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Parikh vectors
5MUY_1 4FCH_1 7EZN_1 Letter Amino acid
9 3 4 H Histidine
40 8 19 L Leucine
26 14 13 K Lycine
6 8 8 M Methionine
24 14 17 S Serine
17 17 18 D Aspartic acid
2 1 2 C Cysteine
13 3 9 Q Glutamine
18 20 11 T Threonine
2 8 3 W Tryptophan
14 6 10 P Proline
13 11 7 Y Tyrosine
11 15 16 A Alanine
15 4 9 R Arginine
11 18 4 F Phenylalanine
9 12 6 N Asparagine
29 11 13 E Glutamic acid
20 9 9 I Isoleucine
22 24 14 G Glycine
27 15 13 V Valine 

5MUY_1|Chains A, B|L protein|CAS virus (1223561)
>4FCH_1|Chains A, B|Outer membrane protein SusE|Bacteroides thetaiotaomicron (226186)
>7EZN_1|Chain A|Protein-tyrosine-phosphatase|Cryptococcus neoformans var. grubii serotype A (strain H99 / ATCC 208821 / CBS 10515 / FGSC 9487) (235443)
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 \(w=f(c)\)
5MUY , Knot 147 328 0.86 40 203 318 
GPTAVQELNRICDGMILSSILRPQMVEPSLIGDVLKTEPFTSSLEVLKDIKVSGLMRYIKQVLLGRKSYYKFGEEHADGDQNLFDYQFTGTPEEPIKGYWTTTISYRDSKPKISLTIRQEFVEGGVESQAVLATVVGRPHLQDFLLLKRKHLEYSDYPESIDLIEFGDVKVIEKTVGHGQPVVLYMASLFPIGLTKIAEQGIGCHNEKNDATLKSLLKTAILNMTPDDVSLLPRSKLDAIDHVVRNVYLRFNNSSAVSYGLHKLQSFEKILQLNGRIYETKESTLVFHHKKGVLVEDLKGLMSYKTLRVSEWREKQEPEEARVEVLEE
4FCH , Knot 103 221 0.83 40 154 215 
GHMASTSSLTPPKTMFIVGSMLDTDWKVWKPMAGVYGMDGQFYSMIYFDANSEFKFGTKENEYIGINDNRVTVTDKAGAGVSGSDNFVVENAGWYLFYVKAAVKGDDYQFTITFYPAEVYLFGNTTGGSWAFNDEWKFTVPATKDGNFVSPAMTASGEVRMCFKTDLDWWRTEFTLHDGEIFYRDFNLIDSWTEKGDGYSIQGSAGNVIHLNFTAGTGEKK
7EZN , Knot 92 205 0.79 40 144 197 
KEAMGHMQEVVDGLWVGDLVAANDDDELEKNGIKNILSALRPSLKFSDKYAVYPLEIDDSADTDLLSHLPSCVAWIKEILDLRQKAAEPSSQKNGTENGESLKRSPDIDTVAQPGKPGGVLVHCQAGMSRSASIVAAYLMSQYDLDPMEAMTMIREKRPVVEPSATFWHQLGLFYTTDGKVSLKDRSTRQYYMERTTTQFINGDG

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 \(c\).

\(|P_{f(5MUY_1)}(2) \setminus P_{f(4FCH_1)}(2)|=105\), \(|P_{f(4FCH_1)}(2) \setminus P_{f(5MUY_1)}(2)|=56\). 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 1:1101100100100111100110101101011101100011000101100101011100100111100000011000101000110001010100110101000100000010101010001101110001111011101010011110000100000100101101101011000110101111011011111100110011100000001010011001110101001011100010110011001010100001100110010010011010101000000011100001111001011100001010010000010010101100
Pair \(Z_2\) Length of longest common subsequence
5MUY_1,4FCH_1 161 5
5MUY_1,7EZN_1 177 4
4FCH_1,7EZN_1 162 3

Newick tree

 
[
	7EZN_1:86.22,
	[
		5MUY_1:80.5,4FCH_1:80.5
	]:5.72
]

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 \((0.85)(0.8)(\frac{549 }{\log_{20} 549}-\frac{221}{\log_{20}221})=93.8\)
Status Protein1 Protein2 d d1/2
Query variables 5MUY_1 4FCH_1 119 99
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]