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Parikh vectors
6DAM_1 7VNX_1 3KPT_1 Letter Amino acid
33 5 27 N Asparagine
30 24 31 E Glutamic acid
38 16 48 K Lycine
52 10 30 T Threonine
22 18 31 I Isoleucine
53 11 35 V Valine
4 0 0 C Cysteine
7 6 3 H Histidine
38 27 17 L Leucine
16 15 10 F Phenylalanine
72 24 28 G Glycine
11 4 4 M Methionine
28 8 13 P Proline
42 12 14 S Serine
44 12 17 A Alanine
14 15 4 R Arginine
34 8 21 D Aspartic acid
26 4 9 Q Glutamine
23 0 1 W Tryptophan
30 4 12 Y Tyrosine 

6DAM_1|Chain A|Lanthanide-dependent methanol dehydrogenase XoxF|Methylomicrobium buryatense 5G (675511)
>7VNX_1|Chain A|TkArkI|Thermococcus kodakarensis (311400)
>3KPT_1|Chains A, B|Collagen adhesion protein|Bacillus cereus ATCC 14579 (226900)
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 \(w=f(c)\)
6DAM , Knot 244 617 0.84 40 268 569 
MKKPVKSWLIASTVAALLAVPAVSQANSEVEKLTKNPANWATWGGNYHGTRYSELKQINTSNVKNLQPAWTFSTGVLRGHEGGPLVVNDVMYIHTPFPNTVYAIDQKTQAVIWEYTPQQDADVTIPVMCCDTVNRGLAYGDGKIFLQQSDTVLTALDAKTGKRVWSVQNGDPKLGMTNTQAPLVVKDKVITGISGGEFGVRGFLAAYNIRTGELDWKGYSMGPDADTLINPTKTTTWKDGKVQPVGKDSSTSTWEGDQWKIGGGTTWGWYSYDPELNLVYYGSGNPSTWNPAQRPGDNKWSMSLWARNADTGEVKWVYQMTPHDEWDYDGINEVALVDQEIKGQMRKTAVHFDRNGFGYTLDRVTGELLVAEKFDKAVNWASHVDMKSGRPQVVSQYSTEYNGEDVNTEGVCPAALGSKNQQPVSYSPQTGYFYISGNHVCMDYEPFEVEYTAGQPYVGATLSMFPAGKDVITGKEDGSNNLGQFTAWDATTGKIIWSNKEQFSVWSGSLATAGGVVFYGTLEGYLKAVDAKTGKELYRFKTPSGIIGNVNTWEYEGKQYVGVLSGVGGWAGIGIAAGLDSGEESSNSEGLGAVGAYRSLSSYTKLGGTLTVFALPN
7VNX , Knot 102 223 0.82 36 142 210 
ENLYFQGMTFEHIISSKRLEGFLRHLAEEGVGGVEPLAKGTTSLVFTGVLGGRKVVIKLQRPDSPRSNFEKEAELTKIASTFGVTPPIIGLGEFEGLPYLIREFAEGEPILFADVEKEHLFRIVEKTALLDRLGIDHGQIQGGKHIIIGEDVYLIDFEKAGFRKPNNLTSAMAMIFIGENAISKRVREKFGLDEKFREEMKDALRHYKRTGSLSRLLSLLSGL
3KPT , Knot 146 355 0.80 38 184 329 
GSNEIKRGAVDLIKTGVNEKAMAGAVFSLFKKDGTEVKKELATDANGHIRVQGLEYGEYYFQETKAPKGYVIDPTKREFFVKNSGTINEDGTITSGTVVKMEVKNNEEPTIDKKINGKLEALPINPLTNYNYDIKTLIPEDIKEYKKYVVTDTLDNRLVIQGKPIVKIDGAEVNANVVEVAIEGQKVTATVKDFTKMDGKKEFHLQIKSQVKEGVPSGSEILNTAKIHFTNKNDVIGEKESKPVVVIPTTGIIELTKIDSANKNKMKGAEFVLKDNNGKIVVVAGKEVTGVSDENGVIKWSNIPYGDYQIFETKAPTYTKEDGTKTSYQLLKDPIDVKISENNQTVKLTIENNKS

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 \(c\).

\(|P_{f(6DAM_1)}(2) \setminus P_{f(7VNX_1)}(2)|=158\), \(|P_{f(7VNX_1)}(2) \setminus P_{f(6DAM_1)}(2)|=32\). 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 1:10011001111001111111111001000100100011011011100010000010010000100101110100111010011111100110100111001011000001111000100010101111000010011101010111000001101101001001101001010111000011111000110110110111011111001001010101001110100110100000100101011100000001010010111100111000010101100101010010110011000101011100100101011001010001000110011110001010100011010001110010010101111001001101100101001010110000000010010001101111100000110001001010101001010001101000110101110101111100110100010001101011010010111000001011010110111111010101010110100100100100101111010010001000111101111111111111100100000001111111000100000111010111110
Pair \(Z_2\) Length of longest common subsequence
6DAM_1,7VNX_1 190 4
6DAM_1,3KPT_1 146 4
7VNX_1,3KPT_1 140 5

Newick tree

 
[
	6DAM_1:89.08,
	[
		3KPT_1:70,7VNX_1:70
	]:19.08
]

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 \((0.85)(0.8)(\frac{840 }{\log_{20} 840}-\frac{223}{\log_{20}223})=170.\)
Status Protein1 Protein2 d d1/2
Query variables 6DAM_1 7VNX_1 215 143.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]