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Parikh vectors
3KKA_1 6VTV_1 4IXN_1 Letter Amino acid
6 15 24 I Isoleucine
1 3 3 W Tryptophan
2 21 25 A Alanine
5 14 17 R Arginine
1 12 15 N Asparagine
6 24 22 E Glutamic acid
7 19 22 G Glycine
5 4 17 T Threonine
3 5 7 Y Tyrosine
2 17 22 V Valine
2 6 12 K Lycine
4 5 7 M Methionine
3 5 13 F Phenylalanine
2 20 10 P Proline
6 16 13 S Serine
3 7 25 D Aspartic acid
2 5 3 C Cysteine
4 13 14 Q Glutamine
9 12 8 H Histidine
9 35 39 L Leucine 

3KKA_1|Chains A, B|EPHRIN TYPE-A RECEPTOR 1|HOMO SAPIENS (9606)
>6VTV_1|Chains A, B|Gamma-glutamyl-gamma-aminobutyrate hydrolase PuuD|Escherichia coli (strain K12) (83333)
>4IXN_1|Chains A, B|Uncharacterized GTP-binding protein YjiA|Escherichia coli (83333)
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 \(w=f(c)\)
3KKA , Knot 45 82 0.80 40 69 77 
MHHHHHHSSGRENLYFQGDGIPYRTVSEWLESIRMKRYILHFHSAGLDTMECVLELTAEDLTQMGITLPGHQKRILCSIQGF
6VTV , Knot 114 258 0.81 40 167 246 
MRLSMENIMNNPVIGVVMCRNRLKGHATQTLQEKYLNAIIHAGGLPIALPHALAEPSLLEQLLPKLDGIYLPGSPSNVQPHLYGENGDEPDADPGRDLLSMAIINAALERRIPIFAICRGLQELVVATGGSLHRKLCEQPELLEHREDPELPVEQQYAPSHEVQVEEGGLLSALLPECSNFWVNSLHGQGAKVVSPRLRVEARSPDGLVEAVSVINHPFALGVQWHPEWNSSEYALSRILFEGFITACQHHIAEKQRL
4IXN , Knot 139 318 0.84 40 199 303 
MNPIAVTLLTGFLGAGKTTLLRHILNEQHGYKIAVIANEFGEVSVDDQLIGDRATQIKTLTNGCIAASRSNELEDALLDLLDNLDKGNIQFDRLVIECTGMADPGPIIQTFFSHEVLCQRYLLDGVIALVDAVHADEQMNQFTIAQSQVGYADRILLTKTDVAGEAEKLHERLARINARAPVYTVTHGDIDLGLLFNTNGFMLEENVVSTKPRFHFIADKQNDISSIVVELDYPVDISEVSRVMENLLLESADKLLRYKGMLWIDGEPNRLLFQGVQRLYSADWDRPWGDEKPHSTMVFIGIQLPEEEIRAAFAGLRK

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 \(c\).

\(|P_{f(3KKA_1)}(2) \setminus P_{f(6VTV_1)}(2)|=31\), \(|P_{f(6VTV_1)}(2) \setminus P_{f(3KKA_1)}(2)|=129\). 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 1:1000000001000101010111000100110010100011010011100100110101001001110111000011001011
Pair \(Z_2\) Length of longest common subsequence
3KKA_1,6VTV_1 160 3
3KKA_1,4IXN_1 180 3
6VTV_1,4IXN_1 156 4

Newick tree

 
[
	3KKA_1:87.39,
	[
		6VTV_1:78,4IXN_1:78
	]:9.39
]

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 \((0.85)(0.8)(\frac{340 }{\log_{20} 340}-\frac{82}{\log_{20}82})=80.9\)
Status Protein1 Protein2 d d1/2
Query variables 3KKA_1 6VTV_1 100 65
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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