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Parikh vectors
4NKI_1 2KOM_1 6FZY_1 Letter Amino acid
16 4 16 A Alanine
7 8 8 R Arginine
7 5 17 Q Glutamine
3 2 11 M Methionine
12 2 11 P Proline
7 5 9 N Asparagine
6 5 18 D Aspartic acid
6 7 19 E Glutamic acid
22 15 12 G Glycine
11 9 24 K Lycine
10 3 8 Y Tyrosine
22 9 14 V Valine
5 0 1 C Cysteine
8 8 7 H Histidine
6 8 21 I Isoleucine
19 12 39 L Leucine
33 10 17 S Serine
21 6 12 T Threonine
7 3 15 F Phenylalanine
4 0 0 W Tryptophan 

4NKI_1|Chain A[auth H]|Fab light chain|Homo sapiens (9606)
>2KOM_1|Chain A|Partitioning defective 3 homolog|Homo sapiens (9606)
>6FZY_1|Chains A, B|Peroxisome proliferator-activated receptor gamma|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 \(w=f(c)\)
4NKI , Knot 100 232 0.78 40 142 215 
EVQLLESGGGLVQPGGSLRLSCAASGFTFSSYIMMWVRQAPGKGLEWVSSIYPSGGITFYADTVKGRFTISRDNSKNTLYLQMNSLRAEDTAVYYCARIKLGTVTTVDYWGQGTLVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKRVEPKSCAAAHHHHHH
2KOM , Knot 58 121 0.76 36 93 112 
MRGSHHHHHHHHGSENLYFQGSYNTKKIGKRLNIQLKKGTEGLGFSITSRDVTIGGSAPIYVKNILPRGAAIQDGRLKAGDRLIEVNGVDLVGKSQEEVVSLLRSTKMEGTVSLLVFRQED
6FZY , Knot 124 279 0.83 38 178 267 
GSHMQLNPESADLRALAKHLYDSYIKSFPLTKAKARAILTGKTTDKSPFVIYDMNSLMMGEDKIKFKHITPLQEQSKEVAIRIFQGCQFRSVEAVQEITEYAKSIPGFVNLDLNDQVTLLKYGVHEIIYTMLASLMNKDGVLISEGQGFMTREFLKSLRKPFGDFMEPKFEFAVKFNALELDDSDLAIFIAVIILSGDRPGLLNVKPIEDIQDNLLQALELQLKLNHPESSQLFAKLLQKMTDLRQIVMEHVQLLQVIKKTETDMSLHPLLQEIYKDLY

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 \(c\).

\(|P_{f(4NKI_1)}(2) \setminus P_{f(2KOM_1)}(2)|=92\), \(|P_{f(2KOM_1)}(2) \setminus P_{f(4NKI_1)}(2)|=43\). 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 1:0101100111110111010100110110100011111001110110110010101110101001010101000000001010100101000110001010110100100110101101001000110111111000000110111101100011011010100111001100111110001100100110110001100001001000100001000101000111000000
Pair \(Z_2\) Length of longest common subsequence
4NKI_1,2KOM_1 135 6
4NKI_1,6FZY_1 178 4
2KOM_1,6FZY_1 167 3

Newick tree

 
[
	6FZY_1:91.70,
	[
		4NKI_1:67.5,2KOM_1:67.5
	]:24.20
]

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 \((0.85)(0.8)(\frac{353 }{\log_{20} 353}-\frac{121}{\log_{20}121})=71.1\)
Status Protein1 Protein2 d d1/2
Query variables 4NKI_1 2KOM_1 86 64
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]