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Parikh vectors
7GAK_1 1WSR_1 2HSL_1 Letter Amino acid
4 34 4 G Glycine
9 17 0 K Lycine
9 22 0 S Serine
11 37 0 V Valine
4 11 0 N Asparagine
5 19 0 D Aspartic acid
0 8 4 C Cysteine
7 15 0 Q Glutamine
6 18 0 P Proline
3 30 4 A Alanine
3 11 0 H Histidine
8 10 0 I Isoleucine
10 44 0 L Leucine
4 14 0 M Methionine
4 9 0 Y Tyrosine
9 24 0 R Arginine
12 19 0 E Glutamic acid
7 8 0 F Phenylalanine
6 22 0 T Threonine
0 3 0 W Tryptophan 

7GAK_1|Chain A|Microtubule-associated proteins 1A/1B light chain 3B|Homo sapiens (9606)
>1WSR_1|Chains A, B|Aminomethyltransferase|Homo sapiens (9606)
>2HSL_1|Chain A|5'-D(*CP*CP*AP*AP*AP*GP*(D1P)P*AP*CP*CP*GP*GP*G)-3'|synthetic construct (32630)
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 \(w=f(c)\)
7GAK , Knot 59 121 0.78 36 90 118 
SMPSEKTFKQRRTFEQRVEDVRLIREQHPTKIPVIIERYKGEKQLPVLDKTKFLVPDHVNMSELIKIIRRRLQLNANQAFFLLVNGHSMVSVSTPISEVYESEKDEDGFLYMVYASQETFG
1WSR , Knot 158 375 0.83 40 213 357 
AQEVLRRTPLYDFHLAHGGKMVAFAGWSLPVQYRDSHTDSHLHTRQHCSLFDVSHMLQTKILGSDRVKLMESLVVGDIAELRPNQGTLSLFTNEAGGILDDLIVTNTSEGHLYVVSNAGCWEKDLALMQDKVRELQNQGRDVGLEVLDNALLALQGPTAAQVLQAGVADDLRKLPFMTSAVMEVFGVSGCRVTRCGYTGEDGVEISVPVAGAVHLATAILKNPEVKLAGLAARDSLRLEAGLCLYGNDIDEHTTPVEGSLSWTLGKRRRAAMDFPGAKVIVPQLKGRVQRRRVGLMCEGAPMRAHSPILNMEGTKIGTVTSGCPSPSLKKNVAMGYVPCEYSRPGTMLLVEVRRKQQMAVVSKMPFVPTNYYTLK
2HSL , Knot 7 13 0.46 8 9 11 
CCAAAGXACCGGG

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 \(c\).

\(|P_{f(7GAK_1)}(2) \setminus P_{f(1WSR_1)}(2)|=34\), \(|P_{f(1WSR_1)}(2) \setminus P_{f(7GAK_1)}(2)|=157\). 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 1:0110000100000100010010110000100111110000100011110000111100101001101100010101001111110100110100110010000000011101101000011
Pair \(Z_2\) Length of longest common subsequence
7GAK_1,1WSR_1 191 3
7GAK_1,2HSL_1 99 1
1WSR_1,2HSL_1 214 2

Newick tree

 
[
	1WSR_1:11.56,
	[
		7GAK_1:49.5,2HSL_1:49.5
	]:64.06
]

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 \((0.85)(0.8)(\frac{496 }{\log_{20} 496}-\frac{121}{\log_{20}121})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 7GAK_1 1WSR_1 142 93
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]