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Parikh vectors
1BDO_1 1LQS_1 6OPC_1 Letter Amino acid
2 7 42 N Asparagine
8 16 73 E Glutamic acid
7 11 51 I Isoleucine
4 2 23 M Methionine
0 5 3 W Tryptophan
11 17 64 V Valine
5 10 41 P Proline
5 23 40 S Serine
5 9 81 A Alanine
4 5 69 D Aspartic acid
1 4 4 C Cysteine
6 11 59 G Glycine
1 10 11 H Histidine
3 15 69 L Leucine
4 18 39 T Threonine
1 8 18 Y Tyrosine
2 11 51 R Arginine
3 11 20 Q Glutamine
5 9 49 K Lycine
3 12 28 F Phenylalanine 

1BDO_1|Chain A|ACETYL-COA CARBOXYLASE|Escherichia coli (562)
>1LQS_1|Chains A[auth R], B[auth S]|INTERLEUKIN-10 RECEPTOR ALPHA CHAIN|Homo sapiens (9606)
>6OPC_1|Chains A, B, C, D, E, F|Cell division control protein 48|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
1BDO , Knot 43 80 0.78 38 64 78 
EISGHIVRSPMVGTFYRTPSPDAKAFIEVGQKVNVGDTLCIVEAMKMMNQIEADKSGTVKAILVESGQPVEFDEPLVVIE
1LQS , Knot 102 214 0.85 40 160 208 
HGTELPSPPSVWFEAEFFHHILHWTPIPQQSESTCYEVALLRYGIESWNSISQCSQTLSYDLTAVTLDLYHSNGYRARVRAVDGSRHSQWTVTNTRFSVDEVTLTVGSVNLEIHNGFILGKIQLPRPKMAPAQDTYESIFSHFREYEIAIRKVPGQFTFTHKKVKHEQFSLLTSGEVGEFCVQVKPSVASRSNKGMWSKEECISLTRQYFTVTN
6OPC , Knot 305 835 0.82 40 274 729 
MGEEHKPLLDASGVDPREEDKTATAILRRKKKDNMLLVDDAINDDNSVIAINSNTMDKLELFRGDTVLVKGKKRKDTVLIVLIDDELEDGACRINRVVRNNLRIRLGDLVTIHPCPDIKYATRISVLPIADTIEGITGNLFDVFLKPYFVEAYRPVRKGDHFVVRGGMRQVEFKVVDVEPEEYAVVAQDTIIHWEGEPINREDEENNMNEVGYDDIGGCRKQMAQIREMVELPLRHPQLFKAIGIKPPRGVLMYGPPGTGKTLMARAVANETGAFFFLINGPEVMSKMAGESESNLRKAFEEAEKNAPAIIFIDEIDSIAPKRDKTNGEVERRVVSQLLTLMDGMKARSNVVVIAATNRPNSIDPALRRFGRFDREVDIGIPDATGRLEVLRIHTKNMKLADDVDLEALAAETHGYVGADIASLCSEAAMQQIREKMDLIDLDEDEIDAEVLDSLGVTMDNFRFALGNSNPSALRETVVESVNVTWDDVGGLDEIKEELKETVEYPVLHPDQYTKFGLSPSKGVLFYGPPGTGKTLLAKAVATEVSANFISVKGPELLSMWYGESESNIRDIFDKARAAAPTVVFLDELDSIAKARGGSLGDAGGASDRVVNQLLTEMDGMNAKKNVFVIGATNRPDQIDPAILRPGRLDQLIYVPLPDENARLSILNAQLRKTPLEPGLELTAIAKATQGFSGADLLYIVQRAAKYAIKDSIEAHRQHEAEKEVKVEGEDVEMTDEGAKAEQEPEVDPVPYITKEHFAEAMKTAKRSVSDAELRRYEAYSQQMKASRGQFSNFNFNDAPLGTTATDNANSNNSAPSGAGAAFGSNAEEDDDLYS

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 \(c\).

\(|P_{f(1BDO_1)}(2) \setminus P_{f(1LQS_1)}(2)|=33\), \(|P_{f(1LQS_1)}(2) \setminus P_{f(1BDO_1)}(2)|=129\). 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 1:01010110011110100010101011101100101100101101101100101000101011110010110100111110
Pair \(Z_2\) Length of longest common subsequence
1BDO_1,1LQS_1 162 3
1BDO_1,6OPC_1 230 4
1LQS_1,6OPC_1 180 4

Newick tree

 
[
	6OPC_1:10.67,
	[
		1BDO_1:81,1LQS_1:81
	]:28.67
]

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 \((0.85)(0.8)(\frac{294 }{\log_{20} 294}-\frac{80}{\log_{20}80})=68.1\)
Status Protein1 Protein2 d d1/2
Query variables 1BDO_1 1LQS_1 88 58.5
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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