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Parikh vectors
1WKE_1 5RWM_1 8HDW_1 Letter Amino acid
32 16 6 R Arginine
8 17 5 N Asparagine
22 31 7 I Isoleucine
17 7 6 M Methionine
17 20 7 P Proline
18 22 11 V Valine
23 21 11 D Aspartic acid
17 23 4 Q Glutamine
14 32 7 K Lycine
30 38 9 S Serine
4 8 4 W Tryptophan
5 6 0 C Cysteine
10 17 1 H Histidine
20 24 6 F Phenylalanine
8 16 1 Y Tyrosine
39 17 13 A Alanine
24 37 5 E Glutamic acid
33 29 14 G Glycine
27 43 10 L Leucine
18 39 15 T Threonine 

1WKE_1|Chain A|TRNA-GUANINE TRANSGLYCOSYLASE|Zymomonas mobilis (542)
>5RWM_1|Chain A|Phosphatidylinositol 3,4,5-trisphosphate 5-phosphatase 1|Homo sapiens (9606)
>8HDW_1|Chains AA[auth 2], A[auth 1], BA[auth Z], B[auth Y], CA[auth m], DA[auth n], O[auth o], P[auth p], Q[auth q], R[auth r], S[auth s], T[auth t], U[auth u], V[auth v], W[auth w], X[auth x], Y[auth y], Z[auth z]|pam3 tube|uncultured cyanophage (215796)
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 \(w=f(c)\)
1WKE , Knot 160 386 0.82 40 215 373 
MVEATAQETDRPRFSFSIAAREGKARTGTIEMKRGVIRTPAFMPVGTAATVKALKPETVRATGADIILGNTYHLMLRPGAERIAKLGGLHSFMGWDRPILTDSGGYQVMSLSSLTKQSEEGVTFKSHLDGSRHMLSPERSIEIQHLLGSDIVMAFAECTPYPATPSRAASSMERSMRWAKRSRDAFDSRKEQAENAALFGIQQGSVFENLRQQSADALAEIGFDGYAVGGLAVGEGQDEMFRVLDFSVPMLPDDKPHYLMGVGKPDDIVGAVERGIDMFDCVLPTRSGRNGQAFTWDGPINIRNARFSEDLKPLDSECHCAVCQKWSRAYIHHLIRAGEILGAMLMTEHNIAFYQQLMQKIRDSISEGRFSQFAQDFRARYFARNS
5RWM , Knot 195 463 0.86 40 252 447 
SMEQPEPDMITIFIGTWNMGNAPPPKKITSWFLSKGQGKTRDDSADYIPHDIYVIGTQEDPLSEKEWLEILKHSLQEITSVTFKTVAIHTLWNIRIVVLAKPEHENRISHICTDNVKTGIANTLGNKGAVGVSFMFNGTSLGFVNSHLTSGSEKKLRRNQNYMNILRFLALGDKKLSPFNITHRFTHLFWFGDLNYRVDLPTWEAETIIQKIKQQQYADLLSHDQLLTERREQKVFLHFEEEEITFAPTYRFERLTRDKYAYTKQKATGMKYNLPSWCDRVLWKSYPLVHVVCQSYGSTSDIMTSDHSPVFATFEAGVTSQFVSKNGPGTVDSQGQIEFLRCYATLKTKSQTKFYLEFHSSCLESFVKSQEGENEEGSEGELVVKFGETLPKLKPIISDPEYLLDQHILISIKSSDSDESYGEGCIALRLEATETQLPIYTPLTHHGELTGHFQGEIKLQTSQ
8HDW , Knot 69 142 0.80 38 112 138 
MAMKAYSMLNVTATLDGRRVIGLMDGDDAITTSPGVDVGTMLVGADGSWLFSQTADKSATVVIKLKPNSPTHRQLTEKWMAQRAGRLVGFPFDFIDSASNEGGTGAEFFIQKAPDDSKGNNAVVREWTIVTGEWTPTIPTLL

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 \(c\).

\(|P_{f(1WKE_1)}(2) \setminus P_{f(5RWM_1)}(2)|=68\), \(|P_{f(5RWM_1)}(2) \setminus P_{f(1WKE_1)}(2)|=105\). 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 1:11010100000101010111001010010101001110011111110110101101001010110111100001110111001101111001111001110001100110100100000011010001010001101000101001110011111100010110100110010001011000001100000010011111100101100100001011101110101111111101000110110101111100010011111010011111001101100111000100101101011101001010001011000000110001001010011011011111110000111000110010001001010011001010011000
Pair \(Z_2\) Length of longest common subsequence
1WKE_1,5RWM_1 173 4
1WKE_1,8HDW_1 185 3
5RWM_1,8HDW_1 206 4

Newick tree

 
[
	8HDW_1:10.40,
	[
		1WKE_1:86.5,5RWM_1:86.5
	]:14.90
]

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 \((0.85)(0.8)(\frac{849 }{\log_{20} 849}-\frac{386}{\log_{20}386})=124.\)
Status Protein1 Protein2 d d1/2
Query variables 1WKE_1 5RWM_1 161 145.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]