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Parikh vectors
2MRP_1 3REZ_1 1QFH_1 Letter Amino acid
3 13 4 R Arginine
7 8 26 G Glycine
7 8 2 H Histidine
6 0 17 K Lycine
3 3 7 Y Tyrosine
3 16 18 A Alanine
10 7 15 D Aspartic acid
7 0 9 I Isoleucine
1 1 11 F Phenylalanine
5 3 17 N Asparagine
1 8 1 C Cysteine
2 1 2 Q Glutamine
3 0 1 M Methionine
6 8 18 T Threonine
5 5 14 E Glutamic acid
15 21 7 L Leucine
1 16 12 P Proline
8 5 10 S Serine
0 2 0 W Tryptophan
1 4 21 V Valine 

2MRP_1|Chain A|DNA damage-inducible protein 1|Saccharomyces cerevisiae (559292)
>3REZ_1|Chains A, B, C, D|Platelet glycoprotein Ib beta chain, Platelet glycoprotein IX|Homo sapiens (9606)
>1QFH_1|Chains A, B|PROTEIN (GELATION FACTOR)|Dictyostelium discoideum (44689)
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 \(w=f(c)\)
2MRP , Knot 49 94 0.79 38 71 88 
MRGSHHHHHHGSDLTISNELTGEIYGPIEVSEDMALTDLIALLQADCGFDKTKHDLYYNMDILDSNRTQSLKELGLKTDDLLLIRGKISNSKLN
3REZ , Knot 60 129 0.75 34 86 121 
ADPCPAPCSCAGTLVDCGRRGLTALPALPARTTELVLTGNNLTSVPPGAFDHLPQLRTAHLGANPWRCDCSLTYLRLWLEDRPERAPYRDLRCVAPPALRGRLLPYLAEDELRAACAPGPLSSHHHHHH
1QFH , Knot 91 212 0.76 38 130 199 
KPAPSAEHSYAEGEGLVKVFDNAPAEFTIFAVDTKGVARTDGGDPFEVAINGPDGLVVDAKVTDNNDGTYGVVYDAPVEGNYNVNVTLRGNPIKNMPIDVKCIEGANGEDSSFGSFTFTVAAKNKKGEVKTYGGDKFEVSITGPAEEITLDAIDNQDGTYTAAYSLVGNGRFSTGVKLNGKHIEGSPFKQVLGNPGKKNPEVKSFTTTRTAN

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 \(c\).

\(|P_{f(2MRP_1)}(2) \setminus P_{f(3REZ_1)}(2)|=52\), \(|P_{f(3REZ_1)}(2) \setminus P_{f(2MRP_1)}(2)|=67\). 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 1:1010000000100101000101010111010001110011111010011000000100010110000000100111000011110101000010
Pair \(Z_2\) Length of longest common subsequence
2MRP_1,3REZ_1 119 7
2MRP_1,1QFH_1 141 3
3REZ_1,1QFH_1 148 3

Newick tree

 
[
	1QFH_1:76.05,
	[
		2MRP_1:59.5,3REZ_1:59.5
	]:16.55
]

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 \((0.85)(0.8)(\frac{223 }{\log_{20} 223}-\frac{94}{\log_{20}94})=41.8\)
Status Protein1 Protein2 d d1/2
Query variables 2MRP_1 3REZ_1 51 44
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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