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Parikh vectors
8HQK_1 3HYN_1 2XWM_1 Letter Amino acid
7 8 14 P Proline
35 18 9 S Serine
2 3 0 W Tryptophan
5 4 8 H Histidine
49 13 25 L Leucine
9 7 1 M Methionine
14 9 5 F Phenylalanine
17 11 8 I Isoleucine
46 11 40 A Alanine
8 18 1 N Asparagine
22 16 14 D Aspartic acid
23 2 4 Q Glutamine
14 12 1 Y Tyrosine
41 9 28 V Valine
0 1 0 C Cysteine
50 8 11 E Glutamic acid
38 15 5 K Lycine
26 9 14 T Threonine
22 7 13 R Arginine
30 8 22 G Glycine 

8HQK_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M|Major head protein|Escherichia phage DT57C (2681606)
>3HYN_1|Chain A|Putative signal transduction protein|Eubacterium rectale ATCC 33656 (515619)
>2XWM_1|Chains A, B|2-C-METHYL-D-ERYTHRITOL 4-PHOSPHATE CYTIDYLYLTRANSFERASE|MYCOBACTERIUM SMEGMATIS (246196)
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 \(w=f(c)\)
8HQK , Knot 181 458 0.80 38 202 426 
MTIDINKLKEELGLGDLAKSLEGLTAAQKAQEAERMRKEQEEKELARMNALVSKAVGEDRQKLEQALELVKSLDEKSKKSAELFAQTVEKQQETIVGLQDEIKSLLTAREGRSFVGDSVAKALYGTQETFEDEVEKLVLLSYVMEKGVFETEHGQKHLKAVNQSSSVEVSSESYETIFSQRIIRDLQKELVVGALFEELPMSSKILTMLVEPDAGRATWVAASAYGSDNTTGSEVTGALTEIHFSTYKLAAKSFITDETEEDAIFSLLPLLRKRLIEAHAVSIEEAFMTGDGSGKPKGLLTLASEDSAKVTTEAKADGSVLVTAKTISKLRRKLGRHGLKLSKLVLIVSMDAYYDLLEDEEWQDVAQVGNDAVKLQGQVGRIYGLPVVVSEYFPAKAAGKEFAVIVYKDNFVMPRQRAVTVERERQAGKQRDAYYVTQRVNLQRYFENGVVSGAYAAS
3HYN , Knot 92 189 0.85 40 143 185 
GMSYQNANYSAFYVSEPFSESNLGANSTHDFVYYNMLRMWKGEDNSFPFNDAHDKTYNVRDGSDWEKTLKPRLHTRLDNSKNIILFLSSITANSRALREEMNYGIGTKGLPVIVIYPDYDKKSDIVDSNGNFKKQIKDLWDKLPAFRDNMSSVATLHIPCTKSVIISALNNEDFMVNTMADAEKYYYKP
2XWM , Knot 92 223 0.74 36 118 202 
MATVAVVPAAGSGERLRAGRPKAFVTLGGTPLLEHALSGLRASGVIDRIVIAVPPALTDESKLVFGGEDSVIVSGGVDRTESVALALEAAGDAEFVLVHDAARALTPPALIARVVAALKEGHSAVVPGLAPADTIKAVDANGAVLGTPERAGLRAVQTPQGFHADVLRRAYARATAGGVTDDASLVEQLGTPVQIVDGDPLAFKITTPLDLVLAEAVLAHHHH

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 \(c\).

\(|P_{f(8HQK_1)}(2) \setminus P_{f(3HYN_1)}(2)|=111\), \(|P_{f(3HYN_1)}(2) \setminus P_{f(8HQK_1)}(2)|=52\). 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 1:10101001000111101100101101100100100100000000110101110011100000100110110010000000101110010000001111000100110100100111001101101000010001001111001100111000010001011000001010000000110001100100011111110011100011011101011010111101010000010010111001010000111001100000001110111110001101011010011101010101011101100001010001010101110100100100011001101001111101010001100001001101100110101011010111111000111011100111110000111100011010000011000010010001010001001110110110
Pair \(Z_2\) Length of longest common subsequence
8HQK_1,3HYN_1 163 4
8HQK_1,2XWM_1 142 4
3HYN_1,2XWM_1 169 3

Newick tree

 
[
	3HYN_1:86.64,
	[
		8HQK_1:71,2XWM_1:71
	]:15.64
]

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 \((0.85)(0.8)(\frac{647 }{\log_{20} 647}-\frac{189}{\log_{20}189})=130.\)
Status Protein1 Protein2 d d1/2
Query variables 8HQK_1 3HYN_1 157 113.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]