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Parikh vectors
1CGI_1 7DDV_1 1TOJ_1 Letter Amino acid
4 5 11 Y Tyrosine
23 17 26 V Valine
9 15 21 D Aspartic acid
6 5 20 F Phenylalanine
28 10 22 S Serine
8 0 5 W Tryptophan
22 27 46 A Alanine
10 1 5 C Cysteine
10 6 16 Q Glutamine
10 12 17 I Isoleucine
14 4 18 K Lycine
9 9 14 P Proline
4 18 22 R Arginine
14 2 21 N Asparagine
5 11 27 E Glutamic acid
23 16 30 G Glycine
2 19 6 H Histidine
19 16 38 L Leucine
2 3 8 M Methionine
23 20 23 T Threonine 

1CGI_1|Chain A[auth E]|ALPHA-CHYMOTRYPSINOGEN|Bos taurus (9913)
>7DDV_1|Chain A|Imidazoleglycerol-phosphate dehydratase|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
>1TOJ_1|Chain A|Aspartate aminotransferase|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 \(w=f(c)\)
1CGI , Knot 111 245 0.83 40 154 236 
CGVPAIQPVLSGLSRIVNGEEAVPGSWPWQVSLQDKTGFHFCGGSLINENWVVTAAHCGVTTSDVVVAGEFDQGSSSEKIQKLKIAKVFKNSKYNSLTINNDITLLKLSTAASFSQTVSAVCLPSASDDFAAGTTCVTTGWGLTRYTNANTPDRLQQASLPLLSNTNCKKYWGTKIKDAMICAGASGVSSCMGDSGGPLVCKKNGAWTLVGIVSWGSSTCSTSTPGVYARVTALVNWVQQTLAAN
7DDV , Knot 92 216 0.76 38 135 200 
MHHHHHHTTTQTAKASRRARIERRTRESDIVIELDLDGTGQVAVDTGVPFYDHMLTALGSHASFDLTVRATGDVEIEAHHTIEDTAIALGTALGQALGDKRGIRRFGDAFIPMDETLAHAAVDLSGRPYCVHTGEPDHLQHTTIAGSSVPYHTVINRHVFESLAANARIALHVRVLYGRDPHHITEAQYKAVARALRQAVEPDPRVSGVPSTKGAL
1TOJ , Knot 168 396 0.84 40 218 372 
MFENITATTADPILGLADLFRADERPGKIDLGIGVYKDETGKTPVLTSVKKAEQYLLENETTKNYLGIDGIPEFGRCTQELLFGKGSALINDKRARTAQTPGGSGALRVAADFLAKNTSVKRVWVSNPSWPNHKSVFNSAGLEVREYAYYDAENHTLDFDALINSLNEAQAGDVVLFHGCCHNPTGIDPTLEQWQTLAQLSVEKGWLPLFDFAYQGFARGLEEDAEGLRAFAAMHKELIVASSYSKNFALYNERVGACTLVAADSETVDRAFGQMKAAIRANYSSPPAHGASVVATILSNDALRAIWEQELTDMRQRIQRMRQLFVNTLQEKGANRDFSFIIKQNGMFSFSGLTKEQVLRLREEFGVYAVASGRVNVAGMTPDNMAPLCEAIVAVL

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 \(c\).

\(|P_{f(1CGI_1)}(2) \setminus P_{f(7DDV_1)}(2)|=94\), \(|P_{f(7DDV_1)}(2) \setminus P_{f(1CGI_1)}(2)|=75\). 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 1:01111101110110011010011110111010100001101011011000111011001100001111101001000001001011011000000010100010110100110100010110110100011110001001111000001001001001011110000000011001001110111011000110011111000011101111101100000000111010101110110001110
Pair \(Z_2\) Length of longest common subsequence
1CGI_1,7DDV_1 169 4
1CGI_1,1TOJ_1 168 4
7DDV_1,1TOJ_1 169 3

Newick tree

 
[
	7DDV_1:84.66,
	[
		1CGI_1:84,1TOJ_1:84
	]:0.66
]

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 \((0.85)(0.8)(\frac{461 }{\log_{20} 461}-\frac{216}{\log_{20}216})=71.2\)
Status Protein1 Protein2 d d1/2
Query variables 1CGI_1 7DDV_1 92 86
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]