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Parikh vectors
4BBN_1 1DMZ_1 3QLL_1 Letter Amino acid
31 11 14 E Glutamic acid
37 15 29 L Leucine
28 14 7 K Lycine
19 3 3 Y Tyrosine
14 3 50 A Alanine
15 9 10 Q Glutamine
9 3 12 H Histidine
22 13 21 I Isoleucine
28 9 10 F Phenylalanine
15 4 12 P Proline
17 7 29 S Serine
18 9 14 V Valine
24 10 18 D Aspartic acid
27 13 24 G Glycine
1 4 2 C Cysteine
13 3 11 M Methionine
14 7 19 T Threonine
9 2 2 W Tryptophan
23 7 21 R Arginine
21 12 8 N Asparagine 

4BBN_1|Chain A|E3 UBIQUITIN-PROTEIN LIGASE NEDD4|HOMO SAPIENS (9606)
>1DMZ_1|Chain A|PROTEIN (PROTEIN KINASE SPK1)|Saccharomyces cerevisiae (4932)
>3QLL_1|Chains A, B, C|Citrate lyase|Yersinia pestis (632)
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 \(w=f(c)\)
4BBN , Knot 167 385 0.86 40 227 372 
GPLGSRDYKRKYEFFRRKLKKQNDIPNKFEMKLRRATVLEDSYRRIMGVKRADFLKARLWIEFDGEKGLDYGGVAREWFFLISKEMFNPYYGLFEYSATDNYTLQINPNSGINPDHLSYFKFIGRVAGMAVYHGKLLDGFFIRPFYKMMLHKPITLHDMESVDSEYYNSLRWILENDPTELDLRFIIDEELFGQTHQHELKNGGSEIVVTNKNKKEYIYLVIQWRFVNRIQKQMAAFKEGFFELIPQDLIKIFDENELELLMSGLGDVDVNDWREHTKYKNGYSANHQVIQWFWKAVLMMDSEKRIRLLQFVTGTSRVPMNGFAELYGSNGPQSFTVEQWGTPEKLPRAHTCFNRLDLPPYESFEELWDKLQMAIENTQGFDGVD
1DMZ , Knot 79 158 0.84 40 128 156 
GNGRFLTLKPLPDSIIQESLEIQQGVNPFFIGRSEDCNCKIEDNRLSRVHCFIFKKRHAVGKSMYESPAQGLDDIWYCHTGTNVSYLNNNRMIQGTKFLLQDGDEIKIIWDKNNKFVIGFKVEINDTTGLFNEGLGMLQEQRVVLKQTAEEKDLVKKL
3QLL , Knot 133 316 0.80 40 174 288 
MGSSHHHHHHSSGLVPRGSHMASMTGGQQMGRGSEFMSNLDTYQTRSWLFTPATRSDRFAKAAENGADVAIIDLEDSVSQADKEQARQKAISYLSSRPATSLPLALRINGLDTRAGIEDIHALLECGSLPDYLVLPKTESAAHLQILDRLMMFAGSDTRLIGIIESVRGLNAVESIAAATPKLAGLIFGAADMAADIGAASTWEPLALARARLVSACAMNGIPAIDAPFFDVHDVSGLQSETLRASDFGFSAKAAIHPAQISTINTLFTPTAAEIRHARAVLAENTKGVGTVNGMMIDEAVARQARRVLTRAGISA

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 \(c\).

\(|P_{f(4BBN_1)}(2) \setminus P_{f(1DMZ_1)}(2)|=137\), \(|P_{f(1DMZ_1)}(2) \setminus P_{f(4BBN_1)}(2)|=38\). 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 1:1111000000000110001000001100101010010110000001111001011010111010100110011110011111000110100111000100000101010011010010010111011111100101101111011001110011010010010000000101110001001010111000111000000100110011100000000101110101100100011110011101110011011000010111011101010010000000010010001101110111110000010110110100011101110101001100101001101001101000100101110001001100101110000110110
Pair \(Z_2\) Length of longest common subsequence
4BBN_1,1DMZ_1 175 4
4BBN_1,3QLL_1 177 4
1DMZ_1,3QLL_1 168 3

Newick tree

 
[
	4BBN_1:89.29,
	[
		1DMZ_1:84,3QLL_1:84
	]:5.29
]

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 \((0.85)(0.8)(\frac{543 }{\log_{20} 543}-\frac{158}{\log_{20}158})=112.\)
Status Protein1 Protein2 d d1/2
Query variables 4BBN_1 1DMZ_1 144 99.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]