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Parikh vectors
7NTU_1 6SJC_1 6SLL_1 Letter Amino acid
0 10 7 M Methionine
1 41 10 P Proline
1 12 6 Y Tyrosine
3 61 12 R Arginine
6 67 13 E Glutamic acid
4 46 16 G Glycine
0 12 6 H Histidine
1 10 9 I Isoleucine
3 22 4 K Lycine
3 18 15 S Serine
2 16 13 T Threonine
0 9 4 W Tryptophan
0 6 6 N Asparagine
0 46 13 V Valine
1 52 13 A Alanine
3 27 10 D Aspartic acid
1 1 3 C Cysteine
0 16 4 Q Glutamine
5 79 11 L Leucine
2 30 11 F Phenylalanine 

7NTU_1|Chains A, C|Prothrombin|Homo sapiens (9606)
>6SJC_1|Chains A, B|Aspartate--tRNA(Asp) ligase|Thermus thermophilus (274)
>6SLL_1|Chains A, B|L-2,4-diaminobutyric acid acetyltransferase|Geobacillus sp. (strain Y412MC10) (481743)
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 \(w=f(c)\)
7NTU , Knot 24 36 0.79 28 34 34 
TFGSGEADCGLRPLFEKKSLEDKTERELLESYIDGR
6SJC , Knot 217 581 0.79 40 221 498 
GMRRTHYAGSLRETHVGEEVVLEGWVNRRRDLGGLIFLDLRDREGLVQLVAHPASPAYATAERVRPEWVVRAKGLVRLRPEPNPRLATGRVEVELSALEVLAEAKTPPFPVDAGWRGEEEKEASEELRLKYRYLDLRRRRMQENLRLRHRVIKAIWDFLDREGFVQVETPFLTKSTPEGARDFLVPYRHEPGLFYALPQSPQLFKQMLMVAGLDRYFQIARCFRDEDLRADRQPDFTQLDLEMSFVEVEDVLELNERLMAHVFREALGVELPLPFPRLSYEEAMERYGSDKPDLRFGLELKEVGPLFRQSGFRVFQEAESVKALALPKALSRKEVAELEEVAKRHKAQGLAWARVEEGGFSGGVAKFLEPVREALLQATEARPGDTLLFVAGPRKVAATALGAVRLRAADLLGLKREGFRFLWVVDFPLLEWDEEEEAWTYMHHPFTSPHPEDLPLLEKDPGRVRALAYDLVLNGVEVGGGSIRIHDPRLQARVFRLLGIGEEEQREKFGFFLEALEYGAPPHGGIAWGLDRLLALMTGSPSIREVIAFPKNKEGKDPLTGAPSPVPEEQLRELGLMVVRP
6SLL , Knot 90 186 0.84 40 137 178 
WSHPQFEKSGMAVDTGTEVVYRRPEARDGTRVWELIRDTGSLDLNSPYCYMLLGDYFNDTCMIAEHEGDIVGFISAFRSPRNPETLFVWQVAVASSHRRQGIAKAMLTGLMNQKACHGVRFIETTVSPSNMASRRLFLGYAEEKSIPSTVTVGYGAEMFPDGTTHEDEPLFVIGPFFNDIGSAWSH

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 \(c\).

\(|P_{f(7NTU_1)}(2) \setminus P_{f(6SJC_1)}(2)|=8\), \(|P_{f(6SJC_1)}(2) \setminus P_{f(7NTU_1)}(2)|=195\). 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 1:011010100110111000010000000110001010
Pair \(Z_2\) Length of longest common subsequence
7NTU_1,6SJC_1 203 3
7NTU_1,6SLL_1 139 3
6SJC_1,6SLL_1 170 3

Newick tree

 
[
	6SJC_1:10.37,
	[
		7NTU_1:69.5,6SLL_1:69.5
	]:30.87
]

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 \((0.85)(0.8)(\frac{617 }{\log_{20} 617}-\frac{36}{\log_{20}36})=175.\)
Status Protein1 Protein2 d d1/2
Query variables 7NTU_1 6SJC_1 208 111
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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