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Parikh vectors
1HHP_1 1QUP_1 5ECH_1 Letter Amino acid
2 5 11 C Cysteine
6 6 20 Q Glutamine
2 8 32 F Phenylalanine
4 6 26 R Arginine
4 11 28 D Aspartic acid
4 15 40 E Glutamic acid
12 13 49 L Leucine
3 16 28 N Asparagine
1 20 43 S Serine
13 16 36 G Glycine
1 5 10 H Histidine
13 18 37 I Isoleucine
6 16 30 K Lycine
2 2 12 M Methionine
6 10 31 P Proline
8 14 36 T Threonine
3 15 34 A Alanine
1 6 22 Y Tyrosine
6 17 46 V Valine
2 3 4 W Tryptophan 

1HHP_1|Chain A|UNLIGANDED HIV-1 PROTEASE|Human immunodeficiency virus type 1 (BRU ISOLATE) (11686)
>1QUP_1|Chains A, B|SUPEROXIDE DISMUTASE 1 COPPER CHAPERONE|Saccharomyces cerevisiae (4932)
>5ECH_1|Chains A, D|Jasmonic acid-amido synthetase JAR1|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
1HHP , Knot 52 99 0.80 40 81 94 
PQITLWQRPLVTIKIGGQLKEALLDTGADDTVLEEMSLPGRWKPKMIGGIGGFIKVRQYDQILIEICGHKAIGTVLVGPTPVNIIGRNLLTQIGCTLNF
1QUP , Knot 103 222 0.83 40 155 216 
TTNDTYEATYAIPMHCENCVNDIKACLKNVPGINSLNFDIEQQIMSVESSVAPSTIINTLRNCGKDAIIRGAGKPNSSAVAILETFQKYTIDQKKDTAVRGLARIVQVGENKTLFDITVNGVPEAGNYHASIHEKGDVSKGVESTGKVWHKFDEPIECFNESDLGKNLYSGKTFLSAPLPTWQLIGRSFVISKSLNHPENEPSSVKDYSFLGVIARSAGVWE
5ECH , Knot 234 575 0.86 40 274 538 
MLEKVETFDMNRVIDEFDEMTRNAHQVQKQTLKEILLKNQSAIYLQNCGLNGNATDPEEAFKSMVPLVTDVELEPYIKRMVDGDTSPILTGHPVPAISLSSGTSQGRPKFIPFTDELMENTLQLFRTAFAFRNRDFPIDDNGKALQFIFSSKQYISTGGVPVGTATTNVYRNPNFKAGMKSITSPSCSPDEVIFSPDVHQALYCHLLSGILFRDQVQYVFAVFAHGLVHAFRTFEQVWEEIVTDIKDGVLSNRITVPSVRTAMSKLLTPNPELAETIRTKCMSLSNWYGLIPALFPNAKYVYGIMTGSMEPYVPKLRHYAGDLPLVSHDYGSSEGWIAANVTPRLSPEEATFAVIPNLGYFEFLPVSETGEGEEKPVGLTQVKIGEEYEVVITNYAGLYRYRLGDVVKVIGFYNNTPQLKFICRRNLILSINIDKNTERDLQLSVESAAKRLSEEKIEVIDFSSYIDVSTDPGHYAIFWEISGETNEDVLQDCCNCLDRAFIDAGYVSSRKCKTIGALELRVVAKGTFRKIQEHFLGLGSSAGQFKMPRCVKPSNAKVLQILCENVVSSYFSTAF

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 \(c\).

\(|P_{f(1HHP_1)}(2) \setminus P_{f(1QUP_1)}(2)|=40\), \(|P_{f(1QUP_1)}(2) \setminus P_{f(1HHP_1)}(2)|=114\). 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 1:101011001110101110100111001100011001011101010111111111010000011101010011101111101101110011001100101
Pair \(Z_2\) Length of longest common subsequence
1HHP_1,1QUP_1 154 3
1HHP_1,5ECH_1 225 4
1QUP_1,5ECH_1 189 4

Newick tree

 
[
	5ECH_1:11.42,
	[
		1HHP_1:77,1QUP_1:77
	]:34.42
]

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 \((0.85)(0.8)(\frac{321 }{\log_{20} 321}-\frac{99}{\log_{20}99})=69.4\)
Status Protein1 Protein2 d d1/2
Query variables 1HHP_1 1QUP_1 88 62.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]