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Parikh vectors
1BGI_1 1GLE_1 1HIC_1 Letter Amino acid
3 1 4 Q Glutamine
6 19 1 I Isoleucine
2 2 0 M Methionine
7 11 4 T Threonine
10 10 4 S Serine
6 19 4 V Valine
11 3 0 R Arginine
8 0 6 C Cysteine
12 16 8 G Glycine
1 2 1 H Histidine
2 7 2 P Proline
6 17 3 K Lycine
3 8 0 F Phenylalanine
6 0 0 W Tryptophan
12 7 0 A Alanine
14 4 4 N Asparagine
7 13 2 D Aspartic acid
2 17 4 E Glutamic acid
8 12 3 L Leucine
3 0 1 Y Tyrosine 

1BGI_1|Chain A|LYSOZYME|Gallus gallus (9031)
>1GLE_1|Chain A[auth F]|GLUCOSE-SPECIFIC PROTEIN IIIGlc|Escherichia coli (562)
>1HIC_1|Chain A|HIRUDIN VARIANT|Hirudo medicinalis (6421)
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 \(w=f(c)\)
1BGI , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
1GLE , Knot 79 168 0.80 34 112 160 
GLFDKLKSLVSDDKKDTGTIEIIAPLSGEIVNIEDVPDVVFAEKIVGDGIAIKPTGNKMVAPVDGTIGKIFETNHAFSIESDSGVELFVHFGIDTVELKGEGFKRIAEEGQRVKVGDTVIEFDLPLLEEKAKSTLTPVVISNMDEIKELIKLSGSVTVGETPVIRIKK
1HIC , Knot 30 51 0.77 30 45 49 
VVYTDCTESGQNLCLCEGSNVCGQGNKCILGSDGEKNQCVTGEGTPKPQSH

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 \(c\).

\(|P_{f(1BGI_1)}(2) \setminus P_{f(1GLE_1)}(2)|=71\), \(|P_{f(1GLE_1)}(2) \setminus P_{f(1BGI_1)}(2)|=79\). 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
1BGI_1,1GLE_1 150 3
1BGI_1,1HIC_1 119 3
1GLE_1,1HIC_1 121 3

Newick tree

 
[
	1GLE_1:70.78,
	[
		1BGI_1:59.5,1HIC_1:59.5
	]:11.28
]

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 \((0.85)(0.8)(\frac{297 }{\log_{20} 297}-\frac{129}{\log_{20}129})=52.1\)
Status Protein1 Protein2 d d1/2
Query variables 1BGI_1 1GLE_1 65 58
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]