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Parikh vectors
8CWH_1 8DWH_1 7LKX_1 Letter Amino acid
14 13 22 N Asparagine
3 6 11 Y Tyrosine
8 4 12 C Cysteine
2 5 10 M Methionine
3 14 17 F Phenylalanine
2 6 13 P Proline
7 14 24 T Threonine
6 17 11 K Lycine
6 3 3 W Tryptophan
12 16 17 A Alanine
11 20 11 R Arginine
7 20 17 D Aspartic acid
3 7 13 Q Glutamine
2 16 9 E Glutamic acid
6 14 27 V Valine
12 17 27 G Glycine
1 4 7 H Histidine
6 17 12 I Isoleucine
8 18 29 L Leucine
10 15 17 S Serine 

8CWH_1|Chain A|Lysozyme C|Gallus gallus (9031)
>8DWH_1|Chain A[auth B]|Gs-mini-Gq chimera|Homo sapiens (9606)
>7LKX_1|Chains A, B|3C-like proteinase|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 \(w=f(c)\)
8CWH , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
8DWH , Knot 109 246 0.81 40 167 234 
MGSTVSAEDKAAAERSKMIDKNLREDGEKARRTLRLLLLGADNSGKSTIVKQMRILHGGSGGSGGTSGIFETKFQVDKVNFHMFDVGGQRDERRKWIQCFNDVTAIIFVVDSSDYNRLQEALNDFKSIWNNRWLRTISVILFLNKQDLLAEKVLAGKSKIEDYFPEFARYTTPEDATPEPGEDPRVTRAKYFIRKEFVDISTASGDGRHICYPHFTCAVDTENARRIFNDCKDIILQMNLREYNLV
7LKX , Knot 136 309 0.84 40 195 293 
SNIGSGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTF

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 \(c\).

\(|P_{f(8CWH_1)}(2) \setminus P_{f(8DWH_1)}(2)|=53\), \(|P_{f(8DWH_1)}(2) \setminus P_{f(8CWH_1)}(2)|=116\). 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 1:011100011111000110000100110110110100010001000000100001110100011000100110001001100111000101010010011001011011111000001001011101001
Pair \(Z_2\) Length of longest common subsequence
8CWH_1,8DWH_1 169 3
8CWH_1,7LKX_1 177 4
8DWH_1,7LKX_1 186 4

Newick tree

 
[
	7LKX_1:92.77,
	[
		8CWH_1:84.5,8DWH_1:84.5
	]:8.27
]

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 \((0.85)(0.8)(\frac{375 }{\log_{20} 375}-\frac{129}{\log_{20}129})=74.8\)
Status Protein1 Protein2 d d1/2
Query variables 8CWH_1 8DWH_1 96 72.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]