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Parikh vectors
6FPS_1 6EUR_1 3HAD_1 Letter Amino acid
2 22 16 D Aspartic acid
2 18 6 Q Glutamine
4 15 26 V Valine
2 18 12 P Proline
0 2 1 W Tryptophan
6 25 29 A Alanine
0 22 10 N Asparagine
5 25 20 G Glycine
2 10 14 H Histidine
0 19 14 F Phenylalanine
6 17 8 R Arginine
7 24 20 E Glutamic acid
8 28 16 I Isoleucine
6 29 23 S Serine
2 14 6 Y Tyrosine
0 4 1 C Cysteine
4 36 25 L Leucine
3 17 31 K Lycine
0 10 7 M Methionine
3 17 23 T Threonine 

6FPS_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R|2-hydroxymuconate tautomerase|Pseudomonas putida (303)
>6EUR_1|Chains A, B, C, D|L-lysine 4-hydroxylase|Flavobacterium sp. (strain CF136) (1144313)
>3HAD_1|Chains A, B|PROTEIN (L-3-HYDROXYACYL COA DEHYDROGENASE)|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 \(w=f(c)\)
6FPS , Knot 36 62 0.79 30 59 60 
PIAQIHIYEGRSDEQKETLIREVSEAISRSLDAPLTSVRVIITEYAKGHAGIGGELASKVRR
6EUR , Knot 157 372 0.83 40 220 352 
MKSQSIMSVERSAETSLTLEIPTSPLIIKITQQERNILSNVGNLLVKAFGNYENPDYIASLHLHAFQLLPERITRILSQFGSDFSAEQYGAIVFQGLIEVDQDDLGPTPPNWQGADYGKLNKYGFICSLLHGAVPSKPVQYYAQRKGGGLLHAVIPDEKMAATQTGSGSKTDLFVHTEDAFLSNQADFLSFLYLRNEERVPSTLYSIRSHGKMNPVMKKLFEPIYQCPKDANYNDEDVANSGPTASVLYGNRELPFIRFDAAEQIFNENAGQTSEALGNLMDFWDEAKTLINSDYIPNSGDLIFVNNHLCAHGRSAFIAGQRIENGEIIKCERRQMLRMMSKTSLIHIRSVTRTDDPYFIMEEHLGKIFDLD
3HAD , Knot 130 308 0.80 40 182 288 
SSSSTASASAKKIIVKHVTVIGGGLMGAGIAQVAAATGHTVVLVDQTEDILAKSKKGIEESLRKVAKKKFAENPKAGDEFVEKTLSTIATSTDAASVVHSTDLVVEAIVENLKVKNELFKRLDKRAAEHTIFASNTSSLQITSIANATTRQDRFAGLHFFNPVPVMKLVEVIKTPMTSQKTFESLVDFSKALGKHPVSCKDTPGFIVNRLLVPYLMEAIRLYERGDASKEDIDTAMKLGAGYPMGPFELLDYVGLDTTKFIVDGWHEMDAENPLHQPSPSLNKLVAENKFGKKTGEGFYKYKHHHHHH

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 \(c\).

\(|P_{f(6FPS_1)}(2) \setminus P_{f(6EUR_1)}(2)|=14\), \(|P_{f(6EUR_1)}(2) \setminus P_{f(6FPS_1)}(2)|=175\). 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 1:11101010010000000011001001100010111001011100010101111101100100
Pair \(Z_2\) Length of longest common subsequence
6FPS_1,6EUR_1 189 3
6FPS_1,3HAD_1 167 3
6EUR_1,3HAD_1 160 4

Newick tree

 
[
	6FPS_1:92.02,
	[
		3HAD_1:80,6EUR_1:80
	]:12.02
]

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 \((0.85)(0.8)(\frac{434 }{\log_{20} 434}-\frac{62}{\log_{20}62})=114.\)
Status Protein1 Protein2 d d1/2
Query variables 6FPS_1 6EUR_1 147 84.5
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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