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Parikh vectors
1JSD_1 3NRG_1 2WAS_1 Letter Amino acid
11 7 10 K Lycine
4 5 0 M Methionine
17 10 3 P Proline
15 9 1 Y Tyrosine
31 10 6 T Threonine
17 16 2 R Arginine
9 1 1 C Cysteine
12 14 10 E Glutamic acid
26 12 10 G Glycine
30 25 6 L Leucine
7 14 4 F Phenylalanine
11 19 15 A Alanine
12 15 6 D Aspartic acid
10 13 4 Q Glutamine
18 15 7 I Isoleucine
29 7 11 S Serine
20 14 16 V Valine
25 7 7 N Asparagine
9 3 2 H Histidine
6 1 1 W Tryptophan 

1JSD_1|Chain A|HAEMAGGLUTININ (HA1 CHAIN)|Influenza A virus (A/swine/Hong Kong/9/98(H9N2)) (145307)
>3NRG_1|Chains A, B, C, D, E|TetR family transcriptional regulator|Chloroflexus aurantiacus (324602)
>2WAS_1|Chains A, D, E|3-OXOACYL-[ACYL-CARRIER-PROTEIN] SYNTHASE|SACCHAROMYCES CEREVISIAE (4932)
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 \(w=f(c)\)
1JSD , Knot 139 319 0.83 40 201 306 
DKICIGYQSTNSTETVDTLTETNVPVTHAKELLHTSHNGMLCATNLGHPLILDTCTIEGLIYGNPSCDLLLGGREWSYIVERPSAVNGMCYPGNVENLEELRSLFSSASSYQRIQIFPDTIWNVSYSGTSSACSDSFYRSMRWLTQKNNAYPIQDAQYTNNRGKSILFMWGINHPPTDTVQTNLYTRTDTTTSVTTEDINRTFKPVIGPRPLVNGLHGRIDYYWSVLKPGQTLRVRSNGNLIAPWYGHILSGESHGRILKTDLNSGNCVVQCQTERGGLNTTLPFHNVSKYAFGNCPKYVGVKSLKLAVGLRNVPARSS
3NRG , Knot 99 217 0.81 40 147 206 
GMPTETFFNLPEEKRSRLIDVLLDEFAQNDYDSVSINRITERAGIAKGSFYQYFADKKDCYLYLIQLGIEQKTAFLRQTPPASTTDMFAYLRWLLDVGIQFQFHNPRLAQIAYKALYDDVPLPAETMQVIRHGSFAYFKQLVEQGIADGSLVPDLDADTAAFVLNVVFTELGNHLIERFAVNPAELLREGGIVLLQPAMRRVIEQVIDILERGMRRR
2WAS , Knot 60 122 0.78 38 91 115 
NGGVGVDVELITSINVENDTFIERNFTPQEIEYCSAQPSVQSSFAGTWSAKEAVFKSLGVKSLGGGAALKDIEIVRVNKNAPAVELHGNAKKAAEEAGVTDVKVSISHDDLQAVAVAVSTKK

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 \(c\).

\(|P_{f(1JSD_1)}(2) \setminus P_{f(3NRG_1)}(2)|=113\), \(|P_{f(3NRG_1)}(2) \setminus P_{f(1JSD_1)}(2)|=59\). 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 1:0010110000000001001000011100100110000011101001101111000010111010100011111001001100101101100110100100100110010000010111001101000100010000100010110000010110010000001001111111001100010001000000000100001000101111101110110101000101101100101000101111101011010001011000100100110000001110001110010001110010011100101111100111000
Pair \(Z_2\) Length of longest common subsequence
1JSD_1,3NRG_1 172 3
1JSD_1,2WAS_1 178 5
3NRG_1,2WAS_1 140 3

Newick tree

 
[
	1JSD_1:92.61,
	[
		3NRG_1:70,2WAS_1:70
	]:22.61
]

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 \((0.85)(0.8)(\frac{536 }{\log_{20} 536}-\frac{217}{\log_{20}217})=91.5\)
Status Protein1 Protein2 d d1/2
Query variables 1JSD_1 3NRG_1 120 98.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]