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Parikh vectors
4PAN_1 8UQN_1 1VTC_1 Letter Amino acid
32 21 0 A Alanine
13 16 0 Q Glutamine
27 32 0 L Leucine
17 20 0 S Serine
14 18 0 Y Tyrosine
15 15 0 N Asparagine
26 28 0 D Aspartic acid
21 16 0 T Threonine
18 12 4 G Glycine
6 6 0 H Histidine
30 22 0 K Lycine
19 23 0 V Valine
18 17 0 F Phenylalanine
4 12 0 P Proline
3 3 0 W Tryptophan
18 27 0 R Arginine
10 5 4 C Cysteine
28 28 0 E Glutamic acid
26 24 0 I Isoleucine
9 10 0 M Methionine 

4PAN_1|Chain A|Guanine nucleotide-binding protein G(i) subunit alpha-1|Rattus norvegicus (10116)
>8UQN_1|Chain A|Guanine nucleotide-binding protein G(q) subunit alpha|Homo sapiens (9606)
>1VTC_1|Chain A|DNA (5'-D(*GP*GP*CP*CP*GP*GP*CP*C)-3')|
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 \(w=f(c)\)
4PAN , Knot 153 354 0.84 40 205 335 
MGCTLSAEDKAAVERSKMIDRNLREDGEKAAREVKLLLLGAGESGKSTIVKQMKIIHEAGYSEEECKQYKAVVYSNTIQSIIAIIRAMGRLKIDFGDAARADDARQLFVLAGAAEEGFMTAELAGVIKRLWKDSGVQACFNRSREYQLNDSAAYYLNDLDRIAQPNYIPTQQDVLRTRVKTTGIVETHFTFKDLHFKMFDVGGQRSERKKWIHCFEGVTAIIFCVALSDYDLVLAEDEEMNRMHESMKLFDSICNNKWFTDTSIILFLNKKDLFEEKIKKSPLTICYPEYAGSNTYEEAAAYIQCQFEDLNKRKDTKEIYTHFTCATDTKNVQFVYDAVTDVIIKNNLKDCGLF
8UQN , Knot 153 355 0.84 40 213 342 
GPMACCLSEEAKEARRINDEIERQLRRDKRDARRELKLLLLGTGESGKSTFIKQMRIIHGSGYSDEDKRGFTKLVYQNIFTAMQAMIRAMDTLKIPYKYEHNKAHAQLVREVDVEKVSAFENPYVDAIKSLWNDPGIQECYDRRREYQLSDSTKYYLNDLDRVADPAYLPTQQDVLRVRVPTTGIIEYPFDLQSVIFRMVDVGGQRSERRKWIHCFENVTSIMFLVALSEYDQVLVESDNENRMEESKALFRTIITYPWFQNSSVILFLNKKDLLEEKIMYSHLVDYFPEYDGPQRDAQAAREFILKMFVDLNPDSDKIIYSHFTCATDTENIRFVFAAVKDTILQLNLKEYNLV
1VTC , Knot 4 8 0.34 4 4 4 
GGCCGGCC

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 \(c\).

\(|P_{f(4PAN_1)}(2) \setminus P_{f(8UQN_1)}(2)|=63\), \(|P_{f(8UQN_1)}(2) \setminus P_{f(4PAN_1)}(2)|=71\). 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 1:110010100011100001100010001001100101111111001000110010110011000000000011100001001111101110101011011010010011111111001110101111100110001101010000000100011001001001101001100001100010001110001010010101101110000000110010110111101110000111100001001000101100100001100001111100001100010001101001001100000011101000100100000000100010010000010110011001110001000111
Pair \(Z_2\) Length of longest common subsequence
4PAN_1,8UQN_1 134 9
4PAN_1,1VTC_1 203 2
8UQN_1,1VTC_1 213 2

Newick tree

 
[
	1VTC_1:11.72,
	[
		4PAN_1:67,8UQN_1:67
	]:46.72
]

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 \((0.85)(0.8)(\frac{709 }{\log_{20} 709}-\frac{354}{\log_{20}354})=97.1\)
Status Protein1 Protein2 d d1/2
Query variables 4PAN_1 8UQN_1 107 105.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]