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Parikh vectors
1NSR_1 8PFK_1 4HLX_1 Letter Amino acid
1 3 2 C Cysteine
3 0 4 M Methionine
10 0 6 P Proline
21 0 10 V Valine
16 3 5 A Alanine
24 0 6 I Isoleucine
25 0 6 K Lycine
3 0 3 W Tryptophan
10 0 2 Y Tyrosine
8 0 16 R Arginine
17 0 3 N Asparagine
28 0 7 D Aspartic acid
16 0 4 Q Glutamine
24 0 7 S Serine
21 0 9 E Glutamic acid
31 4 14 G Glycine
18 0 8 H Histidine
32 0 8 L Leucine
15 0 7 F Phenylalanine
24 0 5 T Threonine 

1NSR_1|Chains A, B|GALACTOSE MUTAROTASE|Lactococcus lactis (1358)
>8PFK_1|Chain A|RNA (12-mer)|synthetic construct (32630)
>4HLX_1|Chains A, B, C, D|K9|Human herpesvirus 8 (37296)
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 \(w=f(c)\)
1NSR , Knot 149 347 0.83 40 198 329 
MSIKIRDFGLGSDLISLTNKAGVTISFTNLGARIVDWQKDGKHLILGFDSAKEYLEKDAYPGATVGPTAGRIKDGLVKISGKDYILNQNEGPQTLHGGEESIHTKLWTYEVTDLGAEVQVKFSLVSNDGTNGYPGKIEMSVTHSFDDDNKWKIHYEAISDKDTVFNPTGHVYFNLNGDASESVENHGLRLAASRFVPLKDQTEIVRGDIVDIKNTDLDFRQEKQLSNAFNSNMEQVQLVKGINHPFLLDQLGLDKEQARLTLDDTSISVFTDQPSIVIFTANFGDLGTLYHEKKQVHHGGITFECQVSPGSEQIPELGDISLKAGEKYQATTIYSLHTKLEHHHHHH
8PFK , Knot 6 12 0.41 8 8 9 
CGCGAAUUAGCG
4HLX , Knot 65 132 0.80 40 102 125 
MHHHHHHSSGVDLGTENLYFQSMGKASIKDWIVCQVNSGKFPGVEWEDEERTRFRIPVTPLADPCFEWRRDGELGVVYIRERGNMPVDASFKGTRGRRRMLAALRRTRGLQEIGKGISQDGHHFLVFRVRKP

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 \(c\).

\(|P_{f(1NSR_1)}(2) \setminus P_{f(8PFK_1)}(2)|=195\), \(|P_{f(8PFK_1)}(2) \setminus P_{f(1NSR_1)}(2)|=5\). 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 1:10101001111001101000111010100111011010001001111100100010001011101110110100111010100011000011001011000100011000100111010101011000100101101010100010000010100011000001101010101010101000100011011100111100000110101101000010100000100110001001011011001111001110000101010000101100010111101011011010000001001110100010110001101101010110000100100100010000000
Pair \(Z_2\) Length of longest common subsequence
1NSR_1,8PFK_1 200 2
1NSR_1,4HLX_1 178 6
8PFK_1,4HLX_1 108 2

Newick tree

 
[
	1NSR_1:10.76,
	[
		4HLX_1:54,8PFK_1:54
	]:50.76
]

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 \((0.85)(0.8)(\frac{359 }{\log_{20} 359}-\frac{12}{\log_{20}12})=114.\)
Status Protein1 Protein2 d d1/2
Query variables 1NSR_1 8PFK_1 146 75.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]