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Parikh vectors
6JMG_1 8VGM_1 8GFK_1 Letter Amino acid
20 20 9 E Glutamic acid
13 25 16 F Phenylalanine
20 32 27 V Valine
20 11 18 A Alanine
16 13 11 R Arginine
6 0 11 C Cysteine
20 10 26 G Glycine
16 29 29 L Leucine
27 11 11 K Lycine
16 17 16 S Serine
10 15 24 T Threonine
9 10 11 Y Tyrosine
14 9 21 N Asparagine
18 11 17 D Aspartic acid
8 8 13 Q Glutamine
11 5 7 H Histidine
20 40 11 I Isoleucine
7 16 10 M Methionine
7 9 13 P Proline
3 5 3 W Tryptophan 

6JMG_1|Chains A, B|DnaJ homolog subfamily C member 27-A|Xenopus laevis (8355)
>8VGM_1|Chains A, D[auth B], E[auth C], F[auth D]|Chimeric Nav1.7-NavAb|Aliarcobacter butzleri RM4018 (367737)
>8GFK_1|Chain A|3C-like proteinase nsp5|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
6JMG , Knot 125 281 0.83 40 192 273 
METNLQKRKDSRKALRIKVISMGNAEVGKSCIIKRYCEKRFVPKYQATIGIDYGVTKVHIKDREIKVNIFDMAGHPFFYEVRNEFYKDTQGVILVYDVGHKETFESLDGWLAEMKQELGPQGNIDNIVFAVCANKIDSTKHRSVDESEGRLWSESKGFLYFETSAQSGEGINEMFQAFYSAIVDLCDNGGKRPVSAINIGFTKEQADSIRRIRNCKDSWDMLGVKPGATRDEVNKAYRKLAVLLHPDKCMAPGSEDAFKAVVNARTALLKNIKLEHHHHHH
8VGM , Knot 130 296 0.83 38 182 281 
MDYKDDDDKGSLVPRGSHMYLRITNIVESSFFTKFIIYLIVLNMVTMMVEKEGQSQHMTEVLYWINVVFIILFTIEIILRIYVHRISFFKDPWSLFDFVVVIISIVGMFLADLIETYFVSPTLFRVIRLARIGRILRLVTAVPQMRKIVSALISVIPGMLSVIALMTLFFYIFAIMATQLFGERFPEWFGTLGESFYTLFQVMTLESWSMGIVRPLMEVYPYAWVFFIPFIFVVTFVMINLVVAIIVDAMAILNQKEEQHIIDEVQSHEDNINNEIIKLREEIVELKELIKTSLKN
8GFK , Knot 131 304 0.82 40 194 287 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSAGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVT

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 \(c\).

\(|P_{f(6JMG_1)}(2) \setminus P_{f(8VGM_1)}(2)|=86\), \(|P_{f(8VGM_1)}(2) \setminus P_{f(6JMG_1)}(2)|=76\). 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 1:10001000000001101011011010110001100000001110001011100110010100001010110111011100100010000011111001100001001011110100011101010011111010010000000100001011000011101000100101100110110011101000110011011011100001001001000000101111011100001001000111110100011110001101110100111001010000000
Pair \(Z_2\) Length of longest common subsequence
6JMG_1,8VGM_1 162 3
6JMG_1,8GFK_1 180 3
8VGM_1,8GFK_1 182 4

Newick tree

 
[
	8GFK_1:93.45,
	[
		6JMG_1:81,8VGM_1:81
	]:12.45
]

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 \((0.85)(0.8)(\frac{577 }{\log_{20} 577}-\frac{281}{\log_{20}281})=83.3\)
Status Protein1 Protein2 d d1/2
Query variables 6JMG_1 8VGM_1 104 102.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]