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Parikh vectors
7DIO_1 3VCD_1 2KDV_1 Letter Amino acid
1 2 8 W Tryptophan
7 25 6 A Alanine
2 2 7 F Phenylalanine
14 6 9 K Lycine
3 8 6 M Methionine
3 3 6 Y Tyrosine
2 6 12 Q Glutamine
10 8 9 E Glutamic acid
7 19 11 G Glycine
9 21 10 L Leucine
3 5 8 P Proline
5 18 11 S Serine
2 5 16 R Arginine
3 2 2 C Cysteine
2 9 1 H Histidine
11 13 7 I Isoleucine
3 7 5 T Threonine
9 21 16 V Valine
10 4 6 N Asparagine
5 8 8 D Aspartic acid 

7DIO_1|Chain A|Glutaredoxin|Plasmodium falciparum (isolate 3D7) (36329)
>3VCD_1|Chains A, B, C, D, E, F, G, H|Propanediol utilization polyhedral body protein PduT|Salmonella enterica (28901)
>2KDV_1|Chain A|RNA pyrophosphohydrolase|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 \(w=f(c)\)
7DIO , Knot 57 111 0.80 40 92 107 
MAGTSEAVKKWVNKIIEENIIAVFAKTECPYCIKAISILKGYNLNSHMHVENIEKNPDMANIQAYLKELTGKSSVPRIFINKDVVGGCDDLVKENDEGKLKERLQKLGLVN
3VCD , Knot 85 192 0.77 40 123 183 
MSQAIGILELTSIAAGMELGDAMLKSANVDLLVSKTISPGKFLLMLGGDIGAIQQAIETGTSQAGELLVDSLVLANIHPSVLPAISGLNSVDKRQAVGIVETWSVAACISAADRAVKGSNVTLVRVHMAFGIGGKCYMVVAGDVSDVALAVTVASSSAGAYGLLVYASLIPRPHEAMWRQMVEGLEHHHHHH
2KDV , Knot 82 164 0.85 40 130 162 
MIDDDGYRPNVGIVICNRQGQVMWARRFGQHSWQFPQGGINPGESAEQAMYRELFEEVGLSRKDVRILASTRNWLRYKLPKRLVRWDTKPVCIGQKQKWFLLQLVSGDAEINMQTSSTPEFDGWRWVSYWYPVRQVVSFKRDVYRRVMKEFASVVMSLQENTPK

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 \(c\).

\(|P_{f(7DIO_1)}(2) \setminus P_{f(3VCD_1)}(2)|=49\), \(|P_{f(3VCD_1)}(2) \setminus P_{f(7DIO_1)}(2)|=80\). 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 1:111000110011001100011111100001001011011010010001010010001011010101001010001101110001111000110000010100010011110
Pair \(Z_2\) Length of longest common subsequence
7DIO_1,3VCD_1 129 3
7DIO_1,2KDV_1 164 3
3VCD_1,2KDV_1 171 4

Newick tree

 
[
	2KDV_1:89.27,
	[
		7DIO_1:64.5,3VCD_1:64.5
	]:24.77
]

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 \((0.85)(0.8)(\frac{303 }{\log_{20} 303}-\frac{111}{\log_{20}111})=60.0\)
Status Protein1 Protein2 d d1/2
Query variables 7DIO_1 3VCD_1 71 56.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]