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Parikh vectors
1AFJ_1 7FCS_1 4BNY_1 Letter Amino acid
3 5 5 P Proline
10 18 35 A Alanine
4 7 10 D Aspartic acid
2 0 0 C Cysteine
4 12 33 G Glycine
2 1 5 F Phenylalanine
1 1 14 I Isoleucine
3 28 27 L Leucine
1 23 14 R Arginine
0 0 13 N Asparagine
3 19 10 Q Glutamine
11 14 25 V Valine
9 8 16 T Threonine
0 4 2 W Tryptophan
1 4 5 Y Tyrosine
3 27 15 E Glutamic acid
0 1 8 H Histidine
9 8 8 K Lycine
1 4 8 M Methionine
5 9 16 S Serine 

1AFJ_1|Chain A|MERP|Shigella flexneri (623)
>7FCS_1|Chain A|Apolipoprotein E|Homo sapiens (9606)
>4BNY_1|Chains A, B, C, D|3-OXOACYL-[ACYL-CARRIER-PROTEIN] REDUCTASE FABG|PSEUDOMONAS AERUGINOSA (208964)
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 \(w=f(c)\)
1AFJ , Knot 39 72 0.77 34 60 68 
ATQTVTLAVPGMTCAACPITVKKALSKVEGVSKVDVGFEKREAVVTFDDTKASVQKLTKATADAGYPSSVKQ
7FCS , Knot 83 193 0.75 36 114 179 
GPKVEQAVETEPEPELRQQTEWQSGQRWELALGRFWDYLRWVQTLSEQVQEELLSSQVTQELRALMDETMKELKAYKSELEEQLTPVAEETRARLSKELQAAQARLGADMEDVVGRLVQYRGEVQAMLGQSTEELRVRLASHLRKLRKRLLRDADDLQKRLAVYQAGAREGAERGLSAIRERLGPLVEQGRVR
4BNY , Knot 113 269 0.78 38 157 252 
MHHHHHHSSGVDLGTENLYFQSMSLQGKVALVTGASRGIGQAIALELGRLGAVVIGTATSASGAEKIAETLKANGVEGAGLVLDVSSDESVAATLEHIQQHLGQPLIVVNNAGITRDNLLVRMKDDEWFDVVNTNLNSLYRLSKAVLRGMTKARWGRIINIGSVVGAMGNAGQTNYAAAKAGLEGFTRALAREVGSRAITVNAVAPGFIDTDMTRELPEAQREALLGQIPLGRLGQAEEIAKVVGFLASDGAAYVTGATVPVNGGMYMS

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 \(c\).

\(|P_{f(1AFJ_1)}(2) \setminus P_{f(7FCS_1)}(2)|=31\), \(|P_{f(7FCS_1)}(2) \setminus P_{f(1AFJ_1)}(2)|=85\). 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 1:100010111111001101101001100101100101110000111010000101001001010110100100
Pair \(Z_2\) Length of longest common subsequence
1AFJ_1,7FCS_1 116 3
1AFJ_1,4BNY_1 139 3
7FCS_1,4BNY_1 135 4

Newick tree

 
[
	4BNY_1:71.66,
	[
		1AFJ_1:58,7FCS_1:58
	]:13.66
]

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 \((0.85)(0.8)(\frac{265 }{\log_{20} 265}-\frac{72}{\log_{20}72})=62.4\)
Status Protein1 Protein2 d d1/2
Query variables 1AFJ_1 7FCS_1 74 50
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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