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Parikh vectors
4EYC_1 7LNP_1 7YPK_1 Letter Amino acid
0 16 0 W Tryptophan
8 16 0 R Arginine
4 37 0 N Asparagine
4 7 0 C Cysteine
8 51 0 K Lycine
4 27 0 F Phenylalanine
7 45 0 V Valine
7 22 0 P Proline
5 35 0 A Alanine
13 38 0 D Aspartic acid
5 14 0 Q Glutamine
6 43 0 G Glycine
8 28 0 L Leucine
3 15 0 M Methionine
3 39 0 E Glutamic acid
1 19 0 H Histidine
3 29 0 I Isoleucine
8 44 0 S Serine
8 38 0 T Threonine
2 36 0 Y Tyrosine 

4EYC_1|Chains A, B|Cathelicidin antimicrobial peptide|Homo sapiens (9606)
>7LNP_1|Chains A, B[auth C], C[auth D], D[auth E]|Exo-alpha-L-galactosidase|Bacteroides plebeius (484018)
>7YPK_1|Chain A[auth S]|alpha-S1-casein|Bos taurus (9913)
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 \(w=f(c)\)
4EYC , Knot 57 107 0.83 38 90 105 
GSHMQVLSYKEAVLRAIDGINQRSSDANLYRLLDLDPRPTMDGDPDTPKPVSFTVKETVCPRTTQQSPEDCDFKKDGLVKRCMGTVTLNQARGSFDISCDKDNKRFA
7LNP , Knot 239 599 0.85 40 286 562 
MGSSHHHHHHSSGLVPRGSHMASNSTYAPNVDSTVSLKGTDVFKADSASIAQNYTIPEWFKDAKFGIFIHWGVYSVPAYGSEWYSRWMYKEGHPINKYHVQTYGPLTKFGYKDFIPMFKAENFNADEWLAVVKSSGAQYIVPVAEHHDGFAMYSSTFNKWNAVDMGPKRDIIGELKEATKKAGLRFGLSSHRCENAWFYEYGMETPSDVQDTTITLYGERLHEPEGQGMTPYCGKYEGSNERSRRQFLMHTYELIDKYQPELIWFNWTVGKYPFQPTFYKFMAYYYNSALDWNKEVVVNTKFGYGDNIQVFDIERGKSDRIREYPWQTDTSVGKKSWSYCVGEENKSPDHIIDDFVDIVSKNGNLLLNIGPKADGTITDEQKNVLAEIGKWLKTNGEAIYGSRPWVIASEGHNAGTAGYMTDNTKTEYTADDIRFTTCDNNLYAVSLAWTDGSVTIKSLATKYCRNVEIESVEMLGSSEKIDYKMTDEGLVVNFPKNKPTEYAHVFKIKLKGVVVSKPLYDKVDNGCLITVRVANHNAEDANVTLKSVVDGNEVSTQVAVKAKSEQWVKMQNKDVKSFDDMSCKFYFNDNLTYENEFKK
7YPK , Knot 2 19 0.10 2 1 1 
XXXXXXXXXXXXXXXXXXX

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 \(c\).

\(|P_{f(4EYC_1)}(2) \setminus P_{f(7LNP_1)}(2)|=18\), \(|P_{f(7LNP_1)}(2) \setminus P_{f(4EYC_1)}(2)|=214\). 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 1:10010110000111011011000000101001101010101010100101101010001010000001000010001110001101010010101010000000011
Pair \(Z_2\) Length of longest common subsequence
4EYC_1,7LNP_1 232 4
4EYC_1,7YPK_1 91 0
7LNP_1,7YPK_1 287 0

Newick tree

 
[
	7LNP_1:14.35,
	[
		4EYC_1:45.5,7YPK_1:45.5
	]:10.85
]

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 \((0.85)(0.8)(\frac{706 }{\log_{20} 706}-\frac{107}{\log_{20}107})=172.\)
Status Protein1 Protein2 d d1/2
Query variables 4EYC_1 7LNP_1 224 130
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]