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Parikh vectors
2ZAA_1 1GRH_1 1LQJ_1 Letter Amino acid
17 21 7 D Aspartic acid
10 11 14 Q Glutamine
5 29 14 E Glutamic acid
21 31 12 S Serine
21 31 12 T Threonine
17 34 24 L Leucine
7 15 3 M Methionine
8 14 11 F Phenylalanine
5 3 6 W Tryptophan
16 42 17 A Alanine
16 17 10 N Asparagine
0 10 1 C Cysteine
10 16 13 H Histidine
8 29 10 I Isoleucine
10 13 5 Y Tyrosine
14 44 17 V Valine
4 17 10 R Arginine
24 43 18 G Glycine
14 34 9 K Lycine
9 24 16 P Proline 

2ZAA_1|Chain A|Alginate lyase|Sphingomonas sp. (90322)
>1GRH_1|Chain A|GLUTATHIONE REDUCTASE|HOMO SAPIENS (9606)
>1LQJ_1|Chains A, B, C, D|URACIL-DNA GLYCOSYLASE|Escherichia coli (562)
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 \(w=f(c)\)
2ZAA , Knot 109 236 0.84 38 156 226 
PAAAPGKNFDLSHWKLQLPDANTTEISSANLGLGYTSQYFYTDTDGAMTFWAPTTGGTTANSSYPRSELREMLDPSNSKVNWGWQGTHTMKLSGKTVQLPSSGKIIVAQINGIMDDGTNAPPLVKAVFQDGQLDMQVKQNSDGTGSDVHNYFTGIKLGDLYNMEIRVTDGVAYVTMNGDTRSVDFVGKDAGWKNLKYYFKAGNFVQDNTSTGGSAIAKLYSLSVSHSNLEHHHHHH
1GRH , Knot 198 478 0.85 40 246 462 
ACRQEPQPQGPPPAAGAVASYDYLVIGGGSGGLASARRAAELGARAAVVESHKLGGTCVNVGCVPKKVMWNTAVHSEFMHDHADYGFPSCEGKFNWRVIKEKRDAYVSRLNAIYQNNLTKSHIEIIRGHAAFTSDPKPTIEVSGKKYTAPHILIATGGMPSTPHESQIPGASLGITSDGFFQLEELPGRSVIVGAGYIAVEMAGILSALGSKTSLMIRHDKVLRSFDSMISTNCTEELENAGVEVLKFSQVKEVKKTLSGLEVSMVTAVPGRLPVMTMIPDVDCLLWAIGRVPNTKDLSLNKLGIQTDDKGHIIVDEFQNTNVKGIYAVGDVCGKALLTPVAIAAGRKLAHRLFEYKEDSKLDYNNIPTVVFSHPPIGTVGLTEDEAIHKYGIENVKTYSTSFTPMYHAVTKRKTKCVMKMVCANKEEKVVGIHMQGLGCDEMLQGFAVAVKMGATKADFDNTVAIHPTSSEELVTLR
1LQJ , Knot 105 229 0.83 40 155 221 
MANELTWHDVLAEEKQQPYFLNTLQTVASERQSGVTIYPPQKDVFNAFRFTELGDVKVVILGQDPYHGPGQAHGLAFSVRPGIAIPPSLLNMYKELENTIPGFTRPNHGYLESWARQGVLLLNTVLTVRAGQAHSHASLGWETFTDKVISLINQHREGVVFLLWGSHAQKKGAIIDKQRHHVLKAPHPSPLSAHRGFFGCNHFVLANQWLEQRGETPIDWMPVLPAESE

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 \(c\).

\(|P_{f(2ZAA_1)}(2) \setminus P_{f(1GRH_1)}(2)|=48\), \(|P_{f(1GRH_1)}(2) \setminus P_{f(2ZAA_1)}(2)|=138\). 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 1:11111100101001010110100001001011110000010000011101111001100100001000100110100001011101000101010010110010111101011100100111110111001010101000001010010001011011010010101001110101010000101110011100100010110110000001101110100101000010000000
Pair \(Z_2\) Length of longest common subsequence
2ZAA_1,1GRH_1 186 3
2ZAA_1,1LQJ_1 181 3
1GRH_1,1LQJ_1 191 4

Newick tree

 
[
	1GRH_1:95.47,
	[
		2ZAA_1:90.5,1LQJ_1:90.5
	]:4.97
]

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 \((0.85)(0.8)(\frac{714 }{\log_{20} 714}-\frac{236}{\log_{20}236})=133.\)
Status Protein1 Protein2 d d1/2
Query variables 2ZAA_1 1GRH_1 174 126.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]