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Parikh vectors
5MLA_1 6BGE_1 5TKP_1 Letter Amino acid
11 18 34 G Glycine
5 10 7 H Histidine
11 31 17 I Isoleucine
8 9 12 Y Tyrosine
12 28 28 E Glutamic acid
4 6 8 M Methionine
4 8 19 P Proline
0 2 2 W Tryptophan
9 19 43 A Alanine
4 21 12 N Asparagine
15 15 14 D Aspartic acid
3 8 5 C Cysteine
16 14 21 V Valine
10 15 16 R Arginine
9 6 10 Q Glutamine
11 26 38 L Leucine
10 25 27 K Lycine
6 19 11 F Phenylalanine
9 27 23 S Serine
12 16 16 T Threonine 

5MLA_1|Chain A|GTPase KRas|Homo sapiens (9606)
>6BGE_1|Chains A, B|VirB11-like protein|Helicobacter pylori (210)
>5TKP_1|Chain A|Fructose-bisphosphate aldolase|Toxoplasma gondii (5811)
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 \(w=f(c)\)
5MLA , Knot 82 169 0.83 38 133 167 
GSHMTEYKLVVVGAVGVGKSALTIQLIQNHFVDEYDPTIEDSYRKQVVIDGETCLLDILDTAGQEEYSAMRDQYMRTGEGFLCVFAINNTKSFEDIHHYREQIKRVKDSEDVPMVLVGNKCDLPSRTVDTKQAQDLARSYGIPFIETSAKTRQGVDDAFYTLVREIRKH
6BGE , Knot 144 323 0.85 40 197 307 
LSAEDKKFLEVERALKEAALNPLRHATEELFGDFLKMENITEICYNGNKVVWVLKNNGEWQPFDVRDRKAFSLSRLMHFARCCASFKKKTIDNYENPILSSNLANGERVQIVLSPVTVNDETISISIRIPSKTTYPHSFFEEQGFYNLLDNKEQAISAIKDGIAIGKNVIVCGGTGSGKTTYIKSIMEFIPKEERIISIEDTEEIVFKHHKNYTQLFFGGNITSADCLKSCLRMRPDRIILGELRSSEAYDFYNVLCSGHKGTLTTLHAGSSEEAFIRLANMSSSNSAARNIKFESLIEGFKDLIDMIVHINHHKQCDEFYIK
5TKP , Knot 153 363 0.82 40 203 345 
MSGYGLPISQEVAKELAENARKIAAPGKGILAADESTGTIKKRFDSIGVENTEANRAFYRDLLFSTKGLGQYISGAILFEETLYQKSPSGVPMVDLLKAEGIIPGIKVDKGLETLPLTDDEKATMGLDGLSERCKKYYEAGARFAKWRAVLSIDPAKGKPTNLSITEVAHGLARYAAICQANRLVPIVEPEILTDGSHDITVCAEVTERVLAAVFKALNDHHVLLEGALLKPNMVTHGSDCPKPASHEEIAFYTVRSLKRTVPPALPGVMFLSGGQSEEDASLNLNEMNKMGPHPFQLSFSYGRALQASCLKAWKGVPENKAKAQQVLMERARANGEAQLGKYGGGAGGALAASSLFEKRYVY

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 \(c\).

\(|P_{f(5MLA_1)}(2) \setminus P_{f(6BGE_1)}(2)|=53\), \(|P_{f(6BGE_1)}(2) \setminus P_{f(5MLA_1)}(2)|=117\). 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 1:1001000011111111110011010110001100001010000000111010001101100110000011000010010111011110000010010000001001000001111111000011000100001001100011111000100001100110011001000
Pair \(Z_2\) Length of longest common subsequence
5MLA_1,6BGE_1 170 3
5MLA_1,5TKP_1 174 4
6BGE_1,5TKP_1 168 4

Newick tree

 
[
	5MLA_1:86.66,
	[
		6BGE_1:84,5TKP_1:84
	]:2.66
]

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 \((0.85)(0.8)(\frac{492 }{\log_{20} 492}-\frac{169}{\log_{20}169})=94.5\)
Status Protein1 Protein2 d d1/2
Query variables 5MLA_1 6BGE_1 124 93
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]