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Parikh vectors
5CBB_1 8HGU_1 6QBS_1 Letter Amino acid
26 9 9 Q Glutamine
36 16 10 E Glutamic acid
14 21 2 H Histidine
26 4 19 K Lycine
16 4 13 Y Tyrosine
62 19 9 D Aspartic acid
19 4 4 M Methionine
48 10 5 T Threonine
10 10 4 W Tryptophan
3 3 8 C Cysteine
60 26 24 G Glycine
65 29 12 L Leucine
36 17 8 P Proline
34 18 16 S Serine
59 21 15 V Valine
29 7 19 N Asparagine
46 20 7 R Arginine
41 8 8 I Isoleucine
23 19 5 F Phenylalanine
88 33 18 A Alanine 

5CBB_1|Chain A|Malate synthase G|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
>8HGU_1|Chains A, B, C, D|Alpha/beta hydrolase|Bosea sp. PAMC 26642 (1792307)
>6QBS_1|Chains A, B|Cathepsin K|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 \(w=f(c)\)
5CBB , Knot 277 741 0.82 40 268 652 
MTDRVSVGNLRIARVLYDFVNNEALPGTDIDPDSFWAGVDKVVADLTPQNQALLNARDELQAQIDKWHRRRVIEPIDMDAYRQFLTEIGYLLPEPDDFTITTSGVDAEITTTAGPQLVVPVLNARFALNAANARWGSLYDALYGTDVIPETDGAEKGPTYNKVRGDKVIAYARKFLDDSVPLSSGSFGDATGFTVQDGQLVVALPDKSTGLANPGQFAGYTGAAESPTSVLLINHGLHIEILIDPESQVGTTDRAGVKDVILESAITTIMDFEDSVAAVDAADKVLGYRNWLGLNKGDLAAAVDKDGTAFLRVLNRDRNYTAPGGGQFTLPGRSLMFVRNVGHLMTNDAIVDTDGSEVFEGIMDALFTGLIAIHGLKASDVNGPLINSRTGSIYIVKPKMHGPAEVAFTCELFSRVEDVLGLPQNTMKIGIMDEERRTTVNLKACIKAAADRVVFINTGFLDRTGDEIHTSMEAGPMVRKGTMKSQPWILAYEDHNVDAGLAAGFSGRAQVGKGMWTMTELMADMVETKIAQPRAGASTAWVPSPTAATLHALHYHQVDVAAVQQGLAGKRRATIEQLLTIPLAKELAWAPDEIREEVDNNCQSILGYVVRWVDQGVGASKVPDIHDVALMEDRATLRISSQLLANWLRHGVITSADVRASLERMAPLVDRQNAGDVAYRPMAPNFDDSIAFLAAQELILSGAQQPNGYTEPILHRRRREFKARAAEKPAPSDRAGDDAAR
8HGU , Knot 133 298 0.84 40 188 288 
MSSDFATPPGLRHRQIAVRDTTLHVAEIGSGGTPVLLLHGWPEFWATWLPLMNRLHDQFHLIAPDLRGFGDSEKSAVPRSDVGANSHADDMAALLGALGLESVGVVGHDVGAYAAQALARRHPQLVDRLLFFNCPTASVGGAWVHHGHVNEVWYQSFQQLGLAEALVGTSRETCALYFRHFLEHWSHRKDAFEPAFELWIDNFMKPGNLRGGFDWYRSQNALRLAAIDGHPTPSVRIHQPTRVHWGRHDPILKSEWSAFVPEHFDDARISFCESAGHFVHVEAPDEAADVLAEFFGGR
6QBS , Knot 99 215 0.82 40 148 209 
APDSVDYRKKGYVTPVKNQGQCGSCWAFSSVGALEGQLKKKTGKLLNLSPQNLVDCVSENDGCGGGYMTNAFQYVQKNRGIDSEDAYPYVGQEESCMYNPTGKAAKCRGYREIPEGNEKALKRAVARVGPVSVAIDASLTSFQFYSKGVYYDESCNSDNLNHAVLAVGYGIQKGNKHWIIKNSWGENWGNKGYILMARNKNNACGIANLASFPKM

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 \(c\).

\(|P_{f(5CBB_1)}(2) \setminus P_{f(8HGU_1)}(2)|=113\), \(|P_{f(8HGU_1)}(2) \setminus P_{f(5CBB_1)}(2)|=33\). 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 1:100010110101101100110001111001010011111001110101000111010001010100100001101101010001100110111010010100011010100011101111110101110110101101001101001110001100110000101001110100110001110010110101101001011111100001110110111001110010011110011010111010001100001110011100110011010001111011001110001111001011111000101110110000000111110101110011110011011000111000100110111011101111101101001011110000101011010101110111000110010011111000101111000000010101010111001111001110001001000101111100101000111110000010111111101010110111010011101100011010111001111010110101100001011110011110001010011011110011111001000100000011101101100111100110100111100010101000111011001110010101010011111000011011001111010001111110011101100101000111000000101011001110001100110
Pair \(Z_2\) Length of longest common subsequence
5CBB_1,8HGU_1 146 4
5CBB_1,6QBS_1 192 3
8HGU_1,6QBS_1 200 3

Newick tree

 
[
	6QBS_1:10.04,
	[
		5CBB_1:73,8HGU_1:73
	]:32.04
]

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 \((0.85)(0.8)(\frac{1039 }{\log_{20} 1039}-\frac{298}{\log_{20}298})=198.\)
Status Protein1 Protein2 d d1/2
Query variables 5CBB_1 8HGU_1 246 172.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]