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Parikh vectors
3VXT_1 9IIS_1 8VNN_1 Letter Amino acid
10 38 0 V Valine
12 69 0 L Leucine
10 33 0 P Proline
33 52 0 S Serine
17 36 0 D Aspartic acid
0 24 0 H Histidine
9 30 0 I Isoleucine
9 16 0 Y Tyrosine
13 19 0 N Asparagine
11 21 0 F Phenylalanine
3 13 0 W Tryptophan
13 19 0 Q Glutamine
8 52 0 E Glutamic acid
11 46 1 G Glycine
12 40 0 K Lycine
4 16 0 M Methionine
11 33 3 A Alanine
6 33 0 R Arginine
4 5 3 C Cysteine
9 22 6 T Threonine 

3VXT_1|Chains A, C|T36-5 TCR alpha chain|Homo sapiens (9606)
>9IIS_1|Chain A|Ubiquitin-like protein SMT3,L-fucokinase/L-fucose-1-P guanylyltransferase|Saccharomyces cerevisiae S288C (559292)
>8VNN_1|Chains A[auth C], C[auth D]|DNA (5'-D(*TP*TP*GP*AP*CP*TP*CP*TP*CP*TP*TP*AP*A)-3')|synthetic construct (32630)
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 \(w=f(c)\)
3VXT , Knot 96 205 0.83 38 140 198 
MQKEVEQNSGPLSVPEGAIASLNCTYSDRGSQSFFWYRQYSGKSPELIMSIYSNGDKEDGRFTAQLNKASQYVSLLIRDSQPSDSATYLWGTYNQGGKLIFGQGTELSVKPNIQNPDPAVYQLRDSKSSDKSVCLFTDFDSQTNVSQSKDSDVYITDKCVLDMRSMDFKSNSAVAWSNKSDFACANAFNNSIIPEDTFFPSPESS
9IIS , Knot 239 617 0.83 40 265 564 
MGSSHHHHHHGSGLVPRGSASMSDSEVNQEAKPEVKPEVKPETHINLKVSDGSSEIFFKIKKTTPLRRLMEAFAKRQGKEMDSLRFLYDGIRIQADQTPEDLDMEDNDIIEAHREQIGGEFMQKLLSLPSNLVQSFHELERVNRTDWFCTSDPVGKKLGSGGGTSWLLEECYNEYSDGATFGEWLEKEKRILLHAGGQSRRLPGYAPSGKILTPVPVFRWERGQHLGQNLLSLQLPLYEKIMSLAPDKLHTLIASGDVYIRSEKPLQSIPEADVVCYGLWVDPSLATHHGVFASDRKHPEQLDFMLQKPSLAELESLSKTHLFLMDIGIWLLSDRAVEILMKRSHKESSEELKYYDLYSDFGLALGTHPRIEDEEVNTLSVAILPLPGGEFYHYGTSKELISSTLSVQNKVYDQRRIMHRKVKPNPAMFVQNAVVRIPLCAENADLWIENSHIGPKWKIASRHIITGVPENDWSLAVPAGVCVDVVPMGDKGFVARPYGLDDVFKGDLRDSKTTLTGIPFGEWMSKRGLSYTDLKGRTDDLQAVSVFPMVNSVEELGLVLRWMLSEPELEEGKNIWLRSEHFSADEISAGANLKRLYAQREEFRKGNWKALAVNHEK
8VNN , Knot 7 13 0.46 8 8 9 
TTGACTCTCTTAA

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 \(c\).

\(|P_{f(3VXT_1)}(2) \setminus P_{f(9IIS_1)}(2)|=32\), \(|P_{f(9IIS_1)}(2) \setminus P_{f(3VXT_1)}(2)|=157\). 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 1:1000100001110110111101000000010001110000010010111010001000010101010010001011100001000100111000011011110100101010100101110010000000001011001000001000000010100001101001010000111100000110101100011100011101000
Pair \(Z_2\) Length of longest common subsequence
3VXT_1,9IIS_1 189 5
3VXT_1,8VNN_1 140 2
9IIS_1,8VNN_1 265 2

Newick tree

 
[
	9IIS_1:12.58,
	[
		3VXT_1:70,8VNN_1:70
	]:56.58
]

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 \((0.85)(0.8)(\frac{822 }{\log_{20} 822}-\frac{205}{\log_{20}205})=171.\)
Status Protein1 Protein2 d d1/2
Query variables 3VXT_1 9IIS_1 214 140
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]