}(2)|=22\), \(|P_{f(5MKV_1)}(2) \setminus P_{f(3REJ_1)}(2)|=172\). 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 1:100000100001101100011001100011101110010000110111001000000001110011100110011001000101000111110010010111110000101101001011100101100101001
Pair \(Z_2\) Length of longest common subsequence
3REJ_1,5MKV_1 194 4
3REJ_1,3QFW_1 146 3
5MKV_1,3QFW_1 162 4

Newick tree

 
[
	5MKV_1:94.18,
	[
		3REJ_1:73,3QFW_1:73
	]:21.18
]

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 \((0.85)(0.8)(\frac{639 }{\log_{20} 639}-\frac{135}{\log_{20}135})=145.\)
Status Protein1 Protein2 d d1/2
Query variables 3REJ_1 5MKV_1 184 114.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]