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Parikh vectors
3FVG_1 6GUN_1 5VRM_1 Letter Amino acid
21 28 29 G Glycine
7 13 7 F Phenylalanine
21 11 9 K Lycine
9 18 13 P Proline
4 8 4 W Tryptophan
13 20 17 V Valine
15 22 11 E Glutamic acid
20 13 20 I Isoleucine
24 27 27 L Leucine
3 4 1 C Cysteine
2 9 6 H Histidine
15 23 17 S Serine
14 12 6 Y Tyrosine
12 22 36 A Alanine
13 23 13 R Arginine
9 11 8 N Asparagine
19 14 13 T Threonine
17 19 15 D Aspartic acid
9 4 10 Q Glutamine
9 7 10 M Methionine

3FVG_1|Chains A, B|Glutamate receptor, ionotropic kainate 1|Homo sapiens (9606)
>6GUN_1|Chains A, B, C, D|EstB from Aspergillus nidulans|Aspergillus nidulans FGSC A4 (227321)
>5VRM_1|Chains A, B, C, D|Enoyl-[acyl-carrier-protein] reductase [NADH]|Mycobacterium tuberculosis (strain CDC 1551 / Oshkosh) (83331)
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 \(w=f(c)\)
3FVG , Knot 117 256 0.84 40 167 245
ANRTLIVTTILEEPYVMYRKSDKPLYGNDRFEGYCLDLLKELSNILGFIYDVKLVPDGKYGAQNDKGEWNGMVKELIDHRADLAVAPLTITYVREKVIDFSKPFMTLGISILYRKGTPIDSADDLAKQTKIEYGAVRDGSTMTFFKKSKISTYEKMWAFMSSRQQTALVRNSDEGIQRVLTTDYALLMESTSIEYVTQRNCNLTQIGGLIDSKGYGVGTPIGSPYRDKITIAILQLQEEGKLHMMKEKWWRGNGCP
6GUN , Knot 138 308 0.85 40 194 298
GSTQIDKPAVVSLPSSEQFYLNNSRGERYLIQVSWPLHWKDHKPDTDRNDVPLIYIVDGNALFLTATEALWRRSADSHYCGGGIVVAIGYPLEGTGKVYHRVRRGFDLTVPTPDSPVEGHGGADILLDFIAETVRPAVRERFPDVSVSREALYGHSYGGLFALHALFTRPSMFDAYIASSPSIWWNGRCILNEAKAFTRKIKENGAYTNGEKKLPSLMMYLGGLEQDPRRWNDEPDESWEGRKRDAEAFNMKVNLLELMGLIRGCTRLHAVSFSEYAGEDHGTVMACSLGRGFSTFMEDWPVPREESV
5VRM , Knot 116 272 0.79 40 169 256
GSHMTGLLDGKRILVSGIITDSSIAFHIARVAQEQGAQLVLTGFDRLRLIQRITDRLPAKAPLLELDVQNEEHLASLAGRVTEAIGAGNKLDGVVHSIGFMPQTGMGINPFFDAPYADVSKGIHISAYSYASMAKALLPIMNPGGSIVGMDFDPSRAMPAYNWMTVAKSALESVNRFVAREAGKYGVRSNLVAAGPIRTLAMSAIVGGALGEEAGAQIQLLEEGWDQRAPIGWNMKDATPVAKTVCALLSDWLPATTGDIIYADGGAHTQLL

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 \(c\).

\(|P_{f(3FVG_1)}(2) \setminus P_{f(6GUN_1)}(2)|=68\), \(|P_{f(6GUN_1)}(2) \setminus P_{f(3FVG_1)}(2)|=95\). 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 1:1000111001100101100000011010001010010110010011111001011101001100001010111001100010111111010010001101001110111011000101100100110000100111001001011000010000011111000000111000001100110000111100001001000000100111110001011101110100001011110100010101100011010101
Pair \(Z_2\) Length of longest common subsequence
3FVG_1,6GUN_1 163 3
3FVG_1,5VRM_1 154 4
6GUN_1,5VRM_1 175 4

Newick tree

 
[
	6GUN_1:86.92,
	[
		3FVG_1:77,5VRM_1:77
	]:9.92
]

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 \((0.85)(0.8)(\frac{564 }{\log_{20} 564}-\frac{256}{\log_{20}256})=87.3\)
Status Protein1 Protein2 d d1/2
Query variables 3FVG_1 6GUN_1 113 101.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]