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Parikh vectors
2AFM_1 7KQR_1 8ZYW_1 Letter Amino acid
28 21 23 S Serine
16 25 20 R Arginine
15 3 10 N Asparagine
19 19 20 D Aspartic acid
22 24 19 P Proline
7 6 16 M Methionine
15 10 13 F Phenylalanine
12 10 2 Y Tyrosine
23 33 17 A Alanine
19 20 31 E Glutamic acid
17 9 17 I Isoleucine
10 5 15 K Lycine
2 3 3 C Cysteine
16 10 9 H Histidine
7 5 2 W Tryptophan
13 23 19 V Valine
16 10 25 Q Glutamine
16 27 21 G Glycine
44 35 26 L Leucine
12 18 13 T Threonine 

2AFM_1|Chains A, B|Glutaminyl-peptide cyclotransferase|Homo sapiens (9606)
>7KQR_1|Chains A, B|Heme-dependent L-tyrosine hydroxylase|Streptomyces sclerotialus (1957)
>8ZYW_1|Chains A, B|PomB|Vibrio alginolyticus (663)
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 \(w=f(c)\)
2AFM , Knot 142 329 0.83 40 203 313 
ASAWPEEKNYHQPAILNSSALRQIAEGTSISEMWQNDLQPLLIERYPGSPGSYAARQHIMQRIQRLQADWVLEIDTFLSQTPYGYRSFSNIISTLNPTAKRHLVLACHYDSKYFSHWNNRVFVGATDSAVPCAMMLELARALDKKLLSLKTVSDSKPDLSLQLIFFDGEEAFLHWSPQDSLYGSRHLAAKMASTPHPPGARGTSQLHGMDLLVLLDLIGAPNPTFPNFFPNSARWFERLQAIEHELHELGLLKDHSLEGRYFQNYSYGGVIQDDHIPFLRRGVPVLHLIPSPFPEVWHTMDDNEENLDESTIDNLNKILQVFVLEYLHL
7KQR , Knot 133 316 0.80 40 168 292 
GHMNTGTGTVLTELPDHGRWDFGDFPYGLEPLTLPEPGSLEAADSGSVPAEFTLTCRHIAAIAAGGGPAERVQPADSSDRLYWFRWITGHQVTFILWQLLSRELARLPEEGPERDAALKAMTRYVRGYCAMLLYTGSMPRTVYGDVIRPSMFLQHPGFSGTWAPDHKPVQALFRGKKLPCVRDSADLAQAVHVYQVIHAGIAARMVPSGRSLLQEASVPSGVQHPDVLGVVYDNYFLTLRSRPSSRDVVAQLLRRLTAIALDVKDNALYPDGREAGSELPEELTRPEVTGHERDFLAILSEVAEEATGSPALASDR
8ZYW , Knot 138 321 0.82 40 193 302 
MDDEDNKCDCPPPGLPLWMGTFADLMSLLMCFFVLLLSFSEMDVLKFKQIAGSMKFAFGVQNQLEVKDIPKGTSIIAQEFRPGRPEPTPIDVIMQQTMDITQQTLEFHEGESDRAGGTKRDEGKLTGGQSPATSTQNNESAEADMQQQQSKEMSQEMETLMESIKKALEREIEQGAIEVENLGQQIVIRMREKGAFPEGSAFLQPKFRPLVRQIAELVKDVPGIVRVSGHTDNRPLDSELYRSNWDLSSQRAVSVAQEMEKVRGFSHQRLRVRGMADTEPLLPNDSDDNRALNRRVEISIMQGEPLYSEEVPVIQHHHHHH

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 \(c\).

\(|P_{f(2AFM_1)}(2) \setminus P_{f(7KQR_1)}(2)|=105\), \(|P_{f(7KQR_1)}(2) \setminus P_{f(2AFM_1)}(2)|=70\). 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 1:10111000000011110001100110100100110001011110001101100110001100100101011101001100010100010011001010100011110000000100100011111000111011110110110001101001000010101011110100111010100010100011101100101111010001011011111011111010110111001011001011000100111100001010010000011110000111100111110111011101100100000010000100100110111100101
Pair \(Z_2\) Length of longest common subsequence
2AFM_1,7KQR_1 175 4
2AFM_1,8ZYW_1 156 5
7KQR_1,8ZYW_1 169 3

Newick tree

 
[
	7KQR_1:88.52,
	[
		2AFM_1:78,8ZYW_1:78
	]:10.52
]

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 \((0.85)(0.8)(\frac{645 }{\log_{20} 645}-\frac{316}{\log_{20}316})=91.2\)
Status Protein1 Protein2 d d1/2
Query variables 2AFM_1 7KQR_1 118 113
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]