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Parikh vectors
4ADZ_1 1VAQ_1 9IXG_1 Letter Amino acid
10 0 22 E Glutamic acid
7 0 15 I Isoleucine
10 0 22 D Aspartic acid
12 0 24 L Leucine
7 0 4 K Lycine
2 0 18 P Proline
10 0 17 R Arginine
11 2 23 G Glycine
7 0 12 H Histidine
2 0 5 M Methionine
1 0 3 F Phenylalanine
2 0 5 Y Tyrosine
2 2 2 C Cysteine
1 0 5 N Asparagine
8 0 6 Q Glutamine
6 0 10 S Serine
11 2 12 T Threonine
0 0 5 W Tryptophan
9 0 28 V Valine
18 2 43 A Alanine 

4ADZ_1|Chains A, B|CSOR|STREPTOMYCES LIVIDANS (1916)
>1VAQ_1|Chains A, B, C, D|5'-D(*TP*TP*GP*GP*CP*CP*AP*A)-3'|synthetic construct (32630)
>9IXG_1|Chains A, B|N(omega)-hydroxy-L-arginine amidinohydrolase|Streptomyces lavendulae (1914)
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 \(w=f(c)\)
4ADZ , Knot 67 136 0.80 38 102 131 
GSHMTTTEAGASAPSPAVDGAVNQTARQAEADGTDIVTDHDRGVHGYHKQKAEHLKRLRRIEGQIRGLQRMVDEDVYCIDILTQVSASTKALQSFALQLLEEHLRHCVADAALKGGTEIDAKVEEATKAIGRLLRT
1VAQ , Knot 5 8 0.43 8 7 6 
TTGGCCAA
9IXG , Knot 121 281 0.81 40 153 262 
MIDLIVSQGRVADRAAWMIEGAARTARALEERYGLKGHYVGEPAPHADDDWSVALPQARETLVAVREAATESIKGDNLTVLVNNTCSVSLATLPVVAREHPDAVVLYIDGHGDFNTPETTDTGYLGGMVLSGACGLWDSGHGAGLRPEQAVLVGSRDIDEGERELIRKAGVRVIPPGEATAQAVLDAVKDAPVWIHIDWDVLEPGSIPADYTVPDGMLPAQIRAVFEAIPAERLIGVELAELNAPADSERAEQAVAVILDMVAPAFDAAAARPLEHHHHHH

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 \(c\).

\(|P_{f(4ADZ_1)}(2) \setminus P_{f(1VAQ_1)}(2)|=99\), \(|P_{f(1VAQ_1)}(2) \setminus P_{f(4ADZ_1)}(2)|=4\). 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 1:1001000011101101110111000100101010011000001101000001001001001010101100110001001011001010001100111011000100011011101100101010010011101100
Pair \(Z_2\) Length of longest common subsequence
4ADZ_1,1VAQ_1 103 2
4ADZ_1,9IXG_1 147 3
1VAQ_1,9IXG_1 152 2

Newick tree

 
[
	9IXG_1:81.04,
	[
		4ADZ_1:51.5,1VAQ_1:51.5
	]:29.54
]

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 \((0.85)(0.8)(\frac{144 }{\log_{20} 144}-\frac{8}{\log_{20}8})=51.1\)
Status Protein1 Protein2 d d1/2
Query variables 4ADZ_1 1VAQ_1 65 34
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]