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Parikh vectors
5LBT_1 5CSX_1 8HMO_1 Letter Amino acid
13 10 6 F Phenylalanine
18 21 11 G Glycine
14 19 15 K Lycine
11 12 10 P Proline
14 25 8 S Serine
12 14 6 Q Glutamine
21 25 16 L Leucine
10 19 22 I Isoleucine
15 12 14 T Threonine
16 16 13 V Valine
8 18 6 R Arginine
11 15 18 D Aspartic acid
4 3 2 C Cysteine
15 16 16 E Glutamic acid
12 9 14 H Histidine
5 10 7 M Methionine
6 5 1 W Tryptophan
8 8 5 Y Tyrosine
18 14 15 A Alanine
6 11 8 N Asparagine 

5LBT_1|Chains A, B|Ribosyldihydronicotinamide dehydrogenase [quinone]|Homo sapiens (9606)
>5CSX_1|Chain A|Serine/threonine-protein kinase B-raf|Homo sapiens (9606)
>8HMO_1|Chains A, B, C, D, E, F|Metal-dependent hydrolase|Bacillus smithii (1479)
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 \(w=f(c)\)
5LBT , Knot 110 237 0.84 40 171 228 
MAGKKVLIVYAHQEPKSFNGSLKNVAVDELSRQGCTVTVSDLYAMNLEPRATDKDITGTLSNPEVFNYGVETHEAYKQRSLASDITDEQKKVREADLVIFQFPLYWFSVPAILKGWMDRVLCQGFAFDIPGFYDSGLLQGKLALLSVTTGGTAEMYTKTGVNGDSRYFLWPLQHGTLHFCGFKVLAPQISFAPEIASEEERKGMVAAWSQRLQTIWKEEPIPCTAHWHFGQHHHHHH
5CSX , Knot 129 282 0.86 40 188 274 
GPRDSSDDWEIPDGQITVGQRIGSGSFGTVYKGKWHGDVAVKMLNVTAPTPQQLQAFKNEVGVLRKTRHVNILLFMGYSTKPQLAIVTQWCEGSSLYHHLHASETKFEMKKLIDIARQTARGMDYLHAKSIIHRDLKSNNIFLHEDNTVKIGDFGLATVKSRWSGSHQFEQLSGSILWMAPEVIRMQDSNPYSFQSDVYAFGIVLYELMTGQLPYSNINNRDQIIEMVGRGSLSPDLSKVRSNCPKRMKRLMAECLKKKRDERPSFPRILAEIEELARELSG
8HMO , Knot 100 213 0.84 40 149 202 
MKIIDITAPIYEGMPVYKNKPEKQPSITTQTNGHVTESRICMDVHTGTHVDAPLHMMNDGKTIETISIEKLVRPCKVIDLTHVHEKITKSDVENADIQKDDFILLKTKNSFDKEFNFDFIYLAEDAARYLAEIGIAGVGIDSLGIERAQPEHPTHRALMDKDIVVIEGLQLADVEEGSYFMIAAPLNIQGTDASPARVLLLDNWKLEHHHHHH

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 \(c\).

\(|P_{f(5LBT_1)}(2) \setminus P_{f(5CSX_1)}(2)|=72\), \(|P_{f(5CSX_1)}(2) \setminus P_{f(5LBT_1)}(2)|=89\). 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 1:111001111010001001010100111001000100101001011010101000010101001011001100001000001100100000010010111101110110111110111001100111101111000111010111101001101010000110100001111100101010110111101011101100000011111100010011000111001010110000000
Pair \(Z_2\) Length of longest common subsequence
5LBT_1,5CSX_1 161 3
5LBT_1,8HMO_1 160 6
5CSX_1,8HMO_1 175 4

Newick tree

 
[
	5CSX_1:85.38,
	[
		5LBT_1:80,8HMO_1:80
	]:5.38
]

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 \((0.85)(0.8)(\frac{519 }{\log_{20} 519}-\frac{237}{\log_{20}237})=80.8\)
Status Protein1 Protein2 d d1/2
Query variables 5LBT_1 5CSX_1 102 93.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]