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Parikh vectors
5HAS_1 8ITX_1 4FQA_1 Letter Amino acid
13 5 7 H Histidine
55 13 13 L Leucine
28 8 8 K Lycine
10 3 4 M Methionine
37 14 18 A Alanine
29 7 7 D Aspartic acid
26 6 4 E Glutamic acid
21 19 16 G Glycine
1 1 4 W Tryptophan
10 7 7 Y Tyrosine
44 10 15 S Serine
23 9 7 R Arginine
17 14 8 N Asparagine
25 5 9 Q Glutamine
27 10 4 I Isoleucine
31 7 4 T Threonine
9 1 6 C Cysteine
16 8 2 F Phenylalanine
18 21 6 P Proline
28 14 13 V Valine 

5HAS_1|Chains A, B, C, D|Sec7|Thielavia terrestris (2587410)
>8ITX_1|Chains A, B|Galectin-3|Homo sapiens (9606)
>4FQA_1|Chain A|toxic effector Tse1|Pseudomonas aeruginosa PAO1 (208964)
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 \(w=f(c)\)
5HAS , Knot 187 468 0.82 40 228 437 
MGSSHHHHHHMSSLKFVVSSLDIIAAQAGRNKQLAELAEKALAAIKENDQQLPDPEVVFAPLQLATKSGTIPLTTTALDCIGKLISYSYFSAPSSSATQDGTEQTPLIERAIDTICDCFQGETTLVEIQLQIVKSLLAAVLNDKIIVHGAGLLKAVRQVYNIFLLSRSTANQQVAQGTLTQMVGTVFERVKTRLHMKEARANLGRLKASRSSLAVDRSDDQDSQAGKVDGEDATVETVSDATPSESVDKAGGGKLTLKDLEHRKSFDDSHMGDGPTMVSQVKPMKKASRSVSEQSLQESPQDETPESLDAEDEAYIRDAYLVFRSFCNLSTKILPPDQLYDLRGQPMRSKLISLHLIHTLLNNHITVFTSPLCTIRNTKNNEPTNFLQAIKYYLCLSITRNGASSVDRVFDICCEIFWLMLKYMRSSFKNEIEVFLNEIYLALLARRNAPLSQKLTFVGILKRLCEDP
8ITX , Knot 83 182 0.79 40 131 175 
GAPAPGVYPGPPSGPGAYPSSGQPSATGAYPATGHMGPYGAPAGPLIVPYNLPLPGGVVPRMLITILGTVKPNANRIALDFQRGNDVAFHFNPRFNENNRRVIVCNTKLDNNWGREERQSVFPFESGKPFKIQVLVEPDHFKVAVNDAHLLQYNHRVKKLNEISKLGISGDIDLTSASYTMI
4FQA , Knot 79 162 0.82 40 116 152 
MDSLDQCIVNACKNSWDKSYLAGTPNKDNCSGFVQSVAAELGVPMPRGNANAMVDGLEQSWTKLASGAEAAQKAAQGFLVIAGLKGRTYGHVAVVISGPLYRQKYPMCWCGSIAGAVGQSQGLKSVGQVWNRTDRDRLNYYVYSLASCSLPRASLEHHHHHH

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 \(c\).

\(|P_{f(5HAS_1)}(2) \setminus P_{f(8ITX_1)}(2)|=136\), \(|P_{f(8ITX_1)}(2) \setminus P_{f(5HAS_1)}(2)|=39\). 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 1:110000000010010111001011110110000110110011111000000110101111110110001011100011001101100001011000100010000111001100100010100011010101100111111000111011111011001001111000010001101010011101100100010100101011010100001110000000001101010010100100101000100111101010010000010000110110110010110010001000010001000010010100010100101110010010001111001001010110001101011001100010110011001000000010011011000101010001100100110100011111100100010001011100101111100011100010111110010001
Pair \(Z_2\) Length of longest common subsequence
5HAS_1,8ITX_1 175 4
5HAS_1,4FQA_1 194 6
8ITX_1,4FQA_1 161 3

Newick tree

 
[
	5HAS_1:96.00,
	[
		8ITX_1:80.5,4FQA_1:80.5
	]:15.50
]

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 \((0.85)(0.8)(\frac{650 }{\log_{20} 650}-\frac{182}{\log_{20}182})=133.\)
Status Protein1 Protein2 d d1/2
Query variables 5HAS_1 8ITX_1 168 116
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]