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Parikh vectors
3NQG_1 8XIL_1 6GRI_1 Letter Amino acid
22 61 29 I Isoleucine
9 61 11 F Phenylalanine
7 63 14 T Threonine
3 0 7 C Cysteine
16 62 19 E Glutamic acid
17 87 34 L Leucine
7 72 24 K Lycine
0 8 6 W Tryptophan
20 37 13 R Arginine
7 67 13 N Asparagine
19 61 17 D Aspartic acid
2 23 10 H Histidine
10 59 31 S Serine
25 71 7 A Alanine
3 34 7 Q Glutamine
21 62 12 G Glycine
16 60 17 V Valine
10 22 6 M Methionine
10 33 9 P Proline
4 49 9 Y Tyrosine 

3NQG_1|Chains A, B|Orotidine 5'-phosphate decarboxylase|Methanothermobacter thermautotrophicus (145262)
>8XIL_1|Chains A, B|Cellodextrin phosphorylase|Acetivibrio thermocellus (1515)
>6GRI_1|Chains A[auth 1], B[auth 2]|Microcin B17-processing protein McbB|Escherichia coli (562)
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 \(w=f(c)\)
3NQG , Knot 98 228 0.77 38 146 215 
MRSRRVDVMDVMNRLILAMDLMNRDDALRVTGEVREYIDTVKIGYPLVLSEGMDIIAEFRKRFGCRIIADFKVADIPETNEKICRATFKAGADAIIVHGFPGADSVRACLNVAEEMGREVFLLTEMSHPGAEMFIQGAADEIARMGVDLGVKNYDGPSTRPERLSRLREIIGQDSFLISPGVGAQGGDPGETLRFADAIIVGRSIYLADNPAAAAAGIIESIKDLLNP
8XIL , Knot 368 992 0.85 38 304 893 
MITKVTARNNKITPVELLNQKFGNKINLGNFADAVFTDAAFKNVAGIANLPMKAPVMQVLMENSIVSKYLKQFVPDRSVSFVEEGQKFYIVLEDGQKIEVPEDVNKALKATVSDVKHWAGYLTEDGEHVIDLLKPAPGPHFYVNLLIGNRLGFKRTLQTTPKSVVDRFGRGSFRSHAATQVLATRFDMRQEENGFPANRQFYLYEDGKQIFYSALIDDNIVEATSKHSSNRTVIKYKTASNLEITRTIFLVPHKKGFPLATELQRIEIKNASDKARNLSITYTGMFGTGAVHAIFEDVTYTNVIMQSAALYNDKGEFIGITPDYYPEEFKQDTRFVTMIVRNGDEKSFPQSFSTDYNDFVGTGTLEHPAGGSNLNNKLNRKGPGFFALGAPFTVEPGKTVIIDTFTGLSSSKDNENYSDAVMLRELDNLLRYFEKSESVEETLNEIINFHENYGKYFQFNTGNKLFDSGFNRNLAFQVLYQTFMSRSFGQTQKGYREIGFREIQDLFASMYYFINIGYQDFVKELLFEWTANVYKMGYANHNFYWVGKQPGLYSDDSLWLLQAYYRYIIYTKDTSVLNEEVPVADGNNEKRAVRETLKAIIQYSASISVGDHGLPLLDLADWNDSLKIDSNSIDGATKEKLYYEQLKKTNGKYGDRFMSDYSESVMNAFLLKLAIDHLAEIATLDNDTQLAQQMSELSKEVTDRIQKHAWKENFFARVLINRYKDGSYTYLGAKGDKLSADPNIDGVYFLNSFAWSVLSDVATDEQIAIMVDVIKKHLLTPYGLRLVTPADLNKIANDTATGHYFFGDRENGAVFKHASMMAVAALIKAAKKVKDNELAKEMARIAYFMIDLVLPYKNLENPFQVAGNPRISTQYINTDTGENIGPLLSGTATWLNLNLISLAGIEYTRDGISFNPILREEETQLNFTLKAPKSSYKFSITKPVGFARMESSEYELFVDGQKIDNTVIPMYTDEKEHIVTLKFKLEHHHHHH
6GRI , Knot 130 295 0.83 40 185 281 
MVLPDIKKGKDMINILPFEIISRNTKTLLITYISSVDITHEGMKKVLESLRSKQGIISEYLLDKLLDESLIDKDKGKEFLITTGVINKTKTSPLWVNSVIISDVPHLFSNAREQWKCDGVFVSHIIDIKDNNINVSDSTLIWLHLENYHSDIVKRIYSKFESNPGVAFIQSYYLKESFRIDGVYSPDLGTPCHFCHIERWLSREEKSFRRNEMSWANLLQLLKKYQMTLPALALGESERGFSYHLIKRRLQELTGTSLVKSHVDNFMSSVSADLITCILCKEPVIHWQACSCLER

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 \(c\).

\(|P_{f(3NQG_1)}(2) \setminus P_{f(8XIL_1)}(2)|=15\), \(|P_{f(8XIL_1)}(2) \setminus P_{f(3NQG_1)}(2)|=173\). 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 1:100001011011001111101100001101010100010010110111100110111010001100111010110110000010010101110111101111100101010110011001111001001110111011100110111011100001100010010010011100011101111101101100101101111100101100111111111001001101
Pair \(Z_2\) Length of longest common subsequence
3NQG_1,8XIL_1 188 4
3NQG_1,6GRI_1 173 3
8XIL_1,6GRI_1 175 5

Newick tree

 
[
	8XIL_1:92.19,
	[
		3NQG_1:86.5,6GRI_1:86.5
	]:5.69
]

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 \((0.85)(0.8)(\frac{1220 }{\log_{20} 1220}-\frac{228}{\log_{20}228})=264.\)
Status Protein1 Protein2 d d1/2
Query variables 3NQG_1 8XIL_1 333 203
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]