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Parikh vectors
1ZYC_1 5DXM_1 2EBY_1 Letter Amino acid
23 14 7 K Lycine
16 9 8 N Asparagine
25 13 4 I Isoleucine
14 10 2 G Glycine
9 15 3 M Methionine
12 7 3 F Phenylalanine
9 9 3 P Proline
22 20 2 S Serine
11 11 9 T Threonine
17 15 11 A Alanine
18 13 5 D Aspartic acid
17 13 8 V Valine
20 15 13 E Glutamic acid
34 48 16 L Leucine
3 3 2 W Tryptophan
14 4 3 Y Tyrosine
1 4 0 C Cysteine
10 9 3 Q Glutamine
19 12 9 R Arginine
9 13 2 H Histidine 

1ZYC_1|Chains A, B, C, D|Serine/threonine-protein kinase GCN2|Saccharomyces cerevisiae (4932)
>5DXM_1|Chains A, B|Estrogen receptor|Homo sapiens (9606)
>2EBY_1|Chains A, B|Putative HTH-type transcriptional regulator ybaQ|Escherichia coli (562)
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 \(w=f(c)\)
1ZYC , Knot 131 303 0.82 40 183 291 
SLRYASDFEEIAVLGQGAFGQVVKARNALDSRYYAIKKIRHTEEKLSTILSEVMLLASLNHQYVVRYYAAWLERRNFVKPMTAVKKKSTLFIQMEYCENRTLYDLIHSENLNQQRDEYWRLFRQILEALSYIHSQGIIHRDLKPMNIFIDESRNVKIGDFGLAKNVHRSLDILKLDSQNLPGSSDNLTSAIGTAMYVATEVLDGTGHYNEKIDMYSLGIIFFEMIYPFSTGMERVNILKKLRSVSIEFPPDFDDNKMKVEKKIIRLLIDHDPNKRPGARTLLNSGWLPVKHQDEVIKEALKSL
5DXM , Knot 117 257 0.84 40 165 246 
IKRSKKNSLALSLTADQMVSALLDAEPPILYSEYDPTRPFSEASMMGLLTNLADRELVHMINWAKRVPGFVDLTLHDQVHLLECAWLEILMIGLVWRSMEHPGKLLFAPNLLLDRNQGKCVEGMVEIFDMLLATSSRFRMMNLQGEEFVCLKSIILLNSGVYTFLSSTLKSLEEKDHIHRVLDKITDTLIHLMAKAGLTLQQQHQRLAQLLLILSHIRHMSNKGMEHLYSMKCKNVVPLSDLLLEMLDAHRLHAPTS
2EBY , Knot 58 113 0.80 38 90 110 
MKQATRKPTTPGDILLYEYLEPLDLKINELAELLHVHRNSVSALINNNRKLTTEMAFRLAKVFDTTVDFWLNLQAAVDLWEVENNMRTQEELGRIETVAEYLARREERAKKVA

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 \(c\).

\(|P_{f(1ZYC_1)}(2) \setminus P_{f(5DXM_1)}(2)|=87\), \(|P_{f(5DXM_1)}(2) \setminus P_{f(1ZYC_1)}(2)|=69\). 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 1:010010010011111011110110100110000011001000000100110011111010000110001111000011011011000001110100000001001100001000000010110011011001000111000101101110000010110111100100010110100001110000100111011011001101010000010100111111011011001100101100100101011101000010100011011100010001110011001111100000110011001
Pair \(Z_2\) Length of longest common subsequence
1ZYC_1,5DXM_1 156 5
1ZYC_1,2EBY_1 163 3
5DXM_1,2EBY_1 161 4

Newick tree

 
[
	2EBY_1:81.97,
	[
		1ZYC_1:78,5DXM_1:78
	]:3.97
]

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 \((0.85)(0.8)(\frac{560 }{\log_{20} 560}-\frac{257}{\log_{20}257})=85.9\)
Status Protein1 Protein2 d d1/2
Query variables 1ZYC_1 5DXM_1 106 98
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]