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Parikh vectors
3FGU_1 2MKV_1 8BAV_1 Letter Amino acid
25 2 11 A Alanine
9 3 10 P Proline
37 3 17 V Valine
16 0 10 H Histidine
42 8 21 L Leucine
25 3 19 K Lycine
31 3 11 S Serine
30 7 9 R Arginine
27 5 20 D Aspartic acid
12 0 2 C Cysteine
48 3 16 E Glutamic acid
3 1 1 W Tryptophan
12 4 10 Y Tyrosine
42 9 24 G Glycine
17 4 14 I Isoleucine
23 0 8 M Methionine
13 3 16 N Asparagine
16 1 9 Q Glutamine
19 4 12 F Phenylalanine
23 1 18 T Threonine 

3FGU_1|Chain A|Glucokinase|Homo sapiens (9606)
>2MKV_1|Chain A|Sodium/potassium-transporting ATPase subunit gamma|Homo sapiens (9606)
>8BAV_1|Chains A, B|Green fluorescent protein,Synaptosomal-associated protein 25|Aequorea victoria (6100)
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 \(w=f(c)\)
3FGU , Knot 192 470 0.83 40 241 440 
MGHHHHHHENLYFQGMKKEKVEQILAEFQLQEEDLKKVMRRMQKEMDRGLRLETHEEASVKMLPTYVRSTPEGSEVGDFLSLDLGGTNFRVMLVKVGEGEEGQWSVKTKHQMYSIPEDAMTGTAEMLFDYISECISDFLDKHQMKHKKLPLGFTFSFPVRHEDIDKGILLNWTKGFKASGAEGNNVVGLLRDAIKRRGDFEMDVVAMVNDTVATMISCYYEDHQCEVGMIVGTGCNACYMEEMQNVELVEGDEGRMCVNTEWGAFGDSGELDEFLLEYDRLVDESSANPGQQLYEKLIGGKYMGELVRLVLLRLVDENLLFHGEASEQLRTRGAFETRFVSQVESDTGDRKQIYNILSTLGLRPSTTDCDIVRRACESVSTRAAHMCSAGLAGVINRMRESRSEDVMRITVGVDGSVYKLHPSFKERFHASVRRLTPSCEITFIESEEGSGRGAALVSAVACKKACMLGQ
2MKV , Knot 37 64 0.80 34 56 62 
LDRWYLGGSPKGDVDPFYYDYETVRNGGLIFAGLAFIVGLLILLSRRFRSGGNKKRRQINEDEP
8BAV , Knot 115 258 0.82 40 161 244 
SMASKGEELFTGVVPILVELDGDVNGHKFSVSGEGEGDATYGKLTLKFICTTGKLPVPWPTLVTTLTYGVQCFSRYPDHMKRHDFFKSAMPEGYVQERTIFFKDDGNYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNYNSHNVYIMADKQKNGIKVNFKTRHNIEDGSVQLADHYQQNTPIGDGPVLLPDNHYLSTQSALSKDPNEKRDHMVLLEFVTAAGTMAGIIGNLRHMALDMGNEIDTQNRQID

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 \(c\).

\(|P_{f(3FGU_1)}(2) \setminus P_{f(2MKV_1)}(2)|=201\), \(|P_{f(2MKV_1)}(2) \setminus P_{f(3FGU_1)}(2)|=16\). 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 1:11000000001010110000100111010100001001100100010011010000010101110010001010011011010111001011110110100101010000010011001101010111001000100110000100001111101011100001001111010011010110100111110011000101010111110001101100000000001111110100100100100101101001010100011111001010011100001100001011001000111100110110111101100011101010001000111000110010000100001001100111010000001100100010001101001111111001000000011010111010100101010001010100101000101100001010111110111000101110
Pair \(Z_2\) Length of longest common subsequence
3FGU_1,2MKV_1 217 4
3FGU_1,8BAV_1 176 4
2MKV_1,8BAV_1 169 3

Newick tree

 
[
	3FGU_1:10.10,
	[
		8BAV_1:84.5,2MKV_1:84.5
	]:18.60
]

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 \((0.85)(0.8)(\frac{534 }{\log_{20} 534}-\frac{64}{\log_{20}64})=141.\)
Status Protein1 Protein2 d d1/2
Query variables 3FGU_1 2MKV_1 177 99.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]