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Parikh vectors
1ATK_1 4FLG_1 6SEC_1 Letter Amino acid
4 2 26 W Tryptophan
13 1 24 Y Tyrosine
18 5 111 A Alanine
7 4 72 R Arginine
8 6 65 P Proline
4 2 14 M Methionine
16 1 58 S Serine
8 0 5 C Cysteine
10 4 71 E Glutamic acid
24 13 106 G Glycine
2 1 28 H Histidine
19 7 16 K Lycine
15 7 89 V Valine
19 3 22 N Asparagine
9 4 69 D Aspartic acid
9 5 36 Q Glutamine
5 8 47 T Threonine
8 14 26 I Isoleucine
12 10 90 L Leucine
5 2 35 F Phenylalanine 

1ATK_1|Chain A|CATHEPSIN K|Homo sapiens (9606)
>4FLG_1|Chains A, B|HIV-1 protease|Human immunodeficiency virus 1 (11676)
>6SEC_1|Chain A|Beta-galactosidase|Arthrobacter sp. 32cB (1492190)
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 \(w=f(c)\)
1ATK , Knot 99 215 0.82 40 148 209 
APDSVDYRKKGYVTPVKNQGQCGSCWAFSSVGALEGQLKKKTGKLLNLSPQNLVDCVSENDGCGGGYMTNAFQYVQKNRGIDSEDAYPYVGQEESCMYNPTGKAAKCRGYREIPEGNEKALKRAVARVGPVSVAIDASLTSFQFYSKGVYYDESCNSDNLNHAVLAVGYGIQKGNKHWIIKNSWGENWGNKGYILMARNKNNACGIANLASFPKM
4FLG , Knot 53 99 0.82 38 77 96 
PQITLWKRPLVTIKIGGQLKEALLDTGADDTVIEEMSLPGRWKPKMVGGIGGFIKVRQYDQIIIEIAGHKAIGTVLVGPTPVNIIGRNLLTQIGATLNF
6SEC , Knot 360 1010 0.82 40 292 868 
MSVETPSALADSSPHTAPGSAGRSLELGAADIQDLESFEAGRGALPARAYLQSDAPRLSLNGEWQFRLSPGSRVAPDDGWQLGEALNGFESLPVPSSWPMHGHGAPAYTNVQFPFAVEPPHVPEANPIGDHLVVFEAGPEFFPHALLRFDGIESAGTVWLNGVELGTTRGSRLAHEFDVSGILEQGENTLAVRVAQFSAASYVEDQDMWWLPGIFRDVTLQARPAAGIDDVFVHAGYDHITGEGILKVEASRGGQAIDAVVRVPELALELAAGTEVRVPAVEPWSAEVPKLYEAAVSAAGESVALQIGFRSIAIEDAQFKVNGRRILLRGVNRHEHHPRLGRVVPRDVVEAELRLMKQHNINAIRTSHYPPHPQFLALADQLGFYVVLECDLETHGFESAGWAQNPSDDPQWEDALVDRMRRTVERDKNHASVVMWSLGNEAGTGRNLAAMSRWTKDRDPSRPIHYEGDWSSEHVDVYSRMYASQAETALIGQGIEPALNDAALDARRRAMPFVLCEYVHAMGNGPGGMSEYQALFEKYPRLMGGFVWEWLEHGITVSTADGVDHYGYGGDFGEEVHDGNFVTDGLVDADRRPRPGLLDFKKVIEPLRIDVARDWTGFTLRNGQDFADTSAFSFRYEVEADGGALDGGTVDVAPVAPQSETVVELPGSVAALAAGLSDGRPAVLTVRAVLGADSAWADAGHEVAWGQSVREPGAPVPPAPVEPVQVQDSELTLGPVVFSRATGMPTSIGGVPVEKLGLTLWWAPTDNDLGREWGGADERPLATQWKDAGLNRLHTRLLGISANPGQDGGETLTVRTRVSAADKQYGVLVDYTWSTDGETVGLRTQVRRDGTWVNRGFEVEWARIGLEFVLGEETELVSWFGQGPHQSYPDTGQGARAGWFSLPLAKMDVEYVRPQECGARSGSRSAALQLGGRTLEICGDPFALTVRPYSQDVLDAAAHRPDLKADGRTYLYVDHALRGVGTAACGPGVLEQYRLKPRDADFILTLKVRS

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 \(c\).

\(|P_{f(1ATK_1)}(2) \setminus P_{f(4FLG_1)}(2)|=113\), \(|P_{f(4FLG_1)}(2) \setminus P_{f(1ATK_1)}(2)|=42\). 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 1:11001000001010110001001001110011110101000010110101001100100001011101001100100001100001010110000010010101100010001101000110011101111011101010010100011000000000010011111101100100011100011001100101111000001011101101101
Pair \(Z_2\) Length of longest common subsequence
1ATK_1,4FLG_1 155 4
1ATK_1,6SEC_1 216 4
4FLG_1,6SEC_1 235 4

Newick tree

 
[
	6SEC_1:12.38,
	[
		1ATK_1:77.5,4FLG_1:77.5
	]:44.88
]

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 \((0.85)(0.8)(\frac{314 }{\log_{20} 314}-\frac{99}{\log_{20}99})=67.3\)
Status Protein1 Protein2 d d1/2
Query variables 1ATK_1 4FLG_1 86 61.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]