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Parikh vectors
5LOK_1 5OMM_1 2YHU_1 Letter Amino acid
3 5 7 C Cysteine
10 18 7 F Phenylalanine
10 29 12 P Proline
23 30 13 T Threonine
5 21 17 N Asparagine
12 12 10 D Aspartic acid
17 30 24 L Leucine
10 7 10 K Lycine
27 22 27 V Valine
9 17 10 Q Glutamine
16 13 15 E Glutamic acid
22 25 22 G Glycine
6 6 13 H Histidine
15 20 25 S Serine
4 9 6 Y Tyrosine
28 20 35 A Alanine
13 12 13 R Arginine
11 10 14 I Isoleucine
11 4 7 M Methionine
1 5 1 W Tryptophan 

5LOK_1|Chains A, B, C, D, E, F|Uridine phosphorylase|Vibrio cholerae (666)
>5OMM_1|Chains A, B|Capsid protein|Norovirus GII.10 (747305)
>2YHU_1|Chains A, B, C, D|PTERIDINE REDUCTASE|TRYPANOSOMA BRUCEI (5691)
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 \(w=f(c)\)
5LOK , Knot 113 253 0.82 40 164 238 
MTKTVFHLGVTEADLNGATLAIIPGDPARVQKIAELMDNPVFLASHREYTVYRAELDGQSVVVCSTGIGGPSTSIAVEELAQLGVRTFLRVGTTGAIQPHVNVGDMIVTTGSVRLDGASLHFAPMEFPAVPDFDVATAMKAAAQESGATVHMGVTASSDTFYPGQERYDTFTGRVVRRFQGSMKEWQDMGVLNFEMESATLLTMCASSGLKAGCVAGVIINRTQKEIPDHATLKETEARSIKVVVEAARKMLK
5OMM , Knot 134 315 0.81 40 191 300 
SKPFTLPILTLGELTNSRFPLPIDVLYTNPNESAIVQCQNGRCTLDGELQGTTQLLPTGICAFRGKVTQQVQDEHRGTHWNMTVTNLNGTPFDPTEDVPAPLGTPDFSGQIYGVISQRNTNTVPGEGNLPANRAHEAVIATYSPKFTPKLGNIQFSTWETQDVSSGQPTKFTPVGLASVDANSHFDQWTLPSYSGALTLNMNLAPSVAPVFPGECLLFFRSFIPLKGGYGNPAIDCLMPQEWVQHLYQESAPSLSDVALVRYVNPETGRTLFEAKLHRNGFLTVARNSAGPVVAPTNGYFRFDSWVNQFYTLAPM
2YHU , Knot 127 288 0.83 40 179 275 
MGSSHHHHHHSSGLVPRGSHMEAPAAVVTGAAKRIGRAIAVKLHQTGYRVVIHYHNSAEAAVSLADELNKERSNTAVVCQADLTNSNVLPASCEEIINSCFRAFGRCDVLVNNASAFYPTPLVQGDHEDNSNGKTVETQVAELIGTNAIAPFLLTMSFAQRQKGTNPNCTSSNLSIVNLCDAMVDQPCMAFSLYNMGKHALVGLTQSAALELAPYGIRVNGVAPGVSLLPVAMGEEEKDKWRRKVPLGRREASAEQIADAVIFLVSGSAQYITGSIIKVDGGLSLVHA

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 \(c\).

\(|P_{f(5LOK_1)}(2) \setminus P_{f(5OMM_1)}(2)|=68\), \(|P_{f(5OMM_1)}(2) \setminus P_{f(5LOK_1)}(2)|=95\). 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 1:1000110111001010110111111011010011011001111100000010010101001110001111100011100110111001101100111010101101110010101011010111101111101011011011100011010111010000101100000010101100101010010011110101001011010100110110111111000000110010100001001011101100110
Pair \(Z_2\) Length of longest common subsequence
5LOK_1,5OMM_1 163 3
5LOK_1,2YHU_1 157 4
5OMM_1,2YHU_1 184 4

Newick tree

 
[
	5OMM_1:89.53,
	[
		5LOK_1:78.5,2YHU_1:78.5
	]:11.03
]

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 \((0.85)(0.8)(\frac{568 }{\log_{20} 568}-\frac{253}{\log_{20}253})=89.3\)
Status Protein1 Protein2 d d1/2
Query variables 5LOK_1 5OMM_1 111 101
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]