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Parikh vectors
5EIU_1 5OJQ_1 5YWM_1 Letter Amino acid
13 10 15 Q Glutamine
21 11 22 E Glutamic acid
5 15 8 T Threonine
2 4 23 Y Tyrosine
9 9 16 A Alanine
10 4 22 R Arginine
2 6 15 N Asparagine
4 11 23 D Aspartic acid
8 11 17 G Glycine
11 11 27 K Lycine
6 2 9 M Methionine
2 7 13 F Phenylalanine
7 13 23 V Valine
4 4 2 C Cysteine
4 14 16 S Serine
7 6 14 H Histidine
6 9 19 I Isoleucine
16 10 35 L Leucine
3 10 15 P Proline
1 3 5 W Tryptophan 

5EIU_1|Chains A, B[auth D]|TRIM protein-E3 ligase Chimera|Macaca mulatta (9544)
>5OJQ_1|Chains AA[auth 2], A[auth 1], CA[auth 4], C[auth 3], E[auth 5], G[auth 6], I[auth L], KA[auth N], K[auth M], MA[auth O], OA[auth P], QA[auth Q], SA[auth S], S[auth R], UA[auth U], U[auth T], W[auth V], Y[auth W]|Haemolysin co-regulated protein|Vibrio cholerae (666)
>5YWM_1|Chain A[auth X]|Casein kinase II subunit alpha'|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 \(w=f(c)\)
5EIU , Knot 70 141 0.82 40 109 137 
EEGQKVDHCARHGEKLLLFCQEDSKVICWLCERSQEHRGHHTFLMEEVAQEYHVKLQTALEMLRQKQQEAETERNQVAKRVPKAPPEEKEALIARGKALGEQTQYMRELISELEHRLQGSMMDLLQGVDGIIKRIENMTLK
5OJQ , Knot 77 170 0.77 40 126 163 
PTPCYISIEGQTQGLITAGACTADSIGDSFVEGHEDEMLVQQFDHVVTVPTDPQSGQPSGQRVHKPFKFTVALNKAVPLLYNALSSGEKLKTVELKWYRTSIEGKQENFFTTKLENASIVDIHCEMPHCQDPAKSDFTQNVTVSLSYRKITWDHVNAGTSGSDDWRKPIE
5YWM , Knot 146 339 0.83 40 201 323 
GPLGSMPGPAAGSRARVYAEVNSLRSREYWDYEAHVPSWGNQDDYQLVRKLGRGKYSEVFEAINITNNERVVVKILKPVKKKKIKREVKILENLRGGTNIIKLIDTVKDPVSKTPALVFEYINNTDFKQLYQILTDFDIRFYMYELLKALDYCHSKGIMHRDVKPHNVMIDHQQKKLRLIDWGLAEFYHPAQEYNVRVASRYFKGPELLVDYQMYDYSLDMWSLGCMLASMIFRREPFFHGQDNYDQLVRIAKVLGTEELYGYLKKYHIDLDPHFNDILGQHSRKRWENFIHSENRHLVSPEALDLLDKLLRYDHQQRLTAKEAMEHPYFYPVVKEQSQ

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 \(c\).

\(|P_{f(5EIU_1)}(2) \setminus P_{f(5OJQ_1)}(2)|=70\), \(|P_{f(5OJQ_1)}(2) \setminus P_{f(5EIU_1)}(2)|=87\). 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 1:001001000100100111100000011011000000001000111001100001010011011000000100000011001101110000111101011100000100110010001010110110110111001001010
Pair \(Z_2\) Length of longest common subsequence
5EIU_1,5OJQ_1 157 4
5EIU_1,5YWM_1 182 4
5OJQ_1,5YWM_1 197 3

Newick tree

 
[
	5YWM_1:99.67,
	[
		5EIU_1:78.5,5OJQ_1:78.5
	]:21.17
]

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 \((0.85)(0.8)(\frac{311 }{\log_{20} 311}-\frac{141}{\log_{20}141})=52.3\)
Status Protein1 Protein2 d d1/2
Query variables 5EIU_1 5OJQ_1 66 60.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]