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Parikh vectors
4JPB_1 5QBT_1 1NVT_1 Letter Amino acid
18 1 5 R Arginine
6 9 2 Q Glutamine
36 23 25 V Valine
0 2 0 C Cysteine
10 13 10 P Proline
19 49 4 S Serine
1 5 1 W Tryptophan
15 29 29 A Alanine
14 6 19 N Asparagine
20 21 18 D Aspartic acid
25 38 25 G Glycine
39 18 36 I Isoleucine
28 19 21 L Leucine
9 20 8 F Phenylalanine
3 13 9 Y Tyrosine
32 5 22 E Glutamic acid
5 1 4 H Histidine
24 11 30 K Lycine
6 0 9 M Methionine
9 47 10 T Threonine 

4JPB_1|Chain A|Chemotaxis protein CheA|Thermotoga maritima (243274)
>5QBT_1|Chain A|Endothiapepsin|Cryphonectria parasitica (5116)
>1NVT_1|Chains A, B|Shikimate 5'-dehydrogenase|Methanocaldococcus jannaschii (2190)
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 \(w=f(c)\)
4JPB , Knot 136 319 0.82 38 170 304 
GSHMVPISFVFNRFPRMVRDLAKKMNKEVNFIMRGEDTELDRTFVEEIGEPLLHLLRNAIDHGIEPKEERIAKGKPPIGTLILSARHEGNNVVIEVEDDGRGIDKEKIIRKAIEKGLIDESKAATLSDQEILNFLVPGFSTKEKVSEVSGRGVGMDVVKNVVESLNGSISIESEKDKGTKVTIRLPLTLAIIQALLVKVNNLVYAIPIANIDTILSISKEDIQRVQDRDVIVIRGEVIPVYRLWEVLQIEHKEELEEMEAVIVRVGNRKYGIVVDDLLGQDDIVIKSLGKVFSEVKEFSGAAILGDGSIALIINVSGIV
5QBT , Knot 132 330 0.77 38 160 297 
STGSATTTPIDSLDDAYITPVQIGTPAQTLNLDFDTGSSDLWVFSSETTASEVDGQTIYTPSKSTTAKLLSGATWSISYGDGSSSSGDVYTDTVSVGGLTVTGQAVESAKKVSSSFTEDSTIDGLLGLAFSTLNTVSPTQQKTFFDNAKASLDSPVFTADLGYHAPGTYNFGFIDTTAYTGSITYTAVSTKQGFWEWTSTGYAVGSGTFKSTSIDGIADTGTTLLYLPATVVSAYWAQVSGAKSSSSVGGYVFPCSATLPSFTFGVGSARIVIPGDYIDFGPISTGSSSCFGGIQSSAGIGINIFGDVALKAAFVVFNGATTPTLGFASK
1NVT , Knot 125 287 0.82 38 152 264 
GPLGSMINAKTKVIGLIGHPVEHSFSPIMHNAAFKDKGLNYVYVAFDVLPENLKYVIDGAKALGIVGFNVTIPHKIEIMKYLDEIDKDAQLIGAVNTIKIEDGKAIGYNTDGIGARMALEEEIGRVKDKNIVIYGAGGAARAVAFELAKDNNIIIANRTVEKAEALAKEIAEKLNKKFGEEVKFSGLDVDLDGVDIIINATPIGMYPNIDVEPIVKAEKLREDMVVMDLIYNPLETVLLKEAKKVNAKTINGLGMLIYQGAVAFKIWTGVEPNIEVMKNAIIDKITK

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 \(c\).

\(|P_{f(4JPB_1)}(2) \setminus P_{f(5QBT_1)}(2)|=83\), \(|P_{f(5QBT_1)}(2) \setminus P_{f(4JPB_1)}(2)|=73\). 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 1:1001111011100110110011001000101110100001000110011011101100110011010000110101111011101000100111010001011000011001100111000011010000110111111000001001010111101100110010101010000001001010111011110111101001101111101001101000010010000111101011110011011010000010010111101100001111001110001110011011001001011111101011111010111
Pair \(Z_2\) Length of longest common subsequence
4JPB_1,5QBT_1 156 4
4JPB_1,1NVT_1 132 4
5QBT_1,1NVT_1 150 4

Newick tree

 
[
	5QBT_1:79.71,
	[
		4JPB_1:66,1NVT_1:66
	]:13.71
]

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 \((0.85)(0.8)(\frac{649 }{\log_{20} 649}-\frac{319}{\log_{20}319})=91.4\)
Status Protein1 Protein2 d d1/2
Query variables 4JPB_1 5QBT_1 113 112.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]