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Parikh vectors
7YMZ_1 8XNB_1 3ZGB_1 Letter Amino acid
65 36 49 P Proline
90 70 46 T Threonine
94 64 55 V Valine
90 96 70 A Alanine
44 45 70 R Arginine
35 0 12 C Cysteine
70 40 37 F Phenylalanine
70 49 63 D Aspartic acid
49 31 84 E Glutamic acid
50 25 55 K Lycine
142 58 53 S Serine
76 52 34 Q Glutamine
118 89 110 L Leucine
20 3 23 M Methionine
9 7 14 W Tryptophan
74 33 31 Y Tyrosine
76 48 25 N Asparagine
99 96 54 G Glycine
24 13 34 H Histidine
74 23 53 I Isoleucine 

7YMZ_1|Chains A, B, C|Spike glycoprotein|Human betacoronavirus 2c EMC/2012 (1235996)
>8XNB_1|Chains A, B|Outer membrane protein assembly factor BamA|Deinococcus radiodurans R1 = ATCC 13939 = DSM 20539 (243230)
>3ZGB_1|Chains A, B|PHOSPHOENOLPYRUVATE CARBOXYLASE|Flaveria pringlei (4226)
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 \(w=f(c)\)
7YMZ , Knot 482 1369 0.84 40 335 1188 
MDSWFILVLLGSGLICVSASYVDVGPDSVKSACIEVDIQQTFFDKTWPRPIDVSKADGIIYPQGRTYSNITITYQGLFPYQGDHGDMYVYSAGHATGTTPQKLFVANYSQDVKQFANGFVVRIGAAANSTGTVIISPSTSATIRKIYPAFMLGSSVGNFSDGKMGRFFNHTLVLLPDGCGTLLRAFYCILEPRSGNHCPAGNSYTSFATYHTPATDCSDGNYNRNASLNSFKEYFNLRNCTFMYTYNITEDEILEWFGITQTAQGVHLFSSRYVDLYGGNMFQFATLPVYDTIKYYSIIPHSIRSIQSDRKAWAAFYVYKLQPLTFLLDFSVDGYIRRAIDCGFNDLSQLHCSYESFDVESGVYSVSSFEAKPSGSVVEQAEGVECDFSPLLSGTPPQVYNFKRLVFTNCNYNLTKLLSLFSVNDFTCSQISPAAIASNCYSSLILDYFSYPLSMKSDLSVSSAGPISQFNYKQSFSNPTCLILATVPHNLTTITKPLKYSYINKCSRLLSDDRTEVPQLVNANQYSPCVSIVPSTVWEDGDYYRKQLSPLEGGGWLVASGSTVAMTEQLQMGFGITVQYGTDTNSVCPKLEFANDTKIASQLGNCVEYSLYGVSGRGVFQNCTAVGVRQQRFVYDAYQNLVGYYSDDGNYYCLRACVSVPVSVIYDKETKTHATLFGSVACEHISSTMSQYSRSTRSMLKRRDSTYGPLQTPVGCVLGLVNSSLFVEDCKLPLGQSLCALPDTPSTLTPASVGSVPGEMRLASIAFNHPIQVDQLNSSYFKLSIPTNFSFGVTQEYIQTTIQKVTVDCKQYVCNGFQKCEQLLREYGQFCSKINQALHGANLRQDDSVRNLFASVKSSQSSPIIPGFGGDFNLTLLEPVSISTGSRSARSAIEDLLFDKVTIADPGYMQGYDDCMQQGPASARDLICAQYVAGYKVLPPLMDVNMEAAYTSSLLGSIAGVGWTAGLSSFAAIPFAQSIFYRLNGVGITQQVLSENQKLIANKFNQALGAMQTGFTTTNEAFQKVQDAVNNNAQALSKLASELSNTFGAISASIGDIIQRLDPPEQDAQIDRLINGRLTTLNAFVAQQLVRSESAALSAQLAKDKVNECVKAQSKRSGFCGQGTHIVSFVVNAPNGLYFMHVGYYPSNHIEVVSAYGLCDAANPTNCIAPVNGYFIKTNNTRIVDEWSYTGSSFYAPEPITSLNTKYVAPQVTYQNISTNLPPPLLGNSTGIDFQDELDEFFKNVSTSIPNFGSLTQINTTLLDLTYEMLSLQQVVKALNESYIDLKELGNYTYEFGSGGYIPEAPRDGQAYVRKDGEWVLLSTFLKGQDNSADIQHSGRPLESRGPFEQKLISEEDLNMHTGHHHHHH
8XNB , Knot 317 878 0.81 38 250 741 
MKYLLPTAAAGLLLLAAQPAMAGSDYKDDDDKSGHHHHHHHHHHGSDEVDAGSQQAGTVQDISVVGTTDLLSEFLKTTLTVQPGAALSSVNLRQVEQEVLASGYFKTATAELRTTGGRDTLVVTVTANPVIKDVQATGLTFLPADAFKQSVAELLNIAPGAVLNTQRLEQAKEALAQNYRQEGFPFVPSISADTKTNKDGTATVTFVVDESAPLSRIEVTGATLLPQATVQNIFRPLQTSKRFTTQAFFAASDALQAAYEAAGYFQAGIDPRSVTLENGVLKLSVIESRVASVDLSPLGTLAQTPALQTKAGQPLRLAQLQADVRALANQTGQPVGFALQADPQNPSQVTVLFGAADVESGPVKSIAFVGNTKVPTAQLQAAIKTKPGDVYSPQLAQDDFMALRNVYRQAGYEISTRDAITFQNGVLTYNLREVRLAGYELAWQGQHRTKDRVILRELPAAGQTFNSKDVQAALGRVSALGYVTINDVRVKSDPQNPENVTYVIALSEGRTGIPVNLSLGYDSLQGGWSGDVAYTNSNAFGLGHSFGVRLGAVQNQAGQNWVGSLNYTIPWLDLDFADFRKNRTSLSFGVGSDVGGNIALLDANKEDTGRDYTTRTNGFSLGLGRNITPNLTASANVAFNNRTSYLEPKQEGETSNLDDAAATALLPATSLTTRLSGNLNYDNTDNVNFPGRGVRAYGALGYNVGRAGDAPLSWTDGEIGVSGYYGFGGRIKRSFGLETYRQVLAARANTGTTTGTFPDGTGYFIGGSNPLASRELRGLEDGQLFGTNYFSSSLEYRYDFGLSGGVAQGLYGVLFADYGGVWNSGEAFRSAYGVGAGVQLNLGFGGAQLPSLRFDYGYSGQNAQKPNGRFHFRIGNFW
3ZGB , Knot 358 972 0.84 40 321 867 
MGHHHHHHHHHLEKLASIDAQLRLLVPGKVSEDDKLIEYDALLLDKFLDILQDLHGEDLKEAVQECYELSAEYEGKHDPKKLEELGSVLTSLDPGDSIVIAKAFSHMLNLANLAEEVQIAYRRRIKLKRGDFADEANATTESDIEETFKKLVLKLNKSPEEVFDALKNQTVDLVLTAHPTQSVRRSLLQKHGRIRNCLAQLYAKDITPDDKQELDEALHREIQAAFRTDEIRRTPPTPQDEMRAGMSYFHETIWKGVPKFLRRVDTALKNIGINERVPYNAPLIQFSSWMGGDRDGNPRVTPEVTRDVCLLARMMASNMYFSQIEDLMFEMSMWRCNSELRVRAEELYRTARRDVKHYIEFWKQVPPTEPYRVILGDVRDKLYNTRERSRHLLAHGISDIPEEAVYTNVEQFLEPLELCYRSLCDCGDRVIADGSLLDFLRQVSTFGLSLVKLDIRQESDRHTDVLDAITQHLEIGSYREWSEEKRQEWLLAELSGKRPLFGSDLPKTEEVKDVLDTFNVLAELPSDCFGAYIISMATSPSDVLAVELLQRECHVKHPLRVVPLFEKLADLEAAPAAMARLFSIDWYRNRIDGKQEVMIGYSDSGKDAGRFSAAWQLYKAQEEIIKVAKEFGVKLVIFHGRGGTVGRGGGPTHLAILSQPPDTIHGSLRVTVQGEVIEQSFGEEHLCFRTLQRFCAATLEHGMNPPISPRPEWRELMDQMAVVATEEYRSIVFKEPRFVEYFRLATPELEYGRMNIGSRPSKRKPSGGIESLRAIPWIFAWTQTRFHLPVWLGFGAAFKHAIKKDSKNLQMLQEMYKTWPFFRVTIDLVEMVFAKGDPGIAALNDKLLVSEDLWPFGESLRANYEETKDYLLKIAGHRDLLEGDPYLKQRIRLRDSYITTLNVCQAYTLKRIRDPNYHVTLRPHISKEYAAEPSKPADELIHLNPTSEYAPGLEDTLILTMKGIAAGMQNTG

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 \(c\).

\(|P_{f(7YMZ_1)}(2) \setminus P_{f(8XNB_1)}(2)|=97\), \(|P_{f(8XNB_1)}(2) \setminus P_{f(7YMZ_1)}(2)|=12\). 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 1:1001111111101110101001011100100101010100011000110110100101110101000001010001111001001010100110101001001111000001001101111011111000101110100010100101111110011010010110110001111101010110110011010010001110000011000011000001000001010010001010000110000100001101111000101101100001010110110110111000100001110010010000011111010010110111010101010011001100100100000010100110010010101010110010110001011101011010010011100000010011011010010000101111100000011100100110100010100111100100000100100111101100100100110000100000110000001101101000010101110011001000000101101111111010011100010111110100100000101010110000110011001000101101011100001111000011001000111000001000010101011101100000000101110110001000100000000011000000011100111011111000111000011110010111001001011011011101011011100110100100001010110010111000010001001010000010011000001100010100010011011010000010011101000000111111110101011011010010001001100111001011011010100001001110100110100111001111110101011000011101111110111001111111001100101111000110000011100100111110011000001100100110001011001100100011110101101100101100010100110101001011110011000011101011000100010100000110101001101110110110110110010001011010110011010001111010110000001100100010010110110010000111010000100011111110001101000100110010001101101001000110100011010011011000010100110000011011011011001010100010111100110100001010001011000111000110000101001000000
Pair \(Z_2\) Length of longest common subsequence
7YMZ_1,8XNB_1 109 7
7YMZ_1,3ZGB_1 96 7
8XNB_1,3ZGB_1 119 10

Newick tree

 
[
	8XNB_1:59.76,
	[
		7YMZ_1:48,3ZGB_1:48
	]:11.76
]

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 \((0.85)(0.8)(\frac{2247 }{\log_{20} 2247}-\frac{878}{\log_{20}878})=329.\)
Status Protein1 Protein2 d d1/2
Query variables 7YMZ_1 8XNB_1 426 341
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]