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Parikh vectors
5CMO_1 1BUX_1 7ZBP_1 Letter Amino acid
3 7 16 N Asparagine
3 1 3 Q Glutamine
1 3 6 H Histidine
11 13 21 L Leucine
10 11 9 K Lycine
5 9 17 P Proline
4 6 16 T Threonine
8 18 12 V Valine
7 10 19 S Serine
3 3 2 Y Tyrosine
7 8 13 R Arginine
4 5 15 D Aspartic acid
10 13 14 E Glutamic acid
13 16 18 G Glycine
10 8 18 I Isoleucine
2 3 4 M Methionine
14 12 21 A Alanine
0 0 2 C Cysteine
9 8 10 F Phenylalanine
1 1 3 W Tryptophan 

5CMO_1|Chains A, B, C|Holo-[acyl-carrier-protein] synthase|Neisseria meningitidis serogroup C / serotype 2a (strain ATCC 700532 / DSM 15464 / FAM18) (272831)
>1BUX_1|Chains A, B, C|NUCLEOSIDE DIPHOSPHATE KINASE|Dictyostelium discoideum (44689)
>7ZBP_1|Chains A, B, C, D|Marasmius rotula UPO|Marasmius rotula (182057)
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 \(w=f(c)\)
5CMO , Knot 63 125 0.81 38 92 119 
MIYGIGTDIVSLKRIIRLNKKFGQAFAGRILTPEELLEFPQAGKPVNYLAKRFAAKEAFAKAVGTGIRGAVSFRNIGIGHDALGKPEFFYGPALSKWLEEQGISRVSLSMSDEEDTVLAFVVAEK
1BUX , Knot 75 155 0.81 38 115 147 
MSTNKVNKERTFLAVKPDGVARGLVGEIIARYEKKGFVLVGLKQLVPTKDLAESHYAEHKERPFFGGLVSFITSGPVVAMVFEGKGVVASARLMIGVTNPLASAPGSIRGDFGVDVGRNIIHGSDSVESANREIALWFKPEELLTEVKPNPNLYE
7ZBP , Knot 110 239 0.84 40 159 232 
AVDFSAHPWKAPGPNDSRGPCPGLNTLANHGFLPRNGRNISVPMIVKAGFEGYNVQSDILILAGKIGMLTSREADTISLEDLKLHGTIEHDASLSREDVAIGDNLHFNEAIFTTLANSNPGADVYNISSAAQVQHDRLADSLARNPNVTNTDLTATIRSSESAFFLTVMSAGDPLRGEAPKKFVNVFFREERMPIKEGWKRSTTPITIPLLGPIIERITELSDWKPTGDNCGAIVLSPE

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 \(c\).

\(|P_{f(5CMO_1)}(2) \setminus P_{f(1BUX_1)}(2)|=53\), \(|P_{f(1BUX_1)}(2) \setminus P_{f(5CMO_1)}(2)|=76\). 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 1:11011100110100110100011011110110100110110110110011001110011101110110111010011110011101011011110011000110010101000000111111100
Pair \(Z_2\) Length of longest common subsequence
5CMO_1,1BUX_1 129 5
5CMO_1,7ZBP_1 135 3
1BUX_1,7ZBP_1 140 4

Newick tree

 
[
	7ZBP_1:70.12,
	[
		5CMO_1:64.5,1BUX_1:64.5
	]:5.62
]

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 \((0.85)(0.8)(\frac{280 }{\log_{20} 280}-\frac{125}{\log_{20}125})=48.4\)
Status Protein1 Protein2 d d1/2
Query variables 5CMO_1 1BUX_1 59 53.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]