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Parikh vectors
5HBP_1 5YHR_1 5YQR_1 Letter Amino acid
8 12 12 D Aspartic acid
1 3 6 M Methionine
18 16 21 A Alanine
5 1 22 N Asparagine
8 5 19 K Lycine
3 1 14 F Phenylalanine
1 2 5 W Tryptophan
2 4 12 Y Tyrosine
3 4 15 V Valine
5 2 20 R Arginine
6 6 9 Q Glutamine
9 11 14 E Glutamic acid
10 1 3 H Histidine
10 6 17 I Isoleucine
14 4 25 L Leucine
8 6 15 S Serine
1 1 2 C Cysteine
6 5 15 G Glycine
6 0 7 P Proline
3 7 23 T Threonine 

5HBP_1|Chain A|Inner membrane protein YgaP|Escherichia coli (562)
>5YHR_1|Chains A, B|Anti-CRISPR protein 30|Pseudomonas phage D3112 (10708)
>5YQR_1|Chain A|Endolysin/Membrane-anchored lipid-binding protein LAM6 fusion protein|Enterobacteria phage T4 (10665)
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 \(w=f(c)\)
5HBP , Knot 61 127 0.77 40 95 120 
MKSSHHHHHHENLYFQSNAALTTISPHDAQELIARGAKLIDIRDADEYLREHIPEADLAPLSVLEQSGLPAKLRHEQIIFHCQAGKRTSNNADKLAAIAAPAEIFLLEDGIDGWKKAGLPVAVNKSQ
5YHR , Knot 50 97 0.78 38 75 92 
GMTKTAQMIAQQHKDTVAACEAAEAIAIAKDQVWDGEGYTKYTFDDNSVLIQSGTTQYAMDADDADSIKGYADWLDDEARSAEASEIERLLESVEEE
5YQR , Knot 121 276 0.82 40 180 264 
GSNIFEMLRIDEGLRLKIYKNTEGYYTIGIGHLLTKSPSLNAAKSELDKAIGRNTNGVITKDEAEKLFNQDVDAAVRGILRNAKLKPVYDSLDAVRRAALINMVFQMGETGVAGFTNSLRMLQQKRWDEAAVNLAKSRWYNQTPNRAKRVITTFRTGTWDAYGSEANKKFRQMFKPLAPNTRLITDYFCYFHREFPYQGRIYLSNTHLCFNSTVLNWMAKLQIPLNEIKYLDKVTTNSSAISVETVTNRYTFSGFIARDEVFQLITRVWSKENLTN

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 \(c\).

\(|P_{f(5HBP_1)}(2) \setminus P_{f(5YHR_1)}(2)|=73\), \(|P_{f(5YHR_1)}(2) \setminus P_{f(5HBP_1)}(2)|=53\). 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 1:1000000000001010001110010100100111011011010010001000110101111011000111101000011100011000000100111111110111100110110011111110000
Pair \(Z_2\) Length of longest common subsequence
5HBP_1,5YHR_1 126 3
5HBP_1,5YQR_1 159 3
5YHR_1,5YQR_1 177 3

Newick tree

 
[
	5YQR_1:90.06,
	[
		5HBP_1:63,5YHR_1:63
	]:27.06
]

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 \((0.85)(0.8)(\frac{224 }{\log_{20} 224}-\frac{97}{\log_{20}97})=41.1\)
Status Protein1 Protein2 d d1/2
Query variables 5HBP_1 5YHR_1 50 44.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]