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Parikh vectors
3INN_1 4KGI_1 3MCQ_1 Letter Amino acid
29 29 36 L Leucine
9 6 8 M Methionine
7 12 21 S Serine
12 10 11 H Histidine
20 8 23 I Isoleucine
23 12 27 G Glycine
14 12 14 T Threonine
7 7 3 N Asparagine
18 9 18 Q Glutamine
10 9 10 F Phenylalanine
22 10 16 P Proline
0 3 6 W Tryptophan
7 10 4 Y Tyrosine
29 13 16 V Valine
21 14 26 D Aspartic acid
1 2 3 C Cysteine
19 15 12 E Glutamic acid
5 14 7 K Lycine
33 18 45 A Alanine
28 10 13 R Arginine 

3INN_1|Chains A, B, C, D|Pantothenate synthetase|Brucella melitensis (29459)
>4KGI_1|Chains A, B, C, D|Glutathione S-transferase|Shigella flexneri (198215)
>3MCQ_1|Chain A|Thiamine-monophosphate kinase|Methylobacillus flagellatus (265072)
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 \(w=f(c)\)
3INN , Knot 133 314 0.81 38 176 297 
MAHHHHHHMGTLEAQTQGPGSMQIIHTIEELRQALAPARQQGKKIGFVPTMGYLHKGHLELVRRARVENDVTLVSIFVNPLQFGANEDLGRYPRDLERDAGLLHDAQVDYLFAPTVSDMYPRPMQTVVDVPPLGNQIEGEARPGHFAGVATVVSKLFNIVGPDAAYFGEKDFQQLVIIRRMVDDMAIPVRIVGVETVREDDGLACSSRNVYLTPEQRRAAIIVPQALDEADRLYRSGMDDPDALEAAIRTFIGRQPLAVPEVIAIRDPETLERLPALQGRPILVALFVRVGATRLLDNRVIGHAAPQITQERAA
4KGI , Knot 101 223 0.81 40 150 215 
MHHHHHHSSGVDLGTENLYFQSMKLFYKPGACSLASHITLRESGKDFTLVSVDLMKKRLENGDDYFSVNPKGQVPALLLDDGTLLTEGVAIMQYLADSVPDRQLLAPVNSISRYKTIEWLNYIATELHKGFTPLFRPDTPEEYKPTVRAQLDKKLQYVNEALKDEHWICGQRFTIADAYLFTVLRWAYAVKLNLEGLEHIAAFMQRMAERPEVQDALSAEGLK
3MCQ , Knot 134 319 0.80 40 170 292 
GMASEFDLIQRYFRRAHPSAVLGVGDDAALIQPSPGMELAVSADMLVANTHFYPNIDPWLIGWKSLAVNISDMAAMGAQPRWATLTIALPEADEDWISKFAAGFFACAAQFDIALIGGDTTRGPLTISVQIMGETPPGASLLRSTARADDDIWVSGPLGDAALALAAIQGRYPLSDTELAACGKALHQPQPRVVLGQALRGLAHSALDISDGLLADLGHILEHSQVGAEVWLKAIPKSEVVSAHSQEVAIQKMILSGGDDYELCFTASTQHRQQIADIGRQLSLDMAVIGRITDTQQLVIHGLDDAPLTLKEHGFDHFA

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 \(c\).

\(|P_{f(3INN_1)}(2) \setminus P_{f(4KGI_1)}(2)|=84\), \(|P_{f(4KGI_1)}(2) \setminus P_{f(3INN_1)}(2)|=58\). 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 1:11000000110101000111010110010010011111000100111110110100101011001010001011011101101110001100100100011110010100111101001010110011011111001010101101111101100110111101101100010011110011001111101111001000011100000101010000111111011001001000110010110111001110011111011110010010011110101111111101110011000111011101000011
Pair \(Z_2\) Length of longest common subsequence
3INN_1,4KGI_1 142 6
3INN_1,3MCQ_1 154 5
4KGI_1,3MCQ_1 162 4

Newick tree

 
[
	3MCQ_1:81.52,
	[
		3INN_1:71,4KGI_1:71
	]:10.52
]

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 \((0.85)(0.8)(\frac{537 }{\log_{20} 537}-\frac{223}{\log_{20}223})=90.0\)
Status Protein1 Protein2 d d1/2
Query variables 3INN_1 4KGI_1 107 92
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]