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Parikh vectors
1LZE_1 7ORB_1 3MBR_1 Letter Amino acid
6 10 7 K Lycine
3 14 7 F Phenylalanine
12 12 21 A Alanine
7 9 19 D Aspartic acid
12 16 20 G Glycine
1 7 6 H Histidine
2 0 2 M Methionine
11 10 21 R Arginine
14 17 7 N Asparagine
6 7 10 I Isoleucine
8 12 27 L Leucine
10 15 10 S Serine
4 15 9 Y Tyrosine
6 16 18 V Valine
8 8 0 C Cysteine
3 6 6 Q Glutamine
2 7 12 E Glutamic acid
2 10 15 P Proline
7 12 19 T Threonine
5 2 7 W Tryptophan 

1LZE_1|Chain A|HEN EGG WHITE LYSOZYME|Gallus gallus (9031)
>7ORB_1|Chains A[auth R], F[auth X]|Spike protein S1|Severe acute respiratory syndrome coronavirus 2 (2697049)
>3MBR_1|Chain A[auth X]|Glutamine cyclotransferase|Xanthomonas campestris (340)
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 \(w=f(c)\)
1LZE , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRYWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
7ORB , Knot 94 205 0.81 38 142 195 
ETGHHHHHHTNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYRYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGKK
3MBR , Knot 107 243 0.80 38 159 231 
RDPVPTQGYRVVKRYPHDTTAFTEGLFYLRGHLYESTGETGRSSVRKVDLETGRILQRAEVPPPYFGAGIVAWRDRLIQLTWRNHEGFVYDLATLTPRARFRYPGEGWALTSDDSHLYMSDGTAVIRKLDPDTLQQVGSIKVTAGGRPLDNLNELEWVNGELLANVWLTSRIARIDPASGKVVAWIDLQALVPDADALTDSTNDVLNGIAFDAEHDRLFVTGKRWPMLYEIRLTPLPHAAAGK

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 \(c\).

\(|P_{f(1LZE_1)}(2) \setminus P_{f(7ORB_1)}(2)|=52\), \(|P_{f(7ORB_1)}(2) \setminus P_{f(1LZE_1)}(2)|=90\). 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 1:011100011111000110000100110110110100010001000000100001110100001000100110001001100111000101010010011001011011111000001001011101001
Pair \(Z_2\) Length of longest common subsequence
1LZE_1,7ORB_1 142 3
1LZE_1,3MBR_1 161 4
7ORB_1,3MBR_1 159 4

Newick tree

 
[
	3MBR_1:82.78,
	[
		1LZE_1:71,7ORB_1:71
	]:11.78
]

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 \((0.85)(0.8)(\frac{334 }{\log_{20} 334}-\frac{129}{\log_{20}129})=63.0\)
Status Protein1 Protein2 d d1/2
Query variables 1LZE_1 7ORB_1 75 61
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]