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Parikh vectors
7TKB_1 3WFF_1 4ETN_1 Letter Amino acid
10 15 14 A Alanine
10 8 16 G Glycine
2 15 6 P Proline
1 9 8 D Aspartic acid
12 37 16 L Leucine
6 14 7 F Phenylalanine
6 17 11 V Valine
2 11 5 N Asparagine
1 3 1 C Cysteine
1 16 8 Q Glutamine
2 19 10 K Lycine
3 14 9 T Threonine
0 4 0 W Tryptophan
1 9 11 R Arginine
1 19 12 E Glutamic acid
0 6 13 H Histidine
9 12 8 I Isoleucine
3 11 8 M Methionine
5 25 18 S Serine
1 11 3 Y Tyrosine 

7TKB_1|Chains A[auth 0], B[auth 1], C[auth 2], D[auth 3], E[auth 4], F[auth 5], G[auth 6], H[auth 7], I[auth 8], J[auth 9]|ATP synthase subunit 9, mitochondrial|Saccharomyces cerevisiae (4932)
>3WFF_1|Chain A|Mineralocorticoid receptor|Homo sapiens (9606)
>4ETN_1|Chain A|Low molecular weight protein-tyrosine-phosphatase ywlE|Bacillus subtilis (1423)
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 \(w=f(c)\)
7TKB , Knot 40 76 0.76 36 57 72 
MQLVLAAKYIGAGISTIGLLGAGIGIAIVFAALINGVSRNPSIKDTVFPMAILGFALSEATGLFCLMVSFLLLFGV
3WFF , Knot 120 275 0.81 40 179 260 
GSAPAKEPSVNTALVPQLSTISRALTPSPVMVLENIEPEIVYAGYDSSKPDTAENLLSTLNRLAGKQMIQVVKWAKVLPGFKNLPLEDQITLIQYSWMSLLSFALSWRSYKHTNSQFLYFAPDLVFNEEKMHQSAMYELCQGMHQISLQFVRLQLTFEEYTIMKVLLLLSTIPKDGLKSQAAFEEMRTNYIKELRKMVTKCPNNSGQSWQRFYQLTKLLDSMHDLVSDLLEFCFYTFRESHALKVEFPAMLVEIISDQLPKVESGNVKPLYFHRK
4ETN , Knot 85 184 0.80 38 130 174 
MGSSHHHHHHSSGLVPRGSHMASMTGGQQMGRGSMDIIFVCTGNTSRSPMAEALFKSIAEREGLNVNVRSAGVFASPNGKATPHAVEALFEKHIALNHVSSPLTEELMESADLVLAMTHQHKQIIASQFGRYRDKVFTLKEYVTGSHGDVLDPFGGSIDIYKQTRDELEELLRQLAKQLKKDRR

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 \(c\).

\(|P_{f(7TKB_1)}(2) \setminus P_{f(3WFF_1)}(2)|=26\), \(|P_{f(3WFF_1)}(2) \setminus P_{f(7TKB_1)}(2)|=148\). 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 1:1011111001111100111111111111111111011000101000111111111110010111011101111111
Pair \(Z_2\) Length of longest common subsequence
7TKB_1,3WFF_1 174 4
7TKB_1,4ETN_1 135 4
3WFF_1,4ETN_1 175 3

Newick tree

 
[
	3WFF_1:92.90,
	[
		7TKB_1:67.5,4ETN_1:67.5
	]:25.40
]

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 \((0.85)(0.8)(\frac{351 }{\log_{20} 351}-\frac{76}{\log_{20}76})=86.2\)
Status Protein1 Protein2 d d1/2
Query variables 7TKB_1 3WFF_1 108 67.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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