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Parikh vectors
8HFK_1 3LNL_1 6HSB_1 Letter Amino acid
1 2 0 C Cysteine
15 39 10 I Isoleucine
17 28 19 L Leucine
5 12 0 M Methionine
13 5 11 R Arginine
11 24 5 N Asparagine
17 27 6 D Aspartic acid
8 17 5 Q Glutamine
20 15 8 S Serine
1 2 1 W Tryptophan
30 28 9 V Valine
14 26 1 K Lycine
14 14 5 F Phenylalanine
11 12 6 P Proline
27 21 20 A Alanine
14 26 8 E Glutamic acid
27 27 10 G Glycine
16 12 10 H Histidine
13 17 8 T Threonine
5 16 3 Y Tyrosine 

8HFK_1|Chains A, B, C, D|Versicolorin reductase|Cercospora sp. JNU001 (2979285)
>3LNL_1|Chains A, B|UPF0135 protein SA1388|Staphylococcus aureus subsp. aureus (158879)
>6HSB_1|Chain A|3-dehydroquinate dehydratase|Acidithiobacillus caldus (strain SM-1) (990288)
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 \(w=f(c)\)
8HFK , Knot 121 279 0.81 40 171 267 
MAMSPPTQDLHIPGRLDGKVALVTGSGRGIGAAVAVHLGLLGAKVVVNYANSPTHAQKVVDEIKQLGSDAIAIKADVRQVPEIVRLFDEAVAHFGQLDIAVSNSGVVSFGHLKDVTEEEFDRVFSLNTRGQFFVAREAYKHLNNGGRIIMTSSNTSRDFSVPKFSLYSGSKGAIDSFVRIFSKDCGDKKITVNAVAPGGTVTDMFHDVSQHYIPNGETYTPEERQKMAAHASPLHRNGFPEDIARVVGFLVSAEGEWINGKVLTVDGGAAALEHHHHHH
3LNL , Knot 154 370 0.82 40 204 346 
AMDPMKIADLMTLLDHHVPFSTAESWDNVGLLIGDEDVEVTGVLTALDCTLEVVNEAIEKGYNTIISHHPLIFKGVTSLKANGYGLIIRKLIQHDINLIAMHTNLDVNPHGVNMMLAKVMGLKNISIINNQQDVYYKVQTYIPKDNVGPFKDKLSENGLAQEGNYEYCFFESEGRGQFKPVGEANPTIGQIDKIEDVDEVKIEFMIDAYQKSRAEQLIKQYHPYETPVFDFIEIKQTSLYGLGVMAEVDNQMTLEDFAADIKSKLNIPSVRFVGESNQKIKRIAIIGGSGIGYEYQAVQQGADVFVTGDIKHHDALDAKIHGVNLIDINHYSEYVMKEGLKTLLMNWFNIEKINIDVEASTINTDPFQYI
6HSB , Knot 69 145 0.79 36 104 141 
AQIHVLVLHGPNLNLLGRREPDHYGRTTLAEIDARLKTEAEGRGWLLHSLQSNAEHILVDAVQQAPTQGVTHIVLNPAAFTHTSVALRDALAAVAIPFIEVHLSNIHAREPFRRHSYFSDIASGLITGLGAEGYSLALDAIARRF

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 \(c\).

\(|P_{f(8HFK_1)}(2) \setminus P_{f(3LNL_1)}(2)|=66\), \(|P_{f(3LNL_1)}(2) \setminus P_{f(8HFK_1)}(2)|=99\). 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 1:111011000101110101011110101011111111011111101110010010010011001001100111101010011011011001110110101110001110110100100001001101000101111001000100110111000000001011010100100111001101100001000101011111101001100100001101000010000011101011000111001101111110101011010110101111110000000
Pair \(Z_2\) Length of longest common subsequence
8HFK_1,3LNL_1 165 4
8HFK_1,6HSB_1 157 4
3LNL_1,6HSB_1 180 4

Newick tree

 
[
	3LNL_1:88.78,
	[
		8HFK_1:78.5,6HSB_1:78.5
	]:10.28
]

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 \((0.85)(0.8)(\frac{649 }{\log_{20} 649}-\frac{279}{\log_{20}279})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 8HFK_1 3LNL_1 128 112.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]