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Parikh vectors
3UGP_1 3WAQ_1 7WSG_1 Letter Amino acid
8 13 45 D Aspartic acid
0 0 8 C Cysteine
13 2 36 Q Glutamine
5 5 16 H Histidine
14 12 46 V Valine
21 13 52 A Alanine
21 13 37 R Arginine
20 10 42 K Lycine
4 4 38 F Phenylalanine
12 5 44 T Threonine
3 3 25 W Tryptophan
29 14 57 E Glutamic acid
26 13 74 L Leucine
11 3 40 P Proline
9 3 57 S Serine
4 4 50 N Asparagine
15 5 41 G Glycine
20 3 35 I Isoleucine
4 3 28 M Methionine
6 8 35 Y Tyrosine 

3UGP_1|Chain A|RNA polymerase sigma factor|Thermus aquaticus (271)
>3WAQ_1|Chain A|Hemerythrin-like domain protein DcrH|Desulfovibrio vulgaris (882)
>7WSG_1|Chain A|Angiotensin-converting enzyme|Mammalia (40674)
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 \(w=f(c)\)
3UGP , Knot 108 245 0.80 38 146 234 
GPHMTSDPVRQYLHEIGQVPLLTLEEEIDLARKVEEGMEAIKKLSEATGLDQELIREVVRAKILGTARIQKIPGLKEKPDPKTVEEVDGKLKSLPKELKRYLHIAREGEAARQHLIEANLRLVVSIAKKYTGRGLSFLDLIQEGNQGLIRAVEKFEYKRRFKFSTYATWWIRQAINRAIADQARTIRIPVHMVETINKLSRTARQLQQELGREPSYEEIAEAMGPGWDAKRVEETLKIAQEPVSL
3WAQ , Knot 64 136 0.77 38 102 133 
GDADVLVKWSEDLANLPSIDTQHKRLVDYINDLYRAARRRDMDKAREVFDALKNYAVEHFGYEERLFADYAYPEATRHKEIHRRFVETVLKWEKQLAAGDPEVVMTTLRGLVDWLVNHEMKEDKKYEAYLRERGVS
7WSG , Knot 314 806 0.87 40 323 755 
MLGSSWLLLSLAALTAARSTTEDLVKTFLEKFNSEAEELSYQSSLASWNYNTNITDENVQKMNDAGAKWSAFYEEQSKQAKTYPLEEIQDSTVKRQLQALQHSGSSVLSADKSQRLNTILNAMSTIYSTGKACNPNNPQECLLLEPGLDDIMANSRDYNERLWAWEGWRSEVGKQLRPLYEEYVALKNEMARANNYEDYGDYWRGDYEEEWTNGYNYSRDQLIKDVEQTFTQIQPLYEHLHAYVRAKLMDTYPSHMSPTGCLPAHLLGDMWGRFWTNLYPLTVPFGQKPNIDVTDTMVNQSWDARRIFEEAEKFFVSVGLPNMTQGFWENSMLTEPGDSRKVVCHPTAWDLGKHDFRIKMCTKVTMDDFLTAHHEMGHIQYDMAYAAQPFLLRNGANEGFHEAVGEIMSLSAATPKHLKNIGLLPPGFSEDNETDINFLFKQALTIVGTLPFTYMLEKWRWMVFKGEIPKEQWMKKWWEMKRDLVGVVEPLPHDETYCDPASLFHVANDYSFIRYYTRTIYQFQFQEALCQIAKHEGPLHKCDISNSSEAGQTLLQMLKLGRSKPWTLALYRVVGAKNMDVRPLLNYFDPLFTWLKEQNRNSFVGWNTDWSPYADQSIKVRISLKSALGEKAYEWNDNEMYLFRSSIAYAMREYFSKVKNQMIPFVEDNVWVNNLKPRISFTFFVTSPGNMSDIIPRADVEEAIRMSRGRINDAFRLDDNSLEFLGIQPTLEPPYQPPVTIWLIVFGVVMAVVVVGIVLLIFSGIRSRRKNDQATSEENPYASVNLSKGENNPGFQNVDDVQTSSF

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 \(c\).

\(|P_{f(3UGP_1)}(2) \setminus P_{f(3WAQ_1)}(2)|=96\), \(|P_{f(3WAQ_1)}(2) \setminus P_{f(3UGP_1)}(2)|=52\). 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 1:11010001100010011011110100010110010011011001001011000110011010111010100111100010100100101010011001000101100101100011010101110110000101101101100100111011001000001010001011100110011100100101110110010010001001000110010000110111111010010001011001101
Pair \(Z_2\) Length of longest common subsequence
3UGP_1,3WAQ_1 148 3
3UGP_1,7WSG_1 203 4
3WAQ_1,7WSG_1 239 4

Newick tree

 
[
	7WSG_1:12.67,
	[
		3UGP_1:74,3WAQ_1:74
	]:46.67
]

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 \((0.85)(0.8)(\frac{381 }{\log_{20} 381}-\frac{136}{\log_{20}136})=74.2\)
Status Protein1 Protein2 d d1/2
Query variables 3UGP_1 3WAQ_1 90 70
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]