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Parikh vectors
6PRF_1 7YUP_1 2QHF_1 Letter Amino acid
4 71 70 A Alanine
4 14 5 N Asparagine
13 57 43 G Glycine
8 87 28 L Leucine
7 46 8 K Lycine
9 45 20 T Threonine
2 9 3 W Tryptophan
5 50 25 D Aspartic acid
9 41 17 I Isoleucine
4 57 36 R Arginine
4 33 10 Q Glutamine
2 23 11 M Methionine
1 23 7 Y Tyrosine
0 0 1 C Cysteine
4 71 23 E Glutamic acid
1 14 14 H Histidine
4 11 6 F Phenylalanine
7 48 22 P Proline
0 27 19 S Serine
11 66 39 V Valine 

6PRF_1|Chains A, B|Protease|Human immunodeficiency virus type 1 group M subtype (11678)
>7YUP_1|Chains A, B, C, D, E|Lon protease|Meiothermus taiwanensis (172827)
>2QHF_1|Chain A|Chorismate synthase|Mycobacterium tuberculosis (1773)
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 \(w=f(c)\)
6PRF , Knot 52 99 0.80 36 80 95 
PQITLWKRPFVTVKVGGQLKEALLDTGADNTIFEDINLPGRWKPKMVGGIGGFLKVREYDQVPIEIAGHKVIGTVLVGPTPVNVIGRDTMTQIGATLNF
7YUP , Knot 295 793 0.82 38 260 690 
MRLELPVIPLRNTVILPHTTTPVDVGRAKSKRAVEEAMGADRLIFLVAQRDPEVDDPAPDDLYTWGVQAVVKQAMRLPDGTLQVMVEARARAQVTDYIPGPYLRARGEVFSEIFPIDEAVVRVLVEELKEAFEKYVANHKSLRLDRYQLEAVKGTSDPAMLADTIAYHATWTVAEKQEILELTDLEARLKKVLGLLSRDLERFELDKRVAQRVKEQMDTNQREYYLREQMKAIQKELGGEDGLSDLEALRKKIEEVGMPEAVKTKALKELDRLERMQQGSPEATVARTYLDWLTEVPWSKADPEVLDINHTRQVLDEDHYGLKDVKERILEYLAVRQLTQGLDVRNKAPILVLVGPPGVGKTSLGRSIARSMNRKFHRISLGGVRDEAEIRGHRRTYIGAMPGKLIHAMKQVGVINPVILLDEIDKMSSDWRGDPASAMLEVLDPEQNNTFTDHYLDVPYDLSKVFFITTANTLQTIPRPLLDRMEVIEIPGYTNMEKQAIARQYLWPKQVRESGMEGRIEVTDAAILRVISEYTREAGVRGLERELGKIARKGAKFWLEGAWEGLRTIDASDIPTYLGIPRYRPDKAETEPQVGTAQGLAWTPVGGTLLTIEVAAVPGSGKLSLTGQLGEVMKESAQAALTYLRAHTQDYGLPEDFYNKVDLHVHVPDGATPKDGPSAGITMATAIASALSRRPARMDIAMTGEVSLRGKVMPIGGVKEKLLAAHQAGIHKIVLPKDNEAQLEELPKEVLEGLEIKLVEDVGEVLEYLLLPEPTMPPVVQPSDNRQQPGAGA
2QHF , Knot 164 407 0.80 40 193 369 
MLRWITAGESHGRALVAVVEGMVAGVHVTSADIADQLARRRLGYGRGARMTFERDAVTVLSGIRHGSTLGGPIAIEIGNTEWPKWETVMAADPVDPAELADVARNAPLTRPRPGHADYAGMLKYGFDDARPVLERASARETAARVAAGTVARAFLRQALGVEVLSHVISIGASAPYEGPPPRAEDLPAIDASPVRAYDKAAEADMIAQIEAAKKDGDTLGGVVEAVALGLPVGLGSFTSGDHRLDSQLAAAVMGIQAIKGVEIGDGFQTARRRGSRAHDEMYPGPDGVVRSTNRAGGLEGGMTNGQPLRVRAAMKPISTVPRALATVDLATGDEAVAIHQRSDVCAVPAAGVVVETMVALVLARAALEKFGGDSLAETQRNIAAYQRSVADREAPAARVSGHHHHHH

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 \(c\).

\(|P_{f(6PRF_1)}(2) \setminus P_{f(7YUP_1)}(2)|=13\), \(|P_{f(7YUP_1)}(2) \setminus P_{f(6PRF_1)}(2)|=193\). 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 1:101011001110101110100111001100011001011101010111111111010000011101110011101111101101110001001110101
Pair \(Z_2\) Length of longest common subsequence
6PRF_1,7YUP_1 206 5
6PRF_1,2QHF_1 163 3
7YUP_1,2QHF_1 135 4

Newick tree

 
[
	6PRF_1:99.91,
	[
		2QHF_1:67.5,7YUP_1:67.5
	]:32.41
]

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 \((0.85)(0.8)(\frac{892 }{\log_{20} 892}-\frac{99}{\log_{20}99})=223.\)
Status Protein1 Protein2 d d1/2
Query variables 6PRF_1 7YUP_1 274 152.5
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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