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Parikh vectors
5AFX_1 7AIX_1 1LUP_1 Letter Amino acid
11 36 1 N Asparagine
15 17 1 Q Glutamine
18 19 1 Y Tyrosine
19 48 1 G Glycine
5 15 0 H Histidine
38 60 2 L Leucine
20 26 3 K Lycine
13 32 0 P Proline
24 35 3 E Glutamic acid
16 37 0 F Phenylalanine
30 45 1 V Valine
19 16 2 M Methionine
17 52 0 S Serine
23 24 1 T Threonine
27 25 0 A Alanine
19 24 3 R Arginine
30 27 1 D Aspartic acid
9 9 6 C Cysteine
9 25 2 I Isoleucine
5 14 3 W Tryptophan 

5AFX_1|Chains A, B|FARNESYL PYROPHOSPHATE SYNTHASE|TRYPANOSOMA BRUCEI (5691)
>7AIX_1|Chain A|Acetylcholinesterase|Tetronarce californica (7787)
>1LUP_1|Chain A|GsMTx2|Grammostola rosea (432528)
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 \(w=f(c)\)
5AFX , Knot 153 367 0.82 40 215 354 
MPMQMFMQVYDEIQMFLLEELELKFDMDPNRVRYLRKMMDTTCLGGKYNRGLTVIDVAESLLSLSPNNNGEEDDGARRKRVLHDACVCGWMIEFLQAHYLVEDDIMDNSVTRRGKPCWYRHPDVTVQCAINDGLLLKSWTHMMAMHFFADRPFLQDLLCRFNRVDYTTAVGQLYDVTSMFDSNKLDPDVSQPTTTDFAEFTLSNYKRIVKYKTAYYTYLLPLVMGLIVSEALPTVDMGVTEELAMLMGEYFQVQDDVMDCFTPPERLGKVGTDIQDAKCSWLAVTFLAKASSAQVAEFKANYGSGDSEKVATVRRLYEEADLQGDYVAYEAAVAEQVKELIEKLRLCSPGFAASVETLWGKTYKRQK
7AIX , Knot 233 586 0.84 40 274 557 
MNLLVTSSLGVLLHLVVLCQADDHSELLVNTKSGKVMGTRVPVLSSHISAFLGIPFAEPPVGNMRFRRPEPKKPWSGVWNASTYPNNCQQYVDEQFPGFSGSEMWNPNREMSEDCLYLNIWVPSPRPKSTTVMVWIYGGGFYSGSSTLDVYNGKYLAYTEEVVLVSLSYRVGAFGFLALHGSQEAPGNVGLLDQRMALQWVHDNIQFFGGDPKTVTIFGESAGGASVGMHILSPGSRDLFRRAILQSGSPNCPWASVSVAEGRRRAVELGRNLNCNLNSDEELIHCLREKKPQELIDVEWNVLPFDSIFRFSFVPVIDGEFFPTSLESMLNSGNFKKTQILLGVNKDEGSFFLLYGAPGFSKDSESKISREDFMSGVKLSVPHANDLGLDAVTLQYTDWMDDNNGIKNRDGLDDIVGDHNVICPLMHFVNKYTKFGNGTYLYFFNHRASNLVWPEWMGVIHGYEIEFVFGLPLVKELNYTAEEEALSRRIMHYWATFAKTGNPNEPHSQESKWPLFTTKEQKFIDLNTEPMKVHQRLRVQMCVFWNQFLPKLLNATACDGELSSSGTSSSKGIIFYVLFSILYLIF
1LUP , Knot 21 31 0.77 30 30 29 
YCQKWMWTCDEERKCCEGLVCRLWCKRIINM

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 \(c\).

\(|P_{f(5AFX_1)}(2) \setminus P_{f(7AIX_1)}(2)|=62\), \(|P_{f(7AIX_1)}(2) \setminus P_{f(5AFX_1)}(2)|=121\). 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 1:1110111010001011110010101010100100100110000111000011011011001101010001000011000011001010111101101001100011000100010101000101010011001111001001111011100111001100100100001110100100110000101010010000110101000001100001000011111111110011101011100011111100101000110010110011011001001000111101110100101101010010100001101001000101010011001111001001100101001111101001110000000
Pair \(Z_2\) Length of longest common subsequence
5AFX_1,7AIX_1 183 4
5AFX_1,1LUP_1 209 3
7AIX_1,1LUP_1 274 3

Newick tree

 
[
	1LUP_1:13.39,
	[
		5AFX_1:91.5,7AIX_1:91.5
	]:38.89
]

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 \((0.85)(0.8)(\frac{953 }{\log_{20} 953}-\frac{367}{\log_{20}367})=156.\)
Status Protein1 Protein2 d d1/2
Query variables 5AFX_1 7AIX_1 202 165
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]