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Parikh vectors
1PGB_1 7NSY_1 7FDX_1 Letter Amino acid
6 19 6 A Alanine
5 13 11 D Aspartic acid
0 2 1 C Cysteine
4 22 10 G Glycine
0 11 3 H Histidine
2 8 6 F Phenylalanine
11 9 19 T Threonine
3 8 2 Y Tyrosine
4 14 10 V Valine
0 20 1 P Proline
1 4 2 W Tryptophan
0 10 4 R Arginine
3 6 3 N Asparagine
1 10 3 Q Glutamine
3 12 14 L Leucine
1 7 3 M Methionine
5 6 8 E Glutamic acid
1 12 7 I Isoleucine
6 10 14 K Lycine
0 14 6 S Serine 

1PGB_1|Chain A|PROTEIN G|Streptococcus sp. GX7805 (1325)
>7NSY_1|Chains A[auth AAA], B[auth BBB]|Isoform A of Peptidoglycan-recognition protein LB|Drosophila melanogaster (7227)
>7FDX_1|Chain A|Fatty acid-binding protein, heart|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 \(w=f(c)\)
1PGB , Knot 31 56 0.74 30 46 54 
MTYKLILNGKTLKGETTTEAVDAATAEKVFKQYANDNGVDGEWTYDDATKTFTVTE
7NSY , Knot 102 217 0.84 40 155 209 
GPMQQANLGDGVATARLLSRSDWGARLPKSVEHFQGPAPYVIIHHSYMPAVCYSTPDCMKSMRDMQDFHQLERGWNDIGYSFGIGGDGMIYTGRGFNVIGAHAPKYNDKSVGIVLIGDWRTELPPKQMLDAAKNLIAFGVFKGYIDPAYKLLGHRQVRDTESPGGRLFAEISSWPHFTHINDTEGVSSTTAPVVPHVHPQAAAPQKPHQSPPAAPKV
7FDX , Knot 66 133 0.81 40 99 125 
MVDAFLGTWKLVDSKNFDDYMKSLGVGFATRQVASMTKPTTIIEKNGDILTLKTHSTFKNTEISFKLGVEFDETTADDRKVKSIVTLDGGKLVHLQKWDGQETTLVRELIDGKLILTLTHGTAVCTRTYEKEA

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 \(c\).

\(|P_{f(1PGB_1)}(2) \setminus P_{f(7NSY_1)}(2)|=25\), \(|P_{f(7NSY_1)}(2) \setminus P_{f(1PGB_1)}(2)|=134\). 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 1:10001110100101000001101101001100010001101010000100010100
Pair \(Z_2\) Length of longest common subsequence
1PGB_1,7NSY_1 159 3
1PGB_1,7FDX_1 89 4
7NSY_1,7FDX_1 162 3

Newick tree

 
[
	7NSY_1:89.03,
	[
		1PGB_1:44.5,7FDX_1:44.5
	]:44.53
]

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 \((0.85)(0.8)(\frac{273 }{\log_{20} 273}-\frac{56}{\log_{20}56})=70.8\)
Status Protein1 Protein2 d d1/2
Query variables 1PGB_1 7NSY_1 92 56.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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