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Parikh vectors
7BVF_1 2BYJ_1 7XCO_1 Letter Amino acid
83 22 92 T Threonine
29 7 10 M Methionine
130 37 77 A Alanine
25 18 83 N Asparagine
8 6 30 C Cysteine
28 9 61 Q Glutamine
127 46 100 L Leucine
27 17 53 Y Tyrosine
102 35 90 V Valine
69 21 46 R Arginine
21 27 50 E Glutamic acid
47 28 73 I Isoleucine
88 23 65 P Proline
60 23 94 S Serine
30 6 11 W Tryptophan
49 24 61 D Aspartic acid
99 36 79 G Glycine
24 10 26 H Histidine
23 28 63 K Lycine
47 16 76 F Phenylalanine 

7BVF_1|Chain A[auth B]|Probable arabinosyltransferase B|Mycobacterium tuberculosis H37Rv (83332)
>2BYJ_1|Chains A, B, C|ORNITHINE AMINOTRANSFERASE|HOMO SAPIENS (9606)
>7XCO_1|Chains A, B, C|Spike glycoprotein|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 \(w=f(c)\)
7BVF , Knot 394 1116 0.82 40 293 933 
MTQCASRRKSTPNRAILGAFASARGTRWVATIAGLIGFVLSVATPLLPVVQTTAMLDWPQRGQLGSVTAPLISLTPVDFTATVPCDVVRAMPPAGGVVLGTAPKQGKDANLQALFVVVSAQRVDVTDRNVVILSVPREQVTSPQCQRIEVTSTHAGTFANFVGLKDPSGAPLRSGFPDPNLRPQIVGVFTDLTGPAPPGLAVSATIDTRFSTRPTTLKLLAIIGAIVATVVALIALWRLDQLDGRGSIAQLLLRPFRPASSPGGMRRLIPASWRTFTLTDAVVIFGFLLWHVIGANSSDDGYILGMARVADHAGYMSNYFRWFGSPEDPFGWYYNLLALMTHVSDASLWMRLPDLAAGLVCWLLLSREVLPRLGPAVEASKPAYWAAAMVLLTAWMPFNNGLRPEGIIALGSLVTYVLIERSMRYSRLTPAALAVVTAAFTLGVQPTGLIAVAALVAGGRPMLRILVRRHRLVGTLPLVSPMLAAGTVILTVVFADQTLSTVLEATRVRAKIGPSQAWYTENLRYYYLILPTVDGSLSRRFGFLITALCLFTAVFIMLRRKRIPSVARGPAWRLMGVIFGTMFFLMFTPTKWVHHFGLFAAVGAAMAALTTVLVSPSVLRWSRNRMAFLAALFFLLALCWATTNGWWYVSSYGVPFNSAMPKIDGITVSTIFFALFAIAAGYAAWLHFAPRGAGEGRLIRALTTAPVPIVAGFMAAVFVASMVAGIVRQYPTYSNGWSNVRAFVGGCGLADDVLVEPDTNAGFMKPLDGDSGSWGPLGPLGGVNPVGFTPNGVPEHTVAEAIVMKPNQPGTDYDWDAPTKLTSPGINGSTVPLPYGLDPARVPLAGTYTTGAQQQSTLVSAWYLLPKPDDGHPLVVVTAAGKIAGNSVLHGYTPGQTVVLEYAMPGPGALVPAGRMVPDDLYGEQPKAWRNLRFARAKMPADAVAVRVVAEDLSLTPEDWIAVTPPRVPDLRSLQEYVGSTQPVLLDWAVGLAFPCQQPMLHANGIAEIPKFRITPDYSAKKLDTDTWEDGTNGGLLGITDLLLRAHVMATYLSRDWARDWGSLRKFDTLVDAPPAQLELGTATRSGLWSPGKIRIGPHLGGIKAFHHHHHHHHHH
2BYJ , Knot 182 439 0.84 40 236 415 
MFSKLAHLQRFAVLSRGVHSSVASATSVATKKTVQGPPTSDDIFEREYKYGAHNYHPLPVALERGKGIYLWDVEGRKYFDFLSSISAVNQGHCHPKIVNALKSQVDKLTLTSRAFYNNVLGEYEEYITKLFNYHKVLPMNTGVEAGETACKLARKWGYTVKGIQKYKAKIVFAAGNFWGRTLSAISSSTDPTSYDGFGPFMPGFDIIPYNDLPALERALQDPNVAAFMVEPIQGEAGVVVPDPGYLMGVRELCTRHQVLFIADEIQTGLARTGRWLAVDYENVRPDIVLLGKALSGGLYPVSAVLCDDDIMLTIKPGEHGSTYGGNPLGCRVAIAALEVLEEENLAENADKLGIILRNELMKLPSDVVTAVRGKGLLNAIVIKETKDWDAWKVCLRLRDNGLLAKPTHGDIIRFAPPLVIKEDELRESIEIINKTILSF
7XCO , Knot 443 1240 0.84 40 332 1090 
QCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHVISGTNGTKRFDNPVLPFNDGVYFASIEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLDHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPIIVREPEDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFDEVFNATRFASVYAWNRKRISNCVADYSVLYNLAPFFTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGNIADYNYKLPDDFTGCVIAWNSNKLDSKVSGNYNYLYRLFRKSNLKPFERDISTEIYQAGNKPCNGVAGFNCYFPLRSYSFRPTYGVGHQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLKGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEYVNNSYECDIPIGAGICASYQTQTKSHRRARSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLKRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKYFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFKGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGPALQIPFPMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQDVVNHNAQALNTLVKQLSSKFGAISSVLNDIFSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGSAWSHPQFEKHHHHHHHH

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 \(c\).

\(|P_{f(7BVF_1)}(2) \setminus P_{f(2BYJ_1)}(2)|=98\), \(|P_{f(2BYJ_1)}(2) \setminus P_{f(7BVF_1)}(2)|=41\). 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 1:100010000001001111111010100111011111111101101111110001110110010110101111010110101011001101111111111101100100101011111101001010000111101100010010000101000011011011110010111100111010101011111001011111111101010001000100101111111111011111111010010101011011101101100111100111101001010011111111110111100000101111101100110100010111010011110001111100100101110110111111011110001110111110100110111111110111110011010111111011001110001000010111111101110111010111111111111011101110000111011110111111011101111000100110100101011100110000100001111010101000111110110110111111000011011011110111111101111110100110011111111111111001110101101000011111111111110110001110100011110011101011010011111111111011110111011101011011001111111111111111011111100010000110010111110111001110100011110110100101111111111011110101110001101111010011000010110010011101001111011011011111000011000001101101110100101111101110111001101001100111001111111111110111001010010110010110101110111101110010101001111011011010010001100011110111111110001110101110110101010001001000010010011111100111010111001000110011010010011011110101101000111011010111011110110000000000
Pair \(Z_2\) Length of longest common subsequence
7BVF_1,2BYJ_1 139 4
7BVF_1,7XCO_1 111 8
2BYJ_1,7XCO_1 150 4

Newick tree

 
[
	2BYJ_1:77.09,
	[
		7BVF_1:55.5,7XCO_1:55.5
	]:21.59
]

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 \((0.85)(0.8)(\frac{1555 }{\log_{20} 1555}-\frac{439}{\log_{20}439})=284.\)
Status Protein1 Protein2 d d1/2
Query variables 7BVF_1 2BYJ_1 353 245.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]