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Parikh vectors
7OBK_1 4FNP_1 9IWV_1 Letter Amino acid
5 31 10 F Phenylalanine
7 34 30 P Proline
2 15 6 W Tryptophan
13 58 50 R Arginine
17 55 47 G Glycine
20 46 23 S Serine
10 35 29 T Threonine
8 30 9 N Asparagine
17 23 4 K Lycine
33 50 29 E Glutamic acid
3 16 21 H Histidine
10 27 13 I Isoleucine
24 69 43 L Leucine
10 38 6 Y Tyrosine
15 40 25 D Aspartic acid
2 3 3 C Cysteine
9 20 12 M Methionine
12 52 49 V Valine
27 55 64 A Alanine
9 32 14 Q Glutamine 

7OBK_1|Chain A|14-3-3 protein sigma|Homo sapiens (9606)
>4FNP_1|Chains A, B, C, D|Alpha-galactosidase AgaA|Geobacillus stearothermophilus (1422)
>9IWV_1|Chains A, B|Putative epoxidase LasC|Streptomyces lasalocidi (324833)
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 \(w=f(c)\)
7OBK , Knot 114 253 0.83 40 170 244 
GAMGSMERASLIQKAKLAEQAERYEDMAAFMKGAVEKGEELSCEERNLLSVAYKNVVGGQRAAWRVLSSIEQKSNEEGSEEKGPEVREYREKVETELQGVCDTVLGLLDSHLIKEAGDAESRVFYLKMKGDYYRYLAEVATGDDKKRIIDSARSAYQEAMDISKKEMPPTNPIRLGLALNFSVFHYEIANSPEEAISLAKTTFDEAMADLHTLSEDSYKDSTLIMQLLRDNLTLWTADNAGEEGGEAPQEPQS
4FNP , Knot 285 729 0.86 40 291 671 
MSVAYNPQTKQFHLRAGKASYVMQLFRSGYLAHVYWGKAVRDVRGARAFPRLDRAFSPNPDPSDRTFSLDTLLQEYPAYGNTDFRAPAYQVQLENGSTVTDLRYKTHRIYKGKPRLNGLPATYVEHEQEAETLEIVLGDALIGLEVTLQYTAYEKWNVITRSARFENKGGERLKLLRALSMSVDFPTADYDWIHLPGAWGRERWIERRPLVTGVQAAESRRGASSHQQNPFIALVAKNADEHQGEVYGFSFVYSGNFLAQIEVDQFGTARVSMGINPFDFTWLLQPGESFQTPEVVMVYSDQGLNGMSQTYHELYRTRLARGAFRDRERPILINNWEATYFDFNEEKIVNIARTEAELGIELVVLDDGWFGERDDDRRSLGDWIVNRRKLPNGLDGLAKQVNELGLQFGLWVEPEMVSPNSELYRKHPDWCLHVPNRPRSEGRNQLVLDYSREDVCDYIIETISNVLASAPITYVKWDMNRHMTEIGSSALPPERQRETAHRYMLGLYRVMDEITSRLPHILFESCSGGGGRFDPGMLYYMPQTWTSDNTDAVSRLKIQYGTSLVYPISAMGAHVSAVPNHQVGRVASLKTRGHVAMSGNFGYELDITKLTETEKQMMKQQVAFYKDVRRLVQFGTFYRLLSPFEGNEAAWMFVSADRSEALVAYFRVLAEANAPLSYLRLKGLDSNQDYEIEGLGVYGGDELVYAGVALPYRSSDFISMMWRLKAVQQ
9IWV , Knot 193 488 0.81 42 215 435 
MNHKVHHHHHHIEGRHMTNTRSAVVLGGGMAGMLVSSMLARHVGSVTVIDRDAFPAGPDLRKGVPQARHAHILWSGGARIVEELLPGTTDRLLGAGAHRIGIPDGQVSYTAYGWQHRFPEAQFMIACSRALLDWTVREETLREERIALVEKTEVLALLGDAGRVTGVRVRDQESGEEREVPADLVVDTTGRGSPSKRLLAELGLPAPEEEFVDSGMVYATRLFRAPEAAATNFPLVSVHADHRAGRPGCNAVLMPIEDGRWIVTVSGTRGGEPPADDEGFARFARDGVRHPLVGELIAXAQPLTSVERSRSTVNRRLHYDRLATWPEGLVVLGDAVAAFNPVYGHGMSAAAHSVLALRSQLGQRAFQPGLARAAQRAIAVAVDDAWVLATSHDIGYPGCRTQTRDPRLTRHAGERQRVTDLVGLTATRNQVVNRAAVALNTLSAGMASMQDPAVMAAVRRGPEVPAPTEPPLRPDEVARLVSGAGVTA

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 \(c\).

\(|P_{f(7OBK_1)}(2) \setminus P_{f(4FNP_1)}(2)|=28\), \(|P_{f(4FNP_1)}(2) \setminus P_{f(7OBK_1)}(2)|=149\). 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 1:1111010010110010110010000011111011100100100000011011000111100111011001000000010000110100000010001011000111110001100110100011010101000001101101000001100100100011010000111001101111101011000110010011011000100111010010000000001110110001011010011001101100100
Pair \(Z_2\) Length of longest common subsequence
7OBK_1,4FNP_1 177 4
7OBK_1,9IWV_1 169 4
4FNP_1,9IWV_1 150 5

Newick tree

 
[
	7OBK_1:90.03,
	[
		9IWV_1:75,4FNP_1:75
	]:15.03
]

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 \((0.85)(0.8)(\frac{982 }{\log_{20} 982}-\frac{253}{\log_{20}253})=197.\)
Status Protein1 Protein2 d d1/2
Query variables 7OBK_1 4FNP_1 249 166
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]