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Parikh vectors
6EPO_1 8OHE_1 2DPN_1 Letter Amino acid
11 9 16 K Lycine
10 9 12 S Serine
9 16 66 A Alanine
11 6 38 R Arginine
4 7 8 N Asparagine
3 1 2 C Cysteine
14 18 41 E Glutamic acid
11 16 9 I Isoleucine
0 0 14 W Tryptophan
9 2 10 Q Glutamine
11 11 57 G Glycine
11 14 71 L Leucine
11 12 35 T Threonine
15 8 36 V Valine
4 3 5 H Histidine
4 4 5 M Methionine
4 7 26 P Proline
8 3 8 Y Tyrosine
14 5 18 D Aspartic acid
6 3 18 F Phenylalanine 

6EPO_1|Chain A[auth R]|GTPase KRas|Homo sapiens (9606)
>8OHE_1|Chains A, B|Cyclic di-AMP synthase CdaA|Bacillus subtilis subsp. subtilis str. 168 (224308)
>2DPN_1|Chains A, B|Glycerol kinase|Thermus thermophilus (274)
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 \(w=f(c)\)
6EPO , Knot 84 170 0.84 38 136 168 
GMTEYKLVVVGACGVGKSALTIQLIQNHFVDEYDPTIEDSYRKQVVIDGETCLLDILDTAGQEEYSAMRDQYMRTGEGFLCVFAINNTKSFEDIHHYREQIKRVKDSEDVPMVLVGNKSDLPSRTVESRQAQDLARSYGIPFIETSAKTRQGVDDAFYTLVREIRKHKEK
8OHE , Knot 72 154 0.78 38 109 147 
GPTPVEEAQQKTIEAITKAINYMAKRRIGALLTIERDTGMGDYIETGIPLNAKVSSELLINIFIPNTPLHDGAVIMKNNEIAAAACYLPLSESPFISKELGTRHRAAVGISEVTDSLTIIVSEETGGVSVAKNGDLHRELTEEALKEMLEAEFK
2DPN , Knot 188 495 0.78 40 196 426 
MAFLLALDQGTTSSRAILFTLEGRPVAVAKREFRQLYPKPGWVEHDPLEIWETTLWAAREVLRRAGAEAGEVLALGITNQRETTLLWDRKTGKPLHNAIVWQDRRTTPLCEALRAKGLEPLFRERTGLLFDPYFSGTKLVWLLENVPGLKARAEGGGVAFGTVDTWLIWNLTGGKVHATDPTNASRTLLFNLHTLAWDPELLEALGIPAALLPEVRPSDGDFGETLPELLGAPVPIRGVLGDQQAALFGQAALGGGEGKCTYGTGAFLLLNTGKRPVLSEKGLLATVAWSLGGRATYALEGSLFVAGAAVGWLKEVGLIRESAEVEALAASVEDTGDVYFVPAFTGLGAPYWDPYARGTLLGLTRGTSRAHLARAALEGVAFQVRDVVLAMEEEAGVRLKVLKADGGMAQNRLFLKIQADLLGVPVAVPEVTETTALGAALMAGVGAGALSPEDVAGRFREAERFLPTMPEGRREALYRRWREAVERAKGWAREG

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 \(c\).

\(|P_{f(6EPO_1)}(2) \setminus P_{f(8OHE_1)}(2)|=92\), \(|P_{f(8OHE_1)}(2) \setminus P_{f(6EPO_1)}(2)|=65\). 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 1:11000011111101110011010110001100001010000000111010001101100110000011000010010111011110000010010000001001000001111111000011000100001001100011111000100001100110011001000000
Pair \(Z_2\) Length of longest common subsequence
6EPO_1,8OHE_1 157 3
6EPO_1,2DPN_1 170 3
8OHE_1,2DPN_1 163 4

Newick tree

 
[
	2DPN_1:84.79,
	[
		6EPO_1:78.5,8OHE_1:78.5
	]:6.29
]

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 \((0.85)(0.8)(\frac{324 }{\log_{20} 324}-\frac{154}{\log_{20}154})=51.8\)
Status Protein1 Protein2 d d1/2
Query variables 6EPO_1 8OHE_1 68 62.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]