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Parikh vectors
7FQB_1 4OIK_1 7ZEZ_1 Letter Amino acid
9 8 5 R Arginine
5 2 3 H Histidine
12 4 7 I Isoleucine
4 1 2 Y Tyrosine
12 4 10 E Glutamic acid
10 3 7 G Glycine
10 3 3 P Proline
11 6 7 L Leucine
6 7 5 K Lycine
5 2 3 M Methionine
6 4 6 F Phenylalanine
6 2 5 T Threonine
13 4 12 A Alanine
2 6 0 C Cysteine
4 3 1 Q Glutamine
5 1 0 W Tryptophan
11 3 6 V Valine
6 2 3 N Asparagine
13 1 7 D Aspartic acid
9 8 1 S Serine 

7FQB_1|Chain A|Dihydrofolate reductase|Escherichia coli K-12 (83333)
>4OIK_1|Chains A, B|C-C motif chemokine 1|Homo sapiens (9606)
>7ZEZ_1|Chain A|Isoform 3 of Peptidyl-prolyl cis-trans isomerase E|Homo sapiens (9606)
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 \(w=f(c)\)
7FQB , Knot 81 159 0.86 40 124 153 
MISLIAALAVDRVIGMENAMPWNLPADLAWFKRNTLNKPVIMGRHTWESIGRPLPGRKNIILSSQPGTDDRVTWVKSVDEAIAACGDVPEIMVIGGGRVYEQFLPKAQKLYLTHIDAEVEGDTHFPDYEPDDWESVFSEFHDADAQNSHSYCFEILERR
4OIK , Knot 42 74 0.81 40 67 72 
SKSMQVPFSRCCFSFAEQEIPLRAILCYRNTSSICSNEGLIFKLKRGKEACALDTVGWVQRHRKMLRHCPSKRK
7ZEZ , Knot 49 93 0.79 36 79 89 
AGHMATTKRVLYVGGLAEEVDDKVLHAAFIPFGDITDIQIPLDYETEKHRGFAFVEFELAEDAAAAIDNMNESELFGRTIRVNLAKPMRIKEG

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 \(c\).

\(|P_{f(7FQB_1)}(2) \setminus P_{f(4OIK_1)}(2)|=98\), \(|P_{f(4OIK_1)}(2) \setminus P_{f(7FQB_1)}(2)|=41\). 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 1:110111111100111100111101110111100001001111100010011011110001110001100001011001001111010110111111101000111010010100101010100011000100100110010010100000001011000
Pair \(Z_2\) Length of longest common subsequence
7FQB_1,4OIK_1 139 3
7FQB_1,7ZEZ_1 149 3
4OIK_1,7ZEZ_1 116 3

Newick tree

 
[
	7FQB_1:76.15,
	[
		4OIK_1:58,7ZEZ_1:58
	]:18.15
]

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 \((0.85)(0.8)(\frac{233 }{\log_{20} 233}-\frac{74}{\log_{20}74})=52.0\)
Status Protein1 Protein2 d d1/2
Query variables 7FQB_1 4OIK_1 66 48
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]