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Parikh vectors
3IIX_1 7YGL_1 3JSG_1 Letter Amino acid
5 1 8 H Histidine
24 19 3 K Lycine
11 4 3 M Methionine
12 5 4 F Phenylalanine
20 6 8 P Proline
29 11 5 R Arginine
8 4 3 C Cysteine
5 5 6 Q Glutamine
26 12 9 G Glycine
11 5 5 Y Tyrosine
18 10 11 A Alanine
13 5 8 N Asparagine
22 18 6 I Isoleucine
9 11 10 S Serine
21 7 4 T Threonine
4 1 1 W Tryptophan
22 17 8 V Valine
20 10 4 D Aspartic acid
29 20 4 E Glutamic acid
38 13 12 L Leucine 

3IIX_1|Chain A|[FeFe] hydrogenase maturase subunit HydE|Thermotoga maritima MSB8 (243274)
>7YGL_1|Chains A, B|CRISPR system ring nuclease SSO2081|Saccharolobus solfataricus P2 (273057)
>3JSG_1|Chains A, B, C|Macrophage migration inhibitory factor|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 \(w=f(c)\)
3IIX , Knot 151 348 0.84 42 207 329 
MTGREILEKLERREFTREVLKEALSINDRGFNEALFKLADEIRRKYVGDEVHIRAIIEFSNVCRKNCLYCGLRRDNKNLKRYRMTPEEIVERARLAVQFGAKTIVLQSGEDPYXMPDVISDIVKEIKKMGVAVTLSLGEWPREYYEKWKEAGADRYLLRHETANPVLHRKLRPDTSFENRLNCLLTLKELGYETGAGSMVGLPGQTIDDLVDDLLFLKEHDFDMVGIGPFIPHPDTPLANEKKGDFTLTLKMVALTRILLPDSNIPATTAMGTIVPGGREITLRCGANVIMPNWTPSPYRQLYQLYPGKICVFEKDTACIPCVMKMIELLGRKPGRDWGGRKRVFETV
7YGL , Knot 87 184 0.82 40 128 178 
LGSGRPMVKLVATLGTAPGGVIESFLYLVKKGENIDEVRVVTTSNAEVKKAWRIVRLMFVCCIQEKFPKVEISEHPLDIEDIYSEDDLRKVREFVEKQLGEGDYLDITGGRKSMSVAAALAAKNKGVKIITSIIPQDDYNKISKKVRELKEIPEIKNRGECRQEMKETYCSLIVQDARSIEFEI
3JSG , Knot 63 122 0.82 40 101 116 
PMFIVNTNVPRASVPDGFLSELTQQLAQATGKPPQYIAVHVVPDQLMAFGGSSEPCALCSLHSIGKIGGAQNRSYSKLLCGLLAERLRISPDRVYINYYDMNAANVGWNNSTFALEHHHHHH

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 \(c\).

\(|P_{f(3IIX_1)}(2) \setminus P_{f(7YGL_1)}(2)|=121\), \(|P_{f(7YGL_1)}(2) \setminus P_{f(3IIX_1)}(2)|=42\). 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 1:101001100100001000110011010001100111011001000011001010111010010000010011000000100001010011001011101110011100100100110110011001001111101011011000000100111000110000101110001010001000100110100110001110111111001001100111100001011111111101001110000101010101111001111000111001110111110010100110111101010100010010110101100001011011011011100110011100011001
Pair \(Z_2\) Length of longest common subsequence
3IIX_1,7YGL_1 163 4
3IIX_1,3JSG_1 206 3
7YGL_1,3JSG_1 159 3

Newick tree

 
[
	3IIX_1:96.92,
	[
		7YGL_1:79.5,3JSG_1:79.5
	]:17.42
]

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 \((0.85)(0.8)(\frac{532 }{\log_{20} 532}-\frac{184}{\log_{20}184})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 3IIX_1 7YGL_1 128 96
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]