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Parikh vectors
6ZMT_1 7MEP_1 7NQD_1 Letter Amino acid
15 7 11 R Arginine
15 5 12 D Aspartic acid
2 2 1 C Cysteine
24 8 8 T Threonine
10 3 1 W Tryptophan
9 1 20 N Asparagine
15 6 23 I Isoleucine
11 3 26 K Lycine
10 4 17 F Phenylalanine
14 15 8 S Serine
19 5 10 V Valine
4 1 5 H Histidine
21 8 18 L Leucine
7 1 2 M Methionine
20 7 6 P Proline
37 7 7 A Alanine
15 7 5 Q Glutamine
25 6 19 E Glutamic acid
15 8 13 G Glycine
7 3 10 Y Tyrosine 

6ZMT_1|Chain A[auth B]|40S ribosomal protein SA|Homo sapiens (9606)
>7MEP_1|Chains A[auth G], G[auth J], K|RM19R mAb Light chain|Macaca mulatta (9544)
>7NQD_1|Chains A, B|TPR_REGION domain-containing protein|Marinitoga sp. 1137 (1545835)
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 \(w=f(c)\)
6ZMT , Knot 127 295 0.81 40 176 277 
MSGALDVLQMKEEDVLKFLAAGTHLGGTNLDFQMEQYIYKRKSDGIYIINLKRTWEKLLLAARAIVAIENPADVSVISSRNTGQRAVLKFAAATGATPIAGRFTPGTFTNQIQAAFREPRLLVVTDPRADHQPLTEASYVNLPTIALCNTDSPLRYVDIAIPCNNKGAHSVGLMWWMLAREVLRMRGTISREHPWEVMPDLYFYRDPEEIEKEEQAAAEKAVTKEEFQGEWTAPAPEFTATQPEVADWSEGVQVPSVPIQQFPTEDWSAQPATEDWSAAPTAQATEWVGATTDWS
7MEP , Knot 54 107 0.78 40 84 101 
AIRMTQSPAILSLSPGERATLSCRASQSVDSRLAWYQQKPGQSPRLLIYDVSSRATGIPDRFSGSGSGTEFTLTISSLEPEDVAVYFCHQENDWPWTFGQGTKVEIK
7NQD , Knot 101 222 0.82 40 148 214 
KAPSQNAIKRFMTLFSGREDVFSIQYEGGYRPIRRPLNFQDIKNHFSGKKTLGIYLLKKNDTVKFAAYDIDIKKHYLNREDKFVYEENSKKVAKRLSRELNLENITHYFEFTGNRGYHIWIFFDIPVSAYKIKYIMEKILDRIELEEGIDVEIFPKQTSLNGGLGNLIKVPLGVHKKTGKKCLFVDNDFNVIENQIEFLNNIKENKATEINKLFREIFNEND

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 \(c\).

\(|P_{f(6ZMT_1)}(2) \setminus P_{f(7MEP_1)}(2)|=124\), \(|P_{f(7MEP_1)}(2) \setminus P_{f(6ZMT_1)}(2)|=32\). 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 1:1011101101000011011111001110010101000100000011011010001001111101111100110101100000100111011110110111101011010001011100101111001010001100100101101110000011001011110000110011111111100110101010000110111010100010010000011100110000101010111101010010110100110110111001100010101100010111010100111100010
Pair \(Z_2\) Length of longest common subsequence
6ZMT_1,7MEP_1 156 4
6ZMT_1,7NQD_1 188 4
7MEP_1,7NQD_1 154 3

Newick tree

 
[
	6ZMT_1:89.27,
	[
		7MEP_1:77,7NQD_1:77
	]:12.27
]

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 \((0.85)(0.8)(\frac{402 }{\log_{20} 402}-\frac{107}{\log_{20}107})=89.9\)
Status Protein1 Protein2 d d1/2
Query variables 6ZMT_1 7MEP_1 109 73
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]