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Parikh vectors
6FRP_1 3DZM_1 2VAT_1 Letter Amino acid
12 8 19 Q Glutamine
9 1 13 M Methionine
21 10 16 E Glutamic acid
16 31 29 G Glycine
23 8 13 K Lycine
3 3 6 W Tryptophan
19 7 13 N Asparagine
24 15 31 D Aspartic acid
7 2 11 C Cysteine
28 2 26 I Isoleucine
20 10 29 P Proline
16 14 39 S Serine
14 9 22 T Threonine
19 11 14 Y Tyrosine
24 17 43 A Alanine
19 6 32 R Arginine
19 6 12 H Histidine
44 23 29 L Leucine
13 12 21 F Phenylalanine
14 19 26 V Valine 

6FRP_1|Chain A|Mitogen-activated protein kinase 1|Rattus norvegicus (10116)
>3DZM_1|Chains A, B, C|Hypothetical conserved protein|Thermus thermophilus (262724)
>2VAT_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|ACETYL-COA--DEACETYLCEPHALOSPORIN C ACETYLTRANSFERASE|ACREMONIUM CHRYSOGENUM (5044)
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 \(w=f(c)\)
6FRP , Knot 157 364 0.84 40 235 349 
HHHHHHMAAAAAAGPEMVRGQVFDVGPRYTNLSYIGEGAYGMVCSAYDNLNKVRVAIKKISPFEHQTYCQRTLREIKILLRFRHENIIGINDIIRAPTIEQMKDVYIVQDLMETDLYKLLKTQHLSNDHICYFLYQILRGLKYIHSANVLHRDLKPSNLLLNTTCDLKICDFGLARVADPDHDHTGFLTEYVATRWYRAPEIMLNSKGYTKSIDIWSVGCILAEMLSNRPIFPGKHYLDQLNHILGILGSPSQEDLNCIINLKARNYLLSLPHKNKVPWNRLFPNADSKALDLLDKMLTFNPHKRIEVEQALAHPYLEQYYDPSDEPIAEAPFKFDMELDDLPKEKLKELIFEETARFQPGYRS
3DZM , Knot 93 214 0.77 40 131 192 
QHHHHHHAAKFSVEAGAGFYGGFGGQLAVVAEDLAPGLPLGVRLGVGFATSDALDDGYDLGGGTTWGDVKEAGKFSEWGQNVTLSLDVLYKPSGLGLPVEVAPYFGVRYNFFSGGYTDPEDNLTIKAQTISSNQLGLGLGVRAAYPLMPNLSLVGDLGVDYYFQACFTRVEEDDSGNKSQSSVCPGDSGYEDVNKFVTQPEWVLKLRLGAAYRF
2VAT , Knot 188 444 0.86 40 241 424 
MLPSAQVARLKPDPFPPSLSPIPHGAVTFAALAPCHNLPIFSSRQMLRDSLTYSHTSPTMSPQIANRFEASLDAQDIARISLFTLESGVILRDVPVAYKSWGRMNVSRDNCVIVCHTLTSSAHVTSWWPTLFGQGRAFDTSRYFIICLNYLGSPFGSAGPCSPDPDAEGQRPYGAKFPRTTIRDDVRIHRQVLDRLGVRQIAAVVGASMGGMHTLEWAFFGPEYVRKIVPIATSCRQSGWCAAWFETQRQCIYDDPKYLDGEYDVDDQPVRGLETARKIANLTYKSKPAMDERFHMAPGVQAGRNISSQDAKKEINGTDSGNSHRAGQPIEAVSSYLRYQAQKFAASFDANCYIAMTLKFDTHDISRGRAGSIPEALAMITQPALIICARSDGLYSFDEHVEMGRSIPNSRLCVVDTNEGHDFFVMEADKVNDAVRGFLDQSLM

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 \(c\).

\(|P_{f(6FRP_1)}(2) \setminus P_{f(3DZM_1)}(2)|=149\), \(|P_{f(3DZM_1)}(2) \setminus P_{f(6FRP_1)}(2)|=45\). 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 1:0000001111111110110101101110000100110110111001000100101110010110000000001001011101000011110011011010010010110011000100110000100001001100110110010010110001010011100000101001111011010000011100011001001101110001000010110110111011000111110001001001111110100001001101010001101100001110011101000110110011010100010100111010100000100011101110101010011000100111000101011000
Pair \(Z_2\) Length of longest common subsequence
6FRP_1,3DZM_1 194 6
6FRP_1,2VAT_1 166 4
3DZM_1,2VAT_1 194 4

Newick tree

 
[
	3DZM_1:10.23,
	[
		6FRP_1:83,2VAT_1:83
	]:18.23
]

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 \((0.85)(0.8)(\frac{578 }{\log_{20} 578}-\frac{214}{\log_{20}214})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 6FRP_1 3DZM_1 134 103
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]