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Parikh vectors
1PXM_1 5UTI_1 1DKN_1 Letter Amino acid
21 14 14 K Lycine
5 16 5 M Methionine
19 17 29 P Proline
10 8 4 Y Tyrosine
19 36 36 A Alanine
16 24 19 D Aspartic acid
3 5 8 C Cysteine
17 22 21 E Glutamic acid
23 17 24 V Valine
14 29 26 S Serine
17 16 34 T Threonine
4 4 8 W Tryptophan
16 32 22 R Arginine
8 7 16 N Asparagine
9 16 29 Q Glutamine
39 27 53 L Leucine
16 33 29 G Glycine
10 10 8 H Histidine
16 22 14 I Isoleucine
16 20 11 F Phenylalanine 

1PXM_1|Chain A|Cell division protein kinase 2|Homo sapiens (9606)
>5UTI_1|Chain A|Queuine tRNA-ribosyltransferase|Zymomonas mobilis subsp. mobilis ZM4 = ATCC 31821 (264203)
>1DKN_1|Chain A|PHYTASE|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 \(w=f(c)\)
1PXM , Knot 134 298 0.85 40 191 284 
MENFQKVEKIGEGTYGVVYKARNKLTGEVVALKKIRLDTETEGVPSTAIREISLLKELNHPNIVKLLDVIHTENKLYLVFEFLHQDLKKFMDASALTGIPLPLIKSYLFQLLQGLAFCHSHRVLHRDLKPQNLLINTEGAIKLADFGLARAFGVPVRTYTHEVVTLWYRAPEILLGCKYYSTAVDIWSLGCIFAEMVTRRALFPGDSEIDQLFRIFRTLGTPDEVVWPGVTSMPDYKPSFPKWARQDFSKVVPPLDEDGRSLLSQMLHYDPNKRISAKAALAHPFFQDVTKPVPHLRL
5UTI , Knot 159 375 0.83 40 211 361 
DRPRFSFSIAAREGKARTGTIEMKRGVIRTPAFMPVGTAATVKALKPETVRATGADIILGNTYHLMLRPGAERIAKLGGLHSFMGWDRPILTDSGGYQVMSLSSLTKQSEEGVTFKSHLDGSRHMLSPERSIEIQHLLGSDIVMAFDECTPYPATPSRAASSMERSMRWAKRSRDAFDSRKEQAENAALFGIQQGSVFENLRQQSADALAEIGFDGYAVGGLAVGEGQDEMFRVLDFSVPMLPDDKPHYLMGVGKPDDIVGAVERGIDMFDCVLPTRSGRNGQAFTWDGPINIRNARFSEDLKPLDSECHCAVCQKWSRAYIHHLIRAGEILGAMLMTEHNIAFYQQLMQKIRDSISEGRFSQFAQDFRARYFAR
1DKN , Knot 167 410 0.81 40 216 380 
QSEPELKLESVVIVSRHGVRAPTKATQLMQDVTPDAWPTWPVKLGWLTPRGGELIAYLGHYQRQRLVADGLLAKKGCPQSGQVAIIADVDERTRKTGEAFAAGLAPDCAITVHTQTDTSSPDPLFNPLKTGVCQLDNANVTDAILSRAGGSIADFTGHRQTAFRELERVLNFPQSNLCLKREKQDESCSLTQALPSELKVSADNVSLTGAVSLASMLTEIFLLQQAQGMPEPGWGRITDSHQWNTLLSLHNAQFYLLQRTPEVARSRATPLLDLIKTALTPHPPQKQAYGVTLPTSVLFIAGHDTNLANLGGALELNWTLPGQPDNTPPGGELVFERWRRLSDNSQWIQVSLVFQTLQQMRDKTPLSLNTPPGEVKLTLAGCEERNAQGMCSLAGFTQIVNEARIPACSL

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 \(c\).

\(|P_{f(1PXM_1)}(2) \setminus P_{f(5UTI_1)}(2)|=73\), \(|P_{f(5UTI_1)}(2) \setminus P_{f(1PXM_1)}(2)|=93\). 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 1:1001001001101001110010001010111100101000001110011001011001001011011011000001011101100010011010110111111100011011011110000011000101001110001110110111101111110000001101100110111100000011011011011101100011111000100110110011010011111100110001011011000100111110001001100110001000101011110111001001110101
Pair \(Z_2\) Length of longest common subsequence
1PXM_1,5UTI_1 166 4
1PXM_1,1DKN_1 167 4
5UTI_1,1DKN_1 161 4

Newick tree

 
[
	1PXM_1:84.14,
	[
		5UTI_1:80.5,1DKN_1:80.5
	]:3.64
]

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 \((0.85)(0.8)(\frac{673 }{\log_{20} 673}-\frac{298}{\log_{20}298})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 1PXM_1 5UTI_1 130 117.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]