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Parikh vectors
7YNG_1 3OGK_1 8BZM_1 Letter Amino acid
4 12 0 L Leucine
2 6 0 F Phenylalanine
10 10 9 T Threonine
6 13 0 D Aspartic acid
2 10 0 I Isoleucine
4 8 0 S Serine
3 8 0 N Asparagine
0 4 0 R Arginine
0 3 4 C Cysteine
18 4 2 G Glycine
15 13 0 K Lycine
4 4 0 M Methionine
5 4 0 P Proline
19 12 0 V Valine
19 19 2 A Alanine
18 20 0 E Glutamic acid
1 2 0 H Histidine
0 2 0 W Tryptophan
4 2 0 Y Tyrosine
6 4 0 Q Glutamine 

7YNG_1|Chains A, B, C, D, E, F|Alpha-synuclein|Homo sapiens (9606)
>3OGK_1|Chains A, D[auth C], G[auth E], J[auth G], L[auth I], O[auth K], R[auth M], U[auth O]|SKP1-like protein 1A|Arabidopsis thaliana (3702)
>8BZM_1|Chains A[auth D], F[auth G]|DNA|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 \(w=f(c)\)
7YNG , Knot 59 140 0.69 34 86 121 
MDVFMKGLSKAKEGVVAAAEKTKQGVAEAAGKTKEGVLYVGSKTKEGVVHGVATVAEKTKEQVTNVGGAVVTGVTAVAQKTVEGAGSIAAATGFVKKDQLGKNEEGAPQEGILEDMPVDPDNEAYEMPSEEGYQDYEPEA
3OGK , Knot 78 160 0.82 40 117 150 
MSAKKIVLKSSDGESFEVEEAVALESQTIAHMVEDDCVDNGVPLPNVTSKILAKVIEYCKRHVEAAASKAEAVEGAATSDDDLKAWDADFMKIDQATLFELILAANYLNIKNLLDLTCQTVADMIKGKTPEEIRTTFNIKNDFTPEEEEEVRRENQWAFE
8BZM , Knot 8 17 0.44 8 9 13 
TTACTTCCTGTTTACGT

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 \(c\).

\(|P_{f(7YNG_1)}(2) \setminus P_{f(3OGK_1)}(2)|=52\), \(|P_{f(3OGK_1)}(2) \setminus P_{f(7YNG_1)}(2)|=83\). 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 1:10111011001001111110000011101110000111011000001110111011000000100111111011011100010111011110111000011000011100111001110100010011000100000101
Pair \(Z_2\) Length of longest common subsequence
7YNG_1,3OGK_1 135 4
7YNG_1,8BZM_1 91 2
3OGK_1,8BZM_1 122 2

Newick tree

 
[
	3OGK_1:69.48,
	[
		7YNG_1:45.5,8BZM_1:45.5
	]:23.98
]

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 \((0.85)(0.8)(\frac{300 }{\log_{20} 300}-\frac{140}{\log_{20}140})=49.4\)
Status Protein1 Protein2 d d1/2
Query variables 7YNG_1 3OGK_1 63 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]