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Parikh vectors
4WTW_1 9LWA_1 6XZN_1 Letter Amino acid
1 1 7 C Cysteine
6 5 18 Q Glutamine
8 15 54 G Glycine
3 4 14 H Histidine
0 6 12 K Lycine
8 10 12 P Proline
3 2 9 Y Tyrosine
10 13 27 V Valine
5 13 18 T Threonine
4 1 12 N Asparagine
2 6 23 D Aspartic acid
8 8 16 E Glutamic acid
3 2 18 I Isoleucine
1 3 7 M Methionine
2 2 12 W Tryptophan
3 17 26 A Alanine
9 6 18 R Arginine
10 8 30 L Leucine
2 3 8 F Phenylalanine
8 6 32 S Serine 

4WTW_1|Chains A, B|Integrin beta-4|Homo sapiens (9606)
>9LWA_1|Chain A|Head-to-tail stopper|Mycolicibacterium phage Mycofy1 (3349809)
>6XZN_1|Chains A, B|Ultraviolet-B receptor UVR8|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
4WTW , Knot 51 96 0.80 38 79 94 
GSHMVPDTPTRLVFSALGPTSLRVSWQEPRCERPLQGYSVEYQLLNGGELHRLNIPNPAQTSVVVEDLLPNHSYVFRVRAQSQEGWGREREGVITI
9LWA , Knot 65 131 0.80 40 96 128 
MSEEFGGQVVALVSVAETGVPGWGGLKAKARTTTRQPGVHFRPASSTETPDGQTNVATEVWKLTAPPTAAVLAAKPGGELVYDGTEHPESLDLDSAAGRAATFRVDGPIMPKYDLAGQVHHVTIMCKRQAG
6XZN , Knot 143 373 0.75 40 189 321 
GAMAPPRKVLIISAGASHSVALLSGDIVCSWGRGEDGQLGHGDAEDRPSPTQLSALDGHQIVSVTCGADHTVAYSQSGMEVYSWGWGDFGRLSHGNSSDLFTPLPIKALHGIRIKQIACGDSHCLAVTMEGEVQSWGRNQNGQLGLGDTEDSLVPQKIQAFEGIRIKMVAAGAEHTAAVTEDGDLYGWGWGRYGNLGLGDRTDRLVPERVTSTGGEKMSMVACGWRHTISVSYSGALYTYGWSKYGQLGHGDLEDHLIPHKLEALSNSFISQISGGARHTMALTSDGKLYGWGWNKFGQVGVGNNLDQCSPVQVRFPDDQKVVQVSCGWRHTLAVTERNNVFAWGRGTNGQLGIGESVDRNFPKIIEALSVDG

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 \(c\).

\(|P_{f(4WTW_1)}(2) \setminus P_{f(9LWA_1)}(2)|=51\), \(|P_{f(9LWA_1)}(2) \setminus P_{f(4WTW_1)}(2)|=68\). 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 1:100111001001110111100101010010000110100100011011010010110110001110011100001101010000111000011101
Pair \(Z_2\) Length of longest common subsequence
4WTW_1,9LWA_1 119 4
4WTW_1,6XZN_1 176 4
9LWA_1,6XZN_1 159 4

Newick tree

 
[
	6XZN_1:90.53,
	[
		4WTW_1:59.5,9LWA_1:59.5
	]:31.03
]

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 \((0.85)(0.8)(\frac{227 }{\log_{20} 227}-\frac{96}{\log_{20}96})=42.3\)
Status Protein1 Protein2 d d1/2
Query variables 4WTW_1 9LWA_1 53 45
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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