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Parikh vectors
9FGX_1 7FNE_1 2PYT_1 Letter Amino acid
4 8 6 M Methionine
11 21 9 T Threonine
7 11 3 R Arginine
4 15 9 E Glutamic acid
7 19 5 K Lycine
2 5 4 W Tryptophan
3 14 3 N Asparagine
13 7 16 G Glycine
2 10 6 P Proline
13 18 8 A Alanine
3 1 1 C Cysteine
10 14 12 V Valine
3 23 5 I Isoleucine
6 29 8 L Leucine
3 10 9 F Phenylalanine
14 18 11 S Serine
8 7 2 Y Tyrosine
8 14 8 D Aspartic acid
6 9 5 Q Glutamine
1 5 3 H Histidine 

9FGX_1|Chains A[auth C], C[auth E]|anti-Lysozyme Gluebody|Lama glama (9844)
>7FNE_1|Chain A|Pre-mRNA-splicing factor 8|Saccharomyces cerevisiae S288C (559292)
>2PYT_1|Chains A, B|Ethanolamine utilization protein eutQ|Salmonella typhimurium LT2 (99287)
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 \(w=f(c)\)
9FGX , Knot 65 128 0.82 40 108 123 
SDVQLVENGGGCVKAGGSLRLSCAASGSTDSIEYMTWFRQAPGKAREGVAALYTHTGNTYYTDSVKGRFTISQDKAKNMAYLRMDSVKSEDTAIYTCGATRKAVPVRFALDQSSYDYWGQGTQVMVSS
7FNE , Knot 117 258 0.84 40 175 251 
GAMNSSNYAELFNNDIKLFVDDTNVYRVTVHKTFEGNVATKAINGCIFTLNPKTGHLFLKIIHTSVWAGQKRLSQLAKWKTAEEVSALVRSLPKEEQPKQIIVTRKAMLDPLEVHMLDFPNIAIRPTELRLPFSAAMSIDKLSDVVMKATEPQMVLFNIYDDWLDRISSYTAFSRLTLLLRALKTNEESAKMILLSDPTITIKSYHLWPSFTDEQWITIESQMRDLILTEYGRKYNVNISALTQTEIKDIILGQNIKA
2PYT , Knot 66 133 0.81 40 106 126 
GQSLELGTMQPSFTSVTGKGGVKVIDGSSVKFGRFDGAEPHCVGLTDLVTEQDGSSMAAGFMQWDNAFFPWTLNYDEIDMVLEGELHVRHEGETMIAKAGDVMFIPKGSSIEFGTPTSVRFLYVAWPANWQSV

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 \(c\).

\(|P_{f(9FGX_1)}(2) \setminus P_{f(7FNE_1)}(2)|=57\), \(|P_{f(7FNE_1)}(2) \setminus P_{f(9FGX_1)}(2)|=124\). 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 1:00101100111010111010100110100001001011001110100111110000100000001010101000010011010100100000110001100011110111000000011010011100
Pair \(Z_2\) Length of longest common subsequence
9FGX_1,7FNE_1 181 4
9FGX_1,2PYT_1 142 3
7FNE_1,2PYT_1 175 4

Newick tree

 
[
	7FNE_1:94.25,
	[
		9FGX_1:71,2PYT_1:71
	]:23.25
]

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 \((0.85)(0.8)(\frac{386 }{\log_{20} 386}-\frac{128}{\log_{20}128})=78.2\)
Status Protein1 Protein2 d d1/2
Query variables 9FGX_1 7FNE_1 101 74
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]