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Parikh vectors
3KEE_1 3DFO_1 9GQH_1 Letter Amino acid
12 15 3 R Arginine
3 15 4 N Asparagine
14 34 9 L Leucine
5 26 16 K Lycine
18 21 8 S Serine
5 12 4 Y Tyrosine
7 8 0 C Cysteine
9 11 3 H Histidine
12 19 6 P Proline
17 22 6 T Threonine
7 13 7 D Aspartic acid
8 16 2 Q Glutamine
4 24 10 E Glutamic acid
4 3 2 M Methionine
14 42 8 A Alanine
21 30 15 G Glycine
6 20 9 I Isoleucine
3 7 6 F Phenylalanine
2 3 1 W Tryptophan
19 22 9 V Valine 

3KEE_1|Chains A, C[auth B], E[auth C], G[auth D]|Genome polyprotein|Hepatitis C virus (3052230)
>3DFO_1|Chains A, B, C, D|Fructose-bisphosphate aldolase A|Oryctolagus cuniculus (9986)
>9GQH_1|Chains A, B|Peptidyl-prolyl cis-trans isomerase FKBP5|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 \(w=f(c)\)
3KEE , Knot 91 190 0.83 40 136 181 
MAPITAYSQQTRGLLGCIITSLTGRDKNQVEGEVQVVSTATQSFLATCVNGVCWTVYHGAGSKTLAGPKGPITQMYTNVDQDLVGWQAPPGARSLTPCTCGSSDLYLVTRHADVIPVRRRGDSRGSLLSPRPVSYLKGSSGGPLLCPSGHAVGIFRAAVCTRGVAKAVDFVPVESMETTMRSGSHHHHHH
3DFO , Knot 159 363 0.86 40 206 345 
PHSHPALTPEQKKELSDIAHRIVAPGKGILAANESTGSIAKRLQSIGTENTEENRRFYRQLLLTADDRVNPCIGGVILFHETLYQKADDGRPFPQVIKSKGGVVGIKVDKGVVPLAGTNGETTTQGLDGLSERCAQYKKDGADFAKWRCVLKIGEHTPSALAIMENANVLARYASICQQNGIVPIVEPEILPDGDHDLKRCQYVTEKVLAAVYKALSDHHIYLEGTLLKPNMVTPGHACTQKYSHEEIAMATVTALRRTVPPAVTGVTFLSGGQSEEEASINLNAINKCPLLKPWALTFSYGRALQASALKAWGGKKENLKAAQEEYVKRALANSLACQGKYTPSGQAGAAASESLFISNHAY
9GQH , Knot 64 128 0.80 38 103 125 
GAPATVTEQGEDITSKKDRGVLKIVKRVGNGEETPMIGDKVYVHYKGKLSNGKKFDSSHDRNEPFVFSLGKGQVIKAWDIGVATMKKGEIAHLLIKPEYAYGSAGSLPKIPSNATLFFEIELLDFKGE

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 \(c\).

\(|P_{f(3KEE_1)}(2) \setminus P_{f(3DFO_1)}(2)|=51\), \(|P_{f(3DFO_1)}(2) \setminus P_{f(3KEE_1)}(2)|=121\). 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 1:1111010000001111011001010000010101011001000111001011010100111000111101110010001000111101111100101000100010110001011110001000101101011001010011111010101111101110001110110111100100010010000000
Pair \(Z_2\) Length of longest common subsequence
3KEE_1,3DFO_1 172 3
3KEE_1,9GQH_1 157 3
3DFO_1,9GQH_1 161 4

Newick tree

 
[
	3DFO_1:84.83,
	[
		3KEE_1:78.5,9GQH_1:78.5
	]:6.33
]

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 \((0.85)(0.8)(\frac{553 }{\log_{20} 553}-\frac{190}{\log_{20}190})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 3KEE_1 3DFO_1 134 101
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]