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Parikh vectors
8ADM_1 4BXP_1 1MDR_1 Letter Amino acid
14 16 17 D Aspartic acid
7 13 15 Q Glutamine
17 14 14 K Lycine
9 7 12 M Methionine
13 11 15 R Arginine
5 8 8 F Phenylalanine
5 13 19 P Proline
12 14 31 V Valine
14 16 30 G Glycine
3 6 9 H Histidine
10 19 19 T Threonine
29 13 19 E Glutamic acid
7 6 7 N Asparagine
2 0 2 C Cysteine
10 10 21 I Isoleucine
24 8 42 L Leucine
19 6 17 S Serine
2 0 6 W Tryptophan
10 7 9 Y Tyrosine
24 5 47 A Alanine 

8ADM_1|Chain A|14-3-3 protein sigma|Homo sapiens (9606)
>4BXP_1|Chain A|CPAP|DANIO RERIO (7955)
>1MDR_1|Chain A|MANDELATE RACEMASE|Pseudomonas putida (303)
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 \(w=f(c)\)
8ADM , Knot 108 236 0.83 40 164 229 
GAMGSMERASLIQKAKLAEQAERYEDMAAFMKGAVEKGEELSCEERNLLSVAYKNVVGGQRAAWRVLSSIEQKSNEEGSEEKGPEVREYREKVETELQGVCDTVLGLLDSHLIKEAGDAESRVFYLKMKGDYYRYLAEVATGDDKKRIIDSARSAYQEAMDISKKEMPPTNPIRLGLALNFSVFHYEIANSPEEAISLAKTTFDEAMADLHTLSEDSYKDSTLIMQLLRDNLTLWT
4BXP , Knot 82 192 0.74 36 122 172 
GPHMEEIMQSDSKIEKMLPDGGRLVVFPNGTRKELSADGQTVKVMFFNGDVKHTMPDQRVIYYYAEAQTTHITYPDGMEVLQFPNNQTEKHFPDGRKEITFPDQTVKTLHPDGREESVLTDGTIIQLNPDGSKVIQFNTGQREIHTADFKRREYPDGTVKTVYSDGRQETQYPTGRVRLKDPQGKVIMDTKA
1MDR , Knot 146 359 0.79 40 197 333 
MSEVLITGLRTRAVNVPLAYPVHTAVGTVGTAPLVLIDLATSAGVVGHSYLFAYTPVALKSLKQLLDDMAAMIVNEPLAPVSLEAMLAKRFCLAGYTGLIRMAAAGIDMAAWDALGKVHETPLVKLLGANARPVQAYDSHSLDGVKLATERAVTAAELGFRAVKTKIGYPALDQDLAVVRSIRQAVGDDFGIMVDYNQSLDVPAAIKRSQALQQEGVTWIEEPTLQHDYEGHQRIQSKLNVPVQMGENWLGPEEMFKALSIGACRLAMPDAMKIGGVTGWIRASALAQQFGIPMSSHLFQEISAHLLAATPTAHWLERLDLAGSVIEPTLTFEGGNAVIPDLPGVGIIWREKEIGKYLV

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 \(c\).

\(|P_{f(8ADM_1)}(2) \setminus P_{f(4BXP_1)}(2)|=101\), \(|P_{f(4BXP_1)}(2) \setminus P_{f(8ADM_1)}(2)|=59\). 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 1:11110100101100101100100000111110111001001000000110110001111001110110010000000100001101000000100010110001111100011001101000110101010000011011010000011001001000110100001110011011111010110001100100110110001001110100100000000011101100010110
Pair \(Z_2\) Length of longest common subsequence
8ADM_1,4BXP_1 160 3
8ADM_1,1MDR_1 151 4
4BXP_1,1MDR_1 177 3

Newick tree

 
[
	4BXP_1:87.10,
	[
		8ADM_1:75.5,1MDR_1:75.5
	]:11.60
]

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 \((0.85)(0.8)(\frac{428 }{\log_{20} 428}-\frac{192}{\log_{20}192})=69.5\)
Status Protein1 Protein2 d d1/2
Query variables 8ADM_1 4BXP_1 93 80.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]