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Parikh vectors
2KVV_1 9AST_1 3SVU_1 Letter Amino acid
3 19 10 F Phenylalanine
2 4 11 P Proline
3 3 2 W Tryptophan
3 13 11 Y Tyrosine
6 26 9 I Isoleucine
1 26 10 D Aspartic acid
8 27 17 E Glutamic acid
4 27 18 L Leucine
3 9 10 M Methionine
4 21 19 T Threonine
4 32 15 A Alanine
3 16 9 N Asparagine
3 13 7 Q Glutamine
6 17 22 G Glycine
6 18 9 R Arginine
7 6 7 H Histidine
4 30 19 K Lycine
4 18 11 S Serine
4 19 14 V Valine
0 10 5 C Cysteine 

2KVV_1|Chain A|Putative excisionase|Klebsiella pneumoniae (272620)
>9AST_1|Chain A|Guanine nucleotide-binding protein G(i) subunit alpha-1|Homo sapiens (9606)
>3SVU_1|Chains A, B, C, D|mkate S143C|Artificial gene (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 \(w=f(c)\)
2KVV , Knot 40 78 0.74 38 67 73 
MAQIIFNEEWMVEKALMVRTGLGARQIESYRQGAWIEGVHFKRVSPSGEKTLRGTTWYNYPEINKFIRDSLEHHHHHH
9AST , Knot 154 354 0.85 40 205 334 
MGCTLSAEDKAAVERSKMIDRNLREDGEKAAREVKLLLLGAGESGKNTIVKQMKIIHEAGYSEEECKQYKAVVYSNTIQSIIAIIRAMGRLKIDFGDSARADDARQLFVLAGAAEEGFMTAELAGVIKRLWKDSGVQACFNRSREYQLNDSAAYYLNDLDRIAQPNYIPTQQDVLRTRVKTTGIVETHFTFKDLHFKMFDVGAQRSERKKWIHCFEGVTAIIFCVALSDYDLVLAEDEEMNRMHASMKLFDSICNNKWFTDTSIILFLNKKDLFEEKIKKSPLTICYPEYAGSNTYEEAAAYIQCQFEDLNKRKDTKEIYTHFTCSTDTKNVQFVFDAVTDVIIKNNLKDCGLF
3SVU , Knot 111 235 0.86 40 169 228 
AAASMSALITENMHMKLYMEGTVNNHHFKCTSEGEGKPYEGTQTMRIKVVEGGPLPFAFDILATSFMYGSKTFINHTQGIPDFFKQSFPEGFTWERVTTYEDGGVLTATQDTSLQDGCLIYNVKIRGVNFPSNGPVMQKKTLGWEACTEMLYPADGGLEGRSDMALKLVGGGHLICNLKTTYRSKKPAKNLKMPGVYYVDRRLERIKEADKETYVEQHEVAVARYCDLPSKLAHK

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 \(c\).

\(|P_{f(2KVV_1)}(2) \setminus P_{f(9AST_1)}(2)|=29\), \(|P_{f(9AST_1)}(2) \setminus P_{f(2KVV_1)}(2)|=167\). 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 1:110111000111001111001111001000001111011010010101000101001000101001100010000000
Pair \(Z_2\) Length of longest common subsequence
2KVV_1,9AST_1 196 3
2KVV_1,3SVU_1 176 3
9AST_1,3SVU_1 188 3

Newick tree

 
[
	9AST_1:98.54,
	[
		2KVV_1:88,3SVU_1:88
	]:10.54
]

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 \((0.85)(0.8)(\frac{432 }{\log_{20} 432}-\frac{78}{\log_{20}78})=108.\)
Status Protein1 Protein2 d d1/2
Query variables 2KVV_1 9AST_1 136 81.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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