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Parikh vectors
5SSQ_1 3TSY_1 6GRN_1 Letter Amino acid
2 32 7 R Arginine
3 16 20 C Cysteine
13 70 51 G Glycine
5 37 15 F Phenylalanine
5 54 19 P Proline
11 66 50 S Serine
15 88 27 A Alanine
6 28 4 H Histidine
19 93 24 L Leucine
2 25 6 M Methionine
7 23 20 Y Tyrosine
22 81 21 V Valine
9 52 24 D Aspartic acid
13 63 13 K Lycine
0 6 9 W Tryptophan
16 42 30 N Asparagine
6 61 19 E Glutamic acid
6 64 10 I Isoleucine
6 50 46 T Threonine
3 28 19 Q Glutamine 

5SSQ_1|Chains A, B|Non-structural protein 3|Severe acute respiratory syndrome coronavirus 2 (2697049)
>3TSY_1|Chain A|Fusion Protein 4-coumarate--CoA ligase 1, Resveratrol synthase|Arabidopsis thaliana (3702)
>6GRN_1|Chain A|Exoglucanase 1|Trichoderma reesei (51453)
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 \(w=f(c)\)
5SSQ , Knot 80 169 0.81 38 119 162 
SMVNSFSGYLKLTDNVYIKNADIVEEAKKVKPTVVVNAANVYLKHGGGVAGALNKATNNAMQVESDDYIATNGPLKVGGSCVLSGHNLAKHCLHVVGPNVNKGEDIQLLKSAYENFNQHEVLLAPLLSAGIFGADPIHSLRVCVDTVRTNVYLAVFDKNLYDKLVSSFL
3TSY , Knot 357 979 0.83 40 316 875 
MGSSHHHHHHSSGLVPRGSHMARMAPQEQAVSQVMEKQSNNNNSDVIFRSKLPDIYIPNHLSLHDYIFQNISEFATKPCLINGPTGHVYTYSDVHVISRQIAANFHKLGVNQNDVVMLLLPNCPEFVLSFLAASFRGATATAANPFFTPAEIAKQAKASNTKLIITEARYVDKIKPLQNDDGVVIVCIDDNESVPIPEGCLRFTELTQSTTEASEVIDSVEISPDDVVALPYSSGTTGLPKGVMLTHKGLVTSVAQQVDGENPNLYFHSDDVILCVLPMFHIYALNSIMLCGLRVGAAILIMPKFEINLLLELIQRCKVTVAPMVPPIVLAIAKSSETEKYDLSSIRVVKSGAAPLGKELEDAVNAKFPNAKLGQGYGMTEAGPVLAMSLGFAKEPFPVKSGACGTVVRNAEMKIVDPDTGDSLSRNQPGEICIRGHQIMKGYLNNPAATAETIDKDGWLHTGDIGLIDDDDELFIVDRLKELIKYKGFQVAPAELEALLIGHPDITDVAVVAMKEEAAGEVPVAFVVKSKDSELSEDDVKQFVSKQVVFYKRINKVFFTESIPKAPSGKILRKDLRAKLANGLGSGMASVEEFRNAQRAKGPATILAIGTATPDHCVYQSDYADYYFKVTKSEHMTALKKKFNRICDKSMIKKRYIHLTEEMLEEHPNIGAYMAPSLNIRQEIITAEVPKLGKEAALKALKEWGQPKSKITHLVFCTTSGVEMPGADYKLANLLGLEPSVRRVMLYHQGCYAGGTVLRTAKDLAENNAGARVLVVCSEITVVTFRGPSEDALDSLVGQALFGDGSAAVIVGSDPDISIERPLFQLVSAAQTFIPNSAGAIAGNLREVGLTFHLWPNVPTLISENVEKCLTQAFDPLGISDWNSLFWIAHPGGPAILDAVEAKLNLDKKKLEATRHVLSEYGNMSSACVLFILDEMRKKSLKGERATTGEGLDWGVLFGFGPGLTIETVVLHSIPMVTN
6GRN , Knot 177 434 0.82 40 221 399 
QSACTLQSETHPPLTWQKCSSGGTCTQQTGSVVIDANWRWTHATNSSTNCYDGNTWSSTLCPDNETCAKNCCLDGAAYASTYGVTTSGNSLSIGFVTQSAQKNVGARLYLMASDTTYQEFTLLGNEFSFDVDVSQLPCGLNGALYFVSMDADGGVSKYPTNTAGAKYGTGYCDSQCPRDLKFINGQANVEGWEPSSNNANTGIGGHGSCCSEMDIWEANSISEALTPHPCTTVGQEICEGDGCGGTYSDNRYGGTCDPDGCDWNPYRLGNTSFYGPGSSFTLDTTKKLTVVTQFETSGAINRYYVQNGVTFQQPNAELGSYSGNELNDDYCTAEEAEFGGSSFSDKGGLTQFKKATSGGMVLVMSLWDDYYANMLWLDSTYPTNETSSTPGAVRGSCSTSSGVPAQVESQSPNAKVTFSNIKFGPIGSTGNPSG

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 \(c\).

\(|P_{f(5SSQ_1)}(2) \setminus P_{f(3TSY_1)}(2)|=10\), \(|P_{f(3TSY_1)}(2) \setminus P_{f(5SSQ_1)}(2)|=207\). 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 1:0110010101010001010010110010010101110110101001111111100100011010000011001110111001101001100010111101001001011001000100001111111011111101100101010010001011110001000110011
Pair \(Z_2\) Length of longest common subsequence
5SSQ_1,3TSY_1 217 3
5SSQ_1,6GRN_1 186 4
3TSY_1,6GRN_1 155 4

Newick tree

 
[
	5SSQ_1:10.75,
	[
		6GRN_1:77.5,3TSY_1:77.5
	]:30.25
]

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 \((0.85)(0.8)(\frac{1148 }{\log_{20} 1148}-\frac{169}{\log_{20}169})=264.\)
Status Protein1 Protein2 d d1/2
Query variables 5SSQ_1 3TSY_1 336 195.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]