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Parikh vectors
2XEE_1 1TYW_1 8PDI_1 Letter Amino acid
25 39 17 A Alanine
17 40 21 D Aspartic acid
0 8 6 C Cysteine
12 35 22 V Valine
6 8 17 H Histidine
6 37 31 I Isoleucine
2 23 24 F Phenylalanine
4 40 39 T Threonine
0 7 8 W Tryptophan
3 21 16 Y Tyrosine
3 14 23 Q Glutamine
13 19 37 E Glutamic acid
14 64 29 G Glycine
21 46 43 L Leucine
5 21 16 R Arginine
7 31 17 N Asparagine
14 29 32 K Lycine
1 7 7 M Methionine
4 21 20 P Proline
0 44 38 S Serine 

2XEE_1|Chains A, B, C, D|NI3C DARPIN MUTANT5|SYNTHETIC CONSTRUCT (32630)
>1TYW_1|Chain A|TAILSPIKE PROTEIN|Enterobacteria phage P22 (10754)
>8PDI_1|Chain A|Phosphatidylinositol 3,4,5-trisphosphate 5-phosphatase 1|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 \(w=f(c)\)
2XEE , Knot 43 157 0.46 34 66 83 
DLGKKLLEAARAGQDDEVRILMANGADVNAKDKDGYTPLHLAAREGHLEIVEVLLKAGADVNAKDKDGYTPLHLAAREGHLEIVEVLLKAGADVNAKDKDGYTPLHLAAREGHLEIVEVLLKAGADVNAQDKFGKTPFDLAIDNGNEDIAEVLQKAA
1TYW , Knot 220 554 0.83 40 249 518 
YSIEADKKFKYSVKLSDYPTLQDAASAAVDGLLIDRDYNFYGGETVDFGGKVLTIECKAKFIGDGNLIFTKLGKGSRIAGVFMESTTTPWVIKPWTDDNQWLTDAAAVVATLKQSKTDGYQPTVSDYVKFPGIETLLPPNAKGQNITSTLEIRECIGVEVHRASGLMAGFLFRGCHFCKMVDANNPSGGKDGIITFENLSGDWGKGNYVIGGRTSYGSVSSAQFLRNNGGFERDGGVIGFTSYRAGESGVKTWQGTVGSTTSRNYNLQFRDSVVIYPVWDGFDLGADTDMNPELDRPGDYPITQYPLHQLPLNHLIDNLLVRGALGVGFGMDGKGMYVSNITVEDCAGSGAYLLTHESVFTNIAIIDTNTKDFQANQIYISGACRVNGLRLIGIRSTDGQGLTIDAPNSTVSGITGMVDPSRINVANLAEEGLGNIRANSFGYDSAAIKLRIHKLSKTLDSGALYSHINGGAGSGSAYTQLTAISGSTPDAVSLKVNHKDCRGAEIPFVPDIASDDFIKDSSCFLPYWENNSTSLKALVKKPNGELVRLTLATL
8PDI , Knot 195 463 0.86 40 252 447 
SMEQPEPDMITIFIGTWNMGNAPPPKKITSWFLSKGQGKTRDDSADYIPHDIYVIGTQEDPLSEKEWLEILKHSLQEITSVTFKTVAIHTLWNIRIVVLAKPEHENRISHICTDNVKTGIANTLGNKGAVGVSFMFNGTSLGFVNSHLTSGSEKKLRRNQNYMNILRFLALGDKKLSPFNITHRFTHLFWFGDLNYRVDLPTWEAETIIQKIKQQQYADLLSHDQLLTERREQKVFLHFEEEEITFAPTYRFERLTRDKYAYTKQKATGMKYNLPSWCDRVLWKSYPLVHVVCQSYGSTSDIMTSDHSPVFATFEAGVTSQFVSKNGPGTVDSQGQIEFLRCYATLKTKSQTKFYLEFHSSCLESFVKSQEGENEEGSEGELVVKFGETLPKLKPIISDPEYLLDQHILISIKSSDSDESYGEGCIALRLEATETQLPIYTPLTHHGELTGHFQGEIKLQTSQ

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 \(c\).

\(|P_{f(2XEE_1)}(2) \setminus P_{f(1TYW_1)}(2)|=8\), \(|P_{f(1TYW_1)}(2) \setminus P_{f(2XEE_1)}(2)|=191\). 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 1:0110011011011000010111101101010000100110111001010110111011101010000100110111001010110111011101010000100110111001010110111011101010001100110111001000110110011
Pair \(Z_2\) Length of longest common subsequence
2XEE_1,1TYW_1 199 3
2XEE_1,8PDI_1 218 4
1TYW_1,8PDI_1 143 4

Newick tree

 
[
	2XEE_1:11.21,
	[
		1TYW_1:71.5,8PDI_1:71.5
	]:41.71
]

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 \((0.85)(0.8)(\frac{711 }{\log_{20} 711}-\frac{157}{\log_{20}157})=157.\)
Status Protein1 Protein2 d d1/2
Query variables 2XEE_1 1TYW_1 203 115
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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