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Parikh vectors
3ZNY_1 4OAL_1 1JAY_1 Letter Amino acid
14 13 0 Q Glutamine
6 5 5 M Methionine
12 29 8 P Proline
22 18 8 T Threonine
1 3 2 C Cysteine
3 11 2 W Tryptophan
12 13 7 N Asparagine
19 65 17 G Glycine
12 16 15 I Isoleucine
26 50 24 L Leucine
14 13 12 K Lycine
4 27 5 F Phenylalanine
16 44 17 V Valine
11 26 19 E Glutamic acid
15 38 14 R Arginine
15 29 13 D Aspartic acid
2 17 3 H Histidine
19 34 15 S Serine
5 12 2 Y Tyrosine
35 61 24 A Alanine 

3ZNY_1|Chain A|CTX-M-12A ENZYME|KLEBSIELLA PNEUMONIAE (573)
>4OAL_1|Chains A, B|Cytokinin dehydrogenase 4|Zea mays (4577)
>1JAY_1|Chains A, B|Coenzyme F420H2:NADP+ Oxidoreductase (FNO)|Archaeoglobus fulgidus (2234)
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 \(w=f(c)\)
3ZNY , Knot 120 263 0.84 40 170 249 
QTADVQQKLAELERQSGGRLGVALINTADNSQILYRADERFAMCSTSKVMAAAAVLKKSESEPSLLNQRVEIKKSDLVNYNPIAEKHVNGTMSLAELSAAALQYSDNVAMNKLIAHVGGPASVTAFARQLGDETFRLDRTEPTLNTAIPGDPRDTTSPRAMAQTLRNLTLGKALGDSQRAQLVTWMKGNTTGAASIQAGLPASWVVGDKTGSGGYGTTNDIAVIWPKDRAPLILVTYFTQPQPKAESRRDILASAAKIVTDGL
4OAL , Knot 196 524 0.78 40 224 470 
AGHMLPVEPPAELLQLGGGDVGGGRLSVDASDIAEASRDFGGVARAEPMAVFHPRAAGDVAGLVGAAFRSARGFRVSARGHGHSISGQAQAAGGVVVDMSRGRGPGAAVARALPVHSAALGGHYVDVWGGELWVDVLNWTLSHGGLAPRSWTDYLYLSVGGTLSNAGISGQAFHHGPQISNVYELDVVTGKGEVVTCSETENPDLFFGVLGGLGQFGIITRARIALERAPKRVRWIRALYSNFSEFTADQERLISLGSGGGRRFDYVEGFVVAAEGLINNWRSSFFSPQNPVKLTSLKHHSSVLYCLEVTKNYDDETAGSVDQDVDTLLGELNFLPGTVFTTDLPYVDFLDRVHKAELKLRAKGMWEVPHPWLNLFVPASRIADFDRGVFRGVLGGRTAGAGGPVLIYPMNKHKWDPRSSAVTPDEEVFYLVAFLRSALPGAPESLEALARQNQRILDFCAGTGIGAKQYLPGHKARHEWAEHFGAARWDRFARLKAEFDPRAILAAGQGIFRPPGSPALAADS
1JAY , Knot 92 212 0.77 38 136 201 
MRVALLGGTGNLGKGLALRLATLGHEIVVGSRREEKAEAKAAEYRRIAGDASITGMKNEDAAEACDIAVLTIPWEHAIDTARDLKNILREKIVVSPLVPVSRGAKGFTYSSERSAAEIVAEVLESEKVVSALHTIPAARFANLDEKFDWDVPVCGDDDESKKVVMSLISEIDGLRPLDAGPLSNSRLVESLTPLILNIMRFNGMGELGIKFL

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 \(c\).

\(|P_{f(3ZNY_1)}(2) \setminus P_{f(4OAL_1)}(2)|=56\), \(|P_{f(4OAL_1)}(2) \setminus P_{f(3ZNY_1)}(2)|=110\). 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 1:00101000110100001101111110010000110010001110000011111111000000101100010100001100011100010101011010111100000111001110111110101110011000101000010100111101000001011100100101101110000101101101000111010111110111100010110100001111110001111110010010101000001110110110011
Pair \(Z_2\) Length of longest common subsequence
3ZNY_1,4OAL_1 166 4
3ZNY_1,1JAY_1 138 4
4OAL_1,1JAY_1 168 3

Newick tree

 
[
	4OAL_1:87.80,
	[
		3ZNY_1:69,1JAY_1:69
	]:18.80
]

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 \((0.85)(0.8)(\frac{787 }{\log_{20} 787}-\frac{263}{\log_{20}263})=144.\)
Status Protein1 Protein2 d d1/2
Query variables 3ZNY_1 4OAL_1 176 133.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]