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Parikh vectors
1ZUN_1 4LYP_1 3NBE_1 Letter Amino acid
18 29 10 A Alanine
13 27 10 P Proline
19 32 13 S Serine
12 20 1 M Methionine
5 13 2 W Tryptophan
10 26 13 N Asparagine
18 17 3 H Histidine
25 22 14 L Leucine
19 20 12 V Valine
23 19 4 R Arginine
21 29 6 D Aspartic acid
23 39 11 G Glycine
21 19 10 I Isoleucine
22 24 2 K Lycine
14 18 2 F Phenylalanine
15 20 16 T Threonine
9 27 5 Y Tyrosine
1 3 0 C Cysteine
10 18 9 Q Glutamine
27 27 5 E Glutamic acid 

1ZUN_1|Chain A|Sulfate adenylyltransferase subunit 2|Pseudomonas syringae (317)
>4LYP_1|Chains A, B|Exo-beta-1,4-mannosidase|Rhizomucor miehei (4839)
>3NBE_1|Chains A, B|Ricin B-like lectin|Clitocybe nebularis (117024)
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 \(w=f(c)\)
1ZUN , Knot 140 325 0.83 40 196 303 
MGSSHHHHHHSSGLVPRGSHMVDKLTHLKQLEAESIHIIREVAAEFDNPVMLYSIGKDSAVMLHLARKAFFPGKLPFPVMHVDTRWKFQEMYRFRDQMVEEMGLDLITHINPDGVAQGINPFTHGSAKHTDIMKTEGLKQALDKHGFDAAFGGARRDEEKSRAKERVYSFRDSKHRWDPKNQRPELWNVYNGNVNKGESIRVFPLSNWTELDIWQYIYLEGIPIVPLYFAAERDVIEKNGTLIMIDDERILEHLTDEEKSRIVKKKVRFRTLGCYPLTGAVESEATSLTDIIQEMLLTRTSERQGRVIDHDGAGSMEEKKRQGYF
4LYP , Knot 191 449 0.86 40 259 435 
MGSSHHHHHHSSGLVPRGSHMASMTGGQQMGRGSEFAFVKIASDGKGFTRYGEPYLIRGANYWQGMNLGADDCSGGDRKRMELEIKQMAEMGINNLRVMASSEGPDDQPYRMRPSMMPQPGKYNEGVFVGLDYLLDTMDRYNMTAVMTLGNFWQWSGGFGQYVAWITGNQTIPYPVGDVTYDEFTQFAARFYNDSEIAPKANKLFKDHIYTVQNRRNTVNGKIYKEDPVIMSWQIANEPQEAPASWFEEISTFIKKGAPKHLVSAGLESKLDEYDFDRAHDHKNIDYTTCHCWVENWGIYDPADPDGLPHANEYMHDFLESRSKWAAQLNKPIVMEEFGMARDAWRNPEDETYKYLPSTPTSHKDEYYQKAFNQIVSLASNRSFSGSNFWAYGGEGRSTYPPNPYGMVWLGDPPHEPHGWYSVYSNDTTVQIIKDYNANLLKVQKELSK
3NBE , Knot 72 148 0.81 38 113 146 
SITPGTYNITNVAYTNRLIDLTGSNPAENTLIIGHHLNKTPSGYGNQQWTLVQLPHTTIYTMQAVNPQSYVRVRDDNLVDGAALVGSQQPTPVSIESAGNSGQFRIKIPNLGLALTLPSDANSTPIVLGEVDETSTNQLWAFESVSAV

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 \(c\).

\(|P_{f(1ZUN_1)}(2) \setminus P_{f(4LYP_1)}(2)|=61\), \(|P_{f(4LYP_1)}(2) \setminus P_{f(1ZUN_1)}(2)|=124\). 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 1:1100000000001111010011001001001010010110011101001111001100011110110011111011111101000101001001000110011101100101011101101100101000011000110011000110111111000000001000100100000010100001011010010100100101111001001011001010111111101110001100010111100001100100000001100010100110011011100010010011001110000000101100011101000000101
Pair \(Z_2\) Length of longest common subsequence
1ZUN_1,4LYP_1 185 21
1ZUN_1,3NBE_1 177 3
4LYP_1,3NBE_1 204 3

Newick tree

 
[
	4LYP_1:10.14,
	[
		1ZUN_1:88.5,3NBE_1:88.5
	]:11.64
]

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 \((0.85)(0.8)(\frac{774 }{\log_{20} 774}-\frac{325}{\log_{20}325})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 1ZUN_1 4LYP_1 158 132
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]