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Parikh vectors
3INV_1 6PJC_1 1CYZ_1 Letter Amino acid
55 12 0 L Leucine
31 6 0 P Proline
20 1 0 Y Tyrosine
13 1 0 H Histidine
8 2 2 C Cysteine
39 4 0 E Glutamic acid
38 13 3 G Glycine
8 2 0 W Tryptophan
21 4 0 N Asparagine
25 4 0 D Aspartic acid
12 5 0 Q Glutamine
21 7 0 K Lycine
29 8 3 T Threonine
41 4 0 R Arginine
23 13 0 I Isoleucine
13 2 0 M Methionine
21 2 0 F Phenylalanine
31 0 0 S Serine
35 6 0 V Valine
37 3 2 A Alanine 

3INV_1|Chains A, B|Bifunctional dihydrofolate reductase-thymidylate synthase|Trypanosoma cruzi (5693)
>6PJC_1|Chains A, B, C, D|Protease NL4-3|Human immunodeficiency virus 1 (11676)
>1CYZ_1|Chains A, B|5'-D(*GP*AP*AP*CP*TP*GP*GP*TP*TP*C)-3'|
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 \(w=f(c)\)
3INV , Knot 213 521 0.85 40 264 488 
MSLFKIRMPETVAEGTRLALRAFSLVVAVDEHGGIGDGRSIPWNVPEDMKFFRDLTTKLRGKNVKPSPAKRNAVVMGRKTWDSIPPKFRPLPGRLNVVLSSTLTTQHLLDGLPDEEKRNLHADSIVAVNGGLEQALRLLASPNYTPSIETVYCIGGGSVYAEALRPPCVHLLQAIYRTTIRASESSCSVFFRVPESGTEAAAGIEWQRETISEELTSANGNETKYYFEKLIPRNREEEQYLSLVDRIIREGNVKHDRTGVGTLSIFGAQMRFSLRNNRLPLLTTKRVFWRGVCEELLWFLRGETYAKKLSDKGVHIWDDNGSRAFLDSRGLTEYEEMDLGPVYGFQWRHFGAAYTHHDANYDGQGVDQIKAIVETLKTNPDDRRMLFTAWNPSALPRMALPPCHLLAQFYVSNGELSCMLYQRSCDMGLGVPFNIASYALLTILIAKATGLRPGELVHTLGDAHVYSNHVEPCNEQLKRVPRAFPYLVFRREREFLEDYEEGDMEVIDYAPYPPISMKMAV
6PJC , Knot 52 99 0.80 38 79 94 
PQITLWKRPLVTIKIGGQLKEALLDTGADDTVLEEMNLPGRWKPKMIGGIGGFIKVRQYDQILIEICGHKAIGTVLVGPTPVNIIGRNLLTQIGCTLNF
1CYZ , Knot 7 10 0.53 8 9 8 
GAACTGGTTC

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 \(c\).

\(|P_{f(3INV_1)}(2) \setminus P_{f(6PJC_1)}(2)|=207\), \(|P_{f(6PJC_1)}(2) \setminus P_{f(3INV_1)}(2)|=22\). 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 1:10110101100110100111011011111000111101001110110010110010001010010101100011111000100111010111101011100010000110111000000101001111011100110111010001010010011110101011011010110110000101000000111011001001111101000010001001010000001001110000000010110011001010000011101011110101010000111100001110110001111101000100100011011000100111000110000010111101101001111000001000101100101110010001000011101101011101111100111010100101001100000011111110110011101111010110110110011010100001010000100110111011100000110000010101100110111010111
Pair \(Z_2\) Length of longest common subsequence
3INV_1,6PJC_1 229 4
3INV_1,1CYZ_1 261 3
6PJC_1,1CYZ_1 78 2

Newick tree

 
[
	3INV_1:13.95,
	[
		6PJC_1:39,1CYZ_1:39
	]:10.95
]

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 \((0.85)(0.8)(\frac{620 }{\log_{20} 620}-\frac{99}{\log_{20}99})=152.\)
Status Protein1 Protein2 d d1/2
Query variables 3INV_1 6PJC_1 195 113.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]