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Parikh vectors
7GHK_1 8BTN_1 1LRM_1 Letter Amino acid
17 8 21 A Alanine
11 6 18 R Arginine
21 14 8 N Asparagine
17 11 17 D Aspartic acid
9 13 25 E Glutamic acid
29 20 32 L Leucine
11 17 20 K Lycine
16 22 17 S Serine
26 12 19 G Glycine
11 15 15 I Isoleucine
13 3 20 P Proline
3 5 3 W Tryptophan
14 13 15 Q Glutamine
7 10 10 H Histidine
10 4 2 M Methionine
11 16 20 Y Tyrosine
27 7 15 V Valine
12 1 7 C Cysteine
17 9 22 F Phenylalanine
24 5 19 T Threonine 

7GHK_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>8BTN_1|Chains A, B|TIR domain-containing protein|Bacillus cereus MSX-D12 (1053222)
>1LRM_1|Chain A|Phenylalanine-4-hydroxylase|Homo sapiens (9606)
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 \(w=f(c)\)
7GHK , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
8BTN , Knot 94 211 0.79 40 148 202 
MGSSHHHHHHENLYFQGGMTLFTENDLLNNSYKSIQKSYHFSENQAAKNILEQAYKNYDKNKIYDIFLSHSFLDARKILGLKNYIEGLGYSVYVDWIEDKQLDRSKVSKETAGILRERMQSCKSLFFAISENSDHSLWMPWELGYFDGIKQKVAILPVLKSSYDDSYNGQEYLGLYPYVAKGTIINSTQEEIWIHSSQKQYVRFRNWLQQN
1LRM , Knot 139 325 0.82 40 199 310 
GATVHELSRDKKKDTVPWFPRTIQELDRFANQILSYGAELDADHPGFKDPVYRARRKQFADIAYNYRHGQPIPRVEYMEEEKKTWGTVFKTLKSLYKTHACYEYNHIFPLLEKYCGFHEDNIPQLEDVSQFLQTCTGFRLRPVAGLLSSRDFLGGLAFRVFHCTQYIRHGSKPMYTPEPDICHELLGHVPLFSDRSFAQFSQEIGLASLGAPDEYIEKLATIYWFTVEFGLCKQGDSIKAYGAGLLSSFGELQYCLSEKPKLLPLELEKTAIQNYTVTEFQPLYYVAESFNDAKEKVRNFAATIPRPFSVRYDPYTQRIEVLDNT

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 \(c\).

\(|P_{f(7GHK_1)}(2) \setminus P_{f(8BTN_1)}(2)|=122\), \(|P_{f(8BTN_1)}(2) \setminus P_{f(7GHK_1)}(2)|=74\). 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
7GHK_1,8BTN_1 196 3
7GHK_1,1LRM_1 173 3
8BTN_1,1LRM_1 163 3

Newick tree

 
[
	7GHK_1:95.79,
	[
		1LRM_1:81.5,8BTN_1:81.5
	]:14.29
]

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 \((0.85)(0.8)(\frac{517 }{\log_{20} 517}-\frac{211}{\log_{20}211})=88.2\)
Status Protein1 Protein2 d d1/2
Query variables 7GHK_1 8BTN_1 114 96.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]