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Parikh vectors
3QDR_1 1UHI_1 1IEN_1 Letter Amino acid
2 3 4 C Cysteine
6 5 1 H Histidine
8 12 1 L Leucine
3 8 2 F Phenylalanine
10 6 1 P Proline
0 6 1 W Tryptophan
6 19 0 D Aspartic acid
3 5 0 Q Glutamine
6 12 1 I Isoleucine
14 15 2 K Lycine
1 6 3 R Arginine
14 15 0 G Glycine
10 7 0 S Serine
20 17 1 A Alanine
9 9 2 N Asparagine
3 15 0 E Glutamic acid
2 5 0 M Methionine
4 10 0 T Threonine
4 7 0 Y Tyrosine
2 9 0 V Valine 

3QDR_1|Chain A|Protein tolA|Escherichia coli (83333)
>1UHI_1|Chains A, B|Aequorin 2|Aequorea victoria (6100)
>1IEN_1|Chain A|PROTEIN TIA|null
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 \(w=f(c)\)
3QDR , Knot 55 127 0.70 38 81 118 
HHHHHHMSSGKNAPKTGGGAKGNNASPAGSGNTKNNGASGADINNYAGQIKSAIESKFYDASSYAGKTCTLRIKLAPDGMLLDIKPEGGDPALCQAALAAAKLAKIPKPPSQAVYEVFKNAPLDFKP
1UHI , Knot 94 191 0.86 40 145 183 
ANSKLTSDFDNPRWIGRHKHMFNFLDVNHNGKISLDEMVYKASDIVINNLGATPEQAKRHKDAVEAFFGGAGMKYGVETDWPAYIEGWKKLATDELEKYAKNEPTLIRIWGDALFDIVDKDQNGAITLDEWKAYTKAAGIIQSSEDCEETFRVCDIDESGQLDVDEMTRQHLGFWYTMDPACEKLYGGAVP
1IEN , Knot 15 20 0.75 24 19 18 
FNWRCCLIPACRRNHKKFCX

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 \(c\).

\(|P_{f(3QDR_1)}(2) \setminus P_{f(1UHI_1)}(2)|=41\), \(|P_{f(1UHI_1)}(2) \setminus P_{f(3QDR_1)}(2)|=105\). 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 1:0000001001001100111101001011101000001101101000110100110001001000110000101011101111010101101110011111101101101100110011001110101
Pair \(Z_2\) Length of longest common subsequence
3QDR_1,1UHI_1 146 3
3QDR_1,1IEN_1 94 2
1UHI_1,1IEN_1 150 3

Newick tree

 
[
	1UHI_1:81.03,
	[
		3QDR_1:47,1IEN_1:47
	]:34.03
]

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 \((0.85)(0.8)(\frac{318 }{\log_{20} 318}-\frac{127}{\log_{20}127})=59.0\)
Status Protein1 Protein2 d d1/2
Query variables 3QDR_1 1UHI_1 75 60
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]