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Parikh vectors
3HLK_1 7CLX_1 6XTQ_1 Letter Amino acid
19 9 10 H Histidine
53 20 7 L Leucine
16 19 14 S Serine
17 8 21 T Threonine
14 10 11 Y Tyrosine
37 11 38 A Alanine
7 3 6 C Cysteine
17 11 7 I Isoleucine
32 4 22 R Arginine
13 10 4 Q Glutamine
5 2 14 W Tryptophan
33 11 16 E Glutamic acid
46 12 30 G Glycine
18 12 5 K Lycine
8 2 5 M Methionine
16 8 13 F Phenylalanine
37 14 28 P Proline
9 11 11 N Asparagine
17 3 18 D Aspartic acid
32 12 23 V Valine 

3HLK_1|Chains A, B|Acyl-coenzyme A thioesterase 2, mitochondrial|Homo sapiens (9606)
>7CLX_1|Chain A|Dedicator of cytokinesis protein 8|Mus musculus (10090)
>6XTQ_1|Chain A|Formylglycine-generating enzyme|Thermomonospora curvata (strain ATCC 19995 / DSM 43183 / JCM 3096 / NBRC 15933 / NCIMB 10081 / Henssen B9) (471852)
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 \(w=f(c)\)
3HLK , Knot 181 446 0.82 40 221 409 
GSPQLRQVGQIIRVPARMAATLILEPAGRCCWDEPVRIAVRGLAPEQPVTLRASLRDEKGALFQAHARYRADTLGELDLERAPALGGSFAGLEPMGLLWALEPEKPLVRLVKRDVRTPLAVELEVLDGHDPDPGRLLCQTRHERYFLPPGVRREPVRVGRVRGTLFLPPEPGPFPGIVDMFGTGGGLLEYRASLLAGKGFAVMALAYYNYEDLPKTMETLHLEYFEEAMNYLLSHPEVKGPGVGLLGISKGGELCLSMASFLKGITAAVVINGSVANVGGTLRYKGETLPPVGVNRNRIKVTKDGYADIVDVLNSPLEGPDQKSFIPVERAESTFLFLVGQDDHNWKSEFYANEACKRLQAHGRRKPQIICYPETGHYIEPPYFPLCRASLHALVGSPIIWGGEPRAHAMAQVDAWKQLQTFFHKHLGGHEGTIPSKVLEHHHHHH
7CLX , Knot 91 192 0.83 40 142 185 
GSSGSSGPHTVYRNLLYVYPQRLNFASKLASARNITIKIQFMCGEDPSNAMPVIFGKSSGPEFLQEVYTAITYHNKSPDFYEEVKIKLPAKLTVNHHLLFTFYHISCQQKQGASGESLLGYSWLPILLNERLQTGSYCLPVALEKLPPNYSIHSAEKVPLQNPPIKWAEGHKGVFNIEVQAVSSVHTQDNHL
6XTQ , Knot 128 303 0.80 40 174 291 
HMPSFDFDIPRRSPQEIAKGMVAIPGGTFRMGGEDPDAFPEDGEGPVRTVRLSPFLIDRYAVSNRQFAAFVKATGYVTDAERYGWSFVFHAHVAPGTPVMDAVVPEAPWWVAVPGAYWKAPEGPGSSITDRPNHPVVHVSWNDAVAYATWAGKRLPTEAEWEMAARGGLDQARYPWGNELTPRGRHRCNIWQGTFPVHDTGEDGYTGTAPVNAFAPNGYGLYNVAGNVWEWCADWWSADWHATESPATRIDPRGPETGTARVTKGGSFLCHESYCNRYRVAARTCNTPDSSAAHTGFRCAADP

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 \(c\).

\(|P_{f(3HLK_1)}(2) \setminus P_{f(7CLX_1)}(2)|=137\), \(|P_{f(7CLX_1)}(2) \setminus P_{f(3HLK_1)}(2)|=58\). 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 1:10101001101101110111011101110001001101110111100110101010000111101010001001101010011111101111011111111010011101100010011110101101001011011000000001111110001101101010111110111111110111011111000101111011111111000000110010010100100110011001010111111111001101010110110110111110101101110100010011111100001010001010110110011011000011110010001111110000010001010010001010100010110010010010110111001010111101111110101011101011001001100011100101100110000000
Pair \(Z_2\) Length of longest common subsequence
3HLK_1,7CLX_1 195 3
3HLK_1,6XTQ_1 189 4
7CLX_1,6XTQ_1 184 4

Newick tree

 
[
	3HLK_1:97.31,
	[
		6XTQ_1:92,7CLX_1:92
	]:5.31
]

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 \((0.85)(0.8)(\frac{638 }{\log_{20} 638}-\frac{192}{\log_{20}192})=126.\)
Status Protein1 Protein2 d d1/2
Query variables 3HLK_1 7CLX_1 161 115.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]