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Parikh vectors
2ZEL_1 7QYA_1 1VHX_1 Letter Amino acid
18 21 9 D Aspartic acid
2 5 0 C Cysteine
17 17 10 I Isoleucine
10 23 11 K Lycine
24 20 11 A Alanine
16 11 6 H Histidine
22 12 3 P Proline
28 18 8 S Serine
12 22 11 T Threonine
15 15 5 N Asparagine
16 24 4 Q Glutamine
19 29 10 E Glutamic acid
16 20 14 G Glycine
13 31 10 V Valine
16 16 6 R Arginine
44 51 17 L Leucine
7 11 7 M Methionine
15 16 2 F Phenylalanine
7 6 2 W Tryptophan
12 8 4 Y Tyrosine 

2ZEL_1|Chains A, B|Glutaminyl-peptide cyclotransferase|Homo sapiens (9606)
>7QYA_1|Chain A[auth a]|26S proteasome non-ATPase regulatory subunit 13|Homo sapiens (9606)
>1VHX_1|Chains A, B|Putative Holliday junction resolvase|Bacillus subtilis (1423)
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 \(w=f(c)\)
2ZEL , Knot 142 329 0.83 40 203 313 
ASAWPEEKNYHQPAILNSSALRQIAEGTSISEMWQNDLQPLLIERYPGSPGSYAARQHIMQRIQRLQADWVLEIDTFLSQTPYGYRSFSNIISTLNPTAKRHLVLACHYDSKYFSHWNNRVFVGATDSAVPCAMMLELARALDKKLLSLKTVSDSKPDLSLQLIFFDGEEAFLHWSPQDSLYGSRHLAAKMASTPHPPGARGTSQLHGMDLLVLLALIGAPNPTFPNFFPNSARWFERLQAIEHELHELGLLKDHSLEGRYFQNYSYGGVIQDDHIPFLRRGVPVLHLIPSPFPEVWHTMDDNEENLDESTIDNLNKILQVFVLEYLHL
7QYA , Knot 161 376 0.84 40 215 358 
MKDVPGFLQQSQNSGPGQPAVWHRLEELYTKKLWHQLTLQVLDFVQDPCFAQGDGLIKLYENFISEFEHRVNPLSLVEIILHVVRQMTDPNVALTFLEKTREKVKSSDEAVILCKTAIGALKLNIGDLQVTKETIEDVEEMLNNLPGVTSVHSRFYDLSSKYYQTIGNHASYYKDALRFLGCVDIKDLPVSEQQERAFTLGLAGLLGEGVFNFGELLMHPVLESLRNTDRQWLIDTLYAFNSGNVERFQTLKTAWGQQPDLAANEAQLLRKIQLLCLMEMTFTRPANHRQLTFEEIAKSAKITVNEVELLVMKALSVGLVKGSIDEVDKRVHMTWVQPRVLDLQQIKGMKDRLEFWCTDVKSMEMLVEHQAHDILT
1VHX , Knot 73 150 0.81 38 111 143 
MSLRILGLDLGTKTLGVALSDEMGWTAQGIETIKINEAEGDYGLSRLSELIKDYTIDKIVLGFPKNMNGTVGPRGEASQTFAKVLETTYNVPVVLWDERLTTMAAEKMLIAADVSRQKRKKVIDKMAAVMILQGYLDSLNEGGSHHHHHH

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 \(c\).

\(|P_{f(2ZEL_1)}(2) \setminus P_{f(7QYA_1)}(2)|=81\), \(|P_{f(7QYA_1)}(2) \setminus P_{f(2ZEL_1)}(2)|=93\). 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 1:10111000000011110001100110100100110001011110001101100110001100100101011101001100010100010011001010100011110000000100100011111000111011110110110001101001000010101011110100111010100010100011101100101111010001011011111111111010110111001011001011000100111100001010010000011110000111100111110111011101100100000010000100100110111100101
Pair \(Z_2\) Length of longest common subsequence
2ZEL_1,7QYA_1 174 3
2ZEL_1,1VHX_1 170 3
7QYA_1,1VHX_1 174 3

Newick tree

 
[
	7QYA_1:87.65,
	[
		2ZEL_1:85,1VHX_1:85
	]:2.65
]

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 \((0.85)(0.8)(\frac{705 }{\log_{20} 705}-\frac{329}{\log_{20}329})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 2ZEL_1 7QYA_1 133 122.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]