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Parikh vectors
5LPF_1 5DUD_1 6QPZ_1 Letter Amino acid
15 25 46 A Alanine
15 24 15 R Arginine
8 16 19 I Isoleucine
3 7 20 F Phenylalanine
16 14 25 S Serine
11 13 24 T Threonine
19 14 48 V Valine
7 9 15 H Histidine
8 10 26 K Lycine
8 7 15 Y Tyrosine
12 5 3 C Cysteine
15 22 16 Q Glutamine
6 14 25 E Glutamic acid
22 34 51 G Glycine
17 16 34 P Proline
7 10 21 N Asparagine
11 17 21 D Aspartic acid
25 36 31 L Leucine
3 10 11 M Methionine
6 7 2 W Tryptophan 

5LPF_1|Chains A, B|Kallikrein-10|Homo sapiens (9606)
>5DUD_1|Chains A, C|YbgK|Escherichia coli (strain K12) (83333)
>6QPZ_1|Chains A, B[auth I]|Copper-containing nitrite reductase|Ralstonia pickettii (329)
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 \(w=f(c)\)
5LPF , Knot 106 234 0.82 40 156 226 
LDPEAYGSPCARGSQPWQVSLFNGLSFHCAGVLVDQSWVLTAAHCGNKPLWARVGDDHLLLLQGEQLRRTTRSVVHPKYHQGSGPILPRRTDEHDLMLLKLARPVVLGPRVRALQLPYRCAQPGDQCQVAGWGTTAARRVKYNKGLTCSSITILSPKECEVFYPGVVTNNMICAGLDRGQDPCQSDSGGPLVCDETLQGILSWGVYPCGSAQHPAVYTQICKYMSWINKVIRSN
5DUD , Knot 133 310 0.82 40 187 297 
MLKIIRAGMYTTVQDGGRHGFRQSGISHCGALDMPALRIANLLVGNDANAPALEITLGQLTVEFETDGWFALTGAGCEARLDDNAVWTGWRLPMKAGQRLTLKRPQHGMRSYLAVAGGIDVPPVMGSCSTDLKVGIGGLEGRLLKDGDRLPIGKSKRDSMEAQGVKQLLWGNRIRALPGPEYHEFDRASQDAFWRSPWQLSSQSNRMGYRLQGQILKRTTDRELLSHGLLPGVVQVPHNGQPIVLMNDAQTTGGYPRIACIIEADMYHLAQIPLGQPIHFVQCSLEEALKARQDQQRYFEQLAWRLHNEN
6QPZ , Knot 194 468 0.85 40 237 437 
ASAKLPGDFGPPRGEPIHAVLTSPPLVPPPVNRTYPAKVIVELEVVEKEMQISEGVSYTFWTFGGTVPGSFIRVRQGDTVEFHLKNHPSSKMPHNIDLHGVTGPGGGAASSFTAPGHESQFTFKALNEGIYVYHCATAPVGMHIANGMYGLILVEPPEGLPKVDHEYYVMQGDFYTAGKYREKGLQPFDMEKAIDERPSYVLFNGAEGALTGDKALHAKVGETVRIFVGNGGPNLVSSFHVIGAIFDQVRYEGGTNVQKNVQTTLIPAGGAAVVKFTARVPGSYVLVDHSIFRAFNKGAMAILKIDGAENKLVYSGKELDSVELGDRAAPNMSAVTKATQASVSGTLTVQDQVQAGRALFAGTCSVCHQGNGAGLPGVFPPLAKSDFLAADPKRAMNIVLHGLNGKIKVNGQEYDSVMPPMTQLNDDEVANILTYVLNSWDNPGGRVSAEDVKKVRAQPAPAKAVAEH

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 \(c\).

\(|P_{f(5LPF_1)}(2) \setminus P_{f(5DUD_1)}(2)|=68\), \(|P_{f(5DUD_1)}(2) \setminus P_{f(5LPF_1)}(2)|=99\). 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 1:101010101010100110101101101001111100011101100100111101100011110100100000011010000101111100000001111011011111101011011000101100001111100110010000110000101101000011011110001101110010010000011111000010111011101010100111000100010110011000
Pair \(Z_2\) Length of longest common subsequence
5LPF_1,5DUD_1 167 4
5LPF_1,6QPZ_1 173 4
5DUD_1,6QPZ_1 164 4

Newick tree

 
[
	5LPF_1:85.99,
	[
		5DUD_1:82,6QPZ_1:82
	]:3.99
]

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 \((0.85)(0.8)(\frac{544 }{\log_{20} 544}-\frac{234}{\log_{20}234})=88.5\)
Status Protein1 Protein2 d d1/2
Query variables 5LPF_1 5DUD_1 112 98
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]