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Parikh vectors
1FUT_1 9LGL_1 1UTB_1 Letter Amino acid
3 24 17 E Glutamic acid
4 6 2 C Cysteine
5 19 10 Q Glutamine
14 31 18 G Glycine
2 9 16 H Histidine
0 41 44 L Leucine
10 20 16 S Serine
0 4 2 W Tryptophan
9 8 5 Y Tyrosine
5 9 16 D Aspartic acid
10 2 14 N Asparagine
3 3 12 I Isoleucine
1 9 10 K Lycine
5 19 20 P Proline
11 8 19 T Threonine
2 21 22 R Arginine
0 5 12 M Methionine
4 10 17 F Phenylalanine
6 17 18 V Valine
12 22 25 A Alanine 

1FUT_1|Chain A|RIBONUCLEASE F1|Gibberella fujikuroi (5127)
>9LGL_1|Chains A, B|Membrane-associated tyrosine- and threonine-specific cdc2-inhibitory kinase|Homo sapiens (9606)
>1UTB_1|Chain A|LYSR-TYPE REGULATORY PROTEIN|BURKHOLDERIA SP. (233098)
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 \(w=f(c)\)
1FUT , Knot 54 106 0.79 34 83 104 
QSATTCGSTNYSASQVRAAANAACQYYQNDDTAGSSTYPHTYNNYEGFDFPVDGPYQEFPIKSGGVYTGGSPGADRVVINTNCEYAGAITHTGASGNNFVGCSGTN
9LGL , Knot 127 287 0.83 40 168 275 
HQLQPRRVSFRGEASETLQSPGYDPSRPESFFQQSFQRLSRLGHGSYGEVFKVRSKEDGRLYAVKRSMSPFRGPKDRARKLAEVGSHEKVGQHPCCVRLEQAWEEGGILYLQTELCGPSLQQHCEAWGASLPEAQVWGYLRDTLLALAHLHSQGLVHLDVKPANIFLGPRGRCKLGDFGLLVELGTAGAGEVQEGDPRYMAPELLQGSYGTAADVFSLGLTILEVACNMELPHGGEGWQQLRQGYLPPEFTAGLSSELRSVLVMMLEPDPKLRATAEALLALPVLRQ
1UTB , Knot 134 315 0.81 40 195 294 
MNNGEGEVMDLRDIDLNLLVVFNQLLLDRSVSTAGEKLGLTQPAVSNSLKRLRTALNDDLFLRTSKGMEPTPYALHLAEPVIYALNTLQTALTTRDSFDPFASTRTFNLAMTDIGEMYFMPPLMEALAQRAPHIQISTLRPNAGNLKEDMESGAVDLALGLLPELQTGFFQRRLFRHRYVCMFRKDHPSAKSPMSLKQFSELEHVGVVALNTGHGEVDGLLERAGIKRRMRLVVPHFIAIGPILHSTDLIATVPQRFAVRCEVPFGLTTSPHPAKLPDIAINLFWHAKYNRDPGNMWLRQLFVELFSEAHHHHHH

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 \(c\).

\(|P_{f(1FUT_1)}(2) \setminus P_{f(9LGL_1)}(2)|=48\), \(|P_{f(9LGL_1)}(2) \setminus P_{f(1FUT_1)}(2)|=133\). 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 1:0010001000001001011101100000000011000010000000110111011000111001110011011100111000000111100011010011100100
Pair \(Z_2\) Length of longest common subsequence
1FUT_1,9LGL_1 181 3
1FUT_1,1UTB_1 192 3
9LGL_1,1UTB_1 173 4

Newick tree

 
[
	1FUT_1:95.44,
	[
		9LGL_1:86.5,1UTB_1:86.5
	]:8.94
]

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 \((0.85)(0.8)(\frac{393 }{\log_{20} 393}-\frac{106}{\log_{20}106})=87.7\)
Status Protein1 Protein2 d d1/2
Query variables 1FUT_1 9LGL_1 111 76
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]