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Parikh vectors
5MGX_1 9EPL_1 4MFJ_1 Letter Amino acid
0 25 31 G Glycine
0 16 11 F Phenylalanine
0 13 28 P Proline
0 15 15 S Serine
1 23 15 T Threonine
0 21 4 N Asparagine
0 10 32 R Arginine
0 14 8 Q Glutamine
3 9 28 E Glutamic acid
0 7 23 H Histidine
0 29 40 L Leucine
0 17 30 A Alanine
0 11 10 I Isoleucine
0 11 6 K Lycine
1 10 9 M Methionine
2 17 18 D Aspartic acid
0 3 11 W Tryptophan
0 11 9 Y Tyrosine
1 26 17 V Valine
0 12 0 C Cysteine 

5MGX_1|Chains A, B, C, D|yeast HSP90 C-terminus|Saccharomyces cerevisiae (4932)
>9EPL_1|Chain A|Non-structural protein 11|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4MFJ_1|Chain A|Putative uncharacterized protein tcp24|Actinoplanes teichomyceticus (1867)
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 \(w=f(c)\)
5MGX , Knot 7 8 0.60 10 7 6 
DTEMEEVD
9EPL , Knot 132 300 0.83 40 195 285 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVQQC
4MFJ , Knot 140 345 0.79 38 185 321 
MGSSHHHHHHSSGLVPRGSHMMMDPETVRIALGLEERTAAWLTELDELGPPAEPVRLPRGEEARDLLRRLEVPELDAEEIVAAAPDPDRDPALWWLLERTHHAIVRHMGDHRAKPRGGPPLPYEGGAAARYFHVYVFLATVPAVRRFHAERGIPDEVGWETLTQLGELVAIHRRKYGQGGMNMQWWTTYHLRGILYRLGRLQFSLATGKDGTPHLGLHVPEWGGPLLPKAYDESLHRARPFFDRHFPEHGARVAWGSSWMLDPQLEEYLTEDSNIIQLARFWTLTDSAPEPGNADGDSSILEFVFRYNGQPLDELPQRSSLERAVIAHLKAGRHWHMRTGFVKLP

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 \(c\).

\(|P_{f(5MGX_1)}(2) \setminus P_{f(9EPL_1)}(2)|=4\), \(|P_{f(9EPL_1)}(2) \setminus P_{f(5MGX_1)}(2)|=192\). 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 1:00010010
Pair \(Z_2\) Length of longest common subsequence
5MGX_1,9EPL_1 196 2
5MGX_1,4MFJ_1 186 2
9EPL_1,4MFJ_1 186 3

Newick tree

 
[
	9EPL_1:96.36,
	[
		5MGX_1:93,4MFJ_1:93
	]:3.36
]

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 \((0.85)(0.8)(\frac{308 }{\log_{20} 308}-\frac{8}{\log_{20}8})=101.\)
Status Protein1 Protein2 d d1/2
Query variables 5MGX_1 9EPL_1 128 65.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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