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Parikh vectors
2XAX_1 1HYJ_1 5KLX_1 Letter Amino acid
75 2 15 L Leucine
42 5 13 K Lycine
40 5 11 R Arginine
51 3 15 D Aspartic acid
48 4 26 G Glycine
5 1 2 W Tryptophan
40 0 4 Y Tyrosine
16 1 6 M Methionine
49 4 11 S Serine
39 2 13 T Threonine
20 2 9 H Histidine
50 1 11 I Isoleucine
28 3 8 F Phenylalanine
63 6 11 A Alanine
33 3 8 Q Glutamine
46 3 17 E Glutamic acid
39 5 16 V Valine
35 5 4 N Asparagine
11 8 0 C Cysteine
31 2 9 P Proline 

2XAX_1|Chains A, B, C|RIBONUCLEOSIDE-DIPHOSPHATE REDUCTASE 1 SUBUNIT ALPHA|ESCHERICHIA COLI (83333)
>1HYJ_1|Chain A|ENDOSOME-ASSOCIATED PROTEIN|Homo sapiens (9606)
>5KLX_1|Chains A, B, C, D|Chimera protein of Ubiquitin-like protein SMT3 and Peptidyl-prolyl cis-trans isomerase|Saccharomyces cerevisiae (559292)
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 \(w=f(c)\)
2XAX , Knot 292 761 0.84 40 294 703 
MNQNLLVTKRDGSTERINLDKIHRVLDWAAEGLHNVSISQVELRSHIQFYDGIKTSDIHETIIKAAADLISRDAPDYQYLAARLAIFHLRKKAYGQFEPPALYDHVVKMVEMGKYDNHLLEDYTEEEFKQMDTFIDHDRDMTFSYAAVKQLEGKYLVQNRVTGEIYESAQFLYILVAACLFSNYPRETRLQYVKRFYDAVSTFKISLPTPIMSGVRTPTRQFSSCVLIECGDSLDSINATSSAIVKYVSQRAGIGINAGRIRALGSPIRGGEAFHTGCIPFYKHFQTAVKSCSQGGVRGGAATLFYPMWHLEVESLLVLKNNRGVEGNRVRHMDYGVQINKLMYTRLLKGEDITLFSPSDVPGLYDAFFADQEEFERLYTKYEKDDSIRKQRVKAVELFSLMMQERASTGRIYIQNVDHCNTHSPFDPAIAPVRQSNLCLEIALPTKPLNDVNDENGEIALCTLSAFNLGAINNLDELEELAILAVRALDALLDYQDYPIPAAKRGAMGRRTLGIGVINFAYYLAKHGKRYSDGSANNLTHKTFEAIQYYLLKASNELAKEQGACPWFNETTYAKGILPIDTYKKDLDTIANEPLHYDWEALRESIKTHGLRNSTLSALMPSETSSQISNATNGIEPPRGYVSIKASKDGILRQVVPDYEHLHDAYELLWEMPGNDGYLQLVGIMQKFIDQSISANTNYDPSRFPSGKVPMQQLLKDLLTAYKFGVKTLYAQNTRDGAEDAQDDLVPSIQDDGCESGACKI
1HYJ , Knot 38 65 0.81 38 60 63 
RKWAEDNEVQNCMACGKGFSVTVRRHHCRQCGNIFCAECSAKNALTPSSKKPVRVCDACFNDLQG
5KLX , Knot 92 209 0.78 38 147 197 
MGHHHHHHSGEVKPEVKPETHINLKVSDGSSEIFFKIKKTTPLRRLMEAFAKRQGKEMDSLRFLYDGIRIQADQTPEDLDMEDNDIIEAHREQIGGSTVVTTESGLKYEDLTEGSGAEARAGQTVSVHYTGWLTDGQKFDSSKDRNDPFAFVLGGGMVIKGWDEGVQGMKVGGVRRLTIPPQLGYGARGAGGVIPPNATLVFEVELLDV

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 \(c\).

\(|P_{f(2XAX_1)}(2) \setminus P_{f(1HYJ_1)}(2)|=250\), \(|P_{f(1HYJ_1)}(2) \setminus P_{f(2XAX_1)}(2)|=16\). 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 1:10001110000100001010010011011101100101001010001010011000010001101110110001100001110111101000101010111100011011011000001100000001001001100000101001110010100110001010100010110111110110001000010010010011001010110111011001000100011100100100101000111001000111110110101110110110110010111000100110000011101111011011101010011110000110100100100110100110001101001011010011110011110000100100000000010000101101101110001001010100100000001101111110000101011110011001000010111001011011110010010011111101101110000011111001111000111111011001100100000101001000010110001101000110001101110000010111110000001001100110001011000100011000010111100000010010011011010101010001110011100001001001110111001010111110011000101000001001101011100110011010011100101000001100100011101000100011001
Pair \(Z_2\) Length of longest common subsequence
2XAX_1,1HYJ_1 266 3
2XAX_1,5KLX_1 185 4
1HYJ_1,5KLX_1 155 3

Newick tree

 
[
	2XAX_1:12.47,
	[
		5KLX_1:77.5,1HYJ_1:77.5
	]:46.97
]

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 \((0.85)(0.8)(\frac{826 }{\log_{20} 826}-\frac{65}{\log_{20}65})=218.\)
Status Protein1 Protein2 d d1/2
Query variables 2XAX_1 1HYJ_1 279 151
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]