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Parikh vectors
2DHR_1 8URV_1 1EVM_1 Letter Amino acid
2 0 0 W Tryptophan
29 20 0 D Aspartic acid
1 5 0 C Cysteine
25 15 0 K Lycine
12 8 0 M Methionine
19 14 0 F Phenylalanine
22 7 1 T Threonine
10 6 0 Q Glutamine
53 17 0 E Glutamic acid
40 6 3 G Glycine
42 8 0 V Valine
52 8 0 R Arginine
11 1 0 H Histidine
6 5 0 Y Tyrosine
62 7 1 A Alanine
8 13 0 N Asparagine
20 17 0 I Isoleucine
47 13 0 L Leucine
26 7 0 P Proline
12 16 0 S Serine 

2DHR_1|Chains A, B, C, D, E, F|FtsH|Thermus thermophilus (274)
>8URV_1|Chain A|Interleukin-18|Homo sapiens (9606)
>1EVM_1|Chains A[auth M], B[auth N], C[auth O], D[auth P]|DNA (5'-D(*AP*GP*GP*GP*T)-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 \(w=f(c)\)
2DHR , Knot 196 499 0.81 40 209 441 
RNGRAGPSDSAFSFTKSRARVLTEAPKVTFKDVAGAEEAKEELKEIVEFLKNPSRFHEMGARIPKGVLLVGPPGVGKTHLARAVAGEARVPFITASGSDFVEMFVGVGAARVRDLFETAKRHAPCIVFIDEIDAVGRKRGSGVGGGNDEREQTLNQLLVEMDGFEKDTAIVVMAATNRPDILDPALLRPGRFDRQIAIDAPDVKGREQILRIHARGKPLAEDVDLALLAKRTPGFVGADLENLLNEAALLAAREGRRKITMKDLEEAADRVMMLPAKKSLVLSPRDRRITAYHEAGHALAAHFLEHADGVHKVTIVPRGRALGFMMPRREDMLHWSRKRLLDQIAVALAGRAAEEIVFDDVTTGAENDFRQATELARRMITEWGMHPEFGPVAYAVREDTYLGGYDVRQYSEETAKRIDEAVRRLIEEQYQRVKALLLEKREVLERVAETLLERETLTAEEFQRVVEGLPLEAPEEAREEREPPRVVPKVKPGGALGGA
8URV , Knot 87 193 0.79 38 137 187 
MAAEPVEDNCINFVAMKFIDNTLYFIAEDDENLESDYFGKLESKLSVIRNLNDQVLFIDQGNRPLFEDMTDSDCRDNAPRTIFIISMYKDSQPRGMAVTISVKCEKISTLSCENKIISFKEMNPPDNIKDTKSDIIFFQRSVPGHDNKMQFESSSYEGYFLACEKERDLFKLILKKEDELGDRSIMFTVQNED
1EVM , Knot 3 5 0.32 6 3 3 
AGGGT

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 \(c\).

\(|P_{f(2DHR_1)}(2) \setminus P_{f(8URV_1)}(2)|=127\), \(|P_{f(8URV_1)}(2) \setminus P_{f(2DHR_1)}(2)|=55\). 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 1:0010111000110100001011001101010011110010001001101100100100111011011111111111000110111101011110101001101111111101001100100011011110010111000101111100000001001110101100001111111000101101111011010001110110101000110101010111001011111000111111010011001111110010001010010011001111110001110100001010001101111011001011001011101011111110000110100001100111111101100111001001100010010011001100111010111110110000011100100000001001001100110000001011110000110011001100001010010011011110110010000011011101011111111
Pair \(Z_2\) Length of longest common subsequence
2DHR_1,8URV_1 182 4
2DHR_1,1EVM_1 208 3
8URV_1,1EVM_1 140 1

Newick tree

 
[
	2DHR_1:10.34,
	[
		8URV_1:70,1EVM_1:70
	]:35.34
]

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 \((0.85)(0.8)(\frac{692 }{\log_{20} 692}-\frac{193}{\log_{20}193})=140.\)
Status Protein1 Protein2 d d1/2
Query variables 2DHR_1 8URV_1 173 119.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]