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Parikh vectors
3LOU_1 6WPA_1 5ECM_1 Letter Amino acid
10 14 36 T Threonine
29 19 46 V Valine
13 15 20 Q Glutamine
6 6 12 M Methionine
11 17 43 S Serine
9 7 30 K Lycine
18 9 32 F Phenylalanine
4 5 22 Y Tyrosine
35 32 34 A Alanine
5 0 11 C Cysteine
14 7 37 I Isoleucine
14 15 36 G Glycine
14 6 10 H Histidine
24 21 26 R Arginine
4 2 28 N Asparagine
16 21 40 E Glutamic acid
3 1 4 W Tryptophan
21 11 28 D Aspartic acid
28 28 49 L Leucine
14 9 31 P Proline 

3LOU_1|Chains A, B, C, D|Formyltetrahydrofolate deformylase|Burkholderia mallei (243160)
>6WPA_1|Chains A, B, E[auth C], F[auth D], G[auth E], H[auth F], I[auth G], J[auth H]|AvaR1|Streptomyces avermitilis (227882)
>5ECM_1|Chains A, D|Jasmonic acid-amido synthetase JAR1|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
3LOU , Knot 129 292 0.83 40 182 282 
GMSVPQRPHQFVLTLSCPSAAGQVAAVVGLLDRHRCYVDELTVFDDDLSARFFVRCVFHATDDADALRVDALRREFEPIAERFRMQWAIHDVAARPKVLIMVSKLEHCLADLLFRWKMGELKMDIVGIVSNHPDFAPLAAQHGLPFRHFPITADTKAQQEAQWLDVFETSGAELVILARYMQVLSPEASARLANRAINIHHSFLPGFKGAKPYHQAHARGVKLIGATAHFVTDDLDEGPIIEQVVERVDHSYRPEQLLAVGRDVECITLARAVKAFIERRVFLNGDRTVVFQ
6WPA , Knot 106 245 0.79 38 151 230 
GSENLYFQSGSMARQERAIRTRQTILVAAAEVFDEVGYEAATISDVLKRSGVTKGALYFHFTSKQELAQAVLAEQVASLPRVPEQELKLQQSLDEALLLAHLLREGTGDPIVQGSVRLTVDQGSPRDHLNRRVPMQAWTEHTQSLFEEARAKGEILPHADVEALAKLFVGAFTGVQVLSRIMTGRADLAERVADLYRHLMPSFAMPGILVRLDFSPERGSRVYEAAMKQRESAAASTTDAARTLE
5ECM , Knot 234 575 0.86 40 274 538 
MLEKVETFDMNRVIDEFDEMTRNAHQVQKQTLKEILLKNQSAIYLQNCGLNGNATDPEEAFKSMVPLVTDVELEPYIKRMVDGDTSPILTGHPVPAISLSSGTSQGRPKFIPFTDELMENTLQLFRTAFAFRNRDFPIDDNGKALQFIFSSKQYISTGGVPVGTATTNVYRNPNFKAGMKSITSPSCSPDEVIFSPDVHQALYCHLLSGILFRDQVQYVFAVFAHGLVHAFRTFEQVWEEIVTDIKDGVLSNRITVPSVRTAMSKLLTPNPELAETIRTKCMSLSNWYGLIPALFPNAKYVYGIMTGSMEPYVPKLRHYAGDLPLVSHDYGSSEGWIAANVTPRLSPEEATFAVIPNLGYFEFLPVSETGEGEEKPVGLTQVKIGEEYEVVITNYAGLYRYRLGDVVKVIGFYNNTPQLKFICRRNLILSINIDKNTERDLQLSVESAAKRLSEEKIEVIDFSSYIDVSTDPGHYAIFWEISGETNEDVLQDCCNCLDRAFIDAGYVSSRKCKTIGALELRVVAKGTFRKIQEHFLGLGSSAGQFKMPRCVKPSNAKVLQILCENVVSSYFSTAF

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 \(c\).

\(|P_{f(3LOU_1)}(2) \setminus P_{f(6WPA_1)}(2)|=84\), \(|P_{f(6WPA_1)}(2) \setminus P_{f(3LOU_1)}(2)|=53\). 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 1:1101100100111010010111011111111000000100101100010101110011010001011010110001011100101011100111010111110010001101110101101010111110001011111100111100111010001000101101100011011111001011010101011001101000111110110100010101101111010110001001111001100100000100111110010010110110111000111010001110
Pair \(Z_2\) Length of longest common subsequence
3LOU_1,6WPA_1 137 4
3LOU_1,5ECM_1 172 4
6WPA_1,5ECM_1 169 4

Newick tree

 
[
	5ECM_1:90.14,
	[
		3LOU_1:68.5,6WPA_1:68.5
	]:21.64
]

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 \((0.85)(0.8)(\frac{537 }{\log_{20} 537}-\frac{245}{\log_{20}245})=83.3\)
Status Protein1 Protein2 d d1/2
Query variables 3LOU_1 6WPA_1 105 95
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]