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Parikh vectors
5IQM_1 1FYX_1 7BTR_1 Letter Amino acid
16 8 8 A Alanine
2 8 3 R Arginine
2 4 1 C Cysteine
8 3 5 Q Glutamine
3 14 24 E Glutamic acid
10 13 16 L Leucine
10 7 8 V Valine
8 9 14 D Aspartic acid
4 13 14 I Isoleucine
5 5 13 Y Tyrosine
7 10 6 N Asparagine
2 4 5 H Histidine
1 3 4 M Methionine
3 5 4 P Proline
15 9 11 S Serine
18 3 7 T Threonine
1 5 2 W Tryptophan
11 2 10 G Glycine
3 13 3 K Lycine
3 11 7 F Phenylalanine

5IQM_1|Chains A, B[auth G]|Protein FimG|Escherichia coli K-12 (83333)
>1FYX_1|Chain A|TOLL-LIKE RECEPTOR 2|Homo sapiens (9606)
>7BTR_1|Chains A[auth B], E[auth F]|Antirestriction protein ArdA|Enterococcus faecalis EnGen0302 (1158660)
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 \(w=f(c)\)
5IQM , Knot 65 132 0.80 40 95 124
AKPCTVSTTNATVDLGDLYSFSLMSAGAASAWHDVALELTNCPVGTSRVTASFSGAADSTGYYKNQGTAQNIQLELQDDSGNTLNTGATKTVQVDDSSQSAHFPLQVRALTVNGGATQGTIEAVISITYTYS
1FYX , Knot 72 149 0.80 40 111 143
SRNICYDAFVSYSERDAYWVENLMVQELENFNPPFKLCLHKRDFIHGKWIIDNIIDSIEKSHKTVFVLSENFVKSEWCKYELDFSHFRLFDENNDAAILILLEPIEKKAIPQRFCKLRKIMNTKTYLEWPMDEAQREGFWVNLRAAIKS
7BTR , Knot 79 165 0.81 40 121 155
MDDMQVYIANLGKYNEGELVGAWFTFPIDFEEVKEKIGLNDEYEEYAIHDYELPFTVDEYTSIGELNRLWEMVSELPEELQSELSALLTHFSSIEELSEHQEDIIIHSDCDDMYDVARYYIEETGALGEVPASLQNYIDYQAYGRDLDLSGTFISTNHGIFEIVY

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 \(c\).

\(|P_{f(5IQM_1)}(2) \setminus P_{f(1FYX_1)}(2)|=72\), \(|P_{f(1FYX_1)}(2) \setminus P_{f(5IQM_1)}(2)|=88\). 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 1:101001000010101101001011011110110011101000111000101010111000100000101001010100001001001100010100000010111010110101110010101110100000
Pair \(Z_2\) Length of longest common subsequence
5IQM_1,1FYX_1 160 3
5IQM_1,7BTR_1 140 3
1FYX_1,7BTR_1 158 3

Newick tree

 
[
	1FYX_1:82.42,
	[
		5IQM_1:70,7BTR_1:70
	]:12.42
]

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 \((0.85)(0.8)(\frac{281 }{\log_{20} 281}-\frac{132}{\log_{20}132})=46.4\)
Status Protein1 Protein2 d d1/2
Query variables 5IQM_1 1FYX_1 62 57.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]