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Parikh vectors
8SRB_1 4IVO_1 7HET_1 Letter Amino acid
93 11 9 D Aspartic acid
87 49 13 G Glycine
72 13 6 I Isoleucine
146 70 19 L Leucine
98 37 22 V Valine
61 7 16 N Asparagine
43 20 6 H Histidine
83 12 5 F Phenylalanine
100 45 11 S Serine
54 5 2 M Methionine
63 34 5 P Proline
87 17 6 T Threonine
131 48 15 A Alanine
88 33 2 R Arginine
15 8 3 C Cysteine
85 28 6 E Glutamic acid
76 8 13 K Lycine
39 6 7 Y Tyrosine
53 25 3 Q Glutamine
20 7 0 W Tryptophan 

8SRB_1|Chains A, B, C, D|TRPM2 chanzyme|Salpingoeca rosetta (946362)
>4IVO_1|Chain A[auth B]|Protoporphyrinogen oxidase|Homo sapiens (9606)
>7HET_1|Chains A, B|Non-structural protein 3|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
8SRB , Knot 524 1494 0.85 40 351 1298 
MQRARPGELVEVIMFRPTGKARVSNLDESMAMEFTDLRTRAMSSAAMIRQSVAAKTLLIENEDGKGSTRMEVQDFMKRFHMHASEDDKTGSPSTAWGTLRFPTKEATAPYLRLSVNDDPEDALLFVKAMLAQKYGETYDRPSLILSVTGGARNFTLPPRLETAIAKGLRLAAQRTNAWVVTGGTNTGVMKLTGQIMEALSKTQSHFIPPTIGIATYGVIIGGDDMTRGEPPKIGLEYEMHKKDPPKTTPLDDNHNLFLLVDDGSTNKFGKEIKFRAAFENAAGQAFAAPVVTIVVQGGPGTLGTALQAVRQGTPIVVVDGSGLAADVLAYAYNFMHNPLTRFKSYTIDDLRQKVAQTFNPKSSQQLTNLLDSALECVQDPNLVVVYSLQESGIDEFDDCILKAIFSSQGKLGNKLKQAMYFDQLDVAKRALSEASKNGQHNEIAACINDNLMAAMMHNKPHFVELYLGFDAKIYELKPSEEVAKTNITALDELPSFALAIEELYKREAKKPHSHVQRLVSLSNTDVLGRHYRVSTQRGDGTTRRIGRDLANTRAYNVLRMDQIFARLVSKDFSVNRDFTIYDSKYDKVPGIQFRRTAQASHMLFLWAICLDRFRMARHFWLIGDQSIINALVASRILERLSTHRALQGPHLAEERAKMQHNAKKFEELAVGVLGECHGSDSHMASEMLHSKNDMFNKKNAINIAYDAKSLAFLSHPATQSVINADWYGHLKSVTSFWAVLFAFFFPFFVLPFINFSEDHAEQQVEAPRDFFTDAPRSSHSANSTTSGAHRLRRKFAKFYSAPYTRFISDLLSHFVLCVVTSYFVLDKLEDTISAIEWILLVWFVALLLEELRQMIFCDGIAEYISDTWNRLDLIMITLFFVGFFTHASDPSNQDSKVVSKGIHAFLVVVLWLRFMRYYALSKNLGPKLIMMMEMMKDVSTFVFLLLIFLIGYGVAAQSLLSPDEDFSSRTFIGVLFRPYFQIYGELFLDDLNSEANCLGDTPFTECSRETVRMVPFFLAVYILGSNVLLVNLLIAMFNDTYMKVQEAAEDLWRKQNYELCAEYKDRPFLPAPFILLAHVHMLFMRLLRLCGVHTQEHEKIQDDETKRKITTFEELNTDKFLRRWERERQEMLEARVKMTNDNVVQAMGMMDQLLEHMISFRFSLDQQATKIKQEIRDDGLPSTEPTGLVSRTPSQPINRLNSAVAVHGHTAEAAEWYVPPEEYPKSGGVKRYLIDASMVPLSIMCPSYDPVEYTHPSVAAQPVWADPADPRKIKFNVKDEVNGKVVDRTSCHPSGISIDSNTGRPINPWGRTGMTGRGLLGKWGVNQAADTVVTRWKRSPDGSILERDGKKVLEFVAIQRQDNKMWAIPGGFVDNGEDVALTSGREFMEEALGMGTSADLMSAESKDSLAALFSSGTIVARIYCEDPRNTDNAWVETTCVNFHDESGRHAARLKLQGGDDAEHARWMMVHGGLNLFASHRTLLQHVTSALNAYF
4IVO , Knot 187 483 0.79 40 212 432 
HHHHHHMGRTVVVLGGGISGLAASYHLSRAPCPPKVVLVESSERLGGWIRSVRGPNGAIFELGPQGIRPAGALGARTLLLVSELGLDSEVLPVRGDHPAAQNRFLYVGGALHALPTGLRGLLRPSPPFSKPLFWAGLRELTKPRGKEPDETVHSFAQRRLGPEVASLAMDSLCRGVFAGNSRELSIRSCFPSLFQAEQTHRSILLGLLLGAGRTPQPDSALIRQALAERWSQWSLRGGLEMLPQALETHLTSRGVSVLRGQPVCGLSLQAEGRWKVSLRDSSLEADHVISAIPASVLSELLPAEAAPLARALSAITAVSVAVVNLQYQGAHLPVQGFGHLVPSSEDPGVLGIVYDSVAFPEQDGSPPGLRVTVMLGGSWLQTLEASGCVLSQELFQQRAQEAAATQLGLKEMPSHCLVHLHKNCIPQYTLGHWQKLESARQFLTAHRLPLTLAGASYEGVAVNDCIESGRQAAVSVLGTEPNS
7HET , Knot 80 169 0.81 38 119 162 
SMVNSFSGYLKLTDNVYIKNADIVEEAKKVKPTVVVNAANVYLKHGGGVAGALNKATNNAMQVESDDYIATNGPLKVGGSCVLSGHNLAKHCLHVVGPNVNKGEDIQLLKSAYENFNQHEVLLAPLLSAGIFGADPIHSLRVCVDTVRTNVYLAVFDKNLYDKLVSSFL

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 \(c\).

\(|P_{f(8SRB_1)}(2) \setminus P_{f(4IVO_1)}(2)|=158\), \(|P_{f(4IVO_1)}(2) \setminus P_{f(8SRB_1)}(2)|=19\). 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 1:100101101101111010101010010001110100100011001111000111001110000101000101001100101010000001010011101011000101101010100010011111011110001000001011101011100101110100111011011100001111011000111010101101100000011110111100111111001001011011100010000110001100000111110010000110010101110011101111111011101111011011011001011111010111101110100110011001000010010001100101000001001100110010010111100100011001000110111000101100100110100101100110010001000011101000111111000101101011101010010100011000101100110111110010000100100010011010000111000010000101000011001100010011010011101100010100010100000001111010001010011111110100101100111110001101111001100100001101101100010100010010011111110001000011001100000110000110110010011110011000110101010100100111111111111111111010000100010110011001100000100000110010001101001100011001100111011000111001000101101111111111110010011100111001000100101111011111110010010000001100110111111111011000110001110111110110010011111111111011110011010001000011111101010101011100100010011001100000001011111111011100111101111110000101001100110000001010000011111111111010111101101011000000010000000010010010000110010000001101010100001101111100110011010101000100100010001110001011100010011001001111010010110101110001001110001101011110110100011000010111011110110100101010001010110000001011010000101101110011010111101110011001100100010101100010011011110000001111111110010011100100110011111001011010000011111001011101000010000011100001010000100110101011001001011110111011100001100100110101
Pair \(Z_2\) Length of longest common subsequence
8SRB_1,4IVO_1 177 5
8SRB_1,7HET_1 244 4
4IVO_1,7HET_1 187 4

Newick tree

 
[
	7HET_1:11.63,
	[
		8SRB_1:88.5,4IVO_1:88.5
	]:26.13
]

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 \((0.85)(0.8)(\frac{1977 }{\log_{20} 1977}-\frac{483}{\log_{20}483})=371.\)
Status Protein1 Protein2 d d1/2
Query variables 8SRB_1 4IVO_1 486 315.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]