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Parikh vectors
2ARS_1 6HLP_1 4XVI_1 Letter Amino acid
15 33 43 S Serine
9 24 11 Y Tyrosine
10 20 18 N Asparagine
23 17 32 D Aspartic acid
0 13 15 C Cysteine
2 10 45 Q Glutamine
21 29 40 E Glutamic acid
13 18 11 M Methionine
19 41 56 A Alanine
5 9 23 H Histidine
2 10 8 W Tryptophan
10 35 27 F Phenylalanine
13 24 34 T Threonine
20 48 50 V Valine
21 22 34 R Arginine
17 27 33 G Glycine
15 43 40 I Isoleucine
25 49 74 L Leucine
17 26 44 K Lycine
5 22 28 P Proline 

2ARS_1|Chain A|Lipoate-protein ligase A|Thermoplasma acidophilum (2303)
>6HLP_1|Chain A|Substance-P receptor,Substance-P receptor|Homo sapiens (9606)
>4XVI_1|Chain A|DNA polymerase nu|Homo sapiens (9606)
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 \(w=f(c)\)
2ARS , Knot 119 262 0.84 38 173 254 
MEGRLLLLETPGNTRMSLAYDEAIYRSFQYGDKPILRFYRHDRSVIIGYFQVAEEEVDLDYMKKNGIMLARRYTGGGAVYHDLGDLNFSVVRSSDDMDITSMFRTMNEAVVNSLRILGLDARPGELNDVSIPVNKKTDIMAGEKKIMGAAGAMRKGAKLWHAAMLVHTDLDMLSAVLKVPDEKFRDKIAKSTRERVANVTDFVDVSIDEVRNALIRGFSETLHIDFREDTITEKEESLARELFDKKYSTEEWNMGLLRKEVV
6HLP , Knot 210 520 0.84 40 259 495 
MDNVLPVDSDLSPNISTNTSEPNQFVQPAWQIVLWAAAYTVIVVTSVVGNVVVMWIILAHKRMRTVTNYFLVNAAFAEASMAAFNTVVNFTYAVHNEWYYGLFYCKFHNFFPIAAIFASIYSMTAVAFDRYMAIIHPLQPRLSLTATKVVICVIWVLALLLAFPQGYYSTTETMPSRVVCKIEWPEHPNKIYEKVYHICVTVLIYFLPLLVIGYLYTVVGITLRASGIDCSFWNESYLTGSRDERKKSLLSKFGMDEGVTFMFIGRFDRGQKGVDVLLKAIEILSSKKEFQEMRFIIIGKGDPELEGWARSLEEKHGNVKVITEMLSREFVRELYGSVDFVIIPSYFEPFGLVALEAMCLGAIPIASAVGGLRDIITNETGILVKAGDPGELANAILKALELSRSDLSKFRENCKKRAMSFSEQVSAARKVVKMMIVVVCTFAICWLPFHIFFLLPYINPDLYLKKFIQQVYLAIMWLAMSSTMYNPIIYCCLNDRFRLGFKHAFRCCPFISAGDYEGLE
4XVI , Knot 256 666 0.83 40 276 618 
KKHFCDIRHLDDWAKSQLIEMLKQAAALVITVMYTDGSTQLGADQTPVSSVRGIVVLVKRQAEGGHGCPDAPACGPVLEGFVSDDPCIYIQIEHSAIWDQEQEAHQQFARNVLFQTMKCKCPVICFNAKDFVRIVLQFFGNDGSWKHVADFIGLDPRIAAWLIDPSDATPSFEDLVEKYCEKSITVKVNSTYGNSSRNIVNQNVRENLKTLYRLTMDLCSKLKDYGLWQLFRTLELPLIPILAVMESHAIQVNKEEMEKTSALLGARLKELEQEAHFVAGERFLITSNNQLREILFGKLKLHLLSQRNSLPRTGLQKYPSTSEAVLNALRDLHPLPKIILEYRQVHKIKSTFVDGLLACMKKGSISSTWNQTGTVTGRLSAKHPNIQGISKHPIQITTPKNFKGKEDKILTISPRAMFVSSKGHTFLAADFSQIELRILTHLSGDPELLKLFQESERDDVFSTLTSQWKDVPVEQVTHADREQTKKVVYAVVYGAGKERLAACLGVPIQEAAQFLESFLQKYKKIKDFARAAIAQCHQTGCVVSIMGRRRPLPRIHAHDQQLRAQAERQAVNFVVQGSAADLCKLAMIHVFTAVAASHTLTARLVAQIHDELLFEVEDPQIPECAALVRRTMESLEQVQALELQLQVPLKVSLSAGRSWGHLVPLQ

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 \(c\).

\(|P_{f(2ARS_1)}(2) \setminus P_{f(6HLP_1)}(2)|=39\), \(|P_{f(6HLP_1)}(2) \setminus P_{f(2ARS_1)}(2)|=125\). 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 1:1010111100110001011000110001001001110100000011110101100010100100011111000011111000110101011000001010011001001110010111101011010010111000001111000111111110011011011111000101101110110001000110000001101001101010010011101100010101000010000001100110000000010111100011
Pair \(Z_2\) Length of longest common subsequence
2ARS_1,6HLP_1 164 4
2ARS_1,4XVI_1 173 4
6HLP_1,4XVI_1 157 5

Newick tree

 
[
	2ARS_1:86.12,
	[
		6HLP_1:78.5,4XVI_1:78.5
	]:7.62
]

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 \((0.85)(0.8)(\frac{782 }{\log_{20} 782}-\frac{262}{\log_{20}262})=143.\)
Status Protein1 Protein2 d d1/2
Query variables 2ARS_1 6HLP_1 184 136
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]