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Parikh vectors
3BCH_1 2YVY_1 5MNK_1 Letter Amino acid
2 0 12 C Cysteine
18 11 25 G Glycine
8 5 2 M Methionine
8 3 3 F Phenylalanine
10 1 16 N Asparagine
23 40 14 L Leucine
13 13 8 P Proline
15 13 10 T Threonine
4 2 4 W Tryptophan
26 27 14 A Alanine
16 21 2 R Arginine
9 7 10 Q Glutamine
11 5 3 H Histidine
14 8 15 I Isoleucine
10 11 14 K Lycine
8 11 10 Y Tyrosine
11 25 6 D Aspartic acid
17 37 4 E Glutamic acid
16 7 34 S Serine
14 31 17 V Valine 

3BCH_1|Chain A|40S ribosomal protein SA|Homo sapiens (9606)
>2YVY_1|Chain A|Mg2+ transporter MgtE|Thermus thermophilus (300852)
>5MNK_1|Chain A|Cationic trypsin|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 \(w=f(c)\)
3BCH , Knot 113 253 0.82 40 158 238 
MGSSHHHHHHSSGLVPRGSGTENLYFQGHMASGSSGALDVLQMKEEDVLKFLAAGTHLGGTNLDFQMEQYIYKRKSDGIYIINLKRTWEKLLLAARAIVAIENPADVSVISSRNTGQRAVLKFAAATGATPIAGRFTPGTFTNQIQAAFREPRLLVVTDPRADHQPLTEASYVNLPTIALCNTDSPLRYVDIAIPCNNKGAHSVGLMWWMLAREVLRMRGTISREHPWEVMPDLYFYRDPEEIEKEEQAAAEK
2YVY , Knot 115 278 0.77 38 139 251 
GSHMEEKLAVSLQEALQEGDTRALREVLEEIHPQDLLALWDELKGEHRYVVLTLLPKAKAAEVLSHLSPEEQAEYLKTLPPWRLREILEELSLDDLADALQAVRKEDPAYFQRLKDLLDPRTRAEVEALARYEEDEAGGLMTPEYVAVREGMTVEEVLRFLRRAAPDAETIYYIYVVDEKGRLKGVLSLRDLIVADPRTRVAEIMNPKVVYVRTDTDQEEVARLMADYDFTVLPVVDEEGRLVGIVTVDDVLDVLEAEATEDIHKLGAVDVPDLVYSE
5MNK , Knot 101 223 0.81 40 140 216 
IVGGYTCGANTVPYQVSLNSGYHFCGGSLINSQWVVSAAHCYKSGIQVRLGEDNINVVEGNEQFISASKSIVHPSYNSNTLNNDIMLIKLKSAASLNSRVASISLPTSCASAGTQCLISGWGNTKSSGTSYPDVLKCLKAPILSDSSCKSAYPGQITSNMFCAGYLEGGKDSCQGDSGGPVVCSGKLQGIVSWGSGCAQKNKPGVYTKVCNYVSWIKQTIASN

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 \(c\).

\(|P_{f(3BCH_1)}(2) \setminus P_{f(2YVY_1)}(2)|=74\), \(|P_{f(2YVY_1)}(2) \setminus P_{f(3BCH_1)}(2)|=55\). 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 1:1100000000001111010100010101011010011101101000011011111001110010101000100000011011010001001111101111100110101100000100111011110110111101011010001011100101111001010001100100101101110000011001011110000110011111111100110101010000110111010100010010000011100
Pair \(Z_2\) Length of longest common subsequence
3BCH_1,2YVY_1 129 4
3BCH_1,5MNK_1 192 4
2YVY_1,5MNK_1 179 3

Newick tree

 
[
	5MNK_1:10.48,
	[
		3BCH_1:64.5,2YVY_1:64.5
	]:35.98
]

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 \((0.85)(0.8)(\frac{531 }{\log_{20} 531}-\frac{253}{\log_{20}253})=79.2\)
Status Protein1 Protein2 d d1/2
Query variables 3BCH_1 2YVY_1 92 90.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]