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Parikh vectors
1LVZ_1 5ZUW_1 2BRU_1 Letter Amino acid
0 8 5 Y Tyrosine
1 17 27 I Isoleucine
0 9 11 M Methionine
0 26 21 P Proline
1 20 22 K Lycine
0 50 51 A Alanine
1 22 14 N Asparagine
0 19 18 Q Glutamine
1 23 33 G Glycine
2 46 30 L Leucine
1 19 14 F Phenylalanine
1 27 19 S Serine
0 3 3 W Tryptophan
1 14 18 D Aspartic acid
0 2 2 C Cysteine
1 29 30 E Glutamic acid
0 25 37 V Valine
1 14 15 R Arginine
0 7 7 H Histidine
0 13 24 T Threonine 

1LVZ_1|Chain A|Guanine nucleotide-binding protein G(T), alpha-1 subunit|Bos taurus (9913)
>5ZUW_1|Chain A|Mannose-6-phosphate isomerase|Salmonella typhimurium (strain LT2 / SGSC1412 / ATCC 700720) (99287)
>2BRU_1|Chains A, B|NAD(P) TRANSHYDROGENASE SUBUNIT ALPHA|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 \(w=f(c)\)
1LVZ , Knot 10 11 0.72 20 10 9 
IRENLKDSGLF
5ZUW , Knot 165 393 0.83 40 210 366 
ASMQKLINSVQNYAWGSKTALTELYGIANPQQQPMAELWMGAHPKSSSRITTANGETVSLRDAIEKNKTAMLGEAVANRFGELPFLFKVLCAAQPLSIQVQPNKRNSEIGFAKENAAGIPMDAAERNYKDPNHKPELVFALTPFLAMNAFREFSDIVSLLQPVAGAHSAIAHFLQVPNAERLSQLFASLLNMQGEEKSRALAVLKAALNSQQGEPWQTIRVISEYYPDDSGLFSPLLLNVVKLNPGEAMFLFAETPHAYLQGVALEVMANSDNVLRAGLTPKYIDIPELVANVKFEPKPAGELLTAPVKSGAELDFPIPVDDFAFSLHDLALQETSIGQHSAAILFCVEGEAVLRKDEQRLVLKPGESAFIGADESPVNASGTGRLARVYNKL
2BRU , Knot 159 401 0.79 40 194 373 
MHHHHHHGRIGIPRERLTNETRVAATPKTVEQLLKLGFTVAVESGAGQLASFDDKAFVQAGAEIVEGNSVWQSEIILKVNAPLDDEIALLNPGTTLVSFIWPAQNPELMQKLAERNVTVMAMDSVPRISRAQSLDALSSMANIAGYRAIVEAAHEFGRFFTGQITAAGKVPPAKVMVIGAGVAGLAAIGAANSLGAIVRAFDTRPEVKEQVQSMGAEFLELDFKEEAGSGDGYAKVMSDAFIKAEMELFAAQAKEVDIIVTTALIPGKPAPKLITREMVDSMKAGSVIVDLAAQNGGNCEYTVPGEIFTTENGVKVIGYTDLPGRLPTQSSQLYGTNLVNLLKLLCKEKDGNITVDFDDVVIRGVTVIRAGEITWPAPPIQVSAQPQAAQKAAPEVKTEEK

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 \(c\).

\(|P_{f(1LVZ_1)}(2) \setminus P_{f(5ZUW_1)}(2)|=1\), \(|P_{f(5ZUW_1)}(2) \setminus P_{f(1LVZ_1)}(2)|=201\). 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 1:10001000111
Pair \(Z_2\) Length of longest common subsequence
1LVZ_1,5ZUW_1 202 5
1LVZ_1,2BRU_1 184 2
5ZUW_1,2BRU_1 126 4

Newick tree

 
[
	1LVZ_1:10.45,
	[
		2BRU_1:63,5ZUW_1:63
	]:42.45
]

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 \((0.85)(0.8)(\frac{404 }{\log_{20} 404}-\frac{11}{\log_{20}11})=127.\)
Status Protein1 Protein2 d d1/2
Query variables 1LVZ_1 5ZUW_1 159 81.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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