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Parikh vectors
6MVI_1 7CPX_1 5AJO_1 Letter Amino acid
24 299 37 A Alanine
16 89 25 N Asparagine
15 186 36 D Aspartic acid
39 200 49 G Glycine
12 219 43 V Valine
2 189 35 E Glutamic acid
14 312 49 L Leucine
38 219 33 S Serine
20 35 13 C Cysteine
2 105 14 H Histidine
4 124 21 I Isoleucine
11 87 34 K Lycine
2 80 16 M Methionine
13 120 21 F Phenylalanine
8 79 18 Y Tyrosine
8 175 43 R Arginine
15 132 25 Q Glutamine
21 167 26 P Proline
22 188 19 T Threonine
7 41 14 W Tryptophan 

6MVI_1|Chain A|Endoglucanase V|Neurospora crassa (strain ATCC 24698 / 74-OR23-1A / CBS 708.71 / DSM 1257 / FGSC 987) (367110)
>7CPX_1|Chains A, B|Lovastatin nonaketide synthase, polyketide synthase component|Aspergillus terreus (33178)
>5AJO_1|Chain A|POLYPEPTIDE N-ACETYLGALACTOSAMINYLTRANSFERASE 2|HOMO SAPIENS (9606)
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 \(w=f(c)\)
6MVI , Knot 123 293 0.79 40 166 267 
MRSSTILQTGLVAALPFAVQAASGSGQSTRYWDCCKPSCSWSGKAPVNRPVLACDANNNPLSDASVKSGCDGGSAYTCANNSPWAVNDQLSYGFAATKLSGGTESSWCCACYALTFTSGPVAGKTLVVQSTSTGGDLGSNHFDINMPGGGVGLFDGCKRQFGGLPGAQYGGISSRSQCDSFPAALKPGCQWRFDWFQNADNPNFTFKQVQCPSELTSRTGCKRNDDSQFPVFTPPSGGGSNPSTPTTPPSSGGGSGCTADKYAQCGGSGWSGCTNCPSGSTCKTINDYYHQCA
7CPX , Knot 992 3046 0.87 40 385 2256 
MAQSMYPNEPIVVVGSGCRFPGDANTPSKLWELLQHPRDVQSRIPKERFDVDTFYHPDGKHHGRTNAPYAYVLQDDLGAFDAAFFNIQAGEAESMDPQHRLLLETVYEAVTNAGMRIQDLQGTSTAVYVGVMTHDYETVSTRDLESIPTYSATGVAVSVASNRISYFFDWHGPSMTIDTACSSSLVAVHLAVQQLRTGQSSMAIAAGANLILGPMTFVLESKLSMLSPSGRSRMWDAGADGYARGEAVCSVVLKTLSQALRDGDTIECVIRETGVNQDGRTTGITMPNHSAQEALIKATYAQAGLDITKAEDRCQFFEAHGTGTPAGDPQEAEAIATAFFGHEQVARSDGNERAPLFVGSAKTVVGHTEGTAGLAGLMKASFAVRHGVIPPNLLFDKISPRVAPFYKNLRIPTEATQWPALPPGQPRRASVNSFGFGGTNAHAIIEEYMEPEQNQLRVSNNEDCPPMTGVLSLPLVLSAKSQRSLKIMMEEMLQFLQSHPEIHLHDLTWSLLRKRSVLPFRRAIVGHSHETIRRALEDAIEDGIVSSDFTTEVRGQPSVLGIFTGQGAQWPGMLKNLIEASPYVRNIVRELDDSLQSLPEKYRPSWTLLDQFMLEGEASNVQYATFSQPLCCAVQIVLVRLLEAARIRFTAVVGHSSGEIACAFAAGLISASLAIRIAYLRGVVSAGGARGTPGAMLAAGMSFEEAQEICELDAFEGRICVAASNSPDSVTFSGDANAIDHLKGMLEDESTFARLLKVDTAYHSHHMLPCADPYMQALEECGCAVADAGSPAGSVPWYSSVDAENRQMAARDVTAKYWKDNLVSPVLFSHAVQRAVVTHKALDIGIEVGCHPALKSPCVATIKDVLSGVDLAYTGCLERGKNDLDSFSRALAYLWERFGASSFDADEFMRAVAPDRPCMSVSKLLPAYPWDRSRRYWVESRATRHHLRGPKPHLLLGKLSEYSTPLSFQWLNFVRPRDIEWLDGHALQGQTVFPAAGYIVMAMEAALMIAGTHAKQVKLLEILDMSIDKAVIFDDEDSLVELNLTADVSRNAGEAGSMTISFKIDSCLSKEGNLSLSAKGQLALTIEDVNPRTTSASDQHHLPPPEEEHPHMNRVNINAFYHELGLMGYNYSKDFRRLHNMQRADLRASGTLDFIPLMDEGNGCPLLLHPASLDVAFQTVIGAYSSPGDRRLRCLYVPTHVDRITLVPSLCLATAESGCEKVAFNTINTYDKGDYLSGDIVVFDAEQTTLFQVENITFKPFSPPDASTDHAMFARWSWGPLTPDSLLDNPEYWATAQDKEAIPIIERIVYFYIRSFLSQLTLEERQQAAFHLQKQIEWLEQVLASAKEGRHLWYDPGWENDTEAQIEHLCTANSYHPHVRLVQRVGQHLLPTVRSNGNPFDLLDHDGLLTEFYTNTLSFGPALHYARELVAQIAHRYQSMDILEIGAGTGGATKYVLATPQLGFNSYTYTDISTGFFEQAREQFAPFEDRMVFEPLDIRRSPAEQGFEPHAYDLIIASNVLHATPDLEKTMAHARSLLKPGGQMVILEITHKEHTRLGFIFGLFADWWAGVDDGRCTEPFVSFDRWDAILKRVGFSGVDSRTTDRDANLFPTSVFSTHAIDATVEYLDAPLASSGTVKDSYPPLVVVGGQTPQSQRLLNDIKAIMPPRPLQTYKRLVDLLDAEELPMKSTFVMLTELDEELFAGLTEETFEATKLLLTYASNTVWLTENAWVQHPHQASTIGMLRSIRREHPDLGVHVLDVDAVETFDATFLVEQVLRLEEHTDELASSTTWTQEPEVSWCKGRPWIPRLKRDLARNNRMNSSRRPIYEMIDSSRAPVALQTARDSSSYFLESAETWFVPESVQQMETKTIYVHFSCPHALRVQALGFFYLVQGHVQEGNREVPVVALAERNASIVHVRPDYIYTEADNNLSEGGGSLMVTVLAAAVLAETVISTAKCLGVTDSILVLNPPSICGQMLLHAGEEIGLQVHLATTSGNRSSVSAGDAKSWLTLHARDTDWHLRRVLPRGVQALVDLSADQSCEGLTQRMMKVLMPGCAHYRAADLFTDTVSTELHSGSRHQASLPAAYWEHVVSLARQGLPSVSEGWEVMPCTQFAAHADKTRPDLSTVISWPRESDEATLPTRVRSIDAETLFAADKTYLLVGLTGDLGRSLGRWMVQHGACHIVLTSRNPQVNPKWLAHVEELGGRVTVLSMDVTSQNSVEAGLAKLKDLHLPPVGGIAFGPLVLQDVMLNNMELPMMEMVLNPKVEGVRILHEKFSDPTSSNPLDFFVMFSSIVAVMGNPGQANYSAANCYLQALAQQRVASGLAASTIDIGAVYGVGFVTRAELEEDFNAIRFMFDSVEEHELHTLFAEAVVAGRRAVHQQEQQRKFATVLDMADLELTTGIPPLDPALKDRITFFDDPRIGNLKIPEYRGAKAGEGAAGSKGSVKEQLLQATNLDQVRQIVIDGLSAKLQVTLQIPDGESVHPTIPLIDQGVDSLGAVTVGTWFSKQLYLDLPLLKVLGGASITDLANEAAARLPPSSIPLVAATDGGAESTDNTSENEVSGREDTDLSAAATITEPSSADEDDTEPGDEDVPRSHHPLSLGQEYSWRIQQGAEDPTVFNNTIGMFMKGSIDLKRLYKALRAVLRRHEIFRTGFANVDENGMAQLVFGQTKNKVQTIQVSDRAGAEEGYRQLVQTRYNPAAGDTLRLVDFFWGQDDHLLVVAYHRLVGDGSTTENIFVEAGQLYDGTSLSPHVPQFADLAARQRAMLEDGRMEEDLAYWKKMHYRPSSIPVLPLMRPLVGNSSRSDTPNFQHCGPWQQHEAVARLDPMVAFRIKERSRKHKATPMQFYLAAYQVLLARLTDSTDLTVGLADTNRATVDEMAAMGFFANLLPLRFRDFRPHITFGEHLIATRDLVREALQHARVPYGVLLDQLGLEVPVPTSNQPAPLFQAVFDYKQGQAESGTIGGAKITEVIATRERTPYDVVLEMSDDPTKDPLLTAKLQSSRYEAHHPQAFLESYMSLLSMFSMNPALKLAHVHHHHHH
5AJO , Knot 232 571 0.86 40 283 526 
MRRRSRMLLCFAFLWVLGIAYYMYSGGGSALAGGAGGGAGRKEDWNEIDPIKKKDLHHSNGEEKAQSMETLPPGKVRWPDFNQEAYVGGTMVRSGQDPYARNKFNQVESDKLRMDRAIPDTRHDQCQRKQWRVDLPATSVVITFHNEARSALLRTVVSVLKKSPPHLIKEIILVDDYSNDPEDGALLGKIEKVRVLRNDRREGLMRSRVRGADAAQAKVLTFLDSHCECNEHWLEPLLERVAEDRTRVVSPIIDVINMDNFQYVGASADLKGGFDWNLVFKWDYMTPEQRRSRQGNPVAPIKTPMIAGGLFVMDKFYFEELGKYDMMMDVWGGENLEISFRVWQCGGSLEIIPCSRVGHVFRKQHPYTFPGGSGTVFARNTRRAAEVWMDEYKNFYYAAVPSARNVPYGNIQSRLELRKKLSCKPFKWYLENVYPELRVPDHQDIAFGALQQGTNCLDTLGHFADGVVGVYECHNAGGNQEWALTKEKSVKHMDLCLTVVDRAPGSLIKLQGCREDDSRQKWEQIEGNSKLRHVGSNLCLDSRTAKSGGLSVEVCGPALSQQWKFTLNLQQ

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 \(c\).

\(|P_{f(6MVI_1)}(2) \setminus P_{f(7CPX_1)}(2)|=4\), \(|P_{f(7CPX_1)}(2) \setminus P_{f(6MVI_1)}(2)|=223\). 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 1:10000110011111111110110101000001000010001010111001111001000110010100100110100010001111000100111100101100001001001101001111100111000001101100010101111111110100001111111001110000000011111011001010110010010101001001001000010000000011110110111001001001100111010010001001101101000010100000100000001
Pair \(Z_2\) Length of longest common subsequence
6MVI_1,7CPX_1 227 4
6MVI_1,5AJO_1 203 5
7CPX_1,5AJO_1 114 4

Newick tree

 
[
	6MVI_1:11.88,
	[
		5AJO_1:57,7CPX_1:57
	]:62.88
]

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 \((0.85)(0.8)(\frac{3339 }{\log_{20} 3339}-\frac{293}{\log_{20}293})=733.\)
Status Protein1 Protein2 d d1/2
Query variables 6MVI_1 7CPX_1 943 513
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]