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Parikh vectors
5GWA_1 7CAH_1 8BTX_1 Letter Amino acid
14 3 12 A Alanine
25 8 13 L Leucine
22 16 10 S Serine
9 4 8 Y Tyrosine
2 2 6 R Arginine
23 3 6 N Asparagine
2 2 1 C Cysteine
10 7 10 F Phenylalanine
8 6 7 P Proline
4 2 3 W Tryptophan
20 3 14 V Valine
4 10 10 Q Glutamine
16 7 10 G Glycine
8 1 5 H Histidine
38 6 10 K Lycine
3 0 7 M Methionine
19 3 12 D Aspartic acid
14 4 21 E Glutamic acid
19 10 8 I Isoleucine
20 8 10 T Threonine 

5GWA_1|Chain A|Beta-lactamase|Serratia marcescens (615)
>7CAH_1|Chain A[auth D]|Light chain of H014 Fab|Homo sapiens (9606)
>8BTX_1|Chain A|Protein archease|Homo sapiens (9606)
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 \(w=f(c)\)
5GWA , Knot 121 280 0.81 40 164 267 
AKGTDSLKNSIEKYLKDKKAKVGVAVLGIEDNFKLNVNEKHHYPMQSTYKFHLALAVLDKLDKENISVDKKLFVKKSDLQPNTWSPLKDKYPNGNLELSFSEIIKSTVSHSDNNGCDILFRFVGGTNKVHNFISKLGVKNISIKATEEEMHKAWNVQYTNWTTPDATVQLLKKFYKNEILSKNSYDFLLNTMIETTTGPKRLKGLLPDGTVVAHKTGSSDTNNKGITAATNDIGIITLPNGKHFAIAVYVSDSSEKSDVNEKIIAEICKSVWDYLVKDGK
7CAH , Knot 53 105 0.78 38 80 95 
IVLTQSPFQSVSPKEKVTITCRASQSISSNLHWYQQKPDQSPKLLIKYASQSISGIPSRFSGSGSGTDFTLTINSLEAEDFGIYFCQQTNFWPYIFGQGTKLEIL
8BTX , Knot 86 183 0.81 40 141 178 
GSHMMKGGSRVSNPAVMAQEEEDVRDYNLTEEQKAIKAKYPPVNRKYEYLDHTADVQLHAWGDTLEEAFEQCAMAMFGYMTDTGTVEPLQTVEVETQGDDLQSLLFHFLDEWLYKFSADEFFIPREVKVLSIDQRNFKLRSIGWGEEFSLSKHPQGTEVKAITYSAMQVYNEENPEVFVIIDI

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 \(c\).

\(|P_{f(5GWA_1)}(2) \setminus P_{f(7CAH_1)}(2)|=128\), \(|P_{f(7CAH_1)}(2) \setminus P_{f(5GWA_1)}(2)|=44\). 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 1:1010001000100010000101111111100010101000000110000010111111001000010100011100001010010110000101010101001100010000001001110111100010011001110010101000010011010000100101010110010000110000001110011000011001011110101110001000000011011000111101101001111101000000001000111010001100110010
Pair \(Z_2\) Length of longest common subsequence
5GWA_1,7CAH_1 172 3
5GWA_1,8BTX_1 171 4
7CAH_1,8BTX_1 157 3

Newick tree

 
[
	5GWA_1:88.03,
	[
		8BTX_1:78.5,7CAH_1:78.5
	]:9.53
]

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 \((0.85)(0.8)(\frac{385 }{\log_{20} 385}-\frac{105}{\log_{20}105})=85.7\)
Status Protein1 Protein2 d d1/2
Query variables 5GWA_1 7CAH_1 106 72.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]