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Parikh vectors
5XKF_1 9JJW_1 2RJY_1 Letter Amino acid
16 14 2 Q Glutamine
27 23 0 I Isoleucine
29 21 4 T Threonine
34 17 4 V Valine
36 11 5 A Alanine
21 18 3 R Arginine
32 47 9 L Leucine
20 13 5 P Proline
37 34 5 E Glutamic acid
13 10 1 H Histidine
19 28 7 K Lycine
20 22 5 F Phenylalanine
4 7 1 W Tryptophan
27 23 5 D Aspartic acid
12 7 0 C Cysteine
10 17 1 M Methionine
23 35 3 S Serine
19 11 0 Y Tyrosine
16 19 4 N Asparagine
36 22 3 G Glycine 

5XKF_1|Chains A, C|Tubulin alpha-1B chain|Sus scrofa (9823)
>9JJW_1|Chain A|LOC432253 protein|Xenopus laevis (8355)
>2RJY_1|Chain A|Villin-1|Gallus gallus (9031)
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 \(w=f(c)\)
5XKF , Knot 192 451 0.86 40 258 426 
MRECISIHVGQAGVQIGNACWELYCLEHGIQPDGQMPSDKTIGGGDDSFNTFFSETGAGKHVPRAVFVDLEPTVIDEVRTGTYRQLFHPEQLITGKEDAANNYARGHYTIGKEIIDLVLDRIRKLADQCTGLQGFLVFHSFGGGTGSGFTSLLMERLSVDYGKKSKLEFSIYPAPQVSTAVVEPYNSILTTHTTLEHSDCAFMVDNEAIYDICRRNLDIERPTYTNLNRLISQIVSSITASLRFDGALNVDLTEFQTNLVPYPRIHFPLATYAPVISAEKAYHEQLSVAEITNACFEPANQMVKCDPRHGKYMACCLLYRGDVVPKDVNAAIATIKTKRSIQFVDWCPTGFKVGINYQPPTVVPGGDLAKVQRAVCMLSNTTAIAEAWARLDHKFDLMYAKRAFVHWYVGEGMEEGEFSEAREDMAALEKDYEEVGVDSVEGEGEEEGEEY
9JJW , Knot 167 399 0.83 40 223 373 
AEPDHPIQLVWTDVNGRLSLDLSGAHDCFLNTRYSNYPVFILCIIGEKRRGKSFLMNYIMRALRSMEMDEEISLGADDEPLKGFKWSPGTETTTKGIWMWNRPFLLNHKGGKIAVFLLDTEGSLDIESDRETCIKLSALSLFISSHLIFNVASNLKETELDYMEMYMNMGEECGPKNLQHLDILVRDWYHSKKWDRDVARSYISREIEKLEKLNSYPKVLWSLKSNQTRCFLLPHPGKGITGESEGRLQDMDEDFQESLRSYVSKVVKGICTHIKTNIDGELLTSAHVFSMLQEFTEVLNLQIYGFSSPMEMFYAIKNQKLMGEIENEFQDFLKNQSSLTLPPTMRVKVSQKFSELLEKFMQFVQGSNTSSHDAMLKDLEVRLLEIQEKFCNDFTTRFT
2RJY , Knot 38 67 0.79 34 61 65 
PTKLETFPLDVLVNTAAEDLPRGVDPSRKENHLSDEDFKAVFGMTRSAFANLPLWKQQNLKKEKGLF

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 \(c\).

\(|P_{f(5XKF_1)}(2) \setminus P_{f(9JJW_1)}(2)|=105\), \(|P_{f(9JJW_1)}(2) \setminus P_{f(5XKF_1)}(2)|=70\). 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 1:1000101011011101101010100100110101011000011110001001100011100110111101010110010010000110100110100011000101000110011011100100110000110111110011110101100111001010010000101010111010011101000110000010000011110001100100001010010000100110011001010101011101010010001110101011110011110100100001011010010101100110001001001100110010111001011110100000101101010110111000110111110110100110110000111011101000101101001110101101100101001000111100000011100101010001000
Pair \(Z_2\) Length of longest common subsequence
5XKF_1,9JJW_1 175 4
5XKF_1,2RJY_1 229 3
9JJW_1,2RJY_1 206 4

Newick tree

 
[
	2RJY_1:11.15,
	[
		5XKF_1:87.5,9JJW_1:87.5
	]:27.65
]

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 \((0.85)(0.8)(\frac{850 }{\log_{20} 850}-\frac{399}{\log_{20}399})=120.\)
Status Protein1 Protein2 d d1/2
Query variables 5XKF_1 9JJW_1 160 147.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]