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Parikh vectors
6SWP_1 8FSY_1 1XFY_1 Letter Amino acid
11 12 32 A Alanine
4 14 58 N Asparagine
11 7 44 D Aspartic acid
3 3 27 Q Glutamine
4 12 40 G Glycine
5 3 36 F Phenylalanine
7 3 34 Y Tyrosine
6 11 20 R Arginine
5 2 81 E Glutamic acid
6 2 8 M Methionine
1 7 37 T Threonine
2 8 0 C Cysteine
6 1 19 H Histidine
10 8 67 L Leucine
12 6 101 K Lycine
6 10 49 S Serine
1 6 6 W Tryptophan
3 6 63 I Isoleucine
6 2 22 P Proline
6 6 33 V Valine 

6SWP_1|Chain A[auth AAA]|Bromodomain-containing protein 2|Homo sapiens (9606)
>8FSY_1|Chain A|Lysozyme C|Gallus gallus (9031)
>1XFY_1|Chains A, B, C, D, E, F|Calmodulin-sensitive adenylate cyclase|Bacillus anthracis (1392)
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 \(w=f(c)\)
6SWP , Knot 59 115 0.81 40 94 113 
GSMGKLSEQLKHCNGILKELLSKKHAAYAWPFYKPVDASALGLHDYHDIIKHPMDLSTVKRKMENRDYRDAQEFAADVRLMFSNCYKYNPPDHDVVAMARKLQDVFEFRYAKMPD
8FSY , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
1XFY , Knot 284 777 0.81 38 253 689 
MHHHHHHAAAMNEHYTESDIKRNHKTEKNKTEKEKFKDSINNLVKTEFTNETLDKIQQTQDLLKKIPKDVLEIYSELGGEIYFTDIDLVEHKELQDLSEEEKNSMNSRGEKVPFASRFVFEKKRETPKLIINIKDYAINSEQSKEVYYEIGKGISLDIISKDKSLDPEFLNLIKSLSDDSDSSDLLFSQKFKEKLELNNKSIDINFIKENLTEFQHAFSLAFSYYFAPDHRTVLELYAPDMFEYMNKLEKGGFEKISESLKKEGVEKDRIDVLKGEKALKASGLVPEHADAFKKIARELNTYILFRPVNKLATNLIKSGVATKGLNVHGKSSDWGPVAGYIPFDQDLSKKHGQQLAVEKGNLENKKSITEHEGEIGKIPLKLDHLRIEELKENGIILKGKKEIDNGKKYYLLESNNQVYEFRISDENNEVQYKTKEGKITVLGEKFNWRNIEVMAKNVEGVLKPLTADYDLFALAPSLTEIKKQIPQKEWDKVVNTPNSLEKQKGVTNLLIKYGIERKPDSTKGTLSNWQKQMLDRLNEAVKYTGYTGGDVVNHGTEQDNEEFPEKDNEIFIINPEGEFILTKNWEMTGRFIEKNITGKDYLYYFNRSYNKIAPGNKAYIEWTDPITKAKINTIPTSAEFIKNLSSIRRSSNVGVYKDSGDKDEFAKKESVKKIAGYLSDYYNSANHIFSQEKKRKISIFRGIQAYNEIENVLKSKQIAPEYKNYFQYLKERITNQVQLLLTHQKSNIEFKLLYKQLNFTENETDNFEVFQKIIDEK

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 \(c\).

\(|P_{f(6SWP_1)}(2) \setminus P_{f(8FSY_1)}(2)|=62\), \(|P_{f(8FSY_1)}(2) \setminus P_{f(6SWP_1)}(2)|=72\). 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 1:1011010001000011100110000110111100110101111000001100110100100010000000100111010111000000011000111110010011010010110
Pair \(Z_2\) Length of longest common subsequence
6SWP_1,8FSY_1 134 3
6SWP_1,1XFY_1 211 5
8FSY_1,1XFY_1 207 4

Newick tree

 
[
	1XFY_1:11.30,
	[
		6SWP_1:67,8FSY_1:67
	]:47.30
]

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 \((0.85)(0.8)(\frac{244 }{\log_{20} 244}-\frac{115}{\log_{20}115})=41.0\)
Status Protein1 Protein2 d d1/2
Query variables 6SWP_1 8FSY_1 52 49.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]