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Parikh vectors
9IJK_1 7QGQ_1 3KXY_1 Letter Amino acid
15 5 9 F Phenylalanine
20 35 13 A Alanine
18 7 8 R Arginine
18 43 8 D Aspartic acid
8 0 4 H Histidine
13 25 4 I Isoleucine
11 37 2 K Lycine
9 5 2 M Methionine
19 36 6 S Serine
3 9 1 Y Tyrosine
14 19 12 E Glutamic acid
27 26 9 G Glycine
35 29 28 L Leucine
13 10 2 P Proline
16 20 3 N Asparagine
3 0 2 C Cysteine
10 6 6 Q Glutamine
13 25 5 T Threonine
6 0 2 W Tryptophan
17 36 7 V Valine 

9IJK_1|Chains A, B, C, D|2-hydroxy-6-oxononadienedioate/2-hydroxy-6-oxononatrienedioate hydrolase|Escherichia coli K-12 (83333)
>7QGQ_1|Chains A[auth B], C[auth D], E[auth J], F[auth K], K[auth L], L[auth M], M[auth N], N[auth T], S[auth U], T[auth V], U[auth W], X|Precursor of the S-layer proteins|Clostridioides difficile 630 (272563)
>3KXY_1|Chains A, B, D[auth C], E[auth D], G[auth E], H[auth F], J[auth G], K[auth H], M[auth I], N[auth J], P[auth K], Q[auth L]|Exoenzyme S synthesis protein C|Pseudomonas aeruginosa (287)
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 \(w=f(c)\)
9IJK , Knot 130 288 0.85 40 176 278 
MSYQPQTEAATSRFLNVEEAGKTLRIHFNDCGQGDETVVLLHGSGPGATGWANFSRNIDPLVEAGYRVILLDCPGWGKSDSVVNSGSRSDLNARILKSVVDQLDIAKIHLLGNSMGGHSSVAFTLKWPERVGKLVLMGGGTGGMSLFTPMPTEGIKRLNQLYRQPTIENLKLMMDIFVFDTSDLTDALFEARLNNMLSRRDHLENFVKSLEANPKQFPDFGPRLAEIKAQTLIVWGRNDRFVPMDAGLRLLSGIAGSELHIFRDCGHWAQWEHADAFNQLVLNFLARP
7QGQ , Knot 148 373 0.78 34 164 345 
NDTIASQDTPAKVVIKANKLKDLKDYVDDLKTYNNTYSNVVTVAGEDRIETAIELSSKYYNSDDKNAITDKAVNDIVLVGSTSIVDGLVASPLASEKTAPLLLTSKDKLDSSVKSEIKRVMNLKSDTGINTSKKVYLAGGVNSISKDVENELKNMGLKVTRLSGEDRYETSLAIADEIGLDNDKAFVVGGTGLADAMSIAPVASQLKDGDATPIVVVDGKAKEISDDAKSFLGTSDVDIIGGKNSVSKEIEESIDSATGKTPDRISGDDRQATNAEVLKEDDYFTDGEVVNYFVAKDGSTKEDQLVDALAAAPIAGRFKESPAPIILATDTLSSDQNVAVSKAVPKDGGTNLVQVGKGIASSVINKMKDLLDM
3KXY , Knot 64 133 0.78 40 88 123 
MDLTSKVNRLLAEFAGRIGLPSLSLDEEGMASLLFDEQVGVTLLLLAERERLLLEADVVGIDVLGEGIFRQLASFNRHWHRFDLHFGFDELTGKVQLYAQILAAQLTLECFEATLANLLDHAEFWQRLLPCAS

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 \(c\).

\(|P_{f(9IJK_1)}(2) \setminus P_{f(7QGQ_1)}(2)|=83\), \(|P_{f(7QGQ_1)}(2) \setminus P_{f(9IJK_1)}(2)|=71\). 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 1:100010001100011010011001010100010100011110101111011101000101110110011110011110000110010000101011001100101101011100111000111010110011011111110111011011100110010010001010010111011110000100111010100110000010011001010100110111011010100111110000111101110110111100101100010110100101100111011101
Pair \(Z_2\) Length of longest common subsequence
9IJK_1,7QGQ_1 154 4
9IJK_1,3KXY_1 152 3
7QGQ_1,3KXY_1 158 4

Newick tree

 
[
	7QGQ_1:78.66,
	[
		9IJK_1:76,3KXY_1:76
	]:2.66
]

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 \((0.85)(0.8)(\frac{661 }{\log_{20} 661}-\frac{288}{\log_{20}288})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 9IJK_1 7QGQ_1 128 114.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]