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Parikh vectors
1VDS_1 2LBZ_1 1ZYQ_1 Letter Amino acid
3 0 0 F Phenylalanine
2 0 0 P Proline
10 4 0 S Serine
1 0 0 H Histidine
8 4 0 L Leucine
2 0 0 M Methionine
12 1 7 G Glycine
3 0 0 Y Tyrosine
14 1 0 N Asparagine
7 1 0 D Aspartic acid
3 0 0 Q Glutamine
6 2 0 W Tryptophan
6 3 0 V Valine
11 0 0 R Arginine
6 0 0 K Lycine
7 4 1 T Threonine
6 0 0 I Isoleucine
12 6 7 A Alanine
8 4 7 C Cysteine
2 1 0 E Glutamic acid 

1VDS_1|Chain A|Lysozyme C|Gallus gallus (9031)
>2LBZ_1|Chain A|Thuricin17|Bacillus thuringiensis (1428)
>1ZYQ_1|Chain A[auth P]|5'-D(*CP*GP*AP*AP*AP*AP*CP*GP*AP*CP*GP*GP*CP*CP*AP*GP*TP*GP*CP*CP*AP*(DDG))-3'|
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 \(w=f(c)\)
1VDS , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
2LBZ , Knot 18 31 0.66 22 26 29 
DWTCWSCLVCAACSVELLNLVTAATGASTAS
1ZYQ , Knot 10 22 0.46 8 11 14 
CGAAAACGACGGCCAGTGCCAG

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 \(c\).

\(|P_{f(1VDS_1)}(2) \setminus P_{f(2LBZ_1)}(2)|=92\), \(|P_{f(2LBZ_1)}(2) \setminus P_{f(1VDS_1)}(2)|=14\). 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 1:011100011111000110000100110110110100010001000000100001110100011000100110001001100111000101010010011001011011111000001001011101001
Pair \(Z_2\) Length of longest common subsequence
1VDS_1,2LBZ_1 106 4
1VDS_1,1ZYQ_1 107 3
2LBZ_1,1ZYQ_1 27 3

Newick tree

 
[
	1VDS_1:60.99,
	[
		2LBZ_1:13.5,1ZYQ_1:13.5
	]:47.49
]

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 \((0.85)(0.8)(\frac{160 }{\log_{20} 160}-\frac{31}{\log_{20}31})=45.8\)
Status Protein1 Protein2 d d1/2
Query variables 1VDS_1 2LBZ_1 58 35
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]