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Parikh vectors
8BTV_1 2DER_1 5YXH_1 Letter Amino acid
39 14 9 A Alanine
25 22 10 G Glycine
27 0 14 L Leucine
3 0 3 M Methionine
19 0 5 R Arginine
3 27 0 C Cysteine
7 0 9 Q Glutamine
15 0 4 P Proline
9 0 16 S Serine
18 0 15 V Valine
7 0 6 N Asparagine
10 0 17 I Isoleucine
6 0 13 K Lycine
6 0 7 F Phenylalanine
23 0 9 T Threonine
3 0 1 W Tryptophan
9 0 8 Y Tyrosine
22 0 8 D Aspartic acid
11 0 14 E Glutamic acid
4 0 3 H Histidine 

8BTV_1|Chain A|Beta-lactamase|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
>2DER_1|Chains A[auth C], B[auth D]|tRNA|null
>5YXH_1|Chains A, B, C, D|GTP-binding protein Rheb|Homo sapiens (9606)
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 \(w=f(c)\)
8BTV , Knot 116 266 0.81 40 155 247 
GDLADRFAELERRYDARLGVYVPATGTTAAIEYRADERFAFCSTFKAPLVAAVLHQNPLTHLDKLITYTSDDIRSISPVAQQHVQTGMTIGQLCDAAIRYSDGTAANLLLADLGGPGGGTAAFTGYLRSLGDTVSRLDAEEPELNRDPPGDERNTTTPHAIALVLQQLVLGNALPPDKRALLTDWMARNTTGAKRIRAGFPADWKVIDKTGTGDYGRANDIAVVWSPTGVPYVVAVMSDRAGGGYDAEPREALLAEAATCVAGVLA
2DER , Knot 24 76 0.45 8 15 40 
GUCCCCUUCGUCUAGAGGCCCAGGACACCGCCCUUUCACGGCGGUAACAGGGGUUCGAAUCCCCUAGGGGACGCCA
5YXH , Knot 80 171 0.80 38 127 166 
SPQSKSRKIAILGYRSVGKSSLTIQFVEGQFVDSYDPTIENTFTKLITVNGQEYHLQLVDTAGQDEYSIFPQTYSIDINGYILVYSVTSIKSFEVIKVIHGKLLDMVGKVQIPIMLVGNKKDLHMERVISYEEGKALAESWNAAFLESSAKENQTAVDVFRRIILEAEKLE

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 \(c\).

\(|P_{f(8BTV_1)}(2) \setminus P_{f(2DER_1)}(2)|=151\), \(|P_{f(2DER_1)}(2) \setminus P_{f(8BTV_1)}(2)|=11\). 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 1:10110011010000010111011101001110001000111000101111111100011001001100000010010111000100110110100111000010110111101111111011101010011001001010010100011100000001011111100111101111000111001110000110010111110101100010100101001111101011101111100011110010100111101100111111
Pair \(Z_2\) Length of longest common subsequence
8BTV_1,2DER_1 162 4
8BTV_1,5YXH_1 152 4
2DER_1,5YXH_1 138 2

Newick tree

 
[
	8BTV_1:81.47,
	[
		5YXH_1:69,2DER_1:69
	]:12.47
]

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 \((0.85)(0.8)(\frac{342 }{\log_{20} 342}-\frac{76}{\log_{20}76})=83.6\)
Status Protein1 Protein2 d d1/2
Query variables 8BTV_1 2DER_1 112 67
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]