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Parikh vectors
2XIS_1 1DYW_1 1YUR_1 Letter Amino acid
28 9 13 E Glutamic acid
10 4 1 H Histidine
39 8 13 L Leucine
9 2 1 Y Tyrosine
19 14 6 V Valine
10 9 3 N Asparagine
11 4 4 Q Glutamine
11 16 13 K Lycine
35 6 5 R Arginine
37 26 3 G Glycine
11 11 6 S Serine
6 1 1 W Tryptophan
1 4 0 C Cysteine
11 9 4 I Isoleucine
7 5 2 M Methionine
23 12 5 F Phenylalanine
19 7 2 P Proline
16 9 6 T Threonine
47 8 4 A Alanine
37 9 6 D Aspartic acid 

2XIS_1|Chain A|XYLOSE ISOMERASE|Streptomyces rubiginosus (1929)
>1DYW_1|Chain A|CYCLOPHILIN 3|CAENORHABDITIS ELEGANS (6239)
>1YUR_1|Chains A, B|S100 calcium-binding protein A13|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 \(w=f(c)\)
2XIS , Knot 158 387 0.81 40 205 356 
NYQPTPEDRFTFGLWTVGWQGRDPFGDATRRALDPVESVRRLAELGAHGVTFHDDDLIPFGSSDSEREEHVKRFRQALDDTGMKVPMATTNLFTHPVFKDGGFTANDRDVRRYALRKTIRNIDLAVELGAETYVAWGGREGAESGGAKDVRDALDRMKEAFDLLGEYVTSQGYDIRFAIEPKPNEPRGDILLPTVGHALAFIERLERPELYGVNPEVGHEQMAGLNFPHGIAQALWAGKLFHIDLNGQNGIKYDQDLRFGAGDLRAAFWLVDLLESAGYSGPRHFDFKPPRTEDFDGVWASAAGCMRNYLILKERAAAFRADPEVQEALRASRLDELARPTAADGLQALLDDRSAFEEFDVDAAAARGMAFERLDQLAMDHLLGARG
1DYW , Knot 82 173 0.81 40 120 165 
MSRSKVFFDITIGGKASGRIVMELYDDVVPKTAGNFRALCTGENGIGKSGKPLHFKGSKFHRIIPNFMIQGGDFTRGNGTGGESIYGEKFPDENFKEKHTGPGVLSMANAGPNTNGSQFFLCTVKTEWLDGKHVVFGRVVEGLDVVKAVESNGSQSGKPVKDCMIADCGQLKA
1YUR , Knot 50 98 0.78 38 75 93 
MAAEPLTELEESIETVVTTFFTFARQEGRKDSLSVNEFKELVTQQLPHLLKDVGSLDEKMKSLDVNQDSELKFNEYWRLIGELAKEIRKKKDLKIRKK

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 \(c\).

\(|P_{f(2XIS_1)}(2) \setminus P_{f(1DYW_1)}(2)|=131\), \(|P_{f(1DYW_1)}(2) \setminus P_{f(2XIS_1)}(2)|=46\). 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 1:000101000101111011101001110100011011001001101110110100001111100000000010010011000110111100011001110011101000010001100010010111011100011111001100111001001100100110111001000100101110101001010111101101111100100101011010110001111011011101111101101010100110000010111101011111101100110011001010110000101111011101000111000111101010100110100100110101101101110000110010101111011110010011100111101
Pair \(Z_2\) Length of longest common subsequence
2XIS_1,1DYW_1 177 3
2XIS_1,1YUR_1 172 3
1DYW_1,1YUR_1 141 3

Newick tree

 
[
	2XIS_1:92.17,
	[
		1YUR_1:70.5,1DYW_1:70.5
	]:21.67
]

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 \((0.85)(0.8)(\frac{560 }{\log_{20} 560}-\frac{173}{\log_{20}173})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 2XIS_1 1DYW_1 136 98.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]