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Parikh vectors
9FLZ_1 6LDD_1 5GMN_1 Letter Amino acid
22 20 12 H Histidine
24 44 26 L Leucine
20 14 24 K Lycine
9 34 18 S Serine
36 71 13 A Alanine
12 35 10 N Asparagine
15 39 13 E Glutamic acid
20 36 17 P Proline
4 13 7 W Tryptophan
6 31 8 Y Tyrosine
22 53 19 D Aspartic acid
12 12 2 M Methionine
10 31 12 F Phenylalanine
50 56 22 G Glycine
17 41 9 I Isoleucine
18 36 12 T Threonine
41 37 17 V Valine
12 42 7 R Arginine
6 7 1 C Cysteine
10 18 11 Q Glutamine 

9FLZ_1|Chains A, B, C, D|alcohol dehydrogenase|Paracoccus denitrificans (266)
>6LDD_1|Chains A, B, C, D|LacZ1 Beta-galactosidase|Bifidobacterium dentium (strain ATCC 27534 / DSM 20436 / JCM 1195 / Bd1) (401473)
>5GMN_1|Chain A|Carbonic anhydrase 2|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 \(w=f(c)\)
9FLZ , Knot 151 366 0.81 40 200 342 
MGHHHHHHHHHHSSGHIEGRHNMAKTMKAAVVREFGKPLTIDEVPVPEPGPGMIQVRIQASGVCHTDLHAAEGDWPVKPNPPFIPGHEGVGFVSAVGAGVKHVKEGDRVGVPWLYTACGHCRHCLGGWETLCESQLNTGYSVNGGFADYVVADPNYVGHLPKNVDFLDIAPVLCAGVTVYKGLKVTDTKPGDWVVISGIGGLGHMAVQYAKAMGMNVAAVDIDDEKLALARKLGATVTVNAATEPDPAAAIRKQTDGGAQGVLVTAVGRKAFEQAIGMVARGGTVALNGLPPGDFPLDIFGMVLNGITVRGSIVGTRLDLQESLDFAGDGKVKATVHKAKLEDINNIFGQMHKGQIEGRMVLDMAG
6LDD , Knot 260 670 0.84 40 278 622 
NGMLYPQSNDSRIVFPLDGVWDFRTAGEDSYPAEWADAPLPEPLPMAVPGSYNDQNDELNLRAHYGWVVYQRSFAVPSRLVAGQRMILRFDAATHAADVYLNGQLLGSHFGGFLPFEFDVTSALHAGENLLTVAVDNRIGSSTLPVGNDAGTAFMGSDNANVPAVAEAKKHARRQNLPNFDFFNFAGLNRHVELYTTPADAYIADIAITTERLDHIAGDACTAANALIAYDVTFGGDFPSNPTDPNTPIQPSDPAINHADSSESAESDIQRATTYGRQVRISILDGEGTVVAGVTADIERSGDGTAKASGEIAIRDAKLWNPGAAYLYTAVAELLPEGGAESSSRIIDAYRQTFGIRTVEVSGTTFLINGKPFYFKGFGKHEDSYFHGRGTDDVLNVKDVSLIHWLHANSFRTSHYPYAESMYDLCDREGIVIIDEVPAVGMSWLQYANPLVAERHREAIRGMIARDKNHPCIVMWSIANEPGLDGDGERPRQAYDYFRPLYELAHASDPQNRPVTLVCCQNDYTTDITERTMDVVCINRYYGWYNLSGDLDAACHALNIELDFWENIGKPVMFTEYGADTIEGIHGTHGEMFSEEFQRDYYARINAEIDKRPWFIGEQLWNFADFATFQGIIRVEGNRKGILTRDRQPKMAAHWLRERWAGIPDYGYKG
5GMN , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK

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 \(c\).

\(|P_{f(9FLZ_1)}(2) \setminus P_{f(6LDD_1)}(2)|=50\), \(|P_{f(6LDD_1)}(2) \setminus P_{f(9FLZ_1)}(2)|=128\). 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 1:110000000000001010100011001011110011011010011110111111010101011000010110101110101111110011111011111100100100111111001010000011110010000100100101111001110100110110010110111110111010011010000110111101111110111001011110111101000011110011101010110010111110000011101111011100110011111101101110111110111011111101101010111001010001011101010101001010010011101001010101110111
Pair \(Z_2\) Length of longest common subsequence
9FLZ_1,6LDD_1 178 5
9FLZ_1,5GMN_1 166 4
6LDD_1,5GMN_1 192 4

Newick tree

 
[
	6LDD_1:95.54,
	[
		9FLZ_1:83,5GMN_1:83
	]:12.54
]

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 \((0.85)(0.8)(\frac{1036 }{\log_{20} 1036}-\frac{366}{\log_{20}366})=177.\)
Status Protein1 Protein2 d d1/2
Query variables 9FLZ_1 6LDD_1 228 172
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]