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Parikh vectors
1ZIR_1 1YDG_1 1NLO_1 Letter Amino acid
11 6 2 F Phenylalanine
7 9 2 P Proline
7 13 4 V Valine
13 8 3 D Aspartic acid
7 7 1 M Methionine
6 7 2 I Isoleucine
9 16 5 L Leucine
4 18 8 T Threonine
4 4 2 W Tryptophan
5 7 3 N Asparagine
9 8 2 H Histidine
14 14 6 S Serine
16 5 4 Y Tyrosine
21 9 2 R Arginine
9 13 4 E Glutamic acid
9 10 2 Q Glutamine
13 20 7 G Glycine
1 8 2 K Lycine
2 29 3 A Alanine
6 0 0 C Cysteine 

1ZIR_1|Chain A|Gamma crystallin E|Rattus norvegicus (10116)
>1YDG_1|Chains A, B, C, D, E, F, G, H|trp repressor binding protein WrbA|Deinococcus radiodurans (1299)
>1NLO_1|Chain A[auth C]|C-SRC|Gallus gallus (9031)
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 \(w=f(c)\)
1ZIR , Knot 84 173 0.83 40 126 166 
GKITFYEDRGFQGRHYECSTDHSNLQPYFSRCNSVRVDSGCWMLYEQPNFTGCQYFLRRGDYPDYQQWMGFSDSVRSCRLIPHSSSHRIRIYEREDYRGQMVEITDDCPHLQDRFHFSDFHSFHVMEGYWVLYEMPNYRGRQYLLRPGEYRRYHDWGAMNARVGSLRRIMDFY
1YDG , Knot 94 211 0.79 38 139 199 
MSLTAPVKLAIVFYSSTGTGYAMAQEAAEAGRAAGAEVRLLKVRETAPQDVIDGQDAWKANIEAMKDVPEATPADLEWAEAIVFSSPTRFGGATSQMRAFIDTLGGLWSSGKLANKTFSAMTSAQNVNGGQETTLQTLYMTAMHWGAVLTPPGYTDEVIFKSGGNPYGASVTANGQPLLENDRASIRHQVRRQVELTAKLLEGGSHHHHHH
1NLO , Knot 37 64 0.80 38 58 62 
GSHMGGVTTFVALYDYESRTETDLSFKKGERLQIVNNTEGDWWLAHSLTTGQTGYIPSNYVAPS

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 \(c\).

\(|P_{f(1ZIR_1)}(2) \setminus P_{f(1YDG_1)}(2)|=87\), \(|P_{f(1YDG_1)}(2) \setminus P_{f(1ZIR_1)}(2)|=100\). 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 1:10101000011010000000000010101000001010010111000101010001100100100001111000100001110000001010000000101101000010100010100100101101011100110001000110110000000111101011010011010
Pair \(Z_2\) Length of longest common subsequence
1ZIR_1,1YDG_1 187 3
1ZIR_1,1NLO_1 134 2
1YDG_1,1NLO_1 149 3

Newick tree

 
[
	1YDG_1:89.62,
	[
		1ZIR_1:67,1NLO_1:67
	]:22.62
]

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 \((0.85)(0.8)(\frac{384 }{\log_{20} 384}-\frac{173}{\log_{20}173})=63.0\)
Status Protein1 Protein2 d d1/2
Query variables 1ZIR_1 1YDG_1 81 74
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]