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Parikh vectors
4KIB_1 2DQF_1 6OSN_1 Letter Amino acid
15 2 14 H Histidine
13 5 7 P Proline
15 8 5 T Threonine
6 4 3 Y Tyrosine
8 5 4 N Asparagine
25 4 3 E Glutamic acid
6 2 2 W Tryptophan
4 2 6 C Cysteine
36 10 12 G Glycine
13 7 0 I Isoleucine
28 8 5 L Leucine
7 4 2 K Lycine
9 1 5 M Methionine
31 4 3 V Valine
8 7 6 Q Glutamine
29 3 4 R Arginine
23 3 3 D Aspartic acid
22 5 2 F Phenylalanine
23 20 9 S Serine
36 3 3 A Alanine 

4KIB_1|Chains A, B|Methyltransferase MppJ|Streptomyces hygroscopicus (1912)
>2DQF_1|Chains A, D|lysozyme binding Ig kappa chain V23-J2 region|Mus musculus (10090)
>6OSN_1|Chain A|Tyrosine-protein kinase transmembrane receptor ROR2|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 \(w=f(c)\)
4KIB , Knot 152 357 0.83 40 196 332 
MGSSHHHHHHSSGLVPRGSHMSTEVSEAQARRAVADIFNSTLASSAIGAAWELGALDELRENGKLDVSDFAVRHDLHEPAVVGMFTALASVGIVRREGATVVVGPYFDEANHHRSLFHWLNQGSGELFRRMPQVLPNENRTGKFYQRDAGAISYACREISERYFDPAFWAAVDGLGYTPTTVADLGSGSGERLIQIARRFPGVRGLGVDIADGAIAMAEKEVAAKGFGDQISFVRGDARTIDQVSARGEFAEVDLLTCFMMGHDFWPRENCVQTLRKLRAAFPNVRRFLLGDATRTVGIPDRELPVFTLGFEFGHDMMGVYLPTLDEWDGVFEEGGWRCVKKHAIDSLSVSVVFELE
2DQF , Knot 53 107 0.77 40 80 98 
DIVLTQSPATLSVTPGNSVSLSCRASQSIGNNLHWYQQKSHESPRLLIKYASQSISGIPSRFSGSGSGTDFTLSINSVETEDFGMYFCQQSNSWPYTFGGGTKLEIK
6OSN , Knot 48 98 0.74 38 75 89 
MGSHHHHHHLVPRGSHMHQCYNGSGMDYRGTASTTKSGHQCQPWALQHPHSHHLSSTDFPELGGGHAYCRNPGGQMEGPWCFTQNKNVRMELCDVPSC

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 \(c\).

\(|P_{f(4KIB_1)}(2) \setminus P_{f(2DQF_1)}(2)|=149\), \(|P_{f(2DQF_1)}(2) \setminus P_{f(4KIB_1)}(2)|=33\). 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 1:110000000000111101001000100101001110110001100111111011110010001010100111000100111111101110111100011011111010010000011011001010110011011100000101000011110010001000010111111101110010011011010100110110011110111101101111110001110111001011010100100101010110101100111100111000010010010111101001111010001111000111101110110011110110100101110011100100011001010111010
Pair \(Z_2\) Length of longest common subsequence
4KIB_1,2DQF_1 182 5
4KIB_1,6OSN_1 195 8
2DQF_1,6OSN_1 115 3

Newick tree

 
[
	4KIB_1:10.71,
	[
		2DQF_1:57.5,6OSN_1:57.5
	]:46.21
]

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 \((0.85)(0.8)(\frac{464 }{\log_{20} 464}-\frac{107}{\log_{20}107})=107.\)
Status Protein1 Protein2 d d1/2
Query variables 4KIB_1 2DQF_1 134 84.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]