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Parikh vectors
1SUM_1 1QYF_1 6PHW_1 Letter Amino acid
1 1 8 W Tryptophan
9 5 31 R Arginine
32 15 53 E Glutamic acid
19 19 31 K Lycine
9 21 54 G Glycine
15 12 36 I Isoleucine
28 19 83 L Leucine
17 11 39 A Alanine
10 13 47 N Asparagine
12 17 36 D Aspartic acid
4 16 40 T Threonine
4 8 42 Q Glutamine
2 9 26 H Histidine
9 11 31 F Phenylalanine
13 11 54 S Serine
7 10 35 Y Tyrosine
22 16 49 V Valine
3 2 2 C Cysteine
12 4 20 M Methionine
7 10 23 P Proline 

1SUM_1|Chain A[auth B]|Phosphate transport system protein phoU homolog 2|Thermotoga maritima (2336)
>1QYF_1|Chain A|green-fluorescent protein|Aequorea victoria (6100)
>6PHW_1|Chain A|Alpha-galactosidase|Streptococcus pneumoniae serotype 4 (strain ATCC BAA-334 / TIGR4) (170187)
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 \(w=f(c)\)
1SUM , Knot 104 235 0.80 40 148 222 
MNRLLNEKVEEFKKGVLKAGWFIEKMFRNSISSLVERNESLAREVIADEEVVDQMEVEIQEKAMEVLGLFSPIGKPLLTVTAGIRVAELIENIADKCHDIAKNVLELMEEPPLKPLEDIPAMANQTSEMLKFALRMFADVNVEKSFEVCRMDSKVDDLYEKVREELLLYMMESPKYVKRALLLLEIAGNIEIIADYATNIVEVSVYMVQGEAYKCYHDELLLFKKSGGVLFESSD
1QYF , Knot 106 230 0.83 40 154 218 
MASKGEELFTGVVPILVELDGDVNGHKFSVSGEGEGDATYGKLTLKFICTTGKLPVPWPTLVTTLTYGVQCFSRYPDHMKQHDFFKSAMPEGYVQEATISFKDDGNYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNYNSHNVYITADKQKNGIKANFKIRHNIEDGSVQLADHYQQNTPIGDGPVLLPDNHYLSTQSALSKDPNEKRDHMVLLEFVTAAGI
6PHW , Knot 281 740 0.83 40 291 684 
MGSSHHHHHHSSGLVPRGSHMGVRIENNLFYVESKNLSLIIENRNGYLLLKHLGKTIKNYKGSNSVYERDHAFSGNPTATNRTFSLDTQRQIFGQHGLGDFRKPTIQVQHSVTEVTDFRFVEAKILKGQNGPQGLPSPHSMDDTETLVLMLEDSKAQLSLTLYYTTFNNDATIASYSKLDNNSNQEVVIHKDFSFMADFPAADYEIVTLQGAYAREKTVRRQQVEQGIFSISSNRGASGHAQTPALLLCEQGVTEDAGNVFAIQLMYSGNFEAFVQKNQLNEVRVAIGINPENFSWKLAPEEYFETPVALVTHSDQGLTGISHESQNFVLKHIMLSEFSKKERPILINNWEATYFDFQREKLLELADEAKKVGIELFVLDDGWFGNRFDDNRALGDWVVNEEKLGGSLESLISAIHERGLQFGLWLEPEMISVDSDLYRQHPDWAIQVPGYEHTYSRNQLVLNLANPQVVEYLKSVLDQLLFYHDIDYIKWNMNRNITKLGNGLTYLETQMQSHQYMLGLYELVSYLTEKHSHILFESCSGGGGRNDLGMMRYFPQVWASDNTDAIARLPIQYGSSYLYPTISMGAHVSAVPNHQMGRMTPLETRGLVAMMGNLGYELDLTNLSDEEKATIANQVNLYKELRPVVQLGQQYRLINPDTVSNEAAVQFNYGNQTIVTYVRVLSVVETMETTLKLKDLDEEGLYKLQENGEVYSGAELMYAGLTVILSQGDFLSRQYIFRKL

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 \(c\).

\(|P_{f(1SUM_1)}(2) \setminus P_{f(1QYF_1)}(2)|=77\), \(|P_{f(1QYF_1)}(2) \setminus P_{f(1SUM_1)}(2)|=83\). 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 1:1001100010010011101111100110001001100000110011100011001010100011011111011101110101110110110011000001100110110011101100111110000011011101110101000101001000100100010001110110010010011111011101011100100110101011010100000001111000111110000
Pair \(Z_2\) Length of longest common subsequence
1SUM_1,1QYF_1 160 3
1SUM_1,6PHW_1 185 4
1QYF_1,6PHW_1 191 3

Newick tree

 
[
	6PHW_1:98.23,
	[
		1SUM_1:80,1QYF_1:80
	]:18.23
]

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 \((0.85)(0.8)(\frac{465 }{\log_{20} 465}-\frac{230}{\log_{20}230})=68.0\)
Status Protein1 Protein2 d d1/2
Query variables 1SUM_1 1QYF_1 83 83
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]