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Parikh vectors
6CLM_1 4XJE_1 1KXJ_1 Letter Amino acid
1 14 22 G Glycine
0 9 7 H Histidine
0 16 19 L Leucine
0 7 1 W Tryptophan
0 7 8 Y Tyrosine
1 2 2 Q Glutamine
0 13 8 D Aspartic acid
0 21 22 E Glutamic acid
1 4 11 F Phenylalanine
0 9 8 P Proline
0 13 18 R Arginine
0 12 13 I Isoleucine
0 3 5 M Methionine
1 5 13 S Serine
3 2 8 N Asparagine
0 4 3 C Cysteine
0 4 10 K Lycine
0 7 4 T Threonine
0 10 19 V Valine
0 23 4 A Alanine 

6CLM_1|Chain A|GSNQNNF|synthetic construct (32630)
>4XJE_1|Chain A|AadB|Pseudomonas aeruginosa (287)
>1KXJ_1|Chains A, B|Amidotransferase hisH|Thermotoga maritima (2336)
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 \(w=f(c)\)
6CLM , Knot 6 7 0.55 10 6 5 
GSNQNNF
4XJE , Knot 84 185 0.79 40 134 176 
LEHHHHHHMDTTQVTLIHKILAAADERNLPLWIGGGWAIDARLGRVTRKHDDIDLTFPGERRGELEAIVEMLGGRVMEELDYGFLAEIGDELLDCEPAWWADEAYEIAEAPQGSCPEAAEGVIAGRPVRCNSWEAIIWDYFYYADEVPPVDWPTKHIESYRLACTSLGAEKVEVLRAAFRSRYAA
1KXJ , Knot 94 205 0.81 40 138 196 
GHMRIGIISVGPGNIMNLYRGVKRASENFEDVSIELVESPRNDLYDLLFIPGVGHFGEGMRRLRENDLIDFVRKHVEDERYVVGVCLGMQLLFEESEEAPGVKGLSLIEGNVVKLRSRRLPHMGWNEVIFKDTFPNGYYYFVHTYRAVCEEEHVLGTTEYDGEIFPSAVRKGRILGFQFHPEKSSKIGRKLLEKVIECSLSRRGS

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 \(c\).

\(|P_{f(6CLM_1)}(2) \setminus P_{f(4XJE_1)}(2)|=5\), \(|P_{f(4XJE_1)}(2) \setminus P_{f(6CLM_1)}(2)|=133\). 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 1:1000001
Pair \(Z_2\) Length of longest common subsequence
6CLM_1,4XJE_1 138 2
6CLM_1,1KXJ_1 140 2
4XJE_1,1KXJ_1 154 3

Newick tree

 
[
	1KXJ_1:75.04,
	[
		6CLM_1:69,4XJE_1:69
	]:6.04
]

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 \((0.85)(0.8)(\frac{192 }{\log_{20} 192}-\frac{7}{\log_{20}7})=67.0\)
Status Protein1 Protein2 d d1/2
Query variables 6CLM_1 4XJE_1 83 43
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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