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Parikh vectors
1GLG_1 7FZM_1 7UZH_1 Letter Amino acid
31 14 35 K Lycine
7 6 21 F Phenylalanine
5 2 6 W Tryptophan
43 6 43 A Alanine
31 11 36 D Aspartic acid
15 2 18 Q Glutamine
3 2 9 H Histidine
10 1 31 P Proline
12 10 52 S Serine
6 6 33 R Arginine
22 4 16 N Asparagine
24 5 49 L Leucine
7 2 25 Y Tyrosine
26 17 51 V Valine
0 2 6 C Cysteine
12 9 45 E Glutamic acid
21 11 50 G Glycine
16 8 39 I Isoleucine
6 5 22 M Methionine
12 12 30 T Threonine 

1GLG_1|Chain A|GALACTOSE/GLUCOSE-BINDING PROTEIN|Escherichia coli (562)
>7FZM_1|Chain A|Fatty acid-binding protein, adipocyte|Homo sapiens (9606)
>7UZH_1|Chains A, B, C|ATPase H+-transporting V1 subunit A|Rattus norvegicus (10116)
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 \(w=f(c)\)
1GLG , Knot 132 309 0.81 38 166 285 
ADTRIGVTIYKYDDNFMSVVRKAIEQDAKAAPDVQLLMNDSQNDQSKQNDQIDVLLAKGVKALAINLVDPAAAGTVIEKARGQNVPVVFFNKEPSRKALDSYDKAYYVGTDSKESGIIQGDLIAKHWAANQGWDLNKDGQIQFVLLKGEPGHPDAEARTTYVIKELNDKGIKTEQLQLDTAMWDTAQAKDKMDAWLSGPNANKIEVVIANNDAMAMGAVEALKAHNKSSIPVFGVDALPEALALVKSGALAGTVLNDANNQAKATFDLAKNLADGKGAADGTNWKIDNKVVRVPYVGVDKDNLAEFSKK
7FZM , Knot 69 135 0.83 40 104 129 
GSHMCDAFVGTWKLVSSENFDDYMKEVGVGFATRKVAGMAKPNMIISVNGDVITIKSESTFKNTEISFILGQEFDEVTADDRKVKSTITLDGGVLVHVQKWDGKSTTIKRKREDDKLVVECVMKGVTSTRVYERA
7UZH , Knot 241 617 0.83 40 272 573 
MDFSKLPKIRDEDKESTFGYVHGVSGPVVTACDMAGAAMYELVRVGHSELVGEIIRLEGDMATIQVYEETSGVSVGDPVLRTGKPLSVELGPGIMGAIFDGIQRPLSDISSQTQSIYIPRGVNVSALSRDIKWEFIPSKNLRVGSHITGGDIYGIVNENSLIKHKIMLPPRSRGSVTYIAPPGNYDASDVVLELEFEGVKEKLSMVQVWPVRQVRPVTEKLPANHPLLTGQRVLDALFPCVQGGTTAIPGAFGCGKTVISQSLSKYSNSDVIIYVGCGERGNEMSEVLRDFPELTMEVDGKVESIMKRTALVANTSNMPVAAREASIYTGITLSEYFRDMGYHVSMMADSTSRWAEALREISGRLAEMPADSGYPAYLGARLASFYERAGRVKCLGNPEREGSVSIVGAVSPPGGDFSDPVTSATLGIVQVFWGLDKKLAQRKHFPSVNWLISYSKYMRALDEYYDKHFTEFVPLRTKAKEILQEEEDLAEIVQLVGKASLAETDKITLEVAKLIKDDFLQQNGYTPYDRFCPFYKTVGMLSNMISFYDMARRAVETTAQSDNKITWSIIREHMGEILYKLSSMKFKDPVKDGEAKIKADYAQLLEDMQNAFRSLED

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 \(c\).

\(|P_{f(1GLG_1)}(2) \setminus P_{f(7FZM_1)}(2)|=115\), \(|P_{f(7FZM_1)}(2) \setminus P_{f(1GLG_1)}(2)|=53\). 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 1:100011101000000110110011000101110101110000000000000101111011011110110111110110010100111111000100011000001001100000011101011100111001101000101011110101101010100001100100011000010100111001010001011101101001011110001111111011010000011111101110111110011111011001000101010110011010111010010100011011011100001101000
Pair \(Z_2\) Length of longest common subsequence
1GLG_1,7FZM_1 168 3
1GLG_1,7UZH_1 186 3
7FZM_1,7UZH_1 194 4

Newick tree

 
[
	7UZH_1:98.42,
	[
		1GLG_1:84,7FZM_1:84
	]:14.42
]

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 \((0.85)(0.8)(\frac{444 }{\log_{20} 444}-\frac{135}{\log_{20}135})=92.3\)
Status Protein1 Protein2 d d1/2
Query variables 1GLG_1 7FZM_1 116 84.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]