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Parikh vectors
5WDO_1 3DQJ_1 7PCG_1 Letter Amino acid
4 13 8 N Asparagine
13 16 19 E Glutamic acid
11 21 37 L Leucine
15 17 30 V Valine
12 7 16 Q Glutamine
11 12 13 I Isoleucine
3 11 16 P Proline
11 14 10 T Threonine
9 12 13 Y Tyrosine
11 6 31 R Arginine
8 20 4 K Lycine
0 1 1 W Tryptophan
4 7 7 M Methionine
6 13 13 F Phenylalanine
8 9 10 S Serine
11 9 57 A Alanine
14 20 27 D Aspartic acid
3 2 2 C Cysteine
12 24 34 G Glycine
4 9 10 H Histidine 

5WDO_1|Chain A|GTPase HRas|Homo sapiens (9606)
>3DQJ_1|Chain A|Green fluorescent protein|Aequorea victoria (6100)
>7PCG_1|Chains A, B|Ketol-acid reductoisomerase|Burkholderia thailandensis (strain ATCC 700388 / DSM 13276 / CIP 106301 / E264) (271848)
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 \(w=f(c)\)
5WDO , Knot 81 170 0.81 38 133 167 
FQGHMTEYKLVVVGAGGVGKSALTIQLIQNHFVDEYDPTIEDSYRKQVVIDGETCLLDILDTAGQEEYSAMRDQYMRTGEGFLCVFAINNTKSFEDIHQYREQIKRVKDSDDVPMVLVGNKCDLAARTVESRQAQDLARSYGIPYIETSAKTRQGVEDAFYTLVREIRQH
3DQJ , Knot 115 243 0.86 40 162 231 
GDDPMVSKGEELFTGVVPILVELDGDVNGHKFSVSGEGEGDATYGKLTLKFICTTGKLPVPWPTLVTTFGYGLMCFARYPDHMKQHDFFKSAMPEGYVQERTIFFKDDGNYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNYNSHNVYIMADKQKNGIKVNFKIRHNIEDGSVQLADHYQQNTPIGDGPVLLPDNHYLSYQSALSKDPNEKRDHMVLLEFVTAAGITLGMDELYK
7PCG , Knot 145 358 0.79 40 179 327 
GSHMASNDLIYQDEHASLQPLEGRTVAVIGYGIQGRAFAANLRDSGVAVRVGNIDDRYFELARAEGHRVTNIAEAVAHADIVLLLIPDEAHGAVFDVDIAPNLRDGALLCVAHGHSLVQGDVRPLPGRDLAMLAPRMYGDPIRRYYLAGQGAPAYFDIVADHTGRARDRVLAIARAVGFTRAGVMALGYRQETFLDLFQEQFLAPALVDLVETGFQVLVERGFNPKAALLEVYGSGEMGKMMLDGADIGLDEVVALQGSPTCQVGYHRWRGRTLPTAVRELAARVLDQIEGGDFSAYLKEQASNDYASLDDARRAALKRPLNVAHAQVRAAFRFPTEAAGGLYQAAQAPADVEPEAAR

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 \(c\).

\(|P_{f(5WDO_1)}(2) \setminus P_{f(3DQJ_1)}(2)|=65\), \(|P_{f(3DQJ_1)}(2) \setminus P_{f(5WDO_1)}(2)|=94\). 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 1:10101000011111111110011010110001100001010000000111010001101100110000011000010010111011110000010010000001001000001111111000011100100001001100011101000100001100110011001000
Pair \(Z_2\) Length of longest common subsequence
5WDO_1,3DQJ_1 159 3
5WDO_1,7PCG_1 154 4
3DQJ_1,7PCG_1 181 3

Newick tree

 
[
	3DQJ_1:87.73,
	[
		5WDO_1:77,7PCG_1:77
	]:10.73
]

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 \((0.85)(0.8)(\frac{413 }{\log_{20} 413}-\frac{170}{\log_{20}170})=72.2\)
Status Protein1 Protein2 d d1/2
Query variables 5WDO_1 3DQJ_1 92 78.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]