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Parikh vectors
5ANI_1 4NYA_1 3ZJT_1 Letter Amino acid
17 0 66 E Glutamic acid
16 0 35 I Isoleucine
21 0 54 K Lycine
17 0 54 T Threonine
4 0 21 W Tryptophan
16 0 42 R Arginine
39 0 66 L Leucine
19 0 42 P Proline
19 18 79 A Alanine
8 0 39 N Asparagine
3 19 11 C Cysteine
10 0 36 Y Tyrosine
16 0 58 D Aspartic acid
9 0 30 Q Glutamine
16 25 60 G Glycine
10 0 23 H Histidine
5 0 35 M Methionine
16 0 27 F Phenylalanine
14 0 29 S Serine
23 0 73 V Valine 

5ANI_1|Chain A|CYCLIN-DEPENDENT KINASE 2|HOMO SAPIENS (9606)
>4NYA_1|Chains A, B|thiM TPP riboswitch|Escherichia coli (562)
>3ZJT_1|Chain A|LEUCYL-TRNA SYNTHETASE|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 \(w=f(c)\)
5ANI , Knot 134 298 0.85 40 191 284 
MENFQKVEKIGEGTYGVVYKARNKLTGEVVALKKIRLDTETEGVPSTAIREISLLKELNHPNIVKLLDVIHTENKLYLVFEFLHQDLKKFMDASALTGIPLPLIKSYLFQLLQGLAFCHSHRVLHRDLKPQNLLINTEGAIKLADFGLARAFGVPVRTYTHEVVTLWYRAPEILLGCKYYSTAVDIWSLGCIFAEMVTRRALFPGDSEIDQLFRIFRTLGTPDEVVWPGVTSMPDYKPSFPKWARQDFSKVVPPLDEDGRSLLSQMLHYDPNKRISAKAALAHPFFQDVTKPVPHLRL
4NYA , Knot 28 80 0.51 8 16 48 
GGACUCGGGGUGCCCUUCUGCGUGAAGGCUGAGAAAUACCCGUAUCACCUGAUCUGGAUAAUGCCAGCGUAGGGAAGUUC
3ZJT , Knot 330 880 0.84 40 320 795 
MGSSHHHHHHSSGLVPRGSHMQEQYRPEEIESKVQLHWDEKRTFEVTEDESKEKYYCLSMLPYPSGRLHMGHVRNYTIGDVIARYQRMLGKNVLQPIGWDAFGLPAEGAAVKNNTAPAPWTYDNIAYMKNQLKMLGFGYDWSRELATCTPEYYRWEQKFFTELYKKGLVYKKTSAVNWCPNDQTVLANEQVIDGCCWRCDTKVERKEIPQWFIKITAYADELLNDLDKLDHWPDTVKTMQRNWIGRSEGVEITFNVNDYDNTLTVYTTRPDTFMGCTYLAVAAGHPLAQKAAENNPELAAFIDECRNTKVAEAEMATMEKKGVDTGFKAVHPLTGEEIPVWAANFVLMEYGTGAVMAVPGHDQRDYEFASKYGLNIKPVILAADGSEPDLSQQALTEKGVLFNSGEFNGLDHEAAFNAIADKLTAMGVGERKVNYRLRDWGVSRQRYWGAPIPMVTLEDGTVMPTPDDQLPVILPEDVVMDGITSPIKADPEWAKTTVNGMPALRETDTFDTFMESSWYYARYTCPQYKEGMLDSEAANYWLPVDIYIGGIEHAIMHLLYFRFFHKLMRDAGMVNSDEPAKQLLCQGMVLADAFYYVGENGERNWVSPVDAIVERDEKGRIVKAKDAAGHELVYTGMSKMSKSKNNGIDPQVMVERYGADTVRLFMMFASPADMTLEWQESGVEGANRFLKRVWKLVYEHTAKGDVAALNVDALTENQKALRRDVHKTIAKVTDDIGRRQTFNTAIAAIMELMNKLAKAPTDGEQDRALMQEALLAVVRMLNPFTPHICFTLWQELKGEGDIDNAPWPVADEKAMVEDSTLVVVQVNGKVRAKITVPVDATEEQVRERAGQEHLVAKYLDGVTVRKVIYVPGKLLNLVVG

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 \(c\).

\(|P_{f(5ANI_1)}(2) \setminus P_{f(4NYA_1)}(2)|=188\), \(|P_{f(4NYA_1)}(2) \setminus P_{f(5ANI_1)}(2)|=13\). 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 1:1001001001101001110010001010111100101000001110011001011001001011011011000001011101100010011010110111111100011011011110000011000101001110001110110111101111110000001101100110111100000011011011011101100011111000100110110011010011111100110001011011000100111110001001100110001000101011110111001001110101
Pair \(Z_2\) Length of longest common subsequence
5ANI_1,4NYA_1 201 2
5ANI_1,3ZJT_1 191 4
4NYA_1,3ZJT_1 324 3

Newick tree

 
[
	4NYA_1:14.56,
	[
		5ANI_1:95.5,3ZJT_1:95.5
	]:50.06
]

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 \((0.85)(0.8)(\frac{378 }{\log_{20} 378}-\frac{80}{\log_{20}80})=92.5\)
Status Protein1 Protein2 d d1/2
Query variables 5ANI_1 4NYA_1 131 78
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]