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Parikh vectors
5MTF_1 1GOM_1 6MJA_1 Letter Amino acid
9 10 7 R Arginine
20 22 9 G Glycine
5 14 8 P Proline
5 19 22 D Aspartic acid
5 7 4 E Glutamic acid
4 4 4 H Histidine
26 22 16 L Leucine
4 22 16 T Threonine
11 8 2 W Tryptophan
3 23 11 N Asparagine
1 2 6 C Cysteine
6 21 12 Q Glutamine
13 17 8 I Isoleucine
6 13 13 K Lycine
10 3 3 M Methionine
13 9 9 F Phenylalanine
9 17 25 S Serine
19 35 12 A Alanine
6 9 5 Y Tyrosine
15 26 17 V Valine

5MTF_1|Chain A|Rhomboid protease GlpG|Escherichia coli O45:K1 (strain S88 / ExPEC) (585035)
>1GOM_1|Chain A|ENDO-1,4-BETA-XYLANASE|THERMOASCUS AURANTIACUS (5087)
>6MJA_1|Chain A[auth C]|T cell receptor alpha variable 11,T cell receptor alpha variable 11,T cell receptor alpha joining 18,Human nkt tcr alpha chain, CHIMERIC PROTEIN|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 \(w=f(c)\)
5MTF , Knot 87 190 0.80 40 130 183
AALRERAGPVTWVMMIACVVVFIAMQILGDQEVMLWLAWPFDPTLKFEFWRYFTHALMHFSLMHILFNLLWWWYLGGAVEKRLGSGKLIVITLISALLSGYVQQKFSGPWFGGLSGVVYALMGYVWLRGERDPQSGIYLQRGLIIFALIWIVAGWFDLFGMSMANGAHIAGLAVGLAMAFVDSLNARKRK
1GOM , Knot 134 303 0.84 40 180 287
QAAQSVDQLIKARGKVYFGVATDQNRLTTGKNAAIIQADFGQVTPENSMKWDATEPSQGNFNFAGADYLVNWAQQNGKLIRGHTLVWHSQLPSWVSSITDKNTLTNVMKNHITTLMTRYKGKIRAWDVVNEAFNEDGSLRQTVFLNVIGEDYIPIAFQTARAADPNAKLYINDYNLDSASYPKTQAIVNRVKQWRAAGVPIDGIGSQTHLSAGQGAGVLQALPLLASAGTPEVAITELDVAGASPTDYVNVVNACLNVQSCVGITVWGVADPDSWRASTTPLLFDGNFNPKPAYNAIVQDLQQ
6MJA , Knot 98 209 0.83 40 147 204
MKTQVEQSPQSLVVRQGENCVLQCNYSVTPDNHLRWFKQDTGKGLVSLTVLVDQKDKTSNGRYSATLDKDAKHSTLHITATLLDDTATYICVVGDRGSALGRLHFGAGTQLIVIPDIQNPDPAVYQLRDSKSSDKSVCLFTDFDSQTNVSQSKDSDVYITDKCVLDMRSMDFKSNSAVAWSNKSDFACANAFNNSIIPEDTFFPSPESS

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 \(c\).

\(|P_{f(5MTF_1)}(2) \setminus P_{f(1GOM_1)}(2)|=61\), \(|P_{f(1GOM_1)}(2) \setminus P_{f(5MTF_1)}(2)|=111\). 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 1:1110001111011111101111111011100011111111101010101100100111010110111011111011111000110101111011011101010001011111110111011110111010001001101001111111111111110111101101101111111111111001010000
Pair \(Z_2\) Length of longest common subsequence
5MTF_1,1GOM_1 172 4
5MTF_1,6MJA_1 167 4
1GOM_1,6MJA_1 137 4

Newick tree

 
[
	5MTF_1:89.52,
	[
		6MJA_1:68.5,1GOM_1:68.5
	]:21.02
]

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 \((0.85)(0.8)(\frac{493 }{\log_{20} 493}-\frac{190}{\log_{20}190})=88.2\)
Status Protein1 Protein2 d d1/2
Query variables 5MTF_1 1GOM_1 115 91.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]