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Parikh vectors
6GGH_1 2XAM_1 1EWJ_1 Letter Amino acid
13 21 2 N Asparagine
15 45 8 S Serine
3 4 3 W Tryptophan
21 28 11 G Glycine
17 27 3 V Valine
10 24 11 A Alanine
13 20 8 R Arginine
17 26 10 D Aspartic acid
8 8 3 C Cysteine
24 36 7 E Glutamic acid
8 10 2 H Histidine
15 33 6 I Isoleucine
14 14 6 T Threonine
12 16 8 F Phenylalanine
14 18 7 P Proline
13 16 3 Y Tyrosine
11 16 6 Q Glutamine
30 47 17 L Leucine
27 36 1 K Lycine
7 6 4 M Methionine 

6GGH_1|Chains A, B|Tyrosine-protein kinase JAK1|Homo sapiens (9606)
>2XAM_1|Chains A, B|INOSITOL-PENTAKISPHOSPHATE 2-KINASE|Arabidopsis thaliana (3702)
>1EWJ_1|Chains A, B, C, D, E, F, G, H|BLEOMYCIN RESISTANCE DETERMINANT|Klebsiella pneumoniae (573)
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 \(w=f(c)\)
6GGH , Knot 129 292 0.83 40 199 282 
TEVDPTHFEKRFLKRIRDLGEGHFGKVELCRYDPEGDNTGEQVAVKSLKPESGGNHIADLKKEIEILRNLYHENIVKYKGICTEDGGNGIKLIMEFLPSGSLKEYLPKNKNKINLKQQLKYAVQICKGMDYLGSRQYVHRDLAARNVLVESEHQVKIGDFGLTKAIETDKEYYTVKDDRDSPVFWYAPECLMQSKFYIASDVWSFGVTLHELLTYCDSDSSPMALFLKMIGPTHGQMTVTRLVNTLKEGKRLPCPPNCPDEVYQLMRKCWEFQPSNRTSFQNLIEGFEALLK
2XAM , Knot 186 451 0.84 40 241 424 
MEMILEEKDASDWIYRGEGGANLVLAYAGSSPLFVGKVIRIQKARRNDKAIKNSNGVVSVLTSDEQHLWRENNELISSPNKEVLEQRYVQNVIIPLLGPKHVDAGVRVSVSKEFLECVDKKVTKQRPLWRVNAANVDTSHDSALILNDHSLFSQGITSGGDCISVEIKPKCGFLPTSRFIGKENMLKTSVSRFKMHQLLKLEYIEISEESEYDPLDLFSGSKERVLEAIKALYSTPQNNFRVFLNGSLILGGSGESTGRTSPEIGYAFEDALKGFIQSEDGHRTECFLQLVSDAVYGSGVLDRLLEIQKLDKLDIEGAIHCYYDIINQPCPICKEGRPLEAELSLHALPLDESLKIVKEYLIAATAKDCSIMISFQSRNAWDSEPSGDYVSLKPTNQTFDYKVHFIDLSLKPLKRMESYYKLDKKIISFYNRKQKAENTAEQIGNSKPSHS
1EWJ , Knot 65 126 0.83 40 104 123 
MTDQATPNLPSRDFDSTAAFYERLGFGIVFRDAGWMILQRGDLMLEFFAHPGLDPLASWFSCCLRLDDLAEFYRQCKSVGIQETSSGYPRIHAPELQEWGGTMAALVDPDGTLLRLIQNELLAGIS

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 \(c\).

\(|P_{f(6GGH_1)}(2) \setminus P_{f(2XAM_1)}(2)|=60\), \(|P_{f(2XAM_1)}(2) \setminus P_{f(6GGH_1)}(2)|=102\). 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 1:0010100100011001001101011010100001010001001110010100110011010001011001000011000110000110110111011101010001100000101000100110100110011000010001110011100000101101110011000000001000000111101100110001011001101110100110000000011111101111001010100110010010011011001001001100010101000001001101101110
Pair \(Z_2\) Length of longest common subsequence
6GGH_1,2XAM_1 162 4
6GGH_1,1EWJ_1 195 3
2XAM_1,1EWJ_1 215 3

Newick tree

 
[
	1EWJ_1:10.87,
	[
		6GGH_1:81,2XAM_1:81
	]:27.87
]

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 \((0.85)(0.8)(\frac{743 }{\log_{20} 743}-\frac{292}{\log_{20}292})=124.\)
Status Protein1 Protein2 d d1/2
Query variables 6GGH_1 2XAM_1 154 127
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]