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Parikh vectors
2ZZS_1 2ZUR_1 2NQQ_1 Letter Amino acid
9 28 43 L Leucine
2 2 24 P Proline
8 10 25 S Serine
5 8 35 V Valine
2 15 27 E Glutamic acid
1 6 5 H Histidine
5 4 20 I Isoleucine
0 1 4 W Tryptophan
2 2 6 C Cysteine
6 11 20 Q Glutamine
10 11 43 G Glycine
7 9 10 K Lycine
4 3 8 M Methionine
0 8 19 F Phenylalanine
2 6 25 T Threonine
23 15 37 A Alanine
2 11 23 R Arginine
5 6 5 Y Tyrosine
5 6 13 N Asparagine
5 12 19 D Aspartic acid 

2ZZS_1|Chains AA[auth 2], A[auth 1], BA[auth 4], B[auth 3], CA[auth 6], C[auth 5], DA[auth B], D[auth A], EA[auth D], E[auth C], FA[auth F], F[auth E], G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z|Cytochrome c554|Vibrio parahaemolyticus (223926)
>2ZUR_1|Chain A[auth X]|Ferritin light chain|Equus caballus (9796)
>2NQQ_1|Chains A, B, C, D|Molybdopterin biosynthesis protein moeA|Escherichia coli (562)
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 \(w=f(c)\)
2ZZS , Knot 50 103 0.75 36 74 98 
MKRVMIGAVAALTMLSGQALAGDAAAGQAKAAVCAACHGADGNATIPGYPNLKGQNEQYIVSSIKAYKNKERSGGLAAVMQAQASLLSDDDIANLAAYYSSLK
2ZUR , Knot 83 174 0.82 40 127 170 
SSQIRQNYSTEVEAAVNRLVNLYLRASYTYLSLGFYFDRDDVALEGVCHFFRELAEEKREGAERLLKMQNQRGGRALFQDLQKPSQDEWGTTLDAMKAAIVLEKSLNQALLDLHALGSAQADPHLCDFLESHFLDEEVKLIKKMGDHLTNIQRLVGSQAGLGEYLFERLTLKHD
2NQQ , Knot 169 411 0.82 40 211 379 
MEFTTGLMSLDTALNEMLSRVTPLTAQETLPLVQCFGRILASDVVSPLDVPGFDNSAMDGYAVRLADIASGQPLPVAGKSFAGQPYHGEWPAGTCIRIMTGAPVPEGCEAVVMQEQTEQMDNGVRFTAEVRSGQNIQRRGEDISAGAVVFPAGTRLTTAELPVIASLGIAEVPVIRKVRVALFSTGDELQLPGQPLGDGQIYDTNRLAVHLMLEQLGCEVINLGIIRDDPHALRAAFIEADSQADVVISSGGVSVGEADYTKTILEELGEIAFWKLAIKPGKPFAFGKLSNSWFCGLPGNPVSATLTFYQLVQPLLAKLSGNTASGLPARQRVRTASRLKKTPGRLDFQRGVLQRNADGELEVTTTGHQGSHIFSSFSLGNCFIVLERDRGNVEVGEWVEVEPFNALFGGL

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 \(c\).

\(|P_{f(2ZZS_1)}(2) \setminus P_{f(2ZUR_1)}(2)|=40\), \(|P_{f(2ZUR_1)}(2) \setminus P_{f(2ZZS_1)}(2)|=93\). 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 1:1001111111110110101111011110101110110011010101110101010000011001010000000111111101010110000110111000010
Pair \(Z_2\) Length of longest common subsequence
2ZZS_1,2ZUR_1 133 3
2ZZS_1,2NQQ_1 189 3
2ZUR_1,2NQQ_1 162 4

Newick tree

 
[
	2NQQ_1:94.09,
	[
		2ZZS_1:66.5,2ZUR_1:66.5
	]:27.59
]

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 \((0.85)(0.8)(\frac{277 }{\log_{20} 277}-\frac{103}{\log_{20}103})=55.0\)
Status Protein1 Protein2 d d1/2
Query variables 2ZZS_1 2ZUR_1 70 54
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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