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Parikh vectors
4HEC_1 3OWA_1 5MUG_1 Letter Amino acid
0 24 6 N Asparagine
8 19 8 Q Glutamine
8 41 22 I Isoleucine
6 22 15 F Phenylalanine
21 23 13 R Arginine
13 47 10 G Glycine
1 48 14 K Lycine
5 17 4 M Methionine
8 32 18 S Serine
12 30 9 T Threonine
15 59 13 A Alanine
13 64 16 E Glutamic acid
10 10 9 H Histidine
17 59 24 L Leucine
14 16 10 P Proline
2 3 2 C Cysteine
6 1 5 W Tryptophan
5 21 8 Y Tyrosine
11 40 12 V Valine
15 21 13 D Aspartic acid 

4HEC_1|Chains A, B|Putative uncharacterized protein|Mycobacterium tuberculosis (83332)
>3OWA_1|Chains A, B, C, D|Acyl-CoA dehydrogenase|Bacillus anthracis (261594)
>5MUG_1|Chain A|Alpha-tocopherol transfer protein|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 \(w=f(c)\)
4HEC , Knot 86 190 0.79 38 131 179 
MAHHHHHHMGTLEAQTQGPGSMVRYFYDTEFIEDGHTIELISIGVVAEDGREYYAVSTEFDPERAGSWVRTHVLPKLPPPASQLWRSRQQIRLDLEEFLRIDGTDSIELWAWVGAYDHVALCQLWGPMTALPPTVPRFTRELRQLWEDRGCPRMPPRPRDVHDALVDARDQLRRFRLITSTDDAGRGAAR
3OWA , Knot 230 597 0.82 40 236 542 
SNAMEKTVGNAVKGGSFLVDEITIDQVFTPEDFSSEHKMIAKTTEDFIVNEVLPELEYLEQHEFDRSVRLLKEAGELGLLGADVPEEYGGIGLDKVSSALIAEKFSRAGGFAITHGAHVGIGSLPIVLFGNEEQKKKYLPLLATGEKLAAYALTEPGSGSDALGAKTTARLNAEGTHYVLNGEKQWITNSAFADVFIVYAKIDGEHFSAFIVEKDYAGVSTSPEEKKMGIKCSSTRTLILEDALVPKENLLGEIGKGHIIAFNILNIGRYKLGVGTVGSAKRAVEISAQYANQRQQFKQPIARFPLIQEKLANMAAKTYAAESSVYRTVGLFESRMSTLSEEEVKDGKAVAASIAEYAIECSLNKVFGSEVLDYTVDEGVQIHGGYGFMAEYEIERMYRDSRINRIFEGTNEINRLIVPGTFLRKAMKGELPLLQKAQKLQEELMMMMPEEVGDEPLALQKYLVNNAKKIGLMVAGLAAQKYGKALDKEQEILVNIADIVSNLYAMESAVLRTEKAIKTTGLEKNKQKVLYTEVFCQEAFNEIEAHAKETLIAVENGDMLRMMLSSLRKLTRHTPLNVIPKKREIAAKILEDERYTV
5MUG , Knot 106 231 0.83 40 169 224 
TDSFLLRFLRARDFDLDLAWRLLKNYYKWRAECPEISADLHPRSIIGLLKAGYHGVLRSRDPTGSKVLIYRIAHWDPKVFTAYDVFRVSLITSELIVQEVETQRNGIKAIFDLEGWQFSHAFQITPSVAKKIAAVLTDSFPLKVRGIHLINEPVIFHAVFSMIKPFLTEKIKERIHMHGNNYKQSLLQHFPDILPLEYGGEEFSMEDICQEWTNFIMKSEDYLSSISESIQ

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 \(c\).

\(|P_{f(4HEC_1)}(2) \setminus P_{f(3OWA_1)}(2)|=44\), \(|P_{f(3OWA_1)}(2) \setminus P_{f(4HEC_1)}(2)|=149\). 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 1:1100000011010100011101100100001100100101101111100100001100010100110110001110111110011000001010100110101000101111111000111001111101111011010001001100010101110100100111010001001011000001101110
Pair \(Z_2\) Length of longest common subsequence
4HEC_1,3OWA_1 193 4
4HEC_1,5MUG_1 174 4
3OWA_1,5MUG_1 157 4

Newick tree

 
[
	4HEC_1:95.91,
	[
		5MUG_1:78.5,3OWA_1:78.5
	]:17.41
]

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 \((0.85)(0.8)(\frac{787 }{\log_{20} 787}-\frac{190}{\log_{20}190})=166.\)
Status Protein1 Protein2 d d1/2
Query variables 4HEC_1 3OWA_1 206 136
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]