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Parikh vectors
1VPY_1 6UEX_1 7LDZ_1 Letter Amino acid
22 12 4 E Glutamic acid
13 27 13 G Glycine
11 19 4 D Aspartic acid
12 11 8 T Threonine
6 0 2 W Tryptophan
17 17 0 S Serine
15 9 1 Y Tyrosine
18 17 4 N Asparagine
14 10 5 Q Glutamine
17 9 2 F Phenylalanine
11 14 1 H Histidine
13 18 13 I Isoleucine
27 20 12 L Leucine
19 25 7 K Lycine
10 12 2 M Methionine
20 10 3 A Alanine
10 13 4 R Arginine
2 0 2 C Cysteine
14 10 6 P Proline
18 20 6 V Valine 

1VPY_1|Chain A|PROTEIN (hypothetical protein EF0366)|Enterococcus faecalis (226185)
>6UEX_1|Chain A|Regulatory protein MsrR|Staphylococcus aureus (strain N315) (158879)
>7LDZ_1|Chains A, B|Protease|Human immunodeficiency virus 1 (11676)
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 \(w=f(c)\)
1VPY , Knot 130 289 0.85 40 197 274 
MGSDKIHHHHHHMIRLGLTSFSEHDYLTGKKRSTLYEYASHLPLVEMDTAYYGIPPKERVAEWVKAVPENFRFVMKVYSGISCQGEWQTYYASEEEMITAFLESMAPLIESKKLFAFLVQFSGTFGCTKENVAYLQKIRHWFKDLPIAIELRNNSWYQPNFVKQMLQFMKENQFSLVIVDEPQIPTNPVPFYPYVTNPNLVLFRFHGRNAAGWLANDAEWRKKRTLYHYNTQEIADLSEAVLKMSQEAKEVGVIFNNNSGGDAAENALQMQKVLNLSYDDLNPKQLDLF
6UEX , Knot 115 273 0.78 36 166 255 
MGSSHHHHHHHHHHSSGLVPRGSHMDGKISILVLGADKAQGGQSRTDSIMVVQYDFINKKMKMMSVMRDIYADIPGYGKHKINSAYALGGPELLRKTLDKNLGINPEYYAVVDFTGFEKMIDELMPEGVPINVEKDMSKNIGVSLKKGNHRLNGKELLGYARFRHDPEGDFGRVRRQQQVMQTLKKEMVNFRTVVKLPKVAGILRGYVNTNIPDSGIFQTGLSFGIRGEKDVKSLTVPIKNSYEDVNTNTDGSALQINKNTNKQAIKDFLDED
7LDZ , Knot 52 99 0.80 38 79 94 
PQITLWKRPLVTIKIGGQLKEALLDTGADDTVLEEMNLPGRWKPKMIGGIGGFIKVRQYDQILIEICGHKAIGTVLVGPTPVNIIGRNLLTQIGCTLNF

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 \(c\).

\(|P_{f(1VPY_1)}(2) \setminus P_{f(6UEX_1)}(2)|=101\), \(|P_{f(6UEX_1)}(2) \setminus P_{f(1VPY_1)}(2)|=70\). 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 1:1100010000001101110010000010100000100010011110100100111100011011011100101110100110001010000100001101110011111000011111101010110000011010010011001111101000010010110011011000010111100101100111101010010111101010011111100101000001000000011010011101000100111110000110110011010011010000101001011
Pair \(Z_2\) Length of longest common subsequence
1VPY_1,6UEX_1 171 6
1VPY_1,7LDZ_1 178 3
6UEX_1,7LDZ_1 157 3

Newick tree

 
[
	1VPY_1:90.00,
	[
		6UEX_1:78.5,7LDZ_1:78.5
	]:11.50
]

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 \((0.85)(0.8)(\frac{562 }{\log_{20} 562}-\frac{273}{\log_{20}273})=81.6\)
Status Protein1 Protein2 d d1/2
Query variables 1VPY_1 6UEX_1 106 101
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]