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Parikh vectors
7GRP_1 4FYO_1 7OUE_1 Letter Amino acid
7 11 2 H Histidine
24 7 9 T Threonine
11 13 18 R Arginine
17 13 9 D Aspartic acid
17 7 12 F Phenylalanine
16 12 20 S Serine
17 20 14 A Alanine
26 18 15 G Glycine
11 9 25 I Isoleucine
13 10 12 P Proline
21 16 6 N Asparagine
9 25 21 E Glutamic acid
29 28 23 L Leucine
11 25 19 K Lycine
10 15 5 M Methionine
3 6 6 W Tryptophan
11 19 10 Y Tyrosine
27 23 12 V Valine
12 5 0 C Cysteine
14 9 4 Q Glutamine 

7GRP_1|Chains A, B|3C-like proteinase nsp5|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4FYO_1|Chain A|Tyrosine-protein kinase SYK|Homo sapiens (9606)
>7OUE_1|Chains A, C, E, G|N-glycosylase/DNA lyase|Pyrococcus abyssi (strain GE5 / Orsay) (272844)
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 \(w=f(c)\)
7GRP , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
4FYO , Knot 129 291 0.83 40 191 276 
MALEEIRPKEVYLDRKLLTLEDKELGSGNFGTVKKGYYQMKKVVKTVAVKILKNEANDPALKDELLAEANVMQQLDNPYIVRMIGICEAESWMLVMEMAELGPLNKYLQQNRHVKDKNIIELVHQVSMGMKYLEESNFVHRDLAARNVLLVTQHYAKISDFGLSKALRADENYYKAQTHGKWPVKWYAPECINYYKFSSKSDVWSFGVLMWEAFSYGQKPYRGMKGSEVTAMLEKGERMGCPAGCPREMYDLMNLCWTYDVENRPGFAAVELRLRNYYYDVVNEGHHHHHH
7OUE , Knot 106 242 0.80 38 157 228 
GSHMIARIIGEIGIEGARFIEENIDEQFKALRYLSKGIDSETFVKLVIANSLVSYQLTGKGEQWWWEFAKYFYGRDVKSIYLAYKEFLPNSRFNRRLIPQKLSRIRRVETFLSTLTEERIEEYYGDMSSLWGSIARALGVDKESKTVVFSVKMFGYAARIVLSTFNPYPMEIPIPEDSRIVKLTKKLTNEKPRKFWMKIARESGVPPLHIDSILWPLLGGASIDSAPPELRDKLAELIKIIR

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 \(c\).

\(|P_{f(7GRP_1)}(2) \setminus P_{f(4FYO_1)}(2)|=88\), \(|P_{f(4FYO_1)}(2) \setminus P_{f(7GRP_1)}(2)|=83\). 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 1:011001111010101011010010001011110011001001100000110100001110000001110110101011100100011010100101001000110101100101110001010110001101010101011010010111010000101001001011011011001010101111000010111000010101111101111010011100100010010111100000110000101111101001111101010100110011010011101110001011011000011010
Pair \(Z_2\) Length of longest common subsequence
7GRP_1,4FYO_1 171 4
7GRP_1,7OUE_1 195 3
4FYO_1,7OUE_1 164 4

Newick tree

 
[
	7GRP_1:94.70,
	[
		4FYO_1:82,7OUE_1:82
	]:12.70
]

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 \((0.85)(0.8)(\frac{597 }{\log_{20} 597}-\frac{291}{\log_{20}291})=85.7\)
Status Protein1 Protein2 d d1/2
Query variables 7GRP_1 4FYO_1 106 104.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]