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Parikh vectors
7WLK_1 5YSB_1 6LHG_1 Letter Amino acid
76 18 8 N Asparagine
59 0 4 C Cysteine
175 28 27 G Glycine
254 31 16 L Leucine
151 21 14 P Proline
169 28 16 A Alanine
111 12 12 I Isoleucine
70 38 10 K Lycine
63 17 3 M Methionine
107 20 6 F Phenylalanine
183 25 9 S Serine
90 25 19 T Threonine
52 12 15 Y Tyrosine
136 6 24 R Arginine
29 10 11 W Tryptophan
64 16 15 Q Glutamine
123 20 22 E Glutamic acid
65 7 7 H Histidine
131 24 18 V Valine
115 39 16 D Aspartic acid 

7WLK_1|Chain A|Voltage-dependent T-type calcium channel subunit alpha-1I|Homo sapiens (9606)
>5YSB_1|Chains A, B|Lin1841 protein|Listeria innocua serovar 6a (strain ATCC BAA-680 / CLIP 11262) (272626)
>6LHG_1|Chains A, F[auth D]|MHC class I alpha chain 2|Gallus gallus (9031)
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 \(w=f(c)\)
7WLK , Knot 726 2223 0.84 40 373 1657 
MAESASPPSSSAAAPAAEPGVTTEQPGPRSPPSSPPGLEEPLDGADPHVPHPDLAPIAFFCLRQTTSPRNWCIKMVCNPWFECVSMLVILLNCVTLGMYQPCDDMDCLSDRCKILQVFDDFIFIFFAMEMVLKMVALGIFGKKCYLGDTWNRLDFFIVMAGMVEYSLDLQNINLSAIRTVRVLRPLKAINRVPSMRILVNLLLDTLPMLGNVLLLCFFVFFIFGIIGVQLWAGLLRNRCFLEENFTIQGDVALPPYYQPEEDDEMPFICSLSGDNGIMGCHEIPPLKEQGRECCLSKDDVYDFGAGRQDLNASGLCVNWNRYYNVCRTGSANPHKGAINFDNIGYAWIVIFQVITLEGWVEIMYYVMDAHSFYNFIYFILLIIVGSFFMINLCLVVIATQFSETKQREHRLMLEQRQRYLSSSTVASYAEPGDCYEEIFQYVCHILRKAKRRALGLYQALQSRRQALGPEAPAPAKPGPHAKEPRHYHGKTKGQGDEGRHLGSRHCQTLHGPASPGNDHSGRELCPQHSPLDATPHTLVQPIPATLASDPASCPCCQHEDGRRPSGLGSTDSGQEGSGSGSSAGGEDEADGDGARSSEDGASSELGKEEEEEEQADGAVWLCGDVWRETRAKLRGIVDSKYFNRGIMMAILVNTVSMGIEHHEQPEELTNILEICNVVFTSMFALEMILKLAAFGLFDYLRNPYNIFDSIIVIISIWEIVGQADGGLSVLRTFRLLRVLKLVRFMPALRRQLVVLMKTMDNVATFCMLLMLFIFIFSILGMHIFGCKFSLRTDTGDTVPDRKNFDSLLWAIVTVFQILTQEDWNVVLYNGMASTSPWASLYFVALMTFGNYVLFNLLVAILVEGFQAEGDANRSYSDEDQSSSNIEEFDKLQEGLDSSGDPKLCPIPMTPNGHLDPSLPLGGHLGPAGAAGPAPRLSLQPDPMLVALGSRKSSVMSLGRMSYDQRSLSSSRSSYYGPWGRSAAWASRRSSWNSLKHKPPSAEHESLLSAERGGGARVCEVAADEGPPRAAPLHTPHAHHIHHGPHLAHRHRHHRRTLSLDNRDSVDLAELVPAVGAHPRAAWRAAGPAPGHEDCNGRMPSIAKDVFTKMGDRGDRGEDEEEIDYTLCFRVRKMIDVYKPDWCEVREDWSVYLFSPENRFRVLCQTIIAHKLFDYVVLAFIFLNCITIALERPQIEAGSTERIFLTVSNYIFTAIFVGEMTLKVVSLGLYFGEQAYLRSSWNVLDGFLVFVSIIDIVVSLASAGGAKILGVLRVLRLLRTLRPLRVISRAPGLKLVVETLISSLKPIGNIVLICCAFFIIFGILGVQLFKGKFYHCLGVDTRNITNRSDCMAANYRWVHHKYNFDNLGQALMSLFVLASKDGWVNIMYNGLDAVAVDQQPVTNHNPWMLLYFISFLLIVSFFVLNMFVGVVVENFHKCRQHQEAEEARRREEKRLRRLEKKRRKAQRLPYYATYCHTRLLIHSMCTSHYLDIFITFIICLNVVTMSLEHYNQPTSLETALKYCNYMFTTVFVLEAVLKLVAFGLRRFFKDRWNQLDLAIVLLSVMGITLEEIEINAALPINPTIIRIMRVLRIARVLKLLKMATGMRALLDTVVQALPQVGNLGLLFMLLFFIYAALGVELFGKLVCNDENPCEGMSRHATFENFGMAFLTLFQVSTGDNWNGIMKDTLRDCTHDERSCLSSLQFVSPLYFVSFVLTAQFVLINVVVAVLMKHLDDSNKEAQEDAEMDAELELEMAHGLGPGPRLPTGSPGAPGRGPGGAGGGGDTEGGLCRRCYSPAQENLWLDSVSLIIKDSLEGELTIIDNLSGSIFHHYSSPAGCKKCHHDKQEVQLAETEAFSLNSDRSSSILLGDDLSLEDPTACPPGRKDSKGELDPPEPMRVGDLGECFFPLSSTAVSPDPENFLCEMEEIPFNPVRSWLKHDSSQAPPSPFSPDASSPLLPMPAEFFHPAVSASQKGPEKGTGTGTLPKIALQGSWASLRSPRVNCTLLRQATGSDTSLDASPSSSAGSLQTTLEDSLTLSDSPRRALGPPAPAPGPRAGLSPAARRRLSLRGRGLFSLRGLRAHQRSHSSGGSTSPGCTHHDSMDPSDEEGRGGAGGGGAGSEHSETLSSLSLTSLFCPPPPPPAPGLTPARKFSSTSSLAAPGRPHAAALAHGLARSPSWAADRSKDPPGRAPLPMGLGPLAPPPQPLPGELEPGDAASKRKR
5YSB , Knot 165 397 0.83 38 215 372 
MDDANSSDSKVLNVWAMGDEAKSLKELAQKFTKDTGIEVKVQVIPWANAHDKLLTAVASKSGPDVVQMGTTWMPEFVEAGALLDITKDVEKSKNMNSDLFFPGSVKTTQFDGKTYGVPWYAETRVLFYRTDLLKKVGYNEAPKTWDELSDAALKLSKRGKDMYGFAIDPNEQTTGFIFGRQNGSPLFDKDGTPVFNKKPFVDTVTYLDSFIKNGSAPDTDLGLDASQSFGGDGIVPMFMSGPWMVNTLKDTAPDIDGKWATAVLPKKENNESSLGGANLSIFKYSNKKDDALKFMDYMSQPDVQLSWLKDTNSMPARMDAWEDDMLKNDPYYKVFGEQMKTAEPMPLIPQFEEIAQLYGKSWEQIYRGGADVQTQMDTFNDQVEALLKKLEHHHHHH
6LHG , Knot 122 272 0.83 40 185 263 
ELHTLRYIRTAMTDPGPGQPWFVTVGYVDGELFVHYNSTARRYVPRTEWIAANTDQQYWDGQTQIGQLNEQINRENLGIRQRRYNQTGGSHTVQWMFGCDILEDGTIRGYRQSAYDGRDFIALDKDMKTFTAAVPEAVPTKRKWEEESEPERWKNYLEETCVEWLRRYVEYGKAELGRRERPEVRVWGKEADGILTLSCRAHGFYPRPIVVSWLKDGAVRGQDAHSGGIVPNGDGTYHTWVTIEAQPGDGDKYQCRVEHASLPQPGLYSWKL

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 \(c\).

\(|P_{f(7WLK_1)}(2) \setminus P_{f(5YSB_1)}(2)|=166\), \(|P_{f(5YSB_1)}(2) \setminus P_{f(7WLK_1)}(2)|=8\). 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 1:110010110001111110111000011100110011110011011010110101111111010000010010101100111001011111100101110010001001000001101100111111110111011111111000011001001011111111100010100101011001011011011001101011101110011111011110111111111111011111100001100010101011111000100000111100101001111000111100010000100001001111000101011010100000100010101001110100110111111011010111011001101001001101111111101111010111110010000000001110000001000011001011000001100100110010001111001100000111101111101110100100001000101001001100000010111011000010010100011010100110111101100110010000001001011100001001010100111000101011000001100011000000001011111010110000101011100001001111111100101110000010010011010011100111101110111111100100100110011111011011101011101100101101101101111100011111001001101011111111110111101110010100001001100001001111110110110000101110011100011101011111011001110111111101101010100000000000001001001001100010101011110101010101111101111111111101010101111111000001101101000000100000000111100111100000100100011010000110100111101001110011101111001010010011011000000000101000001011011111110101110111111100000101101100110011001001000001000101010011010010100100010101101000101100011100110011111111001011100101011000011101000110111110101011011101100101000101101111110110111011011110111110110110010110110011110111001100101110111100111111111110110101000111000010000001110001100000100110111011111000111011001101111000110000111110110111110111101111111001000000001001000000010010000001001100100000011100100000101110111010110101000001001001100000110011110111011111100110001001011111101111010010101111101011011011011011011011011011100110111011011111111111011111011101100000100110001010011111101101001001011100010000000000100101101101101110101111011111110010000001000101010101011011111101101011111011111111100011100000011000111001011100010101011001010110000011100000000001011000110100000001111001010010101110000010101101101101100111100011010100110010011101100110000001110110101001111111011011101000110010101011011101011010010100011001010000101010001101000100010100010011111111111011101110001010101110101101000000011000110000001010000101111111110000001001010011011111111110110010000011111010111110111001011100000111011111111111111011110101101100000
Pair \(Z_2\) Length of longest common subsequence
7WLK_1,5YSB_1 174 4
7WLK_1,6LHG_1 208 4
5YSB_1,6LHG_1 160 5

Newick tree

 
[
	7WLK_1:10.61,
	[
		5YSB_1:80,6LHG_1:80
	]:20.61
]

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 \((0.85)(0.8)(\frac{2620 }{\log_{20} 2620}-\frac{397}{\log_{20}397})=542.\)
Status Protein1 Protein2 d d1/2
Query variables 7WLK_1 5YSB_1 688 403.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]