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Parikh vectors
2ZOK_1 3MJC_1 3KZL_1 Letter Amino acid
8 23 5 H Histidine
5 8 9 M Methionine
7 7 8 F Phenylalanine
22 33 13 R Arginine
29 27 21 E Glutamic acid
22 54 22 G Glycine
26 67 17 L Leucine
14 25 13 P Proline
12 10 7 W Tryptophan
4 2 3 C Cysteine
4 8 22 I Isoleucine
11 3 16 K Lycine
13 25 22 S Serine
17 26 16 T Threonine
20 91 13 A Alanine
9 4 12 N Asparagine
12 33 14 D Aspartic acid
16 3 5 Y Tyrosine
14 12 6 Q Glutamine
13 35 24 V Valine 

2ZOK_1|Chains A, C, E, G|H-2 class I histocompatibility antigen, D-B alpha chain|Mus musculus (10090)
>3MJC_1|Chains A, B|AmphB|Streptomyces nodosus (40318)
>3KZL_1|Chains A, B, C, D|Aminoglycoside N3-acetyltransferase|Bacillus anthracis (1392)
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 \(w=f(c)\)
2ZOK , Knot 127 278 0.85 40 186 270 
GPHSMRYFETAVSRPGLEEPRYISVGYVDNKEFVRFDSDAENPRYEPRAPWMEQEGPEYWERETQKAKGQEQWFRVSLRNLLGYYNQSAGGSHTLQQMSGCDLGSDWRLLRGYLQFAYEGRDYIALNEDLKTWTAADMAAQITRRKWEQSGAAEHYKAYLEGECVEWLHRYLKNGNATLLRTDSPKAHVTHHPRSKGEVTLRCWALGFYPADITLTWQLNGEELTQDMELVETRPAGDGTFQKWASVVVPLGKEQNYTCRVYHEGLPEPLTLRWERWE
3MJC , Knot 187 496 0.78 40 185 423 
MGSSHHHHHHSSGLVPRGSHMDALRYHIEWNRVAEPGTARPAGRLLAVISPDHAGAPWVTAVLDALGPDTVRFEAKGTDRAAWAAQLAQLREDEGEFHAVVSLLAAAEALHTDHGSVPLGLAQTLLLAQALGDAGLTAPLWCLTRGGVAAGRGDVLSSPVQGALWGLGRVIGLEHPDRWGGLIDLPETVDTRAAARLTGLLADAGGEDQLAIRGSGVLARRLAHAAPAVPGSGKRPPVHGSVLVTGGTGGIGGRVARRLAEQGAAHLVLTSRRGADAPGAAELRAELEQLGVRVTIAACDAADREALAALLAELPEDAPLTAVFHSAGVAHDDAPVADLTLGQLDALMRAKLTAARHLHELTADLDLDAFVLFSSGAAVWGSGGQPGYAAANAYLDALAEHRRSLGLTASSVAWGTWGEVGMATDPEVHDRLVRQGVLAMEPEHALGALDQMLENDDTAAAITLMDWEMFAPAFTANRPSALLSTVPEAVSALSDE
3KZL , Knot 122 268 0.84 40 177 258 
SNAMNDIVASTQLPNTIKTITNDLRKLGLKKGMTVIVHSSLSSIGWISGGAVAVVEALMEVITEEGTIIMPTQSSDLSDPKHWSRPPVPEEWWQIIRDNVPAFEPHITPTRAMGKVVECFRTYPNVVRSNHPLGSFAAWGRHAEEITVNQSLSMSLGEESPLRKIYDLDGYILLIGVGYDSNTSVGLSEVRSGACELIKVGAPIIENGERVWKEFVDMDYDSDKFVEIGVEFEQKGTVTMGKIGNAKCRLMKQRDIVDFGTEWFRKKN

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 \(c\).

\(|P_{f(2ZOK_1)}(2) \setminus P_{f(3MJC_1)}(2)|=78\), \(|P_{f(3MJC_1)}(2) \setminus P_{f(2ZOK_1)}(2)|=77\). 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 1:11001001001100111001001011010000110100010010001011110001100100000010100011010100111000001110001001010011001011010101100100011100010010110111010000100011100001010100101100010010101100001010100010001010100111110110101010101001000101100011101010011011111100000000100011101101010010
Pair \(Z_2\) Length of longest common subsequence
2ZOK_1,3MJC_1 155 4
2ZOK_1,3KZL_1 173 4
3MJC_1,3KZL_1 172 4

Newick tree

 
[
	3KZL_1:88.97,
	[
		2ZOK_1:77.5,3MJC_1:77.5
	]:11.47
]

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 \((0.85)(0.8)(\frac{774 }{\log_{20} 774}-\frac{278}{\log_{20}278})=136.\)
Status Protein1 Protein2 d d1/2
Query variables 2ZOK_1 3MJC_1 161 129.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]