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Parikh vectors
7AKI_1 3NWH_1 5XGT_1 Letter Amino acid
6 7 11 R Arginine
7 5 9 N Asparagine
5 6 5 D Aspartic acid
9 5 7 G Glycine
5 3 1 I Isoleucine
4 3 1 M Methionine
5 2 6 F Phenylalanine
12 14 6 A Alanine
4 0 3 Y Tyrosine
6 3 6 H Histidine
4 11 8 L Leucine
6 8 2 K Lycine
5 7 8 T Threonine
11 8 15 V Valine
8 9 5 Q Glutamine
3 13 7 E Glutamic acid
2 1 3 P Proline
7 4 6 S Serine
2 0 0 W Tryptophan
2 3 2 C Cysteine 

7AKI_1|Chain A|Protein enabled homolog|Homo sapiens (9606)
>3NWH_1|Chains A, B, C, D|Bone marrow stromal antigen 2|Homo sapiens (9606)
>5XGT_1|Chains A, B|Single-stranded DNA-binding protein|Staphylococcus aureus subsp. aureus (1280)
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 \(w=f(c)\)
7AKI , Knot 59 113 0.82 40 97 110 
GSMSEQSICQARAAVMVYDDANKKWVPAGGSTGFSRVHIYHHTGNNTFRVVGRKIQDHQVVINCAIPKGLKYNQATQTFHQWRDARQVYGLNFGSKEDANVFASAMMHALEVL
3NWH , Knot 60 112 0.84 36 91 109 
GIDPFTKANSEACRDGLRAVMECRNVTHLLQQELTEAQKGFQDVEAQAATCNHTVMALMASLDAEKAQGQKKVEELEGEITTLNHKLQDASAEVERLRRENQVLSVRIADKK
5XGT , Knot 56 111 0.79 38 88 104 
MLNRVVLVGRLTKDPEYRTTPSGVSVATFTLAVNRTFTNAQGEREADFINCVVFRRQADNVNNYLSKGSLAGVDGRLQSRNYENQEGRRVFVTEVVCDSVQFLEPHHHHHH

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 \(c\).

\(|P_{f(7AKI_1)}(2) \setminus P_{f(3NWH_1)}(2)|=71\), \(|P_{f(3NWH_1)}(2) \setminus P_{f(7AKI_1)}(2)|=65\). 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 1:10100001001011111000100011111100110010100001000101110010000111001110110000100010010010010110110000101110111011011
Pair \(Z_2\) Length of longest common subsequence
7AKI_1,3NWH_1 136 3
7AKI_1,5XGT_1 131 3
3NWH_1,5XGT_1 107 3

Newick tree

 
[
	7AKI_1:70.63,
	[
		5XGT_1:53.5,3NWH_1:53.5
	]:17.13
]

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 \((0.85)(0.8)(\frac{225 }{\log_{20} 225}-\frac{112}{\log_{20}112})=36.2\)
Status Protein1 Protein2 d d1/2
Query variables 7AKI_1 3NWH_1 47 46.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]