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Parikh vectors
7HAN_1 9LGL_1 3FCD_1 Letter Amino acid
20 3 9 I Isoleucine
7 5 4 M Methionine
1 4 1 W Tryptophan
7 8 4 Y Tyrosine
0 6 3 C Cysteine
9 19 5 Q Glutamine
26 24 9 E Glutamic acid
10 9 10 H Histidine
8 2 4 N Asparagine
15 31 7 G Glycine
10 10 5 F Phenylalanine
5 19 10 P Proline
15 22 9 A Alanine
16 20 7 S Serine
17 8 5 T Threonine
11 17 4 V Valine
8 21 6 R Arginine
16 9 10 D Aspartic acid
19 41 18 L Leucine
17 9 4 K Lycine 

7HAN_1|Chain A|Heat shock protein HSP 90-alpha|Homo sapiens (9606)
>9LGL_1|Chains A, B|Membrane-associated tyrosine- and threonine-specific cdc2-inhibitory kinase|Homo sapiens (9606)
>3FCD_1|Chains A, B|Lyase|uncultured bacterium (77133)
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 \(w=f(c)\)
7HAN , Knot 105 237 0.80 38 157 225 
MDQPMEEEEVETFAFQAEIAQLMSLIINTFYSNKEIFLRELISNSSDALDKIRYESLTDPSKLDSGKELHINLIPNKQDRTLTIVDTGIGMTKADLINNLGTIAKSGTKAFMEALQAGADISMIGQFGVGFYSAYLVAEKVTVITKHNDDEQYAWESSAGGSFTVRTDTGEPMGRGTKVILHLKEDQTEYLEERRIKEIVKKHSQFIGYPITLFVEKERDKEVSDDEAELEHHHHHH
9LGL , Knot 127 287 0.83 40 168 275 
HQLQPRRVSFRGEASETLQSPGYDPSRPESFFQQSFQRLSRLGHGSYGEVFKVRSKEDGRLYAVKRSMSPFRGPKDRARKLAEVGSHEKVGQHPCCVRLEQAWEEGGILYLQTELCGPSLQQHCEAWGASLPEAQVWGYLRDTLLALAHLHSQGLVHLDVKPANIFLGPRGRCKLGDFGLLVELGTAGAGEVQEGDPRYMAPELLQGSYGTAADVFSLGLTILEVACNMELPHGGEGWQQLRQGYLPPEFTAGLSSELRSVLVMMLEPDPKLRATAEALLALPVLRQ
3FCD , Knot 67 134 0.81 40 105 126 
MSLSDIHQITPFLHIPDMQEALTLFCDTLGFELKYRHSNYAYLELSGCGLRLLEEPARKIIPDGIARVAICIDVSDIDSLHTKLSPALENLPADQVEPLKNMPYGQREFQVRMPDGDWLNFTAPLAEGHHHHHH

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 \(c\).

\(|P_{f(7HAN_1)}(2) \setminus P_{f(9LGL_1)}(2)|=81\), \(|P_{f(9LGL_1)}(2) \setminus P_{f(7HAN_1)}(2)|=92\). 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 1:100110000100111010110110111001000001110011000001100100001001001001001010111000000101100111100101100110110010011101101110101110111110010111001011000000000110001110101000010111010011101000000010000100110000011101101110000000100001010000000
Pair \(Z_2\) Length of longest common subsequence
7HAN_1,9LGL_1 173 3
7HAN_1,3FCD_1 164 6
9LGL_1,3FCD_1 159 3

Newick tree

 
[
	7HAN_1:85.81,
	[
		3FCD_1:79.5,9LGL_1:79.5
	]:6.31
]

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 \((0.85)(0.8)(\frac{524 }{\log_{20} 524}-\frac{237}{\log_{20}237})=82.1\)
Status Protein1 Protein2 d d1/2
Query variables 7HAN_1 9LGL_1 104 94
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]