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Parikh vectors
1VNS_1 7YWM_1 2TUN_1 Letter Amino acid
2 1 2 C Cysteine
36 9 6 T Threonine
11 0 2 W Tryptophan
21 3 7 Y Tyrosine
34 9 9 R Arginine
20 7 10 Q Glutamine
38 10 10 E Glutamic acid
29 6 8 I Isoleucine
13 5 0 M Methionine
31 9 4 F Phenylalanine
39 17 14 V Valine
57 18 12 A Alanine
34 5 7 N Asparagine
12 3 3 H Histidine
19 5 6 K Lycine
43 9 4 D Aspartic acid
46 14 11 G Glycine
40 16 19 L Leucine
60 6 10 P Proline
24 10 13 S Serine 

1VNS_1|Chain A|VANADIUM CHLOROPEROXIDASE|Curvularia inaequalis (38902)
>7YWM_1|Chains A, B, C|Phosphopantetheine adenylyltransferase|Mycobacteroides abscessus (36809)
>2TUN_1|Chains A, B, C, D, E, F|TUMOR NECROSIS FACTOR-ALPHA|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 \(w=f(c)\)
1VNS , Knot 242 609 0.85 40 269 575 
MGSVTPIPLPKIDEPEEYNTNYILFWNHVGLELNRVTHTVGGPLTGPPLSARALGMLHLAIHDAYFSICPPTDFTTFLSPDTENAAYRLPSPNGANDARQAVAGAALKMLSSLYMKPVEQPNPNPGANISDNAYAQLGLVLDRSVLEAPGGVDRESASFMFGEDVADVFFALLNDPRGASQEGYHPTPGRYKFDDEPTHPVVLIPVDPNNPNGPKMPFRQYHAPFYGKTTKRFATQSEHFLADPPGLRSNADETAEYDDAVRVAIAMGGAQALNSTKRSPWQTAQGLYWAYDGSNLIGTPPRFYNQIVRRIAVTYKKEEDLANSEVNNADFARLFALVDVACTDAGIFSWKEKWEFEFWRPLSGVRDDGRPDHGDPFWLTLGAPATNTNDIPFKPPFPAYPSGHATFGGAVFQMVRRYYNGRVGTWKDDEPDNIAIDMMISEELNGVNRDLRQPYDPTAPIEDQPGIVRTRIVRHFDSAWELMFENAISRIFLGVHWRFDAAAARDILIPTTTKDVYAVDNNGATVFQNVEDIRYTTRGTREDEEGLFPIGGVPLGIEIADEIFNNGLKPTPPEIQPMPQETPVQKPVGQQPVKGMWEEEQAPVVKEAP
7YWM , Knot 80 162 0.83 38 119 159 
SMTGAVCPGSFDPVTLGHLDVFERAAAQFDEVIVAVLINPNKAGMFTVDERIEMIRESTADLPNLRVESGQGLLVDFVRERGLNAIVKGLRTGTDFEYELQMAQMNKHIAGVDTFFVATAPAYSFVSSSLAKEVATYGGDVSALLPASVHQRLLGKLRGQAQ
2TUN , Knot 76 157 0.81 38 119 154 
VRSSSRTPSDKPVAHVVANPQAEGQLQWLNRRANALLANGVELRDNQLVVPSEGLYLIYSQVLFKGQGCPSTHVLLTHTISRIVVSYQTKVNLLSAIKSPCQRETPEGAEAKPWYEPIYLGGVFQLEKGDRLSAEINRPDYLLFAESGQVYFGIIAL

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 \(c\).

\(|P_{f(1VNS_1)}(2) \setminus P_{f(7YWM_1)}(2)|=173\), \(|P_{f(7YWM_1)}(2) \setminus P_{f(1VNS_1)}(2)|=23\). 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 1:110101111101001000000011110011101001000111110111101011111011100101010110010011010000110011010110010011111110110010101100101011101000101011111000110111110000101111001101111110010110001001011000100010011111110100101101110000111010000011000001110111100010001000011011111111011000000110010110110010011101101000110011100000001100010010110111110110001111010001010110110110001010010111101111100000111011111010101011111101100000101101000010011101110001011000100100101110001111000110010011011100110011111010101111001111000001011000110110010010000010000001111111111110110011001101011010111000110011100110111000011110011
Pair \(Z_2\) Length of longest common subsequence
1VNS_1,7YWM_1 196 3
1VNS_1,2TUN_1 200 4
7YWM_1,2TUN_1 142 4

Newick tree

 
[
	1VNS_1:10.71,
	[
		7YWM_1:71,2TUN_1:71
	]:35.71
]

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 \((0.85)(0.8)(\frac{771 }{\log_{20} 771}-\frac{162}{\log_{20}162})=171.\)
Status Protein1 Protein2 d d1/2
Query variables 1VNS_1 7YWM_1 221 137.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]