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Parikh vectors
1PME_1 2JGU_1 1HPG_1 Letter Amino acid
23 83 4 K Lycine
13 26 4 F Phenylalanine
19 19 9 N Asparagine
7 4 4 C Cysteine
11 15 3 Q Glutamine
27 70 7 I Isoleucine
22 26 25 S Serine
19 47 9 Y Tyrosine
23 38 6 D Aspartic acid
20 88 2 E Glutamic acid
45 68 4 L Leucine
10 10 1 M Methionine
16 51 21 V Valine
21 14 4 H Histidine
21 36 4 P Proline
14 32 19 T Threonine
3 11 1 W Tryptophan
26 43 22 A Alanine
20 44 5 R Arginine
20 50 33 G Glycine 

1PME_1|Chain A|ERK2|Homo sapiens (9606)
>2JGU_1|Chain A|DNA POLYMERASE|PYROCOCCUS FURIOSUS (2261)
>1HPG_1|Chain A|Glutamic acid specific protease|Streptomyces griseus (1911)
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 \(w=f(c)\)
1PME , Knot 162 380 0.84 40 240 365 
MSSSHHHHHHSSGLVPRGSHMAAAAAAGAGPEMVRGQVFDVGPRYTNLSYIGEGAYGMVCSAYDNVNKVRVAIKKISPFEHQTYCQRTLREIKILLRFRHENIIGINDIIRAPTIEQMKDVYLVTHLMGADLYKLLKTQHLSNDHICYFLYQILRGLKYIHSANVLHRDLKPSNLLLNTTCDLKICDFGLARVADPDHDHTGFLTEYVATRWYRAPEIMLNSKGYTKSIDIWSVGCILAEMLSNRPIFPGKHYLDQLNHILGILGSPSQEDLNCIINLKARNYLLSLPHKNKVPWNRLFPNADSKALDLLDKMLTFNPHKRIEVEQALAHPYLEQYYDPSDEPIAEAPFKFDMELDDLPKEKLKELIFEETARFQPGYRS
2JGU , Knot 290 775 0.83 40 272 695 
MILDVDYITEEGKPVIRLFKKENGKFKIEHDRTFRPYIYALLRDDSKIEEVKKITGERHGKIVRIVDVEKVEKKFLGKPITVWKLYLEHPQDVPTLREKVREHPAVVDIFEYDIPFAKRYLIDKGLIPMEGEEELKILAFDIETLYHEGEEFGKGPIIMISYADENEARVITWKNIDLPYVESVSTEKEMIKRFLRIIREKDPDIIVTYNGDSFDFPYLAKRAEKLGIKLTIGRDGSEPKMQRIGDMTAVEVKGRIHFDLYHVIRTTINLPTYTLEAVYEAIFGKPKEKVYADEIAKAWESGENLERVAKYSMEDAKATYELGKEFLPMEIQLSRLVGQPLWDVSRSSTGNLVEWFLLRKAYERNEVAPNKPSEEEYQRRLRESYTGGFVKEPEKGLWENIVYLDYKSLYPSIIITHNVSPDTLNLEGCKNYDIAPQVGHKFCKDIPGFIPSLLGHLLEERQKIKTKMKETQDPIEKILLDYRQKAIKLLANSFYGYYGYAKARWYCKECAESVTAWGRKYIELVWKELEEKFGFKVLYIDTDGLYATIPGGESEEIKKKALEFVKYINSKLPGLLELEYEGFYKRGFFVTKKRYAVIDEEGKVITRGLEIVRRDWSEIAKETQARVLETILKHGDVEEAVRIVKEVIQKLANYEIPPEKLAIYEQITRPLHEYKAIGPHVAVAKKLAAKGVKIKPGMVIGYIVLRGDGPISNRAILAEEYDPKKHKYDAEYYIENQVLPAVLRILEGFGYRKEDLRYQKTRQVGLTSWLNIKKS
1HPG , Knot 78 187 0.72 40 102 165 
VLGGGAIYGGGSRCSAAFNVTKGGARYFVTAGHCTNISANWSASSGGSVVGVREGTSFPTNDYGIVRYTDGSSPAGTVDLYNGSTQDISSAANAVVGQAIKKSGSTTKVTSGTVTAVNVTVNYGDGPVYNMVRTTACSAGGDSGGAHFAGSVALGIHSGSSGCSGTAGSAIHQPVTEALSAYGVTVY

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 \(c\).

\(|P_{f(1PME_1)}(2) \setminus P_{f(2JGU_1)}(2)|=52\), \(|P_{f(2JGU_1)}(2) \setminus P_{f(1PME_1)}(2)|=84\). 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 1:10000000000011110100111111111110110101101110000100110110111001000100101110010110000000001001011101000011110011011010010010110011110100110000100001001100110110010010110001010011100000101001111011010000011100011001001101110001000010110110111011000111110001001001111110100001001101010001101100001110011101000110110011010100010100111010100000100011101110101010011000100111000101011000
Pair \(Z_2\) Length of longest common subsequence
1PME_1,2JGU_1 136 4
1PME_1,1HPG_1 206 4
2JGU_1,1HPG_1 212 4

Newick tree

 
[
	1HPG_1:11.11,
	[
		1PME_1:68,2JGU_1:68
	]:46.11
]

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 \((0.85)(0.8)(\frac{1155 }{\log_{20} 1155}-\frac{380}{\log_{20}380})=203.\)
Status Protein1 Protein2 d d1/2
Query variables 1PME_1 2JGU_1 254 189
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]