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Parikh vectors
6ORQ_1 4PLC_1 6HUS_1 Letter Amino acid
8 23 14 I Isoleucine
3 12 13 M Methionine
12 18 28 S Serine
22 27 23 L Leucine
1 9 21 F Phenylalanine
7 15 22 T Threonine
3 8 18 Y Tyrosine
10 23 41 A Alanine
6 17 35 D Aspartic acid
3 12 5 H Histidine
7 1 10 W Tryptophan
10 13 34 N Asparagine
3 5 2 C Cysteine
10 30 23 G Glycine
5 32 36 K Lycine
3 12 13 P Proline
6 33 35 V Valine
5 11 6 R Arginine
16 8 23 Q Glutamine
8 25 26 E Glutamic acid 

6ORQ_1|Chains A, C, F|RC1 variant of HIV-1 Env glycoprotein gp41|Human immunodeficiency virus 1 (11676)
>4PLC_1|Chains A, B, C, D|lactate dehydrogenase|Apicomplexa (5794)
>6HUS_1|Chain A|ABC transporter substrate-binding protein|Bifidobacterium longum subsp. infantis (1682)
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 \(w=f(c)\)
6ORQ , Knot 70 148 0.78 40 112 142 
AVSLGFLGAAGSTMGAASMTLTVQARNLLSGIVQQQSNLLRAPEPQQHLLKDTHWGIKQLQARVLAVEHYLRDQQLLGIWGCSGKLICCTNVPWNSSWSNRNLSEIWDNMTWLQWDKEISNYTQIIYGLLEESQNQQEKNEQDLLALD
4PLC , Knot 142 334 0.82 40 190 316 
MTGTMTKRKKISLIGSGMIGGTMAYLCAQKELGDVVLFDVVKNMPQGKALDLSHSSSIADTNVKVTGTNSYEDIKGSDVVIITAGLTKAPGKSDKEWSRDDLLPFNAKIMREVGENIKKYCPNAFVIVITNPLDVMVKVLHEHSGLPKNKVCGMAGVLDSSRFRHFIAEKLNVSPRDVQAMVIGAHGDKMVPLTRYVTVNGIPLQEFIKKGRITQEEIDEIVERTKNAGGEIVNLLGQGSAYFAPAASAIEMAEAYLKDKKRVLVCSCYLEGQYGHKDMFVGVPAVIGGNGVEKVIELELTPEEKELFDKSVEEVRKLQKAIKALGLEHHHHHH
6HUS , Knot 178 428 0.84 40 234 410 
KSDVTAQDVENALTDTSKNVELTVWAYSAKQMEPTVKAFEKKYPHIKINFVNTGAAEDHFTKFQNVVQAQKDIPDVVQMSANKFQQFAVSGALLNFANDSIEKAWSKLYTKTAWAQVHYAGGLYGAPQDATPLANYVRKDILDEHNLQVPESWEDIYNEGIKLHKEDSNKYMGILGSDISFFTNLYRSVGARLWKVNSVDDVELTMNSGKAKEFTEFLQKCLKDGVLEGGTVFTDEFNRSINDGRYATFINENWMGNTYKEQNPSLKGKMVVAAPPSWKGQPYQSSSVGSMMSVSAACPKEKQAAALAFINWLDSDKDAIQSWQDTNNGNFFMAASVYQDDENQRNKKETDGYYANDDVNAVYFDSMDKVNTDWEYLPFMSQVEVVFNDVIVPEMNENGDLVGAMAKAQQKLKAYAEDNGFKVTTDAD

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 \(c\).

\(|P_{f(6ORQ_1)}(2) \setminus P_{f(4PLC_1)}(2)|=50\), \(|P_{f(4PLC_1)}(2) \setminus P_{f(6ORQ_1)}(2)|=128\). 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 1:1101111111100111101010101001101110000011011010001100001110010101111000100001111110010110000111000100001001100101101000100000110111000000000000011110
Pair \(Z_2\) Length of longest common subsequence
6ORQ_1,4PLC_1 178 3
6ORQ_1,6HUS_1 192 3
4PLC_1,6HUS_1 168 3

Newick tree

 
[
	6ORQ_1:95.25,
	[
		4PLC_1:84,6HUS_1:84
	]:11.25
]

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 \((0.85)(0.8)(\frac{482 }{\log_{20} 482}-\frac{148}{\log_{20}148})=98.6\)
Status Protein1 Protein2 d d1/2
Query variables 6ORQ_1 4PLC_1 122 87.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]