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Parikh vectors
4WFE_1 2PKR_1 3BWK_1 Letter Amino acid
20 23 17 S Serine
6 5 4 W Tryptophan
8 7 22 Y Tyrosine
12 7 4 Q Glutamine
28 32 23 G Glycine
4 5 3 M Methionine
16 13 8 P Proline
23 41 12 V Valine
29 30 17 A Alanine
4 8 9 C Cysteine
14 14 16 E Glutamic acid
6 6 2 H Histidine
45 30 16 L Leucine
4 25 17 K Lycine
16 15 8 R Arginine
10 34 20 D Aspartic acid
15 21 16 I Isoleucine
18 10 9 F Phenylalanine
18 22 8 T Threonine
3 17 12 N Asparagine 

4WFE_1|Chains A, B|Potassium channel subfamily K member 4|Homo sapiens (9606)
>2PKR_1|Chains A, B, C, D, E[auth H], F[auth I], G[auth L], H[auth M], I[auth O], J[auth P], K[auth Q], L[auth R]|Glyceraldehyde-3-phosphate dehydrogenase Aor|Spinacia oleracea (3562)
>3BWK_1|Chains A, B, C, D|Cysteine protease falcipain-3|Plasmodium falciparum (5833)
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 \(w=f(c)\)
4WFE , Knot 124 299 0.78 40 174 279 
MTTAPQEPPARPLQAGSGAGPAPGRAMRSTTLLALLALVLLYLVSGALVFRALEQPHEQQAQRELGEVREKFLRAHPCVSDQELGLLIKEVADALGGGADPETQSTSQSSHSAWDLGSAFFFSGTIITTIGYGNVALRTDAGRLFCIFYALVGIPLFGILLAGVGDRLGSSLRHGIGHIEAIFLKWHVPPELVRVLSAMLFLLIGCLLFVLTPTFVFCYMEDWSKLEAIYFVIVTLTTVGFGDYVAGADPRQDSPAYQPLVWFWILLGLAYFASVLTTIGNWLRVVSRRTSNSLEVLFQ
2PKR , Knot 154 365 0.83 40 194 353 
KLKVAINGFGRIGRNFLRCWHGRKDSPLDVVVINDTGGVKQASHLLKYDSILGTFDADVKTAGDSAISVDGKVIKVVSDRNPVNLPWGDMGIDLVIEGTGVFVDRDGAGKHLQAGAKKVLITAPGKGDIPTYVVGVNEEGYTHADTIISNASCTTNCLAPFVKVLDQKFGIIKGTMTTTHSYTGDQRLLDASHRDLRRARAACLNIVPTSTGAAKAVALVLPNLKGKLNGIALRVPTPNVSVVDLVVQVSKKTFAEEVNAAFRESADNELKGILSVCDEPLVSIDFRCTDVSSTIDSSLTMVMGDDMVKVIAWYDNEWGYSQRVVDLADIVANKWQGLEGSVASGDPLEDFCKDNPADEECKLYE
3BWK , Knot 113 243 0.85 40 170 237 
YEANYEDVIKKYKPADAKLDRIAYDWRLHGGVTPVKDQALCGSCWAFSSVGSVESQYAIRKKALFLFSEQELVDCSVKNNGCYGGYITNAFDDMIDLGGLCSQDDYPYVSNLPETCNLKRCNERYTIKSYVSIPDDKFKEALRYLGPISISIAASDDFAFYRGGFYDGECGAAPNHAVILVGYGMKDIYNEDTGRMEKFYYYIIKNSWGSDWGEGGYINLETDENGYKKTCSIGTEAYVPLLE

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 \(c\).

\(|P_{f(4WFE_1)}(2) \setminus P_{f(2PKR_1)}(2)|=70\), \(|P_{f(2PKR_1)}(2) \setminus P_{f(4WFE_1)}(2)|=90\). 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 1:10011001110110110111111101100001111111111011011111011001000010001101000110101010000111110011011111101000000000001101101111010110011010111000110110110111111111111111100110010011101011110101110110110111111110111110101110010010010110111101001111001111010000110011111111111101101100110110110000000101110
Pair \(Z_2\) Length of longest common subsequence
4WFE_1,2PKR_1 160 4
4WFE_1,3BWK_1 184 3
2PKR_1,3BWK_1 156 3

Newick tree

 
[
	4WFE_1:88.77,
	[
		2PKR_1:78,3BWK_1:78
	]:10.77
]

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 \((0.85)(0.8)(\frac{664 }{\log_{20} 664}-\frac{299}{\log_{20}299})=101.\)
Status Protein1 Protein2 d d1/2
Query variables 4WFE_1 2PKR_1 129 115
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]