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Parikh vectors
7BDG_1 5EZX_1 9JGM_1 Letter Amino acid
22 22 0 E Glutamic acid
17 7 0 H Histidine
13 19 0 Y Tyrosine
25 34 0 V Valine
14 6 25 C Cysteine
16 15 0 N Asparagine
12 23 0 I Isoleucine
11 20 0 F Phenylalanine
29 18 0 R Arginine
40 29 0 L Leucine
13 14 0 K Lycine
30 30 0 S Serine
21 23 0 T Threonine
13 7 0 W Tryptophan
30 34 31 G Glycine
18 21 0 D Aspartic acid
15 16 0 Q Glutamine
4 8 0 M Methionine
20 19 0 P Proline
20 25 27 A Alanine 

7BDG_1|Chain A|Palmitoleoyl-protein carboxylesterase NOTUM|Homo sapiens (9606)
>5EZX_1|Chain A|Beta-secretase 1|Homo sapiens (9606)
>9JGM_1|Chains A, B, C, D|yybP riboswitch|Escherichia coli K-12 (83333)
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 \(w=f(c)\)
7BDG , Knot 162 383 0.83 40 219 364 
ETGSAQQLNEDLRLHLLLNTSVTCNDGSPAGYYLKESRGSRRWLLFLEGGWYCFNRENCDSRYDTMRRLMSSRDWPRTRTGTGILSSQPEENPYWWNANMVFIPYCSSDVWSGASSKSEKNEYAFMGALIIQEVVRELLGRGLSGAKVLLLAGSSAGGTGVLLNVDRVAEQLEKLGYPAIQVRGLADSGWFLDNKQYRHTDCVDTITCAPTEAIRRGIRYWNGVVPERCRRQFQEGEEWNCFFGYKVYPTLRSPVFVVQWLFDEAQLTVDNVHLTGQPVQEGLRLYIQNLGRELRHTLKDVPASFAPACLSHEIIIRSHWTDVQVKGTSLPRALHCWDRSLHDSHKASKTPLKGCPVHLVDSCPWPHCNPSCPTGTKHHHHHH
5EZX , Knot 166 390 0.84 40 226 375 
RGSFVEMVDNLRGKSGQGYYVEMTVGSPPQTLNILVDTGSSNFAVGAAPHPFLHRYYQRQLSSTYRDLRKGVYVPYTQGKWEGELGTDLVSIPHGPNVTVRANIAAITESDKFFINGSNWEGILGLAYAEIARPDDSLEPFFDSLVKQTHVPNLFSLQLCGAGFPLNQSEVLASVGGSMIIGGIDHSLYTGSLWYTPIRREWYYEVIIVRVEINGQDLKMDCKEYNYDKSIVDSGTTNLRLPKKVFEAAVASIKAASSTEKFPDGFWLGEQLVCWQAGTTPWNIFPVISLYLMGEVTNQSFRITILPQQYLRPVEDVATSQDDCYKFAISQSSTGTVMGAVIMEGFYVVFDRARKRIGFAVSACHVHDEFRTAAVEGPFVTLDMEDCGYN
9JGM , Knot 30 105 0.44 8 16 49 
GGAAGGGGAGUAACUUCAUUGCCGGUCGAUCGUCAUUACGAUGUGUGAAAACACAUCCGGUCACCGGGCAACCUGCCGAAAGGCAGGUUGUAAGUGAGACCUUCC

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 \(c\).

\(|P_{f(7BDG_1)}(2) \setminus P_{f(5EZX_1)}(2)|=88\), \(|P_{f(5EZX_1)}(2) \setminus P_{f(7BDG_1)}(2)|=95\). 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 1:00101001000101011100010000101110010000100011111011100100000000000100110000110000101110001000101101011111000001101100000000011111111001100111011011011111100111011110100110010011011101011100111100000000001001001100110011001011110000001001001001110010101001111101110010101001010101100110101001100100010011101111010001110001001010100110110010001000001000110101101100011100010010100000000
Pair \(Z_2\) Length of longest common subsequence
7BDG_1,5EZX_1 183 3
7BDG_1,9JGM_1 223 3
5EZX_1,9JGM_1 230 3

Newick tree

 
[
	9JGM_1:11.64,
	[
		7BDG_1:91.5,5EZX_1:91.5
	]:28.14
]

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 \((0.85)(0.8)(\frac{773 }{\log_{20} 773}-\frac{383}{\log_{20}383})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 7BDG_1 5EZX_1 135 134.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]