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Parikh vectors
7AUG_1 6EUG_1 4WKG_1 Letter Amino acid
38 23 43 E Glutamic acid
40 38 48 G Glycine
13 8 20 Q Glutamine
15 14 9 M Methionine
21 16 28 F Phenylalanine
2 15 10 W Tryptophan
25 15 42 V Valine
16 15 27 H Histidine
33 22 63 L Leucine
30 15 27 T Threonine
12 20 35 P Proline
21 25 53 A Alanine
17 17 45 R Arginine
18 13 28 N Asparagine
35 24 35 D Aspartic acid
2 4 11 C Cysteine
21 19 52 I Isoleucine
32 30 25 K Lycine
17 24 36 S Serine
14 17 23 Y Tyrosine 

7AUG_1|Chain A|rsGCamP1.3|Aequorea victoria (6100)
>6EUG_1|Chain A|Beta-glucanase|Bacteroides thetaiotaomicron (818)
>4WKG_1|Chains A, B, C, D, E, F|Bifunctional polymyxin resistance protein ArnA|Escherichia coli (83333)
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 \(w=f(c)\)
7AUG , Knot 174 422 0.83 40 223 383 
MVDSSRRKWNKTGHAVRAIGRLSSLENVYIKADKQKNGIKANFKFRHNIEDGGVQLAYHYQQNTPIGDGPVLLPDNHYLSTQSKLSKDPNEKRDHMVLLEFVTAAGITLGMDELYKGGTGGSMVSKGEELFTGVVPILVELDGDVNGHKFSVSGEGEGDATYGKLTLKFICTTGKLPVPWPTLVTTLAYGVLCFSRYPDHMKQHDFFKSAMPEGYIQERTIFFKDDGNYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNTRDQLTEEQIAEFKEAFSLFDKDGDGTITTKELGTVMRSLGQNPTEAELQDMINEVDADGDGTIDFPEFLTMMARKMKYTDSEEEIREAFRVFDKDGNGYISAAELRHVMTNLGEKLTDEEVDEMIREADIDGDGQVNYEEFVQMMTAKHHHHHH
6EUG , Knot 159 374 0.84 40 224 355 
MGSSHHHHHHSSGLVPRGSHMASQKEKGIASGKVWRDTDGNVINAHGGGILFHEGKYYWFGEHRPASGFVTEKGINCYSSTDLYNWKSEGIALAVSEEEGHDIEKGCIMERPKVIYNAKTGKFVMWLHLELKGQGYGPARAAVAVSDSPAGPYRFIRSGRVNPGAYPLNMTRKERKMKWNPEEYKEWWTPKWYEAIAKGMFVKRDLKDGQMSRDMTLFVDDDGKAYHIYSSEDNLTLQIAELADDYLSHTGKYIRIFPGGHNEAPAIFKKEGTYWMITSGCTGWDPNKARLLTADSMLGEWKQLPNPCVGEDADKTFGGQSTYILPLPEKGQFFFMADMWRPKSLADSRYIWLPVQFDDKGVPFIKWMDRWNFD
4WKG , Knot 258 660 0.84 40 283 607 
MKTVVFAYHDMGCLGIEALLAAGYEISAIFTHTDNPGEKAFYGSVARLAAERGIPVYAPDNVNHPLWVERIAQLSPDVIFSFYYRHLIYDEILQLAPAGAFNLHGSLLPKYRGRAPLNWVLVNGETETGVTLHRMVKRADAGAIVAQLRIAIAPDDIAITLHHKLCHAARQLLEQTLPAIKHGNILEIAQRENEATCFGRRTPDDSFLEWHKPASVLHNMVRAVADPWPGAFSYVGNQKFTVWSSRVHPHASKAQPGSVISVAPLLIACGDGALEIVTGQAGDGITMQGSQLAQTLGLVQGSRLNSQPACTARRRTRVLILGVNGFIGNHLTERLLREDHYEVYGLDIGSDAISRFLNHPHFHFVEGDISIHSEWIEYHVKKCDVVLPLVAIATPIEYTRNPLRVFELDFEENLRIIRYCVKYRKRIIFPSTSEVYGMCSDKYFDEDHSNLIVGPVNKPRWIYSVSKQLLDRVIWAYGEKEGLQFTLFRPFNWMGPRLDNLNAARIGSSRAITQLILNLVEGSPIKLIDGGKQKRCFTDIRDGIEALYRIIENAGNRCDGEIINIGNPENEASIEELGEMLLASFEKHPLRHHFPPFAGFRVVESSSYYGKGYQDVEHRKPSIRNAHRCLDWEPKIDMQETIDETLDFFLRTVDLTDKPS

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 \(c\).

\(|P_{f(7AUG_1)}(2) \setminus P_{f(6EUG_1)}(2)|=80\), \(|P_{f(6EUG_1)}(2) \setminus P_{f(7AUG_1)}(2)|=81\). 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 1:11000000100010110111010010010101000001101010100010011101100000001110111111000010000010001000000111101101111011100100110110110010011011111110101010100101010101010010101011000101111110110011011101000100100001100111010100001110001000001010101001100101011010001011100100000001000011010011011000101010000110110011001001010011001010101010110110111001000000001001101100010101011010011001100100001001100101010101000011011010000000
Pair \(Z_2\) Length of longest common subsequence
7AUG_1,6EUG_1 161 6
7AUG_1,4WKG_1 164 4
6EUG_1,4WKG_1 177 3

Newick tree

 
[
	4WKG_1:86.85,
	[
		7AUG_1:80.5,6EUG_1:80.5
	]:6.35
]

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 \((0.85)(0.8)(\frac{796 }{\log_{20} 796}-\frac{374}{\log_{20}374})=114.\)
Status Protein1 Protein2 d d1/2
Query variables 7AUG_1 6EUG_1 141 131.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]