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Parikh vectors
7QGY_1 8ILM_1 6MBQ_1 Letter Amino acid
26 43 11 L Leucine
24 26 10 K Lycine
2 12 3 M Methionine
10 14 4 N Asparagine
9 22 11 I Isoleucine
12 24 6 F Phenylalanine
18 19 8 S Serine
19 31 15 D Aspartic acid
22 45 13 G Glycine
11 12 9 Q Glutamine
8 14 8 Y Tyrosine
1 7 3 C Cysteine
13 29 12 E Glutamic acid
12 16 4 H Histidine
17 22 4 P Proline
12 29 12 T Threonine
7 9 0 W Tryptophan
13 44 9 A Alanine
7 27 10 R Arginine
17 27 15 V Valine 

7QGY_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>8ILM_1|Chains A, B, F, G, H, I, J, K|Ribulose bisphosphate carboxylase large chain|Synechococcus elongatus PCC 6301 (269084)
>6MBQ_1|Chain A|GTPase KRas|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 \(w=f(c)\)
7QGY , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
8ILM , Knot 195 472 0.84 40 255 445 
MPKTQSAAGYKAGVKDYKLTYYTPDYTPKDTDLLAAFRFSPQPGVPADEAGAAIAAESSTGTWTTVWTDLLTDMDRYKGKCYHIEPVQGEENSYFAFIAYPLDLFEEGSVTNILTSIVGNVFGFKAIRSLRLEDIRFPVALVKTFQGPPHGIQVERDLLNKYGRPMLGCTIKPKLGLSAKNYGRAVYECLRGGLDFTKDDENINSQPFQRWRDRFLFVADAIHKSQAETGEIKGHYLNVTAPTCEEMMKRAEFAKELGMPIIMHDFLTAGFTANTTLAKWCRDNGVLLHIHRAMHAVIDRQRNHGIHFRVLAKCLRLSGGDHLHSGTVVGKLEGDKASTLGFVDLMREDHIEADRSRGVFFTQDWASMPGVLPVASGGIHVWHMPALVEIFGDDSVLQFGGGTLGHPWGNAPGATANRVALEACVQARNEGRDLYREGGDILREAGKWSPELAAALDLWKEIKFEFETMDKL
6MBQ , Knot 80 167 0.81 38 130 165 
GGTEYKLVVVGAGGVGKSALTIQLIQNHFVDEYDPTIEDSYRKQVVIDGETCLLDILDTAGQEEYSAMRDQYMRTGEGFLCVFAINNTKSFEDIHHYREQIKRVKDSEDVPMVLVGNKCDLPSRTVDTKQAQDLARSYGIPFIETSAKTRQGVDDAFYTLVREIRKH

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 \(c\).

\(|P_{f(7QGY_1)}(2) \setminus P_{f(8ILM_1)}(2)|=44\), \(|P_{f(8ILM_1)}(2) \setminus P_{f(7QGY_1)}(2)|=123\). 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 1:10001101000110010001111010000110100001000101011010000100101100101101010000001110111101000110101011010101000010000011010110100001011011001011111111101101011100110110010001001010010101111001001001101001111001011110011010000110100101010101001110010110110000101010
Pair \(Z_2\) Length of longest common subsequence
7QGY_1,8ILM_1 167 3
7QGY_1,6MBQ_1 174 4
8ILM_1,6MBQ_1 193 3

Newick tree

 
[
	6MBQ_1:94.49,
	[
		7QGY_1:83.5,8ILM_1:83.5
	]:10.99
]

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 \((0.85)(0.8)(\frac{732 }{\log_{20} 732}-\frac{260}{\log_{20}260})=130.\)
Status Protein1 Protein2 d d1/2
Query variables 7QGY_1 8ILM_1 168 129.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]