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Parikh vectors
5CEO_1 2GWX_1 5IJP_1 Letter Amino acid
9 13 9 Q Glutamine
19 17 9 I Isoleucine
25 14 13 S Serine
21 21 18 K Lycine
12 17 7 F Phenylalanine
6 7 8 Y Tyrosine
14 22 11 A Alanine
8 10 3 N Asparagine
16 12 17 D Aspartic acid
21 17 17 E Glutamic acid
15 9 10 H Histidine
12 11 16 R Arginine
21 9 10 G Glycine
7 7 4 M Methionine
12 15 6 T Threonine
6 3 0 C Cysteine
29 34 17 L Leucine
15 11 8 P Proline
9 2 3 W Tryptophan
23 16 15 V Valine 

5CEO_1|Chain A|Mitogen-activated protein kinase kinase kinase 12|Homo sapiens (9606)
>2GWX_1|Chains A, B|PROTEIN (PPAR-DELTA)|Homo sapiens (9606)
>5IJP_1|Chain A|Putative uncharacterized protein|Chaetomium thermophilum (209285)
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 \(w=f(c)\)
5CEO , Knot 131 300 0.83 40 185 286 
MGSEDLWEVPFEEILDLQWVGSGAQGAVFLGRFHGEEVAVKKVRDLKETDIKHLRKLKHPNIITFKGVCTQAPCYCILMEFCAQGQLYEVLRAGRPVTPSLLVDWSMGIAGGMNYLHLHKIIHRDLKSPNMLITYDDVVKISDFGTSKELSDKSTKMSFAGTVAWMAPEVIRNEPVSEKVDIWSFGVVLWELLTGEIPYKDVDSSAIIWGVGSNSLHLPVPSSCPDGFKILLRQCWNSKPRNRPSFRQILLHLDIASADVLSTPQETYFKSQAEWREEVKLHFEKIKSEGTGNSHHHHHH
2GWX , Knot 124 267 0.86 40 183 259 
LKAFSKHIYNAYLKNFNMTKKKARSILTGKASHTAPFVIHDIETLWQAEKGLVWKQLVNGLPPYKEISVHVFYRCQCTTVETVRELTEFAKSIPSFSSLFLNDQVTLLKYGVHEAIFAMLASIVNKDGLLVANGSGFVTREFLRSLRKPFSDIIEPKFEFAVKFNALELDDSDLALFIAAIILCGDRPGLMNVPRVEAIQDTILRALEFHLQANHPDAQQLFPKLLQKMADLRQLVTEHAQMMQRIKKTETETSLHPLLQEIYKDMY
5IJP , Knot 91 201 0.80 38 135 192 
MKFGQQLRSSIIREYQWHYIDYDGLKADLKRASGPLVASSDPTKPPRREWTEDDESRFVSKLEAELDKVHAKQQVKAMEISRRIAVSEREVQDVVGRLQDRGPGQEGPSEEEFMLLEEDLSDIIADVHDLAKFVQVNYTGFYKIIKKHDKMTGWRLKPVFDTRLKAKPFYKENYDASVVRLSKLYDLVRTRGNLEHHHHHH

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 \(c\).

\(|P_{f(5CEO_1)}(2) \setminus P_{f(2GWX_1)}(2)|=92\), \(|P_{f(2GWX_1)}(2) \setminus P_{f(5CEO_1)}(2)|=90\). 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 1:110001101110011010111011011111101010011100100100001001001001011010110001100011101010101001101101101011101011111110010100110001001011100001101001100001000000101110111111011000110001011011111101101011000100011111110001011110001011011100010001000101001110101101011001000010001010001010100100010100000000
Pair \(Z_2\) Length of longest common subsequence
5CEO_1,2GWX_1 182 3
5CEO_1,5IJP_1 170 6
2GWX_1,5IJP_1 160 4

Newick tree

 
[
	5CEO_1:90.57,
	[
		5IJP_1:80,2GWX_1:80
	]:10.57
]

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 \((0.85)(0.8)(\frac{567 }{\log_{20} 567}-\frac{267}{\log_{20}267})=84.8\)
Status Protein1 Protein2 d d1/2
Query variables 5CEO_1 2GWX_1 106 103.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]