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Parikh vectors
2DQA_1 1QNM_1 1DKH_1 Letter Amino acid
4 2 2 M Methionine
4 9 15 V Valine
3 17 18 A Alanine
14 19 27 G Glycine
13 15 0 K Lycine
3 6 1 W Tryptophan
5 15 11 N Asparagine
3 6 9 F Phenylalanine
7 10 4 P Proline
8 20 18 L Leucine
11 7 23 S Serine
6 9 10 Y Tyrosine
5 4 0 R Arginine
4 8 14 D Aspartic acid
14 2 0 C Cysteine
5 11 4 I Isoleucine
2 7 13 T Threonine
6 9 10 Q Glutamine
4 12 3 E Glutamic acid
3 10 6 H Histidine 

2DQA_1|Chains A, B|Lysozyme|Tapes japonica (129788)
>1QNM_1|Chains A, B|MANGANESE SUPEROXIDE DISMUTASE|Homo sapiens (9606)
>1DKH_1|Chain A|HEME-BINDING PROTEIN A|Serratia marcescens (615)
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 \(w=f(c)\)
2DQA , Knot 66 124 0.85 40 102 121 
SFAPGMVSQKCLLCMCKLESGGCKPIGCRMDVGSLSCGYFQIKQPYWIDCGKPGKDWKSCSNDINCSSKCVQQYMKRYATHYRCPLNCEGFAREHNGGPNGCHSSRTLKYWELLQKIPGCKGVK
1QNM , Knot 93 198 0.82 40 136 189 
KHSLPDLPYDYGALEPHINAQIMQLHHSKHHAAYVNNLNVTEEKYQEALAKGDVTAQIALQPALKFNGGGHINHSIFWTNLSPNGGGEPKGELLEAIKRDFGSFDKFKEKLTAASVGVQGSGWGWLGFNKERGHLQIAACPNNDPLQGTTGLIPLLGIDVWEHAYYLQYKNVRPDYLKAIWNVINWENVTERYMACKK
1DKH , Knot 84 188 0.78 34 116 173 
MAFSVNYDSSFGGYSIHDYLGQWASTFGDVNHTNGNVTDANSGGFYGGSLSGSQYAISSTANQVTAFVAGGNLTYTLFNEPAHTLYGQLDSLSFGDGLSGGDTSPYSIQVPDVSFGGLNLSSLQAQGHDGVVHQVVYGLMSGDTGALETALNGILDDYGLSVNSTFDQVAAATAVGVQHADSPELLAA

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 \(c\).

\(|P_{f(2DQA_1)}(2) \setminus P_{f(1QNM_1)}(2)|=61\), \(|P_{f(1QNM_1)}(2) \setminus P_{f(2DQA_1)}(2)|=95\). 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 1:0111111000011010010011001110010110100101010010110010110010000001000000100010001000001100011100001110100000010010110011100110
Pair \(Z_2\) Length of longest common subsequence
2DQA_1,1QNM_1 156 3
2DQA_1,1DKH_1 166 4
1QNM_1,1DKH_1 138 4

Newick tree

 
[
	2DQA_1:84.03,
	[
		1QNM_1:69,1DKH_1:69
	]:15.03
]

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 \((0.85)(0.8)(\frac{322 }{\log_{20} 322}-\frac{124}{\log_{20}124})=61.1\)
Status Protein1 Protein2 d d1/2
Query variables 2DQA_1 1QNM_1 77 63
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]