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Parikh vectors
6IDW_1 7TWE_1 7UFX_1 Letter Amino acid
15 29 0 D Aspartic acid
4 6 7 C Cysteine
25 23 3 G Glycine
9 18 0 F Phenylalanine
21 24 0 V Valine
33 22 0 N Asparagine
5 11 0 H Histidine
21 28 0 L Leucine
17 11 0 Y Tyrosine
13 16 0 K Lycine
15 16 5 T Threonine
6 9 0 W Tryptophan
13 26 0 E Glutamic acid
15 26 0 I Isoleucine
6 7 0 M Methionine
15 24 0 P Proline
29 20 0 S Serine
31 46 7 A Alanine
16 27 0 R Arginine
13 29 0 Q Glutamine 

6IDW_1|Chains A, B, C, D|Glucanase|Orpinomyces sp. Y102 (884017)
>7TWE_1|Chains A, B|DUF1479 domain-containing protein|Yersinia pestis CO92 (214092)
>7UFX_1|Chain A|DNA (5'-D(A*AP*CP*CP*TP*GP*TP*GP*AP*CP*AP*AP*AP*TP*TP*GP*CP*CP*CP*TP*CP*A)-3')|Escherichia coli (562)
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 \(w=f(c)\)
6IDW , Knot 142 322 0.85 40 202 308 
TSDNFFENELYSNYKFQGEVDQSIQRLSGSLQEKAKKVKYVPTAAWLAWSGATNEVARYLNEAGSKTVVFVLYMIPTRDCNAGGSNGGADNLSTYQGYVNSIYNTINQYPNSRIVMIIEPDTIGNLVTANNANCRNVHDMHKQALSYAISKFGTQKNVRVYLDAAHGGWLNSSADRTAEVIAEILRNAGNGKIRGISTNVSNYQPVYSEYQYHQNLNRALESRGVRGMKFIVDTSRNGRNPSSATWCNLKGAGLGARPQANPDPNMPLLDAYVWIKTPGESDSASSADPVCRNSDSLQGAPAAGSWFHDYFVMLLENANPPF
7TWE , Knot 176 418 0.84 40 235 393 
SNAMASLHIDDIPAAIKAVKQQLRQALPDYQQVFQAVEENIRQQVMEIRRNLAEGKNPVPQLHADDIINGKVTEEQKAQIKQRGCCAILGVFPQEKATAWNREIGDYLDRNNFVERLKNAAEDNYFGTLAASKPQIYGIYWSTPQVEARQDKRMQAVQIFLNNLWQTESNGKQHFDANRVVTYADRTRRRPPKSSSLGLSPHVDGGSIERWLDENFRHVYRHVFSGQWQKYDPFAAEGRPEVREFPSPAVCSMFRTFQGWTALTPQRTHAGTLNVIPIANAMAYILLRALQDDVADDDLCGAAPGRALSASEQWHPLLMEAISPIPDLEAGDTVFWHCDVIHSVENEHNGEFDSNVMYIAAAPWCEKNAAYLPRQLASFIDGRSPPDFAADDFEVDFIGRATIKNLTEIGKQQLGITD
7UFX , Knot 10 22 0.46 8 12 19 
AACCTGTGACAAATTGCCCTCA

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 \(c\).

\(|P_{f(6IDW_1)}(2) \setminus P_{f(7TWE_1)}(2)|=69\), \(|P_{f(7TWE_1)}(2) \setminus P_{f(6IDW_1)}(2)|=102\). 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 1:0000110001000001010100010010101000100100110111111011000110010011000111110111000001110011100100001010010001000100011111010011011010010000100100011001100110000101010110111100010001011101100110101011000100001100000000010011000110110111000001001001010010111111010101010111101011100110000100101100000010111111011000111110010111
Pair \(Z_2\) Length of longest common subsequence
6IDW_1,7TWE_1 171 4
6IDW_1,7UFX_1 206 2
7TWE_1,7UFX_1 233 3

Newick tree

 
[
	7UFX_1:11.97,
	[
		6IDW_1:85.5,7TWE_1:85.5
	]:31.47
]

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 \((0.85)(0.8)(\frac{740 }{\log_{20} 740}-\frac{322}{\log_{20}322})=114.\)
Status Protein1 Protein2 d d1/2
Query variables 6IDW_1 7TWE_1 144 128.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]