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Parikh vectors
4IAO_1 7ORF_1 8BWI_1 Letter Amino acid
23 20 8 A Alanine
49 35 13 L Leucine
21 14 10 T Threonine
32 29 13 V Valine
21 18 5 Y Tyrosine
27 22 9 D Aspartic acid
26 15 10 G Glycine
36 23 7 I Isoleucine
39 28 10 K Lycine
40 20 19 S Serine
17 17 4 R Arginine
21 23 17 E Glutamic acid
11 15 7 M Methionine
22 12 6 F Phenylalanine
3 4 3 W Tryptophan
30 16 13 N Asparagine
10 7 24 C Cysteine
15 17 6 Q Glutamine
20 10 15 H Histidine
29 20 11 P Proline 

4IAO_1|Chains A, B|NAD-dependent histone deacetylase SIR2|Saccharomyces cerevisiae (559292)
>7ORF_1|Chain A|Mitogen-activated protein kinase 10|Homo sapiens (9606)
>8BWI_1|Chain A|Twisted gastrulation protein homolog 1|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 \(w=f(c)\)
4IAO , Knot 203 492 0.85 40 258 469 
MGSSHHHHHHSQDPNSMDPLAVSAASVVSMSNDVLKPETPKGPIIISKNPSNGIFYGPSFTKRESLNARMFLKYYGAHKFLDTYLPEDLNSLYIYYLIKLLGFEVKDQALIGTINSIVHINSQERVQDLGSAISVTNVEDPLAKKQTVRLIKDLQRAINKVLCTRLRLSNFFTIDHFIQKLHTARKILVLTGAGVSTSLGIPDFRSSEGFYSKIKHLGLDDPQDVFNYNIFMHDPSVFYNIANMVLPPEKIYSPLHSFIKMLQMKGKLLRNYTQNIDNLESYAGISTDKLVQCHGSFATATCVTCHWNLPGERIFNKIRNLELPLCPYCYKKRREYFPEGYNNKVGVAASQGSMSERPPYILNSYGVLKPDITFFGEALPNKFHKSIREDILECDLLICIGTSLKVAPVSEIVNMVPSHVPQVLINRDPVKHAEFDLSLLGYCDDIAAMVAQKCGWTIPHKKWNDLKNKNFKSQEKDKGVYVVTSDEHPKTL
7ORF , Knot 157 365 0.84 40 230 353 
SMSKSKVDNQFYSVEVGDSTFTVLKRYQNLKPIGSGAQGIVCAAYDAVLDRNVAIKKLSRPFQNQTHAKRAYRELVLMKCVNHKNIISLLNVFTPQKTLEEFQDVYLVMELMDANLCQVIQMELDHERMSYLLYQMLCGIKHLHSAGIIHRDLKPSNIVVKSDCTLKILDFGLARTAGTSFMMTPYVVTRYYRAPEVILGMGYKENVDIWSVGCIMGEMVRHKILFPGRDYIDQWNKVIEQLGTPCPEFMKKLQPTVRNYVENRPKYAGLTFPKLFPDSLFPADSEHNKLKASQARDLLSKMLVIDPAKRISVDDALQHPYINVWYDPAEVEAPPPQIYDKQLDEREHTIEEWKELIYKEVMNSE
8BWI , Knot 101 210 0.85 40 154 202 
ETGCNKALCASDVSKCLIQELCQCRPGEGNCSCCKECMLCLGALWDECCDCVGMCNPRNYSDTPPTSKSTVEELHEPIPSLFRALTEGDTQLNWNIVSFPVAEELSHHENLVSFLETVNQPHHQNVSVPSNNVHAPYSSDKEHMCTVVYFDDCMSIHQCKISCESMGASKYRWFHNACCECIGPECIDYGSKTVKCMNCMFGTKHHHHHH

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 \(c\).

\(|P_{f(4IAO_1)}(2) \setminus P_{f(7ORF_1)}(2)|=91\), \(|P_{f(7ORF_1)}(2) \setminus P_{f(4IAO_1)}(2)|=63\). 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 1:110000000000010010111101101101000110100101111100010011101101000001010111000110011000110010010100110111101000111101001101000001001101101001001110000101100100110011000101001101001100100100111101111000111101000011000100111001001100011100101100110111110010011001101101010110000001001000111000011000101101001000101110011001001011101000000000110100001111100101000110110001110101011101110010001000110001110110010111100110111001101110001100101010111000011111100011011000100100001000000011011000001001
Pair \(Z_2\) Length of longest common subsequence
4IAO_1,7ORF_1 154 4
4IAO_1,8BWI_1 192 6
7ORF_1,8BWI_1 186 3

Newick tree

 
[
	8BWI_1:99.66,
	[
		4IAO_1:77,7ORF_1:77
	]:22.66
]

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 \((0.85)(0.8)(\frac{857 }{\log_{20} 857}-\frac{365}{\log_{20}365})=132.\)
Status Protein1 Protein2 d d1/2
Query variables 4IAO_1 7ORF_1 166 146.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]