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Parikh vectors
9JGI_1 6ESS_1 7WAK_1 Letter Amino acid
8 5 21 R Arginine
0 0 8 C Cysteine
26 5 35 E Glutamic acid
16 8 46 L Leucine
17 10 27 S Serine
11 6 26 Y Tyrosine
24 25 20 A Alanine
15 21 22 G Glycine
12 2 22 F Phenylalanine
8 4 18 P Proline
1 6 7 W Tryptophan
10 9 40 N Asparagine
18 7 34 D Aspartic acid
16 4 51 I Isoleucine
4 4 15 M Methionine
27 20 25 T Threonine
8 7 17 Q Glutamine
7 2 16 H Histidine
23 5 51 K Lycine
19 9 22 V Valine 

9JGI_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O|tail tube protein|Bacillus subtilis (1423)
>6ESS_1|Chain A|Streptavidin|Streptomyces avidinii (1895)
>7WAK_1|Chain A|Glutamyl-tRNA synthetase|Plasmodium falciparum 3D7 (36329)
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 \(w=f(c)\)
9JGI , Knot 118 270 0.81 38 168 263 
MKTVIQDTADVYFKRKSDGKLVFTAEAQTASFSQAISEEKLRGGIGNKPLYILKSEKEINLTVKNAFFDLEWLAMTQGETIQEETKVKVFDREHGLIVDDTNKVTLKGKPVSDVTFYNKKGLTYKIAVSTDGTYTIPTAFAAAKDKLTAVYQIEKVGRRLAIKASKFSERYEVEYRTIAYNPDTEEVYSDIYIQFPNVSPSGEFEMSLENGNALAPEIKFEALADTDTDEMAVVIEASRDENTAAPVEDTTGSTQSSDLGGTTEHHHHHH
6ESS , Knot 74 159 0.78 38 108 153 
MASMTGGQQMGRDEAGITGTWYNQLGSTFIVTAGADGALTGTYESAVGNAESRYVLTGRYDSAPATDGSGTALGWTVAWKNNYRNAHSATTWSGQYVGGAEARINTQWLLTAGTTEAPAWAMTLVGHDTFTKVKPSAASIDAAKKAGVNNGNPLDAVQQ
7WAK , Knot 215 523 0.85 40 258 492 
GAMANAVIGNVVTRFPPEPSGYLHVGHAKAAFLNNYYAQMYEGKMLLRFDDTNPVLEDIKYEKSIIEDLENLGLKYEKISYSSDHFDLLEKYCIDMIKMNKAYADDTGVEDMRNQRGEGIESINRNNSIEKNLELFNEMRKGTEIGQKNCIRAKINMQSKNKCMRDPVMYRCIVDVPHHKHQFKYKCYPTYDFACPIIDSIEGVTHALRTNEYSDRIEQYNWFISTLNLRKVYIYEFSRLAFVKTVMSKRKLKWFVENNVVDSWVDPRFPTIKGILRRGLTKEALFQFILEQGPSKAGNLMQWDKLWSINKQIIDPIIPRYAAVDKNSSILLILTDLTDQVIQKERDLHMKNKSLGTCNMYYNNKYLIELEDAQTLLENEEITLIKLGNIIIKNIEKENGKIKQINALSNFHGDFKTTKKKIHWLPYLPQQLITCTLYEYDHLITVDKFENDNKDDWTNFINFNSKHETLVYAEPSISSLKVSDKFQFERRGYFILDKIDPHHHLHLIKIPDGKSKNMSIITT

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 \(c\).

\(|P_{f(9JGI_1)}(2) \setminus P_{f(6ESS_1)}(2)|=107\), \(|P_{f(6ESS_1)}(2) \setminus P_{f(9JGI_1)}(2)|=47\). 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 1:100110001010100000101110101001010011000010111100110110000010101001110101111001001000001011000011110000010101011001010000110001110001000110111110001011001001100111010010000010000110010000100010101101010101010100101111010101110000001111101000000111100001000000111000000000
Pair \(Z_2\) Length of longest common subsequence
9JGI_1,6ESS_1 154 4
9JGI_1,7WAK_1 156 4
6ESS_1,7WAK_1 216 3

Newick tree

 
[
	7WAK_1:99.27,
	[
		9JGI_1:77,6ESS_1:77
	]:22.27
]

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 \((0.85)(0.8)(\frac{429 }{\log_{20} 429}-\frac{159}{\log_{20}159})=80.2\)
Status Protein1 Protein2 d d1/2
Query variables 9JGI_1 6ESS_1 99 77
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]