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Parikh vectors
8ONQ_1 1NGR_1 5KCZ_1 Letter Amino acid
1 10 28 A Alanine
0 6 12 R Arginine
0 0 9 M Methionine
1 1 18 F Phenylalanine
0 3 39 V Valine
0 5 17 D Aspartic acid
0 3 7 H Histidine
0 2 30 K Lycine
0 4 25 T Threonine
1 2 2 W Tryptophan
0 2 8 N Asparagine
0 2 14 C Cysteine
0 5 38 G Glycine
0 2 4 Y Tyrosine
0 3 8 Q Glutamine
1 8 21 E Glutamic acid
1 3 24 I Isoleucine
1 14 25 L Leucine
0 3 20 P Proline
0 7 25 S Serine 

8ONQ_1|Chains A, B|Peptide Ac-EFIAWL from hGLP-1|synthetic construct (32630)
>1NGR_1|Chain A|P75 LOW AFFINITY NEUROTROPHIN RECEPTOR|Rattus norvegicus (10116)
>5KCZ_1|Chains A, B|Alcohol dehydrogenase E chain|Equus caballus (9796)
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 \(w=f(c)\)
8ONQ , Knot 7 7 0.64 14 6 5 
XEFIAWL
1NGR , Knot 47 85 0.82 38 74 81 
GNLYSSLPLTKREEVEKLLNGDTWRHLAGELGYQPEHIDSFTHEACPVRALLASWGAQDSATLDALLAALRRIQRADIVESLCSE
5KCZ , Knot 159 374 0.84 40 210 358 
STAGKVIKCKAAVLWEEKKPFSIEEVEVAPPKAHEVRIKMVATGICRTDDHVVSGTLVTPLPVIAGHEAAGIVESIGEGVTTVRPGDKVIPLFTPQCGKCRVCKHPEGNFCLKNDLSMPRGTMQDGTSRFTCRGKPIHHFLGTSTFSQYTVVDEISVAKIDAASPLEKVCLIGCGFSTGYGSAVKVAKVTQGSTCAVFGLGGVGLSVIMGCKAAGAARIIGVDINKDKFAKAKEVGATECVNPQDYKKPIQEVLTEMSNGGVDFSFEVIGRLDTMVTALSCCQEAYGVSVIVGVPPDSQNLSMNPMLLLSGRTWKGAIFGGFKSKDSVPKLVADFMAKKFALDPLITHVLPFEKINEGFDLLRSGESIRTILTF

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 \(c\).

\(|P_{f(8ONQ_1)}(2) \setminus P_{f(1NGR_1)}(2)|=6\), \(|P_{f(1NGR_1)}(2) \setminus P_{f(8ONQ_1)}(2)|=74\). 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 1:0011111
Pair \(Z_2\) Length of longest common subsequence
8ONQ_1,1NGR_1 80 1
8ONQ_1,5KCZ_1 214 2
1NGR_1,5KCZ_1 192 4

Newick tree

 
[
	5KCZ_1:11.07,
	[
		8ONQ_1:40,1NGR_1:40
	]:75.07
]

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 \((0.85)(0.8)(\frac{92 }{\log_{20} 92}-\frac{7}{\log_{20}7})=34.1\)
Status Protein1 Protein2 d d1/2
Query variables 8ONQ_1 1NGR_1 45 24.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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