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Parikh vectors
7QBM_1 6WGT_1 5HJR_1 Letter Amino acid
5 25 43 F Phenylalanine
5 20 39 T Threonine
19 20 24 N Asparagine
9 25 58 D Aspartic acid
2 15 32 H Histidine
15 32 59 V Valine
8 10 58 R Arginine
10 17 46 E Glutamic acid
24 16 60 G Glycine
12 51 90 L Leucine
4 14 21 M Methionine
8 17 58 P Proline
16 28 65 S Serine
4 7 23 W Tryptophan
18 42 49 A Alanine
8 9 7 C Cysteine
9 16 34 Q Glutamine
8 37 34 I Isoleucine
19 32 23 K Lycine
13 15 34 Y Tyrosine 

7QBM_1|Chain A|Cathepsin K|Homo sapiens (9606)
>6WGT_1|Chains A, B, C|5-hydroxytryptamine receptor 2A,Soluble cytochrome b562 fusion|Homo sapiens (9606)
>5HJR_1|Chains A, C|Neutral alpha-glucosidase AB|Mus musculus (10090)
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 \(w=f(c)\)
7QBM , Knot 98 216 0.81 40 148 210 
RAPDSVDYRKKGYVTPVKNQGQCGSCWAFSSVGALEGQLKKKTGKLLNLSPQNLVDCVSENDGCGGGYMTNAFQYVQKNRGIDSEDAYPYVGQEESCMYNPTGKAAKCRGYREIPEGNEKALKRAVARVGPVSVAIDASLTSFQFYSKGVYYDESCNSDNLNHAVLAVGYGIQKGNKHWIIKNSWGENWGNKGYILMARNKNNACGIANLASFPKM
6WGT , Knot 182 448 0.82 40 226 419 
MKTIIALSYIFCLVFADYKDDDDGAPLSLLHLQEKNWSALLTAVVIILTIAGNILVIMAVSLEKKLQNATNYFLMSLAIADMLLGFLVMPVSMLTILYGYRWPLPSKLCAVWIYLDVLFSTASIMHLCAISLDRYVAIQNPIHHSRFNSRTKAFLKIIAVWTISVGISMPIPVFGLQDDSKVFKEGSCLLADDNFVLIGSFVSFFIPATIMVITYFLTIKSLQKEAADLEDNWETLNDNLKVIEKADNAAQVKDALTKMRAAALDAQKATPPKLEDKSPDSPEMKDFRHGFDILVGQIDDALKLANEGKVKEAQAAAEQLKTTRNAYIQKYLTMQSISNEQKACKVLGIVFFLFVVMWCPFFITNIMAVICKESCNEDVIGALLNVFVWIGYASSAVNPLVYTLFNKTYRSAFSRYIQCQYKENKKPGRPLEVLFQGPHHHHHHHHHH
5HJR , Knot 322 857 0.84 40 312 793 
VDRSNFKTCDESSFCKRQRSIRPGLSPYRALLDTLQLGPDALTVHLIHEVTKVLLVLELQGLQKNMTRIRIDELEPRRPRYRVPDVLVADPPTARLSVSGRDDNSVELTVAEGPYKIILTAQPFRLDLLEDRSLLLSVNARGLMAFEHQRAPREPGAWEETFKTHSDSKPYGPTSVGLDFSLPGMEHVYGIPEHADSLRLKVTEGGEPYRLYNLDVFQYELNNPMALYGSVPVLLAHSFHRDLGIFWLNAAETWVDISSNTPQTDIRWMSESGIIDVFLMLGPSVFDVFRQYASLTGTQALPPLFSLGYHQSRWNYRDEADVLEVDQGFDDHNMPCDVIWLDIEHADGKRYFTWDPTRFPQPLNMLEHLASKRRKLVAIVDPHIKVDSGYRVHEELRNHGLYVKTRDGSDYEGWCWPGSASYPDFTNPRMRAWWSNMFSFDNYEGSAPNLYVWNDMNEPSVFNGPEVTMLKDAVHYGGWEHRDIHNIYGLYVHMATADGLIQRSGGIERPFVLSRAFFSGSQRFGAVWTGDNTAEWDHLKISIPMCLSLALVGLSFCGADVGGFFKNPEPELLVRWYQMGAYQPFFRAHAHLDTGRREPWLLASQYQDAIRDALFQRYSLLPFWYTLFYQAHKEGFPVMRPLWVQYPEDMSTFSIEDQFMLGDALLIHPVSDAGAHGVQVYLPGQEEVWYDIQSYQKHHGPQTLYLPVTLSSIPVFQRGGTIVPRWMRVRRSSDCMKDDPITLFVALSPQGTAQGELFLDDGHTFNYQTRHEFLLRRFSFSGSTLVSSSADPKGHLETPIWIERVVIMGAGKPAAVVLQTKGSPESRLSFQHDPETSVLILRKPGVSVASDWSIHLR

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 \(c\).

\(|P_{f(7QBM_1)}(2) \setminus P_{f(6WGT_1)}(2)|=53\), \(|P_{f(6WGT_1)}(2) \setminus P_{f(7QBM_1)}(2)|=131\). 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 1:011001000001010110001001001110011110101000010110101001100100001011101001100100001100001010110000010010101100010001101000110011101111011101010010100011000000000010011111101100100011100011001100101111000001011101101101
Pair \(Z_2\) Length of longest common subsequence
7QBM_1,6WGT_1 184 4
7QBM_1,5HJR_1 226 4
6WGT_1,5HJR_1 166 4

Newick tree

 
[
	7QBM_1:10.89,
	[
		6WGT_1:83,5HJR_1:83
	]:25.89
]

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 \((0.85)(0.8)(\frac{664 }{\log_{20} 664}-\frac{216}{\log_{20}216})=126.\)
Status Protein1 Protein2 d d1/2
Query variables 7QBM_1 6WGT_1 160 117.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]