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Parikh vectors
1VHD_1 6FHL_1 7CNB_1 Letter Amino acid
23 29 4 A Alanine
35 28 3 E Glutamic acid
17 30 7 I Isoleucine
25 23 6 S Serine
12 16 7 Y Tyrosine
1 11 4 Q Glutamine
13 9 2 H Histidine
37 26 9 L Leucine
33 19 7 K Lycine
3 4 1 W Tryptophan
29 21 8 V Valine
23 27 7 T Threonine
15 12 7 F Phenylalanine
17 19 1 P Proline
14 18 6 R Arginine
11 12 11 N Asparagine
19 22 8 D Aspartic acid
2 5 0 C Cysteine
30 28 4 G Glycine
12 16 5 M Methionine 

1VHD_1|Chains A, B|alcohol dehydrogenase, iron-containing|Thermotoga maritima (2336)
>6FHL_1|Chains A, B, C, D, E|Actin, alpha skeletal muscle|Oryctolagus cuniculus (9986)
>7CNB_1|Chain A|DNA packaging protein|Bacillus phage phi29 (10756)
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 \(w=f(c)\)
1VHD , Knot 151 371 0.80 40 192 341 
MSLWEFYMPTDVFFGEKILEKRGNIIDLLGKRALVVTGKSSSKKNGSLDDLKKLLDETEISYEIFDEVEENPSFDNVMKAVERYRNDSFDFVVGLGGGSPMDFAKAVAVLLKEKDLSVEDLYDREKVKHWLPVVEIPTTAGTGSEVTPYSILTDPEGNKRGCTLMFPVYAFLDPRYTYSMSDELTLSTGVDALSHAVEGYLSRKSTPPSDALAIEAMKIIHRNLPKAIEGNREARKKMFVASCLAGMVIAQTGTTLAHALGYPLTTEKGIKHGKATGMVLPFVMEVMKEEIPEKVDTVNHIFGGSLLKFLKELGLYEKVAVSSEELEKWVEKGSRAKHLKNTPGTFTPEKIRNIYREALGVEGGSHHHHHH
6FHL , Knot 159 375 0.83 40 219 360 
DEDETTALVCDNGSGLVKAGFAGDDAPRAVFPSIVGRPRHQGVMVGMGQKDSYVGDEAQSKRGILTLKYPIEHGIITNWDDMEKIWHHTFYNELRVAPEEHPTLLTEAPLNPKANREKMTQIMFETFNVPAMYVAIQAVLSLYASGRTTGIVLDSGDGVTHNVPIYEGYALPHAIMRLDLAGRDLTDYLMKILTERGYSFVTTAEREIVRDIKEKLCYVALDFENEMATAASSSSLEKSYELPDGQVITIGNERFRCPETLFQPSFIGMESAGIHETTYNSIMKCDIDIRKDLYANNVMSGGTTMYPGIADRMQKEITALAPSTMKIKIIAPPERKYSVWIGGSILASLSTFQQMWITKQEYDEAGPSIVHRKCF
7CNB , Knot 59 107 0.86 38 94 105 
DSQVFIEKRSKDSKFVFSIVYNGFTLGVWVDVNQGLMYIDTAHDPSTKNVYTLTTDDLNENMMLITNYKNNYHLRKLASAFMNGYLRFDNQVIRNIAYELFRKMRIQ

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 \(c\).

\(|P_{f(1VHD_1)}(2) \setminus P_{f(6FHL_1)}(2)|=70\), \(|P_{f(6FHL_1)}(2) \setminus P_{f(1VHD_1)}(2)|=97\). 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 1:10110101100111100110001011011100111101000000010100100110000100011001000101001101100000001011111111011011011111100001010010000010011111011001101001010011001010001001111101110100000100010100110110011010100000110011110110110001101101000100011110011111110010011011101100001100101011111111011000110010010011110110110011100011100001001100100100100011010100100100011110110000000
Pair \(Z_2\) Length of longest common subsequence
1VHD_1,6FHL_1 167 3
1VHD_1,7CNB_1 174 4
6FHL_1,7CNB_1 195 3

Newick tree

 
[
	7CNB_1:95.18,
	[
		1VHD_1:83.5,6FHL_1:83.5
	]:11.68
]

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 \((0.85)(0.8)(\frac{746 }{\log_{20} 746}-\frac{371}{\log_{20}371})=101.\)
Status Protein1 Protein2 d d1/2
Query variables 1VHD_1 6FHL_1 132 127.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]