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Parikh vectors
8ACH_1 7UJX_1 5PDM_1 Letter Amino acid
20 15 2 Q Glutamine
10 6 6 M Methionine
37 13 6 P Proline
64 15 8 T Threonine
20 26 9 E Glutamic acid
48 22 6 G Glycine
17 35 14 K Lycine
47 32 13 L Leucine
46 17 11 S Serine
13 6 2 W Tryptophan
16 14 4 Y Tyrosine
43 20 8 V Valine
42 16 6 N Asparagine
39 19 12 D Aspartic acid
33 21 4 I Isoleucine
7 8 8 H Histidine
29 25 8 F Phenylalanine
69 23 4 A Alanine
39 15 5 R Arginine
5 2 2 C Cysteine 

8ACH_1|Chains A, B, C, D|p70|Nudaurelia capensis omega virus (12541)
>7UJX_1|Chain A[auth E]|cAMP-dependent protein kinase catalytic subunit alpha|Mus musculus (10090)
>5PDM_1|Chain A|Bromodomain adjacent to zinc finger domain protein 2B|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 \(w=f(c)\)
8ACH , Knot 246 644 0.82 40 264 569 
MDSNSASGKRRSRNVRIAANTVNVAPKQRQARGRRARSRANNIDNVTAAAQELGQSLDANVITFPTNVATMPEFRSWARGKLDIDQDSIGWYFKYLDPAGATESARAVGEYSKIPDGLVKFSVDAEIREIYNEECPTVSDASIPLDGAQWSLSIISYPMFRTAYFAVANVDNKEISLDVTNDLIVWLNNLASWRDVVDSGQWFTFSDDPTWFVRIRVLHPTYDLPDPTEGLLRTVSDYRLTYKSITCEANMPTLVDQGFWIGGHYALTPIATTQNAVEGSGFVHPFNVTRPGIAAGVTLTWASMPPGGSAPSGDPAWIPDSTTQFQWRHGGFDAPTGVITYTIPRGYTMQYFDTTTNEWNGFANPDDVVTFGQTGGAAGTNATITITAPTVTLTILATTTSAANVINFRNLDAETTAASNRSEVPLPPLTFGQTAPNNPKIEQTLVKDTLGSYLVHSKMRNPVFQLTPASSFGAISFTNPGFDRNLDLPGFGGIRDSLDVNMSTAVCHFRSLSKSCSIVTKTYQGWEGVTNVNTPFGQFAHSGLLKNDEILCLADDLATRLTGVYGATDNFAAAVLAFAANMLTSVLKSEATTSVIKELGNQATGLANQGLARLPGLLASIPGKIAARVRARRDRRRAARMNNN
7UJX , Knot 152 350 0.84 40 209 337 
GNAAAAKKGSEQESVKEFLAKAKEDFLKKWETPSQNTAQLDQFDRIKTLGTGSFGRVMLVKHKESGNHYAMKILDKQKVVKLKQIEHTLNEKRILQAVNFPFLVKLEFSFKDNSNLYMVMEYVAGGEMFSHLRRIGRFSEPHARFYAAQIVLTFEYLHSLDLIYRDLKPENLLIDQQGYIQVTDFGFAKRVKGRTWTLCGTPEYLAPEIILSKGYNKAVDWWALGVLIYEMAAGYPPFFADQPIQIYEKIVSGKVRFPSHFSSDLKDLLRNLLQVDLTKRFGNLKNGVNDIKNHKWFATTDWIAIYQRKVEAPFIPKFKGPGDTSNFDDYEEEEIRVSINEKCGKEFTEF
5PDM , Knot 66 138 0.78 40 109 129 
MHHHHHHSSGVDLGTENLYFQSMSVKKPKRDDSKDLALCSMILTEMETHEDAWPFLLPVNLKLVPGYKKVIKKPMDFSTIREKLSSGQYPNLETFALDVRLVFDNCETFNEDDSDIGRAGHNMRKYFEKKWTDTFKVS

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 \(c\).

\(|P_{f(8ACH_1)}(2) \setminus P_{f(7UJX_1)}(2)|=111\), \(|P_{f(7UJX_1)}(2) \setminus P_{f(8ACH_1)}(2)|=56\). 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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
Pair \(Z_2\) Length of longest common subsequence
8ACH_1,7UJX_1 167 3
8ACH_1,5PDM_1 203 3
7UJX_1,5PDM_1 180 3

Newick tree

 
[
	5PDM_1:99.72,
	[
		8ACH_1:83.5,7UJX_1:83.5
	]:16.22
]

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 \((0.85)(0.8)(\frac{994 }{\log_{20} 994}-\frac{350}{\log_{20}350})=171.\)
Status Protein1 Protein2 d d1/2
Query variables 8ACH_1 7UJX_1 213 166.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]