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Parikh vectors
3NYZ_1 6YKO_1 1MOJ_1 Letter Amino acid
14 68 10 G Glycine
6 38 10 T Threonine
18 92 27 A Alanine
22 39 8 R Arginine
26 63 25 E Glutamic acid
3 25 1 W Tryptophan
9 36 5 Y Tyrosine
5 33 6 Q Glutamine
9 43 2 P Proline
21 37 5 S Serine
30 31 6 I Isoleucine
25 65 20 L Leucine
6 28 4 M Methionine
13 36 6 N Asparagine
0 11 1 C Cysteine
6 13 11 H Histidine
13 71 12 V Valine
11 57 18 D Aspartic acid
16 60 3 K Lycine
8 31 2 F Phenylalanine 

3NYZ_1|Chains A, B|Indole-3-glycerol phosphate synthase|Sulfolobus solfataricus (2287)
>6YKO_1|Chain A|Leucine--tRNA ligase|Neisseria gonorrhoeae (485)
>1MOJ_1|Chains A, B, C, D|Dps-like ferritin|Halobacterium salinarum (2242)
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 \(w=f(c)\)
3NYZ , Knot 114 261 0.81 38 156 250 
MPRYLKGWLKDVVQLSLRRPSFRASRQRPIISLNERILEFNKRNITAIIAAYKRKSPSGLDVERDPIEYSKFMERYAVGLAIATEEKYFNGSYETLRKIASSVSIPILMWDFIVKESQIDDAYNLGADTVALIVKILTERELESLLEYARSYGMEPAIVINDENDLDIALRIGARFIEIASRDLETLEINKENQRKLISMIPSNVVKVAWQGISERNEIEELRKLGVNAFGIGSSLMRNPEKIKEFILGSIEGRGHHHHHH
6YKO , Knot 332 877 0.85 40 313 790 
GMQEHYQPAAIEPAAQKKWDDARISNVSEDASKPKYYCLSMFPYPSGKLHMGHVRNYTIGDVLSRFKLLNGFNVMQPMGWDAFGMPAENAAMKNNVAPAAWTYDNIEYMKTQLKSLGFAVDWEREVATCKPEYYRWEQWLFTKLFEKGIVYRKNGTVNWDPVDQTVLANEQVIDGRGWRSGALIEKREIPMYYFKITDYAEELLNDLDKLEHWPEQVKTMQRNWIGKSRGMTVRFAVSDDSKQGLEGDYAKFLQVYTTRPDTLMGATYVAVAAEHPLATAAAADKPELQAFIAECKAGSVAEADMATMEKKGVPTGRYVVNPLNGDKLEVWIANYVLWGYGDGAVMAVPAHDERDFEFAAKYNLPKKQVIAVGDNAFDANRWQEWYGDKENGVLVNSGDLDGLDFQTAFDAVAAKLQSQGAGEPKTQYRLRDWGISRQRYWGCPIPIVHCEKCGNVPVPADQLPVVLPENVVPDGMGSPLAKMPEFYETSCPCCGGAAKRETDTMDTFIESSWYFFRYMSPKFSDGMVSAESAKYWGAVDQYIGGIEHAILHLLYARFFTKLMRDEGLVNVDEPFERLLTQGMVVCETYYRENDKGGKDWINPADVELTFDDKGRPVSAVLKADGLPVVISGTEKMSKSKNNGVDPQELINAYGADTARLFMMFAAPPEQSLEWSDSGVEGAHRFLRRLWRTVYEYLKQGGAVKAFAGNQDGLSKELKDLRHKLHSTTAKVSDDYGRRQQFNTAIAAVMELLNQYDKTDTGSEQGRAVAQEVLEAAVRLLWPIVPHICETLWSELNGAKLWEAGWPTVDEAALVKSEIEVMVQVNGKLRGKITVAADASKADLEAAALANEGAVKFMEGKPAKKIIVVPGRLVNIVV
1MOJ , Knot 83 182 0.79 40 117 172 
MSTQKNARATAGEVEGSDALRMDADRAEQCVDALNADLANVYVLYHQLKKHHWNVEGAEFRDLHLFLGEAAETAEEVADELAERVQALGGVPHASPETLQAEASVDVEDEDVYDIRTSLANDMAIYGDIIEATREHTELAENLGDHATAHMLREGLIELEDDAHHIEHYLEDDTLVTQGALE

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 \(c\).

\(|P_{f(3NYZ_1)}(2) \setminus P_{f(6YKO_1)}(2)|=13\), \(|P_{f(6YKO_1)}(2) \setminus P_{f(3NYZ_1)}(2)|=170\). 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 1:110010111001101010010101000011101000110100001011111000001011010001100001100011111110000010100001001100101111110111000010010011100111110110000100110010001101111100000101110111011011000100101000000011011100110111011000001001001110111110011001001001111010101000000
Pair \(Z_2\) Length of longest common subsequence
3NYZ_1,6YKO_1 183 4
3NYZ_1,1MOJ_1 159 3
6YKO_1,1MOJ_1 230 4

Newick tree

 
[
	6YKO_1:11.86,
	[
		3NYZ_1:79.5,1MOJ_1:79.5
	]:31.36
]

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 \((0.85)(0.8)(\frac{1138 }{\log_{20} 1138}-\frac{261}{\log_{20}261})=233.\)
Status Protein1 Protein2 d d1/2
Query variables 3NYZ_1 6YKO_1 302 192
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]