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Parikh vectors
5REJ_1 6AIQ_1 1NZN_1 Letter Amino acid
29 5 17 L Leucine
16 2 7 S Serine
12 4 1 C Cysteine
14 5 6 Q Glutamine
7 1 1 H Histidine
11 2 3 I Isoleucine
17 2 4 F Phenylalanine
27 4 9 V Valine
11 2 7 R Arginine
17 4 6 D Aspartic acid
9 4 15 E Glutamic acid
10 1 2 M Methionine
13 6 2 P Proline
24 4 3 T Threonine
17 15 8 A Alanine
21 6 5 N Asparagine
26 7 7 G Glycine
11 5 15 K Lycine
3 3 1 W Tryptophan
11 1 7 Y Tyrosine 

5REJ_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>6AIQ_1|Chain A|High-potential iron-sulfur protein|Thermochromatium tepidum (1050)
>1NZN_1|Chain A|Fission protein Fis1p|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 \(w=f(c)\)
5REJ , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
6AIQ , Knot 44 83 0.78 40 72 81 
AAPANAVTADDPTAIALKYNQDATKSERVAAARPGLPPEEQHCANCQFMQANVGEGDWKGCQLFPGKLINVNGWCASWTLKAG
1NZN , Knot 64 126 0.82 40 93 123 
GSHMEAVLNELVSVEDLLKFEKKFQSEKAAGSVSKSTQFEYAWCLVRTRYNDDIRKGIVLLEELLPKGSKEEQRDYVFYLAVGNYRLKEYEKALKYVRGLLQTEPQNNQAKELERLIDKAMKKDGL

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 \(c\).

\(|P_{f(5REJ_1)}(2) \setminus P_{f(6AIQ_1)}(2)|=154\), \(|P_{f(6AIQ_1)}(2) \setminus P_{f(5REJ_1)}(2)|=30\). 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
5REJ_1,6AIQ_1 184 3
5REJ_1,1NZN_1 175 4
6AIQ_1,1NZN_1 119 4

Newick tree

 
[
	5REJ_1:97.80,
	[
		1NZN_1:59.5,6AIQ_1:59.5
	]:38.30
]

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 \((0.85)(0.8)(\frac{389 }{\log_{20} 389}-\frac{83}{\log_{20}83})=94.6\)
Status Protein1 Protein2 d d1/2
Query variables 5REJ_1 6AIQ_1 118 74.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]