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Parikh vectors
4AZU_1 4XHL_1 7YAD_1 Letter Amino acid
10 43 6 L Leucine
11 31 5 K Lycine
2 20 8 Y Tyrosine
10 14 10 V Valine
10 16 14 T Threonine
7 23 4 N Asparagine
4 25 4 E Glutamic acid
4 24 3 I Isoleucine
10 28 11 S Serine
4 14 0 H Histidine
6 11 3 M Methionine
6 21 4 F Phenylalanine
4 12 3 P Proline
6 10 8 Q Glutamine
11 26 16 G Glycine
1 3 5 W Tryptophan
7 17 8 A Alanine
1 29 7 R Arginine
11 22 5 D Aspartic acid
3 5 2 C Cysteine 

4AZU_1|Chains A, B, C, D|AZURIN|Pseudomonas aeruginosa (287)
>4XHL_1|Chain A|Casein kinase I homolog HRR25|Saccharomyces cerevisiae (4932)
>7YAD_1|Chains A, D[auth C]|S309 neutralizing antibody heavy chain|Homo sapiens (9606)
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 \(w=f(c)\)
4AZU , Knot 68 128 0.86 40 105 126 
AECSVDIQGNDQMQFNTNAITVDKSCKQFTVNLSHPGNLPKNVMGHNWVLSTAADMQGVVTDGMASGLDKDYLKPDDSRVIAHTKLIGSGEKDSVTFDVSKLKEGEQYMFFCTFPGHSALMKGTLTLK
4XHL , Knot 166 394 0.84 40 216 376 
MDLRVGRKFRIGRKIGSGSFGDIYHGTNLISGEEVAIRLESIRSRHPQLDYESRVYRYLSGGVGIPFIRWFGREGEYNAMVIDLLGPSLEDLFNYCHRRFSFKTVIMLALQMFCRIQYIHGRSFIHRDIKPDNFLMGVGRRGSTVHVIDFGLSKKYRDFNTHRHIPYRENKSLTGTARYASVNTHLGIEQSRRDDLESLGYVLIYFCKGSLPWQGLKATTKKQKYDRIMEKKLNVSVETLCSGLPLEFQEYMAYCKNLKFDEKPDYLFLARLFKDLSIKLEYHNDHLFDWTMLRYTKAMVEKQRDLLIEKGDLNANSNAASASNSTDNKSETFNKIKLLAMKKFPTHFHYYKNEDKHNPSPEEIKQQTILNNNAASSLPEELLNALDKGMENLR
7YAD , Knot 60 126 0.76 38 104 123 
QVQLVQSGAEVKKPGASVKVSCKASGYPFTSYGISWVRQAPGQGLEWMGWISTYNGNTNYAQKFQGRVTMTTDTSTTTGYMELRRLRSDDTAVYYCARDYTRGAWFGESLIGGFDNWGQGTLVTVS

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 \(c\).

\(|P_{f(4AZU_1)}(2) \setminus P_{f(4XHL_1)}(2)|=31\), \(|P_{f(4XHL_1)}(2) \setminus P_{f(4AZU_1)}(2)|=142\). 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 1:10001010100010100011010000001010100110110011100111001101011100111011000010100001110001110100001010100100100011100111001110101010
Pair \(Z_2\) Length of longest common subsequence
4AZU_1,4XHL_1 173 4
4AZU_1,7YAD_1 143 3
4XHL_1,7YAD_1 192 3

Newick tree

 
[
	4XHL_1:97.09,
	[
		4AZU_1:71.5,7YAD_1:71.5
	]:25.59
]

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 \((0.85)(0.8)(\frac{522 }{\log_{20} 522}-\frac{128}{\log_{20}128})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 4AZU_1 4XHL_1 148 97
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]