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Parikh vectors
9BBW_1 1WSF_1 1MUM_1 Letter Amino acid
14 8 12 Q Glutamine
9 12 18 E Glutamic acid
26 14 19 G Glycine
16 4 15 S Serine
11 10 18 R Arginine
17 6 18 D Aspartic acid
17 2 10 F Phenylalanine
27 9 21 V Valine
17 16 44 A Alanine
7 5 6 H Histidine
11 10 11 K Lycine
13 5 10 P Proline
24 10 16 T Threonine
3 6 0 W Tryptophan
21 7 11 N Asparagine
11 7 18 I Isoleucine
10 4 8 M Methionine
11 5 10 Y Tyrosine
12 3 2 C Cysteine
29 12 28 L Leucine 

9BBW_1|Chain A|3C-like proteinase nsp5|Severe acute respiratory syndrome coronavirus 2 (2697049)
>1WSF_1|Chains A, B, C, D|Ribonuclease HI|Escherichia coli (562)
>1MUM_1|Chains A, B|2-methylisocitrate lyase|Escherichia coli (562)
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 \(w=f(c)\)
9BBW , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
1WSF , Knot 80 155 0.86 40 128 152 
MLKQVEIFTDGSCLGNPGPGGYGAILRYRGREKTFSAGYTRTTNNRMELMAAIVALEALKEHCEVILSTDSQYVRQGITQWIHNWKARGWKTADKKPVKNVDLWQRLDAALGQHQIKWEWVKGHAGHPENERCAELARAAAMNPTLEDTGYQVEV
1MUM , Knot 130 295 0.83 38 179 283 
SLHSPGKAFRAALTKENPLQIVGTINANHALLAQRAGYQAIYLSGGGVAAGSLGLPDLGISTLDDVLTDIRRITDVCSLPLLVDADIGFGSSAFNVARTVKSMIKAGAAGLHIEDQVGAKRCGHRPNKAIVSKEEMVDRIRAAVDAKTDPDFVIMARTDALAVEGLDAAIERAQAYVEAGAEMLFPEAITELAMYRQFADAVQVPILANITEFGATPLFTTDELRSAHVAMALYPLSAFRAMNRAAEHVYNVLRQEGTQKSVIDTMQTRNELYESINYYQYEEKLDNLFARSQVK

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 \(c\).

\(|P_{f(9BBW_1)}(2) \setminus P_{f(1WSF_1)}(2)|=134\), \(|P_{f(1WSF_1)}(2) \setminus P_{f(9BBW_1)}(2)|=66\). 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 1:011001111010101011010010001011110011001001100000110100001110000001110110101011100100011010100101001000110101100101110001010110001101010101011010010111010000101001001011011011001010101111000010111000010101111101111010011100100010010111100000110000101111101001111101010100110011010011101110001011011000011010
Pair \(Z_2\) Length of longest common subsequence
9BBW_1,1WSF_1 200 3
9BBW_1,1MUM_1 161 4
1WSF_1,1MUM_1 165 4

Newick tree

 
[
	1WSF_1:95.10,
	[
		9BBW_1:80.5,1MUM_1:80.5
	]:14.60
]

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 \((0.85)(0.8)(\frac{461 }{\log_{20} 461}-\frac{155}{\log_{20}155})=90.5\)
Status Protein1 Protein2 d d1/2
Query variables 9BBW_1 1WSF_1 120 90
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]