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Parikh vectors
2IYS_1 9PFG_1 1AQO_1 Letter Amino acid
3 0 0 Y Tyrosine
3 3 0 N Asparagine
3 6 0 Q Glutamine
19 4 9 G Glycine
9 0 0 H Histidine
5 5 0 K Lycine
9 4 0 S Serine
25 6 5 A Alanine
9 9 0 D Aspartic acid
10 15 0 E Glutamic acid
7 3 0 I Isoleucine
2 0 0 F Phenylalanine
11 3 0 T Threonine
21 7 0 R Arginine
0 0 8 C Cysteine
18 10 0 L Leucine
3 7 0 M Methionine
9 0 0 P Proline
18 2 0 V Valine
0 0 0 W Tryptophan 

2IYS_1|Chain A|SHIKIMATE KINASE|MYCOBACTERIUM TUBERCULOSIS (83332)
>9PFG_1|Chains A, C|Synaptosomal-associated protein 25|Rattus norvegicus (10116)
>1AQO_1|Chain A|IRON RESPONSIVE ELEMENT RNA HAIRPIN|
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 \(w=f(c)\)
2IYS , Knot 80 184 0.75 36 113 169 
MAPKAVLVGLPGSGKSTIGRRLAKALGVGLLDTDVAIEQRTGRSIADIFATDGEQEFRRIEEDVVRAALADHDGVLSLGGGAVTSPGVRAALAGHTVVYLEISAAEGVRRTGGNTVRPLLAGPDRAEKYRALMAKRAPLYRRVATMRVDTNRRNPGAVVRHILSRLQVPSPSEAATLEHHHHHH
9PFG , Knot 42 84 0.73 28 63 82 
SMAEDADMRNELEEMQRRADQLADESLESTRRMLQLVEESKDAGIRTLVMLDEQGEQLERIEEGMDQINKDMKEAEKNLTDLGK
1AQO , Knot 11 29 0.42 8 14 20 
GGAGUGCUUCAACAGUGCUUGGACGCUCC

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 \(c\).

\(|P_{f(2IYS_1)}(2) \setminus P_{f(9PFG_1)}(2)|=83\), \(|P_{f(9PFG_1)}(2) \setminus P_{f(2IYS_1)}(2)|=33\). 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 1:1110111111110100011001101111111000111000010011011100100010010001101111000111011111100111011111001101010110110001100101111110010000111100111000110101000000111110011001011010011010000000
Pair \(Z_2\) Length of longest common subsequence
2IYS_1,9PFG_1 116 4
2IYS_1,1AQO_1 119 3
9PFG_1,1AQO_1 75 2

Newick tree

 
[
	2IYS_1:64.29,
	[
		9PFG_1:37.5,1AQO_1:37.5
	]:26.79
]

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 \((0.85)(0.8)(\frac{268 }{\log_{20} 268}-\frac{84}{\log_{20}84})=59.0\)
Status Protein1 Protein2 d d1/2
Query variables 2IYS_1 9PFG_1 71 51.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]