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Parikh vectors
5OGJ_1 3WVN_1 8RJF_1 Letter Amino acid
8 13 7 Y Tyrosine
26 54 42 L Leucine
16 16 15 K Lycine
2 16 12 M Methionine
11 33 26 T Threonine
1 6 6 C Cysteine
14 30 30 E Glutamic acid
17 40 35 G Glycine
14 24 33 I Isoleucine
7 5 1 W Tryptophan
11 39 33 R Arginine
12 19 9 H Histidine
11 14 9 F Phenylalanine
18 35 17 P Proline
29 31 20 S Serine
16 42 38 V Valine
14 62 33 A Alanine
9 12 16 N Asparagine
16 40 27 D Aspartic acid
11 10 15 Q Glutamine 

5OGJ_1|Chains A, B|Carbonic anhydrase 13|Homo sapiens (9606)
>3WVN_1|Chains A, B|Non-ribosomal peptide synthetase|Streptomyces halstedii (1944)
>8RJF_1|Chains A, B, C, D, E, F|Pilus assembly ATPase CpaF|Caulobacter vibrioides NA1000 (565050)
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 \(w=f(c)\)
5OGJ , Knot 116 263 0.82 40 175 257 
MMSRLSWGYREHNGPIHWKEFFPIADGDQQSPIEIKTKEVKYDSSLRPLSIKYDPSSAKIISNSGHSFNVDFDDTENKSVLRGGPLTGSYRLRQVHLHWGSADDHGSEHIVDGVSYAAELHVVHWNSDKYPSFVEAAHEPDGLAVLGVFLQIGEPNSQLQKITDTLDSIKEKGKQTRFTNFDLLSLLPPSWDYWTYPGSLTVPPLLESVTWIVLKQPINISSQQLAKFRSLLCTAEGEAAAFLVSNHRPPQPLKGRKVRASFH
3WVN , Knot 214 541 0.83 40 241 501 
MNHKVHHHHHHIEGRHMYSPDRRAALNSVANMVSDNADKDLRYGGLVHDLLADSGKATPNSDAMEDAFGTWTYQELLNHSQAFSAWLDGKGVARGERIVVQLPNIRQTVAVFYGACRRGVVFVPLNPGMKPFHLRSVIADADPRLVIAEDETAADRLRDVTDLPVYSIDSLWADVERLRDAGAGAEAVEVSPEDLAVLIYTSGSTAAPKAVACPHQQIVFAASSINAVLGYHAEDIVFCRMSVSWDFGLYKVLISTLTGAKLVLAGGEPDIALVKSLRESGATMMPIVPSLASMLTTLIRRDPEGAPTLRMFTNSAAALPQVTIDALRSAFPGAQVVRMYGQTECKRISIMPPHLEHERPDSVGLPLPGTTIEILDEDGTLLPPGEPGEITVTGPHVMAGYWRAPEITARAYRRDETTGAMRLHTGDYGHLDEDGFLYFGGRRDDMFKRKGTRMSTVEIEAAALDIPGVTAAVALPPTATRDLALCVASDLEPHDVLRSLAERLEPAKVPATCRIVNDFPLTLNGKSERKQLARLLDGSDK
8RJF , Knot 176 424 0.83 40 213 396 
DYYHATKTTIFNALLNTIDLSQLAQLDLKQAGEEIRDIVAELVAIKNVSMSVAEQEHLVQDIINDVLGYGPLEPLLARDDIADIMVNGAHRVFIEVGGKVQLTNVRFRDNLQLMNICQRIVSQVGRRVDESSPICDARLPDGSRVNVIAPPLALDGPTLTIRKFKKDKLTMKNLVEFASISPEGARVLGVIGACRCNLVISGGTGSGKTTLLNTMTAFIDPTERVVTCEDAAELQLQQPHVVRLETRPPNLEGSGAVTMRDLVKNCLRMRPERIIVGEVRGPEAFDLLQAMNTGHDGSMGTLHANSPREAISRIESMITMGGYGLPSKTIKEMIVGSVDVIIQAARLRDGSRRITHITEVVGLEGDVIVTQDLFVYEITGEDEHGKVVGKHRSTGIARPRFWDRARYYGLERELAEALDAAEAL

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 \(c\).

\(|P_{f(5OGJ_1)}(2) \setminus P_{f(3WVN_1)}(2)|=48\), \(|P_{f(3WVN_1)}(2) \setminus P_{f(5OGJ_1)}(2)|=114\). 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 1:11001011000001110100111110100001101000010000010110100010010110001001010100000001101111010001001010110100010001101100110101101000001011011001011111111101101000100100010010001000010010110111101001001101011111001011110011010000110100110010101111110000110110100101010
Pair \(Z_2\) Length of longest common subsequence
5OGJ_1,3WVN_1 162 4
5OGJ_1,8RJF_1 156 4
3WVN_1,8RJF_1 142 4

Newick tree

 
[
	5OGJ_1:82.15,
	[
		8RJF_1:71,3WVN_1:71
	]:11.15
]

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 \((0.85)(0.8)(\frac{804 }{\log_{20} 804}-\frac{263}{\log_{20}263})=148.\)
Status Protein1 Protein2 d d1/2
Query variables 5OGJ_1 3WVN_1 184 137
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]