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Parikh vectors
6DLI_1 2VHH_1 7NTL_1 Letter Amino acid
21 22 2 R Arginine
2 23 17 N Asparagine
12 17 7 D Aspartic acid
21 37 9 E Glutamic acid
27 28 23 G Glycine
10 21 7 F Phenylalanine
28 21 9 V Valine
3 7 4 C Cysteine
9 26 5 K Lycine
7 12 8 Y Tyrosine
3 12 9 Q Glutamine
6 25 13 I Isoleucine
3 10 0 M Methionine
12 20 22 S Serine
9 24 11 T Threonine
37 31 26 A Alanine
10 18 8 H Histidine
37 23 12 L Leucine
18 21 24 P Proline
3 7 6 W Tryptophan 

6DLI_1|Chains A, B, C, D|Glutamate racemase|Thermus thermophilus (274)
>2VHH_1|Chains A, B, C, D|CG3027-PA|DROSOPHILA MELANOGASTER (7227)
>7NTL_1|Chain A|LPMO9F|Malbranchea cinnamomea (5041)
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 \(w=f(c)\)
6DLI , Knot 115 278 0.77 40 148 262 
MGSSHHHHHHSSGLVPRGSHMVKDPKAPIGVFDSGVGGLTVLKALRRLLPREEFLYFGDTARVPYGGKPLAMVRRFAWEIAGFLLRQGVKAIVVACNTASSAALPDLAEDLSVPVFGVVEPAARAARGFRKVGLIGTQATVESGAYPRYVDLAWAKACPLFVPLVEEGLWDDPVALLVARHYLEDAPKDLEALILGCTHYPFLKGAIGAVLPGVALLDSAELTAQEVARALEAEGLLNPEGRGRTFHLVTGDPEAYRALAERLGERVEAVRRVSLEEL
2VHH , Knot 175 405 0.86 40 234 389 
MSAFELKNLNDCLEKHLPPDELKEVKRILYGVEEDQTLELPTSAKDIAEQNGFDIKGYRFTAREEQTRKRRIVRVGAIQNSIVIPTTAPIEKQREAIWNKVKTMIKAAAEAGCNIVCTQEAWTMPFAFCTREKFPWCEFAEEAENGPTTKMLAELAKAYNMVIIHSILERDMEHGETIWNTAVVISNSGRYLGKHRKNHIPRVGDFNESTYYMEGNTGHPVFETEFGKLAVNICYGRHHPQNWMMFGLNGAEIVFNPSATIGRLSEPLWSIEARNAAIANSYFTVPINRVGTEQFPNEYTSGDGNKAHKEFGPFYGSSYVAAPDGSRTPSLSRDKDGLLVVELDLNLCRQVKDFWGFRMTQRVPLYAESFKKASEHGFKPQIIKETQFPGDDDDKHHHHHHHHSG
7NTL , Knot 101 222 0.82 38 149 215 
HGYVSSIQANGQTYPGADPHNPNPESPGWQAENTDLGFVEPSAFSTPAIACHKNARAPPAHATVQAGSTIKLTWNTWPESHHGPVLDYIAPCNGDCSSASAGSLNFVKIAEKGLISGSNPGFWAADELIQNGNSWEVTIPANLAPGKYVLRHEIIALHSAGNPNGAQAYPQCINLEVTGGGSATPSGQPATSFYSPNDPGILFNLYQSFDSYPIPGPAVWSG

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 \(c\).

\(|P_{f(6DLI_1)}(2) \setminus P_{f(2VHH_1)}(2)|=40\), \(|P_{f(2VHH_1)}(2) \setminus P_{f(6DLI_1)}(2)|=126\). 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 1:11000000000011110100110010111111001111101101100111000110110010110110111110011101111110011011111000100111101100101111111011101101100111110010100110100101111010111111100111001111111000100110010111110000111011111111111110010101001101101011101010100101101010100111001100101100101001
Pair \(Z_2\) Length of longest common subsequence
6DLI_1,2VHH_1 166 7
6DLI_1,7NTL_1 153 3
2VHH_1,7NTL_1 193 3

Newick tree

 
[
	2VHH_1:94.07,
	[
		6DLI_1:76.5,7NTL_1:76.5
	]:17.57
]

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 \((0.85)(0.8)(\frac{683 }{\log_{20} 683}-\frac{278}{\log_{20}278})=112.\)
Status Protein1 Protein2 d d1/2
Query variables 6DLI_1 2VHH_1 145 117.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]