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Parikh vectors
6GBY_1 3PLH_1 5VNC_1 Letter Amino acid
1 3 7 C Cysteine
10 12 21 Q Glutamine
17 25 37 K Lycine
24 9 32 P Proline
14 12 45 R Arginine
12 4 15 M Methionine
13 10 36 F Phenylalanine
32 26 34 T Threonine
13 10 28 Y Tyrosine
14 14 36 N Asparagine
37 15 49 G Glycine
17 11 31 H Histidine
12 18 34 I Isoleucine
29 20 71 L Leucine
35 26 53 V Valine
46 19 50 A Alanine
20 17 50 D Aspartic acid
22 20 43 E Glutamic acid
6 23 40 S Serine
4 7 8 W Tryptophan 

6GBY_1|Chain A|Copper-containing nitrite reductase|Achromobacter cycloclastes (223)
>3PLH_1|Chain A|Uricase|Aspergillus flavus (5059)
>5VNC_1|Chains A, B, C, D|Glycogen [starch] synthase isoform 2|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
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 \(w=f(c)\)
6GBY , Knot 151 378 0.79 40 203 343 
MTEQLQMTRRTMLAGAALAGAVAPLLHTAQAHAAGAAAAAGAAPVDISTLPRVKVDLVKPPFVHAHDQVAKTGPRVVEFTMTIEEKKLVIDREGTEIHAMTFNGSVPGPLMVVHENDYVELRLINPDTNTLLHNIDFHAATGALGGGALTQVNPGEETTLRFKATKPGVFVYHCAPEGMVPWHVTSGMNGAIMVLPRDGLKDEKGQPLTYDKIYYVGEQDFYVPKDEAGNYKKYETPGEAYEDAVKAMRTLTPTHIVFNGAVGALTGDHALTAAVGERVLVVHSQANRDTRPHLIGGHGDYVWATGKFRNPPDLDQETWLIPGGTAGAAFYTFRQPGVYAYVNHNLIEAFELGAAGHFKVTGEWNDDLMTSVVKPASM
3PLH , Knot 136 302 0.85 42 204 291 
XSAVKAARYGKDNVRVYKVHKDEKTGVQTVYEMTVCVLLEGEIETSYTKADNSVIVATDSIKNTIYITAKQNPVTPPELFGSILGTHFIEKYNHIHAAHVNIVCHRWTRMDIDGKPHPHSFIRDSEEKRNVQVDVVEGKGIDIKSSLSGLTVLKSTNSQFWGFLRDEYTTLKETWDRILSTDVDATWQWKNFSGLQEVRSHVPKFDATWATAREVTLKTFAEDNSASVQATMYKMAEQILARQQLIETVEYSLPNKHYFEIDLSWHKGLQNTGKNAEVFAPQSDPNGLIKCTVGRSSLKSKL
5VNC , Knot 279 720 0.85 40 296 654 
MGSSHHHHHHSSGLVPRGSHMSRDLQNHLLFETATEVANRVGGIYSVLKSKAPITVAQYKDHYHLIGPLNKATYQNEVDILDWKKPEAFSDEMRPVQHALQTMESRGVHFVYGRWLIEGAPKVILFDLDSVRGYSNEWKGDLWSLVGIPSPENDFETNDAILLGYTVAWFLGEVAHLDSQHAIVAHFHQWLAGVALPLCRKRRIDVVTIFTTHATLLGRYLCASGSFDFYNCLESVDVDHEAGRFGIYHRYCIERAAAHSADVFTTVSQITAFEAEHLLKRKPDGILPNGLNVIKFQAFHEFQNLHALKKEKINDFVRGHFHGCFDFDLDNTLYFFIAGRYEYKNKGADMFIEALARLNYRLKVSGSKKTVVAFIVMPAKNNSFTVEALKGQAEVRALENTVHEVTTSIGKRIFDHAIRYPHNGLTTELPTDLGELLKSSDKVMLKRRILALRRPEGQLPPIVTHNMVDDANDLILNKIRQVQLFNSPSDRVKMIFHPEFLNANNPILGLDYDEFVRGCHLGVFPSYYEPWGYTPAECTVMGVPSITTNVSGFGSYMEDLIETNQAKDYGIYIVDRRFKAPDESVEQLVDYMEEFVKKTRRQRINQRNRTERLSDLLDWKRMGLEYVKARQLALRRGYPDQFRELVGEELNDSNMDALAGGKKLKVARPLSVPGSPRDLRSNSTVYMTPGDLGTLQEVNNADDYFSLGVNPAADDDDDGP

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 \(c\).

\(|P_{f(6GBY_1)}(2) \setminus P_{f(3PLH_1)}(2)|=90\), \(|P_{f(3PLH_1)}(2) \setminus P_{f(6GBY_1)}(2)|=91\). 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 1:100010100001111111111111110010101111111111111010011010101101111010001100110110101010000111000100101101010111111110000010101101000011001010110111111110010110000101010011111000110111110100110111111100110000101100001001100010110001100000001101000110110010100111011111101001101111001111000100000101111010011101010011010000111111011111001001110101000110110111110101010100011001101101
Pair \(Z_2\) Length of longest common subsequence
6GBY_1,3PLH_1 181 4
6GBY_1,5VNC_1 161 4
3PLH_1,5VNC_1 160 4

Newick tree

 
[
	6GBY_1:87.44,
	[
		5VNC_1:80,3PLH_1:80
	]:7.44
]

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 \((0.85)(0.8)(\frac{680 }{\log_{20} 680}-\frac{302}{\log_{20}302})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 6GBY_1 3PLH_1 130 118.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]