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Parikh vectors
1FEP_1 4YMP_1 8OEP_1 Letter Amino acid
31 17 6 K Lycine
29 4 5 P Proline
66 7 4 T Threonine
20 0 0 W Tryptophan
51 5 5 A Alanine
50 7 8 L Leucine
11 4 3 M Methionine
37 4 8 R Arginine
56 5 6 N Asparagine
2 0 2 C Cysteine
31 4 4 Q Glutamine
75 6 9 G Glycine
35 9 9 I Isoleucine
48 6 10 S Serine
45 11 11 D Aspartic acid
39 11 7 E Glutamic acid
9 4 3 H Histidine
17 8 3 F Phenylalanine
31 6 3 Y Tyrosine
41 12 8 V Valine 

1FEP_1|Chain A|FERRIC ENTEROBACTIN RECEPTOR|Escherichia coli K12 (83333)
>4YMP_1|Chains A, B[auth C]|Internalin|Bacillus anthracis (1392)
>8OEP_1|Chains A, C|Tyrosine-protein phosphatase non-receptor type 3|Homo sapiens (9606)
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 \(w=f(c)\)
1FEP , Knot 275 724 0.83 40 274 653 
QEPTDTPVSHDDTIVVTAAEQNLQAPGVSTITADEIRKNPVARDVSKIIRTMPGVNLTGNSTSGQRGNNRQIDIRGMGPENTLILIDGKPVSSRNSVRQGWRGERDTRGDTSWVPPEMIERIEVLRGPAAARYGNGAAGGVVNIITKKGSGEWHGSWDAYFNAPEHKEEGATKRTNFSLTGPLGDEFSFRLYGNLDKTQADAWDINQGHQSARAGTYATTLPAGREGVINKDINGVVRWDFAPLQSLELEAGYSRQGNLYAGDTQNTNSDSYTRSKYGDETNRLYRQNYALTWNGGWDNGVTTSNWVQYEHTRNSRIPEGLAGGTEGKFNEKATQDFVDIDLDDVMLHSEVNLPIDFLVNQTLTLGTEWNQQRMKDLSSNTQALTGTNTGGAIDGVSTTDRSPYSKAEIFSLFAENNMELTDSTIVTPGLRFDHHSIVGNNWSPALNISQGLGDDFTLKMGIARAYKAPSLYQTNPNYILYSKGQGCYASAGGCYLQGNDDLKAETSINKEIGLEFKRDGWLAGVTWFRNDYRNKIEAGYVAVGQNAVGTDLYQWDNVPKAVVEGLEGSLNVPVSETVMWTNNITYMLKSENKTTGDRLSIIPEYTLNSTLSWQAREDLSMQTTFTWYGKQQPKKYNYKGQPAVGPETKEISPYSIVGLSATWDVTKNVSLTGGVDNLFDKRLWRAGNAQTTGDLAGANYIAGAGAYTYNEPGRTWYMSVNTHF
4YMP , Knot 63 130 0.78 36 99 124 
ENMAVQSPKKHVFDAVIKAYKDNSDEESYATVYIKDPKLTIENGKRIITATLKDSDFFDYLKVEDSKEPGVFHDVKVLSEDKRKHGTKVIQFEVGELGKRYNMQMHILIPTLGYDKEFKIQFEVNMRTFV
8OEP , Knot 61 114 0.84 38 94 110 
GAMGSSTEDASQYYCDKNDNGDSYLVLIRITPDEDGKFGFNLKGGVDQKMPLVVSRINPESPADTCIPKLNEGDQIVLINGRDISEHTHDQVVMFIKASRESHSRELALVIRRR

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 \(c\).

\(|P_{f(1FEP_1)}(2) \setminus P_{f(4YMP_1)}(2)|=196\), \(|P_{f(4YMP_1)}(2) \setminus P_{f(1FEP_1)}(2)|=21\). 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 1:0010001100000111011000101111001010010001110010011001111010100001001000010101111000111101011000001001101000001000111101100101101111100101111111011000101010101010101100000110000010101111001010101010000101101001000101100100111100111000101110101111001010110000101011000000000000000100000100000110101110011000011000000000110111110010100010001101010011100010111011100010110010000100100000110100011110110000001000101101110001010000110111010000111001011101001110010101111010011010000100110001010010111001010001010001000111010001111110110000000101101111001110010010011011101101010111000111000100110000000100101110001000101010001010001010100010000001011111000010100111101010100010101110011000110110100010111100111111000001100101010001
Pair \(Z_2\) Length of longest common subsequence
1FEP_1,4YMP_1 217 3
1FEP_1,8OEP_1 216 4
4YMP_1,8OEP_1 139 3

Newick tree

 
[
	1FEP_1:11.38,
	[
		8OEP_1:69.5,4YMP_1:69.5
	]:48.88
]

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 \((0.85)(0.8)(\frac{854 }{\log_{20} 854}-\frac{130}{\log_{20}130})=203.\)
Status Protein1 Protein2 d d1/2
Query variables 1FEP_1 4YMP_1 252 147.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]