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Parikh vectors
1YIO_1 3ZZF_1 3QVH_1 Letter Amino acid
7 22 7 K Lycine
7 8 13 P Proline
0 1 5 W Tryptophan
1 12 2 Y Tyrosine
15 21 32 A Alanine
15 12 22 R Arginine
24 33 29 L Leucine
14 24 29 G Glycine
9 8 13 F Phenylalanine
11 18 19 T Threonine
6 13 2 N Asparagine
2 2 3 C Cysteine
15 13 3 Q Glutamine
20 20 14 E Glutamic acid
12 25 9 I Isoleucine
10 21 22 S Serine
16 24 19 V Valine
9 14 22 D Aspartic acid
6 10 14 H Histidine
9 6 7 M Methionine 

1YIO_1|Chain A|response regulatory protein|Pseudomonas fluorescens (294)
>3ZZF_1|Chains A, B, C, D|ACETYLGLUTAMATE KINASE|SACCHAROMYCES CEREVISIAE (4932)
>3QVH_1|Chain A|FomA protein|Streptomyces wedmorensis (43759)
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 \(w=f(c)\)
1YIO , Knot 97 208 0.83 38 141 202 
MTAKPTVFVVDDDMSVREGLRNLLRSAGFEVETFDCASTFLEHRRPEQHGCLVLDMRMPGMSGIELQEQLTAISDGIPIVFITAHGDIPMTVRAMKAGAIEFLPKPFEEQALLDAIEQGLQLNAERRQARETQDQLEQLFSSLTGREQQVLQLTIRGLMNKQIAGELGIAEVTVKVHRHNIMQKLNVRSLANLVHLVEKYESFERGVS
3ZZF , Knot 131 307 0.81 40 179 292 
MGHHHHHHVSSTNGFSATRSTVIQLLNNISTKREVEQYLKYFTSVSQQQFAVIKVGGAIISDNLHELASCLAFLYHVGLYPIVLHGTGPQVNGRLEAQGIEPDYIDGIRITDEHTMAVVRKCFLEQNLKLVTALEQLGVRARPITSGVFTADYLDKDKYKLVGNIKSVTKEPIEASIKAGALPILTSLAETASGQMLNVNADVAAGELARVFEPLKIVYLNEKGGIINGSTGEKISMINLDEEYDDLMKQSWVKYGTKLKIREIKELLDYLPRSSSVAIINVQDLQKELFTDSGAGTMIRRGYKLVK
3QVH , Knot 121 286 0.79 40 164 269 
MGSSHHHHHHSSGLVPRGSHMTPDFLAIKVGGSLFSRKDEPGSLDDDAVTRFARNFARLAETYRGRMVLISGGGAFGHGAIRDHDSTHAFSLAGLTEATFEVKKRWAEKLRGIGVDAFPLQLAAMCTLRNGIPQLRSEVLRDVLDHGALPVLAGDALFDEHGKLWAFSSDRVPEVLLPMVEGRLRVVTLTDVDGIVTDGAGGDTILPEVDARSPEQAYAALWGSSEWDATGAMHTKLDALVTCARRGAECFIMRGDPGSDLEFLTAPFSSWPAHVRSTRITTTASA

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 \(c\).

\(|P_{f(1YIO_1)}(2) \setminus P_{f(3ZZF_1)}(2)|=56\), \(|P_{f(3ZZF_1)}(2) \setminus P_{f(1YIO_1)}(2)|=94\). 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 1:1010101111000101001100110011101001001001100001000101110101111011010001011001111111010101110101101111011101100011101100110101000010000001001100101000011010101110001110111101010100001100101001101101100000100110
Pair \(Z_2\) Length of longest common subsequence
1YIO_1,3ZZF_1 150 4
1YIO_1,3QVH_1 147 3
3ZZF_1,3QVH_1 159 6

Newick tree

 
[
	3ZZF_1:78.50,
	[
		1YIO_1:73.5,3QVH_1:73.5
	]:5.00
]

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 \((0.85)(0.8)(\frac{515 }{\log_{20} 515}-\frac{208}{\log_{20}208})=88.6\)
Status Protein1 Protein2 d d1/2
Query variables 1YIO_1 3ZZF_1 108 91
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]