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Parikh vectors
2XCV_1 8TQI_1 2LNK_1 Letter Amino acid
16 6 2 M Methionine
46 44 2 S Serine
24 11 4 A Alanine
32 20 6 R Arginine
7 14 0 C Cysteine
34 33 1 G Glycine
22 17 0 H Histidine
60 36 4 L Leucine
30 25 3 V Valine
15 13 1 Q Glutamine
23 42 0 I Isoleucine
33 10 1 F Phenylalanine
5 8 0 W Tryptophan
18 31 2 N Asparagine
38 26 3 D Aspartic acid
23 28 2 P Proline
28 12 4 E Glutamic acid
36 22 2 K Lycine
33 35 2 T Threonine
31 21 0 Y Tyrosine 

2XCV_1|Chain A|CYTOSOLIC PURINE 5'-NUCLEOTIDASE|HOMO SAPIENS (9606)
>8TQI_1|Chains A, B|Hemagglutinin-neuraminidase|Human respirovirus 3 (11216)
>2LNK_1|Chain A[auth C]|Myosin heavy chain, non-muscle IIa|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 \(w=f(c)\)
2XCV , Knot 224 554 0.85 40 276 516 
GSSHHHHHHSSGLVPRGSMSTSWSDRLQNAADMPANMDKHALKKYRREAYHRVFVNRSLAMEKIKCFGFNMDYTLAVYKSPEYESLGFELTVERLVSIGYPQELLSFAYDSTFPTRGLVFDTLYGNLLKVDAYGNLLVCAHGFNFIRGPETREQYPNKFIQRDDTERFYILNTLFNLPETYLLACLVDFFTNCPRYTSCETGFKDGDLFMSYRSMFQDVRDAVDWVHYKGSLKEKTVENLEKYVVKDGKLPLLLSRMKEVGKVFLATNSDYKYTDKIMTYLFDFPHGPKPGSSHRPWQSYFDLILVDARKPLFFGEGTVLRQVDTKTGKLKIGTYTGPLQHGIVYSGGSSDTICDLLGAKGKDILYIGDHIFGDILKSKKRQGWRTFLVIPELAQELHVWTDKSSLFEELQSLDIFLAELYKHLDSSSNERPDISSIQRRIKKVTHDMDMCYGMMGSLFRSGSRQTLFASQVMRYADLYAASFINLLYYPFSYLFRAAHVLMPHESTVEHTHVDINEMESPLATRNRTSVDFKDTDYKRHQLTRSISEIKPPNL
8TQI , Knot 186 454 0.83 40 233 427 
EVPPQRITHDVGIKPLNPDDFWRCTSGLPSLMKTPKIRLMPGPGLLAMPTTVDGCVRTPSLVINDLIYAYTSNLITRGCQDIGKSYQVLQIGIITVNSDLVPDLNPRISHTFNINDNRKSCSLALLNTDVYQLCSTPKVDERSDYASSGIEDIVLDIVNHDGSISTTRFKNNNISFDQPYAALYPSVGPGIYYKGKIIFLGYGGLEHPINENAICNTTGCPGKTQRDCNQASHSPWFSDRRMVNSIIVVDKGLNSIPKLKVWTISMRQNYWGSEGRLLLLGNKIYIYTRSTSWHSKLQLGIIDITDYSDIRIKWTWHNVLSRPGNNECPWGHSCPDGCITGVYTDAYPLNPTGSIVSSVILDSQKSRVNPVITYSTSTERVNELAIRNKTLSAGYTTTSCITHYNKGYCFHIVEINHKSLDTFQPMLFKTEIPKSCSGSENLYFQGGSHHHHHH
2LNK , Knot 25 39 0.78 30 35 37 
QRELEDATETADAMNREVSSLKNKLRRGDLPFVVPRRMA

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 \(c\).

\(|P_{f(2XCV_1)}(2) \setminus P_{f(8TQI_1)}(2)|=106\), \(|P_{f(8TQI_1)}(2) \setminus P_{f(2XCV_1)}(2)|=63\). 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 1:10000000000111101010001000100110111010001100000010001110001110010011101000111000100001110101001101101001101100001100111100101011010101011101011011011000000100110000000101100110110001110110110001000000011001011100001100100110110001010000100100011001011111001001101111000000000011001101101101100001100010111101001111101011001000010101100011100111001100001001111010011011001110110000001100111110110010110000011001001011110100010000000101001000100100010100111101100100001110011001010110110110011001101101111000010000101001001110000001010000000001000100101101
Pair \(Z_2\) Length of longest common subsequence
2XCV_1,8TQI_1 169 7
2XCV_1,2LNK_1 251 3
8TQI_1,2LNK_1 220 3

Newick tree

 
[
	2LNK_1:12.22,
	[
		2XCV_1:84.5,8TQI_1:84.5
	]:42.72
]

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 \((0.85)(0.8)(\frac{1008 }{\log_{20} 1008}-\frac{454}{\log_{20}454})=145.\)
Status Protein1 Protein2 d d1/2
Query variables 2XCV_1 8TQI_1 182 166
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]