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Parikh vectors
2WVP_1 5MNX_1 6ZGW_1 Letter Amino acid
5 2 9 M Methionine
27 6 13 D Aspartic acid
13 3 15 H Histidine
7 15 9 I Isoleucine
6 14 6 K Lycine
21 10 11 Y Tyrosine
14 17 31 V Valine
16 2 31 R Arginine
23 4 29 E Glutamic acid
18 3 11 F Phenylalanine
15 10 17 T Threonine
8 8 22 P Proline
10 4 1 W Tryptophan
21 16 5 N Asparagine
8 10 20 Q Glutamine
27 25 33 G Glycine
19 14 47 L Leucine
15 14 51 A Alanine
1 12 8 C Cysteine
12 34 30 S Serine 

2WVP_1|Chain A|OUTER MEMBRANE PROTEIN G|ESCHERICHIA COLI (83333)
>5MNX_1|Chain A|Cationic trypsin|Bos taurus (9913)
>6ZGW_1|Chains A, B, C[auth D], D[auth E]|Galactokinase|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 \(w=f(c)\)
2WVP , Knot 126 286 0.83 40 180 278 
EERNDWHFNIGAMYEIENVEGYGEDMDGLAEPSVYFNAANGPWRIALAYYQEGPVDYSAGKRGTWFDRPELEVHYQFLENDDFSFGLTGGFRNYGYHYVDEPGKDTANMQRWKIAPDWDVKLTDDLRFNGWLSMYKFANDLNTTGYADTRVETETGLQYTFNETVALRVNYYLERGFNMDDSRNNGEFSTQEIRAYLPLTLGNHSVTPCTRIGLDRWSNWDWQDDIEREGHDFNRVGLFYGYDFQNGLSVSLEYAFEWQDHDEGDSDKFHYAGVGVNYSFHHHHHH
5MNX , Knot 101 223 0.81 40 140 216 
IVGGYTCGANTVPYQVSLNSGYHFCGGSLINSQWVVSAAHCYKSGIQVRLGEDNINVVEGNEQFISASKSIVHPSYNSNTLNNDIMLIKLKSAASLNSRVASISLPTSCASAGTQCLISGWGNTKSSGTSYPDVLKCLKAPILSDSSCKSAYPGQITSNMFCAGYLEGGKDSCQGDSGGPVVCSGKLQGIVSWGSGCAQKNKPGVYTKVCNYVSWIKQTIASN
6ZGW , Knot 163 399 0.81 40 205 366 
MAHHHHHHAALRQPQVAELLAEARRAFREEFGAEPELAVSAPGRVNLIGEHTDYNQGLVLPMALELMTVLVGSPRKDGLVSLLTTSEGADEPQRLQFPLPTAQRSLEPGTPRWANYVKGVIQYYPAAPLPGFSAVVVSSVPLGGGLSSSASLEVATYTFLQQLCPDSGTIAARAQVCQQAEHSFAGMPCGIMDQFISLMGQKGHALLIDCRSLETSLVPLSDPKLAVLITNSNVRHSLASSEYPVRRRQCEEVARALGAASLREVQLEELEAARDLVSKEGFRRARHVVGEIRRTAQAAAALRRGDYRAFGRLMVESHRSLRDDYEVSCPELDQLVEAALAVPGVYGSRMTGGGFGGCTVTLLEASAAPHAMRHIQEHYGGTATFYLSQAADGAKVLCL

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 \(c\).

\(|P_{f(2WVP_1)}(2) \setminus P_{f(5MNX_1)}(2)|=115\), \(|P_{f(5MNX_1)}(2) \setminus P_{f(2WVP_1)}(2)|=75\). 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 1:0000010101111001001010100101110101010110111011110000111000110010110010101000110000101110111000100010011000101001011101010100010101110100110010001010001000011000100011101000100110100000010100001010111011000101000111001001010001000100100111101001001101010011010000010000100111110001000000
Pair \(Z_2\) Length of longest common subsequence
2WVP_1,5MNX_1 190 3
2WVP_1,6ZGW_1 177 6
5MNX_1,6ZGW_1 185 4

Newick tree

 
[
	5MNX_1:95.44,
	[
		2WVP_1:88.5,6ZGW_1:88.5
	]:6.94
]

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 \((0.85)(0.8)(\frac{509 }{\log_{20} 509}-\frac{223}{\log_{20}223})=82.3\)
Status Protein1 Protein2 d d1/2
Query variables 2WVP_1 5MNX_1 109 95.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]