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Parikh vectors
3NMM_1 5ERW_1 7LEB_1 Letter Amino acid
2 4 2 M Methionine
11 29 0 S Serine
21 15 3 A Alanine
3 6 4 R Arginine
1 10 5 Q Glutamine
9 2 1 H Histidine
0 6 13 I Isoleucine
18 15 12 L Leucine
2 5 2 W Tryptophan
4 4 4 N Asparagine
8 6 4 D Aspartic acid
11 8 7 K Lycine
7 5 2 F Phenylalanine
7 12 6 P Proline
13 22 6 V Valine
1 5 2 C Cysteine
7 20 13 G Glycine
9 25 8 T Threonine
4 7 4 E Glutamic acid
3 11 1 Y Tyrosine 

3NMM_1|Chains A, C|Hemoglobin subunit alpha|Homo sapiens (9606)
>5ERW_1|Chain A|Anti-HCV E2 Fab HC84-26 heavy chain|Homo sapiens (9606)
>7LEB_1|Chains A, B|Protease|Human immunodeficiency virus 1 (11676)
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 \(w=f(c)\)
3NMM , Knot 69 141 0.80 38 99 136 
VLSPADKTNVKAAWGKVGAHAGEYGAEALERMFLSFPTTKTYFPHFDLSHGSAQVKGWGKKVADALTNAVAHVDDMPNALSALSDLHAHKLRVDPVNFKLLSHCLLVTLAAHLPAEFTPAVHASLDKFLASVSTVLTSKYR
5ERW , Knot 97 217 0.80 40 142 204 
MQVQLVQSGAEVKKPGSSVKVSCEASGGTLSNYVITWVRQAPGQGLEWMGGFIPTFRTAMYAQGFQGRVTITADESTSIAYMELTNLRSEDTAVYYCARGPLSRGYYDYWGQGTLVTVSSAKTTAPSVYPLAPVCGDTTGSSVTLGCLVKGYFPEPVTLTWNSGSLSSGVHTFPAVLQSDLYTLSSSVTVTSSTWPSQSITCNVAHPASSTKVDKKI
7LEB , Knot 52 99 0.80 38 79 94 
PQITLWKRPLVTIKIGGQLKEALLDTGADDTVLEEMNLPGRWKPKMIGGIGGFIKVRQYDQILIEICGHKAIGTVLVGPTPVNIIGRNLLTQIGCTLNF

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 \(c\).

\(|P_{f(3NMM_1)}(2) \setminus P_{f(5ERW_1)}(2)|=50\), \(|P_{f(5ERW_1)}(2) \setminus P_{f(3NMM_1)}(2)|=93\). 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 1:110110000101111011101100110110011101100000110101001010101110011011001110100110110110010100101011010110001110111011101011101010011101001100000
Pair \(Z_2\) Length of longest common subsequence
3NMM_1,5ERW_1 143 3
3NMM_1,7LEB_1 130 3
5ERW_1,7LEB_1 151 4

Newick tree

 
[
	5ERW_1:76.15,
	[
		3NMM_1:65,7LEB_1:65
	]:11.15
]

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 \((0.85)(0.8)(\frac{358 }{\log_{20} 358}-\frac{141}{\log_{20}141})=65.9\)
Status Protein1 Protein2 d d1/2
Query variables 3NMM_1 5ERW_1 82 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]