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Parikh vectors
2FLY_1 5GVK_1 5MOE_1 Letter Amino acid
0 28 20 G Glycine
0 10 16 H Histidine
3 37 23 K Lycine
0 10 2 C Cysteine
1 26 23 E Glutamic acid
1 20 14 F Phenylalanine
0 17 14 P Proline
2 28 21 S Serine
0 20 11 T Threonine
1 29 15 N Asparagine
0 7 10 M Methionine
2 2 5 W Tryptophan
1 32 26 V Valine
3 33 13 A Alanine
1 24 33 D Aspartic acid
0 30 20 I Isoleucine
2 38 30 L Leucine
0 13 23 Y Tyrosine
3 16 24 R Arginine
0 22 9 Q Glutamine 

2FLY_1|Chain A|Proadrenomedullin N-20 terminal peptide|Homo sapiens (9606)
>5GVK_1|Chains A, B, C|Serine hydroxymethyltransferase, putative|Plasmodium vivax (strain Salvador I) (126793)
>5MOE_1|Chains A, B|Casein kinase II subunit alpha|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 \(w=f(c)\)
2FLY , Knot 14 21 0.67 24 19 19 
ARLDVASEFRKKWNKWALSRX
5GVK , Knot 184 442 0.84 40 236 419 
MFNNEPLEQIDKELHDILADEEKRQRETINLIASENLTNGAVRECLGNRVSNKYSEGYPKKRYYGGNDFIDKIEELCQKRALEAFNVSDEEWGVNVQPLSGSAANVQALYALVGVKGKIMGMHLCSGGHLTHGFFDEKKKVSITSDMFESKLYKCNSQGYVDLDAVREMALSFKPKVIICGYTSYPRDIDYQQFRQICDEVNAYLFADISHISSFVACNILNNPFLHADVVTTTTHKILRGPRSALIFFNKKRNPGIEQKINSAVFPSFQGGPHNNKIAAVACQLKEVHSPAFKEYTQQVLLNSKALAKALISKQIDLVTNGTDNHLIVVDLRKFSITGSKLQETCNAINVSLNKNTIPSDVDCVSPSGVRIGTPAMTTRGAKEKDMEFIADVLARAIKITVDLQEQYGKKLVDFKKGLPGNAQLQQLKQEVVTWAGALPFP
5MOE , Knot 153 352 0.85 40 200 331 
GSMDIEFDDDADDDGSGSGSGSGSSGPVPSRARVYTDVNTHRPSEYWDYESHVVEWGNQDDYQLVRKLGRGKYSEVFEAINITNNEKVVVKILKPVKKKKIKREIKILENLRGGPNIITLADIVKDPVSRTPALVFEHVNNTDFKQLYQTLTDYDIRFYMYEILKALDYCHSMGIMHRDVKPHNVMIDHEHRKLRLIDWGLAEFYHPGQEYNVRVASRYFKGPELLVDYQMYDYSLDMWSLGCMLASMIFRKEPFFHGHDNYDQLVRIAKVLGTEDLYDYIDKYNIELDPRFNDILGRHSRKRWERFVHSENQHLVSPEALDFLDKLLRYDHQSRLTAREAMEHPYFYTVVK

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 \(c\).

\(|P_{f(2FLY_1)}(2) \setminus P_{f(5GVK_1)}(2)|=5\), \(|P_{f(5GVK_1)}(2) \setminus P_{f(2FLY_1)}(2)|=222\). 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 1:101011001000100111000
Pair \(Z_2\) Length of longest common subsequence
2FLY_1,5GVK_1 227 3
2FLY_1,5MOE_1 193 3
5GVK_1,5MOE_1 170 4

Newick tree

 
[
	2FLY_1:11.30,
	[
		5MOE_1:85,5GVK_1:85
	]:26.30
]

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 \((0.85)(0.8)(\frac{463 }{\log_{20} 463}-\frac{21}{\log_{20}21})=139.\)
Status Protein1 Protein2 d d1/2
Query variables 2FLY_1 5GVK_1 179 93.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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