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Parikh vectors
1MXF_1 1COZ_1 5NJK_1 Letter Amino acid
7 5 5 H Histidine
4 2 2 M Methionine
8 5 10 F Phenylalanine
9 3 1 N Asparagine
14 11 11 D Aspartic acid
8 1 8 C Cysteine
20 4 14 R Arginine
6 15 17 K Lycine
30 12 7 L Leucine
14 2 3 P Proline
18 3 7 S Serine
11 7 8 T Threonine
6 6 2 Y Tyrosine
13 12 12 E Glutamic acid
23 7 11 G Glycine
12 12 5 I Isoleucine
19 9 14 V Valine
44 4 12 A Alanine
9 6 4 Q Glutamine
1 3 3 W Tryptophan 

1MXF_1|Chains A, B, C, D|PTERIDINE REDUCTASE 2|Trypanosoma cruzi (5693)
>1COZ_1|Chains A, B|PROTEIN (GLYCEROL-3-PHOSPHATE CYTIDYLYLTRANSFERASE)|Bacillus subtilis (1423)
>5NJK_1|Chains A, B, C, D, E, F|Protein numb homolog|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 \(w=f(c)\)
1MXF , Knot 117 276 0.79 40 162 254 
MNETSHEASECPAAVITGGARRIGHSIAVRLHQQGFRVVVHYRHSEGAAQRLVAELNAARAGSAVLCKGDLSLSSSLLDCCEDIIDCSFRAFGRCDVLVNNASAYYPTPLLPGDDTNGAADAKPIDAQVAELFGSNAVAPLFLIRAFARRQGEGGAWRSRNLSVVNLCDAMTDLPLPGFCVYTMAKHALGGLTRAAALELAPRHIRVNAVAPGLSLLPPAMPQETQEEYRRKVPLGQSEASAAQIADAIAFLVSKDAGYITGTTLKVDGGLILARA
1COZ , Knot 66 129 0.82 40 106 125 
MKKVITYGTFDLLHWGHIKLLERAKQLGDYLVVAISTDEFNLQKQKKAYHSYEHRKLILETIRYVDEVIPEKNWEQKKQDIIDHNIDVFVMGDDWEGKFDFLKDQCEVVYLPRTEGISTTKIKEEIAGL
5NJK , Knot 76 156 0.82 40 118 148 
SRPHQWQTDEEGVRTGKCSFPVKYLGHVEVDESRGMHICEDAVKRLKAERKFFKGFFGKTGKKAVKAVLWVSADGLRVVDEKTKDLIVDQTIEKVSFCAPDRNFDRAFSYICRDGTTRRWICHCFMAVKDTGERLSHAVGCAFAACLERKQKREKE

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 \(c\).

\(|P_{f(1MXF_1)}(2) \setminus P_{f(1COZ_1)}(2)|=116\), \(|P_{f(1COZ_1)}(2) \setminus P_{f(1MXF_1)}(2)|=60\). 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 1:100000010001111101110011001110100011011100000011100111010110110111001010100011000001100010111000111001010010111110000111010110101101110011111111011100010111100001011010011001111110100110011111001111011100101011111101111111000000000011110001011011011111100011010100101011111101
Pair \(Z_2\) Length of longest common subsequence
1MXF_1,1COZ_1 176 4
1MXF_1,5NJK_1 170 3
1COZ_1,5NJK_1 148 3

Newick tree

 
[
	1MXF_1:90.29,
	[
		5NJK_1:74,1COZ_1:74
	]:16.29
]

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 \((0.85)(0.8)(\frac{405 }{\log_{20} 405}-\frac{129}{\log_{20}129})=83.3\)
Status Protein1 Protein2 d d1/2
Query variables 1MXF_1 1COZ_1 105 77
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]