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Parikh vectors
3NBW_1 7UPE_1 2MVJ_1 Letter Amino acid
9 35 1 T Threonine
3 5 1 Y Tyrosine
32 27 1 V Valine
9 11 0 N Asparagine
20 29 1 D Aspartic acid
9 2 0 C Cysteine
30 49 3 G Glycine
14 3 3 F Phenylalanine
4 6 2 M Methionine
13 43 0 P Proline
23 34 2 A Alanine
16 19 0 Q Glutamine
15 27 0 E Glutamic acid
12 12 0 H Histidine
27 21 3 L Leucine
12 15 2 I Isoleucine
19 14 2 R Arginine
14 44 4 K Lycine
28 45 1 S Serine
4 0 0 W Tryptophan 

3NBW_1|Chains A, B|Ketohexokinase|Homo sapiens (9606)
>7UPE_1|Chains A, B, C, D, E, F, G, H, I, J|Isoform Tau-F of Microtubule-associated protein tau|Homo sapiens (9606)
>2MVJ_1|Chain A|Stage V sporulation protein M|Bacillus subtilis (224308)
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 \(w=f(c)\)
3NBW , Knot 135 313 0.82 40 188 294 
MGSSHHHHHHSSGLVPRGSQILCVGLVVLDVISLVDKYPKEDSEIRCLSQRWQRGGNASNSCTVLSLLGAPCAFMGSMAPGHVADFLVADFRRRGVDVSQVAWQSKGDTPSSCCIINNSNGNRTIVLHDTSLPDVSATDFEKVDLTQFKWIHIEGRNASEQVKMLQRIDAHNTRQPPEQKIRVSVEVEKPREELFQLFGYGDVVFVSKDVAKHLGFQSAEEALRGLYGRVRKGAVLVCAWAEEGADALGPDGKLLHSDAFPPPRVVDTLGAGDTFNASVIFSLSQGRSVQEALRFGCQVAGKKCGLQGFDGIV
7UPE , Knot 167 441 0.76 38 190 383 
MAEPRQEFEVMEDHAGTYGLGDRKDQGGYTMHQDQEGDTDAGLKESPLQTPTEDGSEEPGSETSDAKSTPTAEDVTAPLVDEGAPGKQAAAQPHTEIPEGTTAEEAGIGDTPSLEDEAAGHVTQARMVSKSKDGTGSDDKKAKGADGKTKIATPRGAAPPGQKGQANATRIPAKTPPAPKTPPSSGEPPKSGDRSGYSSPGSPGTPGSRSRTPSLPTPPTREPKKVAVVRTPPKSPSSAKSRLQTAPVPMPDLKNVKSKIGSTENLKHQPGGGKVQIINKKLDLSNVQSKCGSKDNIKHVPGGGSVQIVYKPVDLSKVTSKCGSLGNIHHKPGGGQVEVKSEKLDFKDRVQSKIGSLDNITHVPGGGNKKIETHKLTFRENAKAKTDHGAEIVYKSPVVSGDTSPRHLSNVSSTGSIDMVDSPQLATLADEVSASLAKQGL
2MVJ , Knot 17 26 0.71 26 23 24 
MKFYTIKLAKFLGGIVRAMLGSFRKD

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 \(c\).

\(|P_{f(3NBW_1)}(2) \setminus P_{f(7UPE_1)}(2)|=74\), \(|P_{f(7UPE_1)}(2) \setminus P_{f(3NBW_1)}(2)|=76\). 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 1:1100000000001111010011011111101101100010000010010001001101000001101111101111011110110111101000110100111000100100001100001000111000011010100100101001011010100100010110010100000110001010101001000110111010111100011001110010011011010100111110111001101111010110001111101100111100101011101001001001101100111000110110111
Pair \(Z_2\) Length of longest common subsequence
3NBW_1,7UPE_1 150 5
3NBW_1,2MVJ_1 183 3
7UPE_1,2MVJ_1 185 3

Newick tree

 
[
	2MVJ_1:97.00,
	[
		3NBW_1:75,7UPE_1:75
	]:22.00
]

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 \((0.85)(0.8)(\frac{754 }{\log_{20} 754}-\frac{313}{\log_{20}313})=120.\)
Status Protein1 Protein2 d d1/2
Query variables 3NBW_1 7UPE_1 145 124
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]