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Parikh vectors
3ZOI_1 6IXH_1 6QGT_1 Letter Amino acid
25 21 27 L Leucine
15 11 8 T Threonine
20 4 9 Y Tyrosine
22 13 17 V Valine
24 15 17 D Aspartic acid
4 3 6 M Methionine
20 7 11 P Proline
5 2 0 W Tryptophan
19 5 10 N Asparagine
17 7 7 Q Glutamine
18 6 16 E Glutamic acid
19 14 22 G Glycine
20 10 15 K Lycine
20 11 15 S Serine
28 19 16 A Alanine
12 9 8 R Arginine
2 1 17 C Cysteine
10 1 7 H Histidine
15 13 19 I Isoleucine
16 6 6 F Phenylalanine 

3ZOI_1|Chain A|ISOPENICILLIN N SYNTHASE|Emericella nidulans (strain FGSC A4 / ATCC 38163 / CBS 112.46 / NRRL 194 / M139) (227321)
>6IXH_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O|Type VI Secretion System TssJ|Escherichia coli (strain 55989 / EAEC) (585055)
>6QGT_1|Chain A[auth G]|Coenzyme F420 hydrogenase subunit gamma|Methanosarcina barkeri MS (1434108)
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 \(w=f(c)\)
3ZOI , Knot 149 331 0.87 40 206 319 
MGSVSKANVPKIDVSPLFGDDQAAKMRVAQQIDAASRDTGFFYAVNHGINVQRLSQKTKEFHMSITPEEKWDLAIRAYNKEHQDQVRAGYYLSIPGKKAVESFCYLNPNFTPDHPRIQAKTPTHEVNVWPDETKHPGFQDFAEQYYWDVFGLSSALLKGYALALGKEENFFARHFKPDDTLASVVLIRYPYLDPYPEAAIKTAADGTKLSFEWHEDVSLITVLYQSNVQNLQVETAAGYQDIEADDTGYLINCGSYMAHLTNNYYKAPIHRVKWVNAERQSLPFFVNLGYDSVIDPFDPREPNGKSDREPLSYGDYLQNGLVSLINKNGQT
6IXH , Knot 82 178 0.79 40 128 169 
MAIIAGKAGYGLIIALFSLSLSGCGLTQRVADGTVSATKSLFYRQIKTLHLDIRAREAINTSAAGIPLSVVVRIYQLKDNRSFDSADYQALFTGDNEILAGDIIAQKDVWLQPGGSVAVDMPLDDAAKFTGVAAMFLEPDQKKNTWRVVLGRDELEPDTPRLIEVSGNTLTLLPVKDK
6QGT , Knot 112 253 0.81 38 164 242 
TNKIKIGHVHMSGCTGCLVSLADNNLGLIKILDDYADLVYCLTLADVRHIPEMDVALVEGSVCLQDHESVEDIKETRKKSKIVVALGSCACYGNITRFSRGGQHNQPQHESYLPIGDLIDVDVYIPGCPPSPELIRNVAVMAYLLLEGNEEQKELAGKYLKPLMDLAKRGTSGCFCDLMYDVINQGLCMGCGTCAASCPVHAITLEFGKPQGERDLCIKCGSCYGACPRSFFNLDVISEFENISEIIAKALKD

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 \(c\).

\(|P_{f(3ZOI_1)}(2) \setminus P_{f(6IXH_1)}(2)|=117\), \(|P_{f(6IXH_1)}(2) \setminus P_{f(3ZOI_1)}(2)|=39\). 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 1:1101001011010101111000110101100101100001110110011010010000001010101000101110100000000101100101110011001001010101001010100100010111000001110011000010111100111010111110000111001010001101111001010101011100110100101010001011011000010010100111000101000101100100110100000011100101101000011111011000110110100101000001100100100111011000100
Pair \(Z_2\) Length of longest common subsequence
3ZOI_1,6IXH_1 156 4
3ZOI_1,6QGT_1 180 4
6IXH_1,6QGT_1 156 3

Newick tree

 
[
	6QGT_1:86.18,
	[
		3ZOI_1:78,6IXH_1:78
	]:8.18
]

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 \((0.85)(0.8)(\frac{509 }{\log_{20} 509}-\frac{178}{\log_{20}178})=96.3\)
Status Protein1 Protein2 d d1/2
Query variables 3ZOI_1 6IXH_1 124 93
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]