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Parikh vectors
4IXJ_1 6DBA_1 3KAI_1 Letter Amino acid
8 7 13 R Arginine
3 4 2 C Cysteine
24 15 17 G Glycine
10 8 5 H Histidine
5 8 8 Q Glutamine
11 11 9 L Leucine
30 5 10 K Lycine
7 3 7 F Phenylalanine
26 5 6 D Aspartic acid
7 2 4 M Methionine
17 11 6 T Threonine
5 4 3 W Tryptophan
14 6 3 Y Tyrosine
15 10 4 V Valine
18 7 11 A Alanine
14 7 5 N Asparagine
17 8 15 E Glutamic acid
18 4 8 I Isoleucine
11 1 7 P Proline
23 16 24 S Serine 

4IXJ_1|Chains A, B|Fimbrial protein (Pilin)|Clostridium difficile (1496)
>6DBA_1|Chains A, B|nanobody VHH R303|Camelus dromedarius (9838)
>3KAI_1|Chain A|Peptidyl-prolyl cis-trans isomerase NIMA-interacting 1|Homo sapiens (9606)
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 \(w=f(c)\)
4IXJ , Knot 128 283 0.85 40 181 271 
MHHHHHHSSGLVPRGSGMKETAAAKFERQHMDSPDLGTDDDDKAMGRNIEKSKAVTCLSNRENIKTQIVIAMAEESSKDKNEVIKEVLENKDGKYFETEPKCKSGGIYSATFDDGYDGITGIESIAKVYVTCTKHPDGIEMARDIHQSMKDLIASFAQDPSIIPGASKGNDDFRKYLLDNKYKNGWPTIPDEFKAKYGLSKDTLYIQPYAYNPTKSDATVVVFANNKTGGNWYTSLVYDYDEGRWYKGKNGISVAGRSWDVDTDSVKSVKTEIHSKEGWGPLN
6DBA , Knot 69 142 0.80 40 105 134 
QVKLEESGGGSVQAGGSLRLSCAASGHTYSTYCMGWFRQVPGKEREGVARINVGGSSTWYADSVRDRFTISQDNAKNTVYLQMNSLKLEDTAIYYCTLHRFCNTWSLGTLNVWGQGTQVTVSSGSEQKLISEEDLNHHHHHH
3KAI , Knot 78 167 0.79 40 122 161 
GSHGMADEEKLPPGWEKAMSRSSGRVYYFNHITNASQWERPSGNSSSGGKNGQGEPARVRCSHLLVKHSQSRRPSSWRQEKITRTKEEALELINGYIQKIKSGEEDFESLASQFSDCSSAKARGDLGAFSRGQMQKPFEDASFALRTGEMSGPVFTDSGIHIILRTE

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 \(c\).

\(|P_{f(4IXJ_1)}(2) \setminus P_{f(6DBA_1)}(2)|=126\), \(|P_{f(6DBA_1)}(2) \setminus P_{f(4IXJ_1)}(2)|=50\). 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 1:1000000001111010110001110100001001011000000111001000011001000001000111111000000000110011000010010001000011100101001001101100110101000001011011001000100111011001011111001000100011000000111011001010011000010101010010000101111100001101000110000010100100110111001010000100100010000111110
Pair \(Z_2\) Length of longest common subsequence
4IXJ_1,6DBA_1 176 6
4IXJ_1,3KAI_1 167 4
6DBA_1,3KAI_1 147 3

Newick tree

 
[
	4IXJ_1:89.49,
	[
		3KAI_1:73.5,6DBA_1:73.5
	]:15.99
]

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 \((0.85)(0.8)(\frac{425 }{\log_{20} 425}-\frac{142}{\log_{20}142})=84.6\)
Status Protein1 Protein2 d d1/2
Query variables 4IXJ_1 6DBA_1 109 80.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]