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Parikh vectors
4ITU_1 7LXP_1 3MQG_1 Letter Amino acid
6 18 7 P Proline
17 0 6 S Serine
1 0 3 W Tryptophan
3 0 5 Y Tyrosine
2 0 7 C Cysteine
8 0 9 E Glutamic acid
17 0 13 I Isoleucine
8 0 3 K Lycine
19 0 10 T Threonine
8 0 5 Q Glutamine
13 0 9 D Aspartic acid
19 0 9 L Leucine
11 1 15 R Arginine
8 0 11 N Asparagine
35 9 23 G Glycine
8 0 7 H Histidine
9 0 3 M Methionine
8 0 6 F Phenylalanine
20 0 23 V Valine
49 0 18 A Alanine 

4ITU_1|Chains A, B, C, D|Short-chain dehydrogenase/reductase SDR|Xanthobacter autotrophicus (78245)
>7LXP_1|Chains A, B, C, D, E, F|Collagen mimetic peptide|synthetic construct (32630)
>3MQG_1|Chains A, B, C, D, E, F|Lipopolysaccharides biosynthesis acetyltransferase|Bordetella petrii (340100)
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 \(w=f(c)\)
4ITU , Knot 112 269 0.77 40 145 245 
MGSSHHHHHHSQDPMSNRLKNEVIAITGGGAGIGLAIASAALREGAKVALIDLDQGLAERSAAMLSTGGAVAKGFGADVTKAADITAAITSAEQTIGSLTGLVNNAGIAGFGSVHDADAAAWDRIMAVNVTGTFLASKAALAGMLERHKGTIVNFGSVAGLVGIPTMAAYCAAKGAIVNLTRQMAADYSGRGVRVNAVCPGTVTSTGMGQQLLGSDTSPEVQARRLAKYPIGRFGTPEDIAEAVIFLLSDQAAFVTGAAFAVDGGMTAI
7LXP , Knot 6 30 0.22 8 7 10 
PPGPPGPPGPPGXPGPRXPPGPPGPPGPPG
3MQG , Knot 91 192 0.83 40 137 183 
GHMATIHPTAIVDEGARIGAHSRIWHWVHICGGAEIGEGCSLGQNVFVGNRVRIGNRVKIQNNVSVYDNVFLEDDVFCGPSMVFTNVYNPRAAIERKSEYRDTIVRQGATLGANCTVVCGATIGRYAFVGAGAVVNKDVPDFALVVGVPARQIGWMSRHGEQLDLPLRGNAEATCPHTGERYILTDGVCRLA

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 \(c\).

\(|P_{f(4ITU_1)}(2) \setminus P_{f(7LXP_1)}(2)|=144\), \(|P_{f(7LXP_1)}(2) \setminus P_{f(4ITU_1)}(2)|=6\). 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 1:11000000000001100010001111011111111111011100110111101001110001111001111101111010011010111001000110101110011111110100101111001111010101110011111110000101101101111111101110011011110100011100010110101101101000111001110000101010011001110110100110111111000111101111110111011
Pair \(Z_2\) Length of longest common subsequence
4ITU_1,7LXP_1 150 2
4ITU_1,3MQG_1 156 3
7LXP_1,3MQG_1 140 2

Newick tree

 
[
	4ITU_1:78.56,
	[
		7LXP_1:70,3MQG_1:70
	]:8.56
]

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 \((0.85)(0.8)(\frac{299 }{\log_{20} 299}-\frac{30}{\log_{20}30})=88.8\)
Status Protein1 Protein2 d d1/2
Query variables 4ITU_1 7LXP_1 108 57
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]