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Parikh vectors
4RPZ_1 1VJY_1 1ZLL_1 Letter Amino acid
26 25 10 L Leucine
8 10 3 M Methionine
14 8 1 P Proline
27 22 2 E Glutamic acid
23 23 8 I Isoleucine
3 6 3 C Cysteine
18 19 2 V Valine
15 10 0 H Histidine
14 8 2 F Phenylalanine
21 18 2 S Serine
19 14 2 T Threonine
19 23 4 R Arginine
21 15 0 D Aspartic acid
13 10 5 Q Glutamine
23 20 0 G Glycine
36 19 2 K Lycine
1 8 0 W Tryptophan
12 10 1 Y Tyrosine
18 24 3 A Alanine
12 11 2 N Asparagine 

4RPZ_1|Chain A|DNA polymerase beta|Homo sapiens (9606)
>1VJY_1|Chain A|TGF-beta receptor type I|Homo sapiens (9606)
>1ZLL_1|Chains A, B, C, D, E|Cardiac phospholamban|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 \(w=f(c)\)
4RPZ , Knot 148 343 0.84 40 201 326 
MGMSKRKAPQETLNGGITDMLTELANFEKNVSQAIHKYNAYRKAASVIAKYPHKIKSGAEAKKLPGVGTKIAEKIDEFLATGKLRKLEKIRQDDTSSSINFLTRVSGIGPSAARKFVDEGIKTLEDLRKNEDKLNHHQRIGLKYFGDFEKRIPREEMLQMQDIVLNEVKKVDSEYIATVCGSFRRGAESSGDMDVLLTHPSFTSESTKQPKLLHQVVEQLQKVHFITDTLSKGETKFMGVCQLPSKNDEKEYPHRRIDIRLIPKDQYYCGVLYFTGSDIFNKNMRAHALEKGFTINEYTIRPLGVTGVAGEPLPVDSEKDIFDYIQWKYREPKDRSEHHHHHH
1VJY , Knot 135 303 0.84 40 200 292 
IARTIVLQESIGKGRFGEVWRGKWRGEEVAVKIFSSREERSWFREAEIYQTVMLRHENILGFIAADNKDNGTWTQLWLVSDYHEHGSLFDYLNRYTVTVEGMIKLALSTASGLAHLHMEIVGTQGKPAIAHRDLKSKNILVKKNGTCCIADLGLAVRHDSATDTIDIAPNHRVGTKRYMAPEVLDDSINMKHFESFKRADIYAMGLVFWEIARRCSIGGIHEDYQLPYYDLVPSDPSVEEMRKVVCEQKLRPNIPNRWQSCEALRVMAKIMRECWYANGAARLTALRIKKTLSQLSQQEGIKM
1ZLL , Knot 32 52 0.81 32 45 49 
MEKVQYLTRSAIRRASTIEMPQQARQKLQNLFINFCLILICLLLICIIVMLL

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 \(c\).

\(|P_{f(4RPZ_1)}(2) \setminus P_{f(1VJY_1)}(2)|=82\), \(|P_{f(1VJY_1)}(2) \setminus P_{f(4RPZ_1)}(2)|=81\). 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 1:1110000110001011100110011010001001100001000110111001001001101001111100110010011101010010010000000010110010111101100110011001001000000100000111001101000110001101001110010010000110101010011000101011100101000000010110011001001011000100100011110011000000001000101011100000011101010011000101011001101000010111101111011110000011001010000100000000000
Pair \(Z_2\) Length of longest common subsequence
4RPZ_1,1VJY_1 163 4
4RPZ_1,1ZLL_1 188 3
1VJY_1,1ZLL_1 193 3

Newick tree

 
[
	1ZLL_1:99.42,
	[
		4RPZ_1:81.5,1VJY_1:81.5
	]:17.92
]

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 \((0.85)(0.8)(\frac{646 }{\log_{20} 646}-\frac{303}{\log_{20}303})=95.3\)
Status Protein1 Protein2 d d1/2
Query variables 4RPZ_1 1VJY_1 119 112.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]