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Parikh vectors
8CPZ_1 6WMG_1 6XZU_1 Letter Amino acid
10 3 2 C Cysteine
125 7 4 K Lycine
229 19 9 A Alanine
148 15 8 Q Glutamine
53 9 6 H Histidine
55 1 4 M Methionine
156 16 12 T Threonine
93 14 1 P Proline
197 17 12 S Serine
40 3 2 W Tryptophan
101 11 6 R Arginine
147 8 4 D Aspartic acid
136 15 5 E Glutamic acid
270 19 10 L Leucine
91 10 2 F Phenylalanine
125 8 7 Y Tyrosine
114 14 11 V Valine
170 3 5 N Asparagine
133 20 13 G Glycine
142 12 4 I Isoleucine

8CPZ_1|Chains A, B, C, D, E|TcdA1|Photorhabdus luminescens (29488)
>6WMG_1|Chain A|SUN domain-containing protein 2|Homo sapiens (9606)
>6XZU_1|Chain A|nanobody hC3Nb1|Lama glama (9844)
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 \(w=f(c)\)
8CPZ , Knot 826 2535 0.85 40 365 1943
MAHHHHHHSSGLEVLFQGPMNESVKEIPDVLKSQCGFNCLTDISHSSFNEFRQQVSEHLSWSETHDLYHDAQQAQKDNRLYEARILKRANPQLQNAVHLAILAPNAELIGYNNQFSGRASQYVAPGTVSSMFSPAAYLTELYREARNLHASDSVYYLDTRRPDLKSMALSQQNMDIELSTLSLSNELLLESIKTESKLENYTKVMEMLSTFRPSGATPYHDAYENVREVIQLQDPGLEQLNASPAIAGLMHQASLLGINASISPELFNILTEEITEGNAEELYKKNFGNIEPASLAMPEYLKRYYNLSDEELSQFIGKASNFGQQEYSNNQLITPVVNSSDGTVKVYRITREYTTNAYQMDVELFPFGGENYRLDYKFKNFYNASYLSIKLNDKRELVRTEGAPQVNIEYSANITLNTADISQPFEIGLTRVLPSGSWAYAAAKFTVEEYNQYSFLLKLNKAIRLSRATELSPTILEGIVRSVNLQLDINTDVLGKVFLTKYYMQRYAIHAETALILCNAPISQRSYDNQPSQFDRLFNTPLLNGQYFSTGDEEIDLNSGSTGDWRKTILKRAFNIDDVSLFRLLKITDHDNKDGKIKNNLKNLSNLYIGKLLADIHQLTIDELDLLLIAVGEGKTNLSAISDKQLATLIRKLNTITSWLHTQKWSVFQLFIMTSTSYNKTLTPEIKNLLDTVYHGLQGFDKDKADLLHVMAPYIAATLQLSSENVAHSVLLWADKLQPGDGAMTAEKFWDWLNTKYTPGSSEAVETQEHIVQYCQALAQLEMVYHSTGINENAFRLFVTKPEMFGAATGAAPAHDALSLIMLTRFADWVNALGEKASSVLAAFEANSLTAEQLADAMNLDANLLLQASIQAQNHQHLPPVTPENAFSCWTSINTILQWVNVAQQLNVAPQGVSALVGLDYIQSMKETPTYAQWENAAGVLTAGLNSQQANTLHAFLDESRSAALSTYYIRQVAKAAAAIKSRDDLYQYLLIDNQVSAAIKTTRIAEAIASIQLYVNRALENVEENANSGVISRQFFIDWDKYNKRYSTWAGVSQLVYYPENYIDPTMRIGQTKMMDALLQSVSQSQLNADTVEDAFMSYLTSFEQVANLKVISAYHDNINNDQGLTYFIGLSETDAGEYYWRSVDHSKFNDGKFAANAWSEWHKIDCPINPYKSTIRPVIYKSRLYLLWLEQKEITWQTGNSKDGYQTETDYRYELKLAHIRYDGTWNTPITFDVNKKISELKLEKNRAPGLYCAGYQGEDTLLVMFYNQQDTLDSYKNASMQGLYIFADMASKDMTPEQSNVYRDNSYQQFDTNNVRRVNNRYAEDYEIPSSVSSRKDYGWGDYYLSMVYNGDIPTINYKAASSDLKIYISPKLRIIHNGYEGQKRNQCNLMNKYGKLGDKFIVYTSLGVNPNNSSNKLMFYPVYQYSGNTSGLNQGRLLFHRDTTYPSKVEAWIPGAKRSLTNQNAAIGDDYATDSLNKPDDLKQYIFMTDSKGTATDVSGPVEINTAISPAKVQIIVKAGGKEQTFTADKDVSIQPSPSFDEMNYQFNALEIDGSGLNFINNSASIDVTFTAFAEDGRKLGYESFSIPVTLKVSTDNALTLHHNENGAQYMQWQSYRTRLNTLFARQLVARATTGIDTILSMETQNIQEPQLGKGFYATFVIPPYNLSTHGDERWFKLYIKHVVDNNSHIIYSGQLTDTNINITLFIPLDDVPLNQDYHAKVYMTFKKSPSDGTWWGPHFVRDDKGIVTINPKSILTHFESVNVLNNISSEPMDFSGANSLYFWELFYYTPMLVAQRLLHEQNFDEANRWLKYVWSPSGYIVHGQIQNYQWNVRPLLEDTSWNSDPLDSVDPDAVAQHDPMHYKVSTFMRTLDLLIARGDHAYRQLERDTLNEAKMWYMQALHLLGDKPYLPLSTTWSDPRLDRAADITTQNAHDSAIVALRQNIPTPAPLSLRSANTLTDLFLPQINEVMMNYWQTLAQRVYNLRHNLSIDGQPLYLPIYATPADPKALLSAAVATSQGGGKLPESFMSLWRFPHMLENARGMVSQLTQFGSTLQNIIERQDAEALNALLQNQAAELILTNLSIQDKTIEELDAEKTVLEKSKAGAQSRFDSYGKLYDENINAGENQAMTLRASAAGLTTAVQASRLAGAAADLVPNIFGFAGGGSRWGAIAEATGYVMEFSANVMNTEADKISQSETYRRRRQEWEIQRNNAEAELKQIDAQLKSLAVRREAAVLQKTSLKTQQEQTQSQLAFLQRKFSNQALYNWLRGRLAAIYFQFYDLAVARCLMAEQAYRWELNDDSARFIKPGAWQGTYAGLLAGETLMLSLAQMEDAHLKRDKRALEVERTVSLAEVYAGLPKDNGPFSLAQEIDKLVSQGSGSAGSGNNNLAFGAGTDTKTSLQASVSFADLKIREDYPASLGKIRRIKQISVTLPALLGPYQDVQAILSYGDKAGLANGCEALAVSHGMNDSGQFQLDFNDGKFLPFEGIAIDQGTLTLSFPNASMPEKGKQATMLKTLNDIILHIRYTIK
6WMG , Knot 106 224 0.85 40 160 219
GPGGSGQGKSAREAAASLSLTLQKEGVIGVTEEQVHHIVKQALQRYSEDRIGLADYALESGGASVISTRCSETYETKTALLSLFGIPLWYHSQSPRVILQPDVHPGNCWAFQGPQGFAVVRLSARIRPTAVTLEHVPKALSPNSTISSAPKDFAIFGFDEDLQQEGTLLGKFTYDQDGEPIQTFHFQAPTMATYQVVELRILTNWGHPEYTCIYRFRVHGEPAH
6XZU , Knot 63 127 0.80 40 103 122
MQVQLVESGGGLVQAGGSLRLSCVVSGSTFSDYAMGWYRQAAGEQRELVAAIYSTGRTNYIDSVKGRFTISRDNAKTTVYLQMNSLKPEDTAVYYCNLLGATTMINTKWGQGTQVTVSSLEHHHHHH

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 \(c\).

\(|P_{f(8CPZ_1)}(2) \setminus P_{f(6WMG_1)}(2)|=210\), \(|P_{f(6WMG_1)}(2) \setminus P_{f(8CPZ_1)}(2)|=5\). 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 1:110000000011011101110001001101100001100100100001001000100010100000100010010000010010110010101001101111110101110000101010001111010011011101001000100101000100100001010011100001010100101000111001000001000001101100101011010001000100110100111001010111111100101111010101011011000100101001000011010110111100100000100001001110100110000000011011100001010100100000001001010111111000010001001001001010100000110001110101000101010010100110111001110101101110101000000011101001101001001010110111001010101000111011100001000110100111100111000000001001001100111010010010001010010010100011001101001011011010000000101000100100101101110100101001011111110100010110000110110010010011000010110111100000000101010011001001101100001011011110111010100001100111110010110111010011011000001100011000001100001110101100001100011011100101111101111100110111100110110111001001111101001010011011010101110101010000011110100110010010011011011001011101101111100100100010010100111110111000010010111000001110000100110111110000010001110001011100001101110101010011001000100111000111010000000001111001100100010101011000110111001000010100100111001001001101011010000100001100111100001100010010000100101110110010010011010000101110000101111000010100100001000000000010110100010100110101000100101000011110011001000111110000001000001010110111011000101000010000000010000100100001000011001000000111000101100101101000110001010101010110010010000000110001011001110001110100000011101100001000110010111000000100101111110001000011110001000100100100011100001010010111010011011010111011100001010001010101010010001011010101101100010101010111001001100010111010100001101000001100101000000100111001110100110011010000100101101101011111001000100011010100110000011001010000101011111001110000010101010001001011110110000111010100110010010110010001101011001011011000111110011000010010011001101010110101000010101110000100011001010111000110001001100101111010010001000010010110101101110010111000100101001101000010001111100011011110100100100111101001110010011001001000101010110111010110101110111100011101100110110110110010111001001100100110000101101110001101110010100001001010001100001110001000101000010110001101010111100110100111111011101111111100111110101011010101100010010000000000001010000101010010101001110001111000010000000000111100010001100110101111010100111100111001001010000101101111010011111100111011010010100000110100010110101111000111011001001100101011010001111110000001010101101010000110110100100101011111110001011100100111101001111001100010101010010111101111001010101101011001001011001001110100010
Pair \(Z_2\) Length of longest common subsequence
8CPZ_1,6WMG_1 215 4
8CPZ_1,6XZU_1 268 6
6WMG_1,6XZU_1 149 4

Newick tree

 
[
	8CPZ_1:13.50,
	[
		6WMG_1:74.5,6XZU_1:74.5
	]:59.00
]

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 \((0.85)(0.8)(\frac{2759 }{\log_{20} 2759}-\frac{224}{\log_{20}224})=625.\)
Status Protein1 Protein2 d d1/2
Query variables 8CPZ_1 6WMG_1 786 426.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]