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Parikh vectors
1SPH_1 7RPG_1 5JFV_1 Letter Amino acid
5 16 12 I Isoleucine
7 23 11 K Lycine
6 12 12 S Serine
6 15 25 V Valine
12 16 23 A Alanine
7 21 19 E Glutamic acid
8 20 26 G Glycine
2 7 10 Y Tyrosine
7 22 35 L Leucine
4 10 10 M Methionine
1 8 16 P Proline
7 15 10 T Threonine
1 7 24 R Arginine
5 17 4 N Asparagine
0 0 6 C Cysteine
1 4 17 H Histidine
6 7 14 D Aspartic acid
2 11 16 Q Glutamine
1 8 12 F Phenylalanine
0 6 6 W Tryptophan 

1SPH_1|Chains A, B|HISTIDINE-CONTAINING PHOSPHOCARRIER PROTEIN HPR|Bacillus subtilis (1423)
>7RPG_1|Chain A|Beta-lactamase|Acinetobacter baumannii (470)
>5JFV_1|Chain A|High affinity nerve growth factor receptor|Homo sapiens (9606)
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 \(w=f(c)\)
1SPH , Knot 47 88 0.79 36 72 85 
MAQKTFKVTADSGIHARPATVLVQTASKYDADVNLEYNGKTVNLKDIMGVMSLGIAKGAEITISASGADENDALNALEETMKSEGLGE
7RPG , Knot 113 245 0.84 38 161 231 
MHISSQQHEKAIKSYFDEAQTQGVIIIKEGKNLSTYGNALARANKEYVPASTFDMLNALIGLENHKATTNEIFKWDGKKRTYPMWEKDMTLGEAMALSAVPVYQELARRTGLELMQKEVKRVNFGNTNIGTQVDNFWLVGPLKITPVQEVNFADDLAHNRLPFKLETQEEVKKMLLIKEVNGSKIYAKSGWGMGVTPQVGWLTGWVEQANGKKIPFSLNLEMKEGMSGSIRNEITYKSLENLGII
5JFV , Knot 134 308 0.83 40 196 295 
MHHHHHHLVPRGSVHHIKRRDIVLKWELGEGAFGKVFLAECHNLLPEQDKMLVAVKALKEASESARQDFQREAELLTMLQHQHIVRFFGVCTEGRPLLMVFEYMRHGDLNRFLRSHGPDAKLLAGGEDVAPGPLGLGQLLAVASQVAAGMVYLAGLHFVHRDLATRNCLVGQGLVVKIGDFGMSRDIYSTDYYRVGGRTMLPIRWMPPESILYRKFTTESDVWSFGVVLWEIFTYGKQPWYQLSNTEAIDCITQGRELERPRACPPEVYAIMRGCWQREPQQRHSIKDVHARLQALAQAPPVYLDVLG

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 \(c\).

\(|P_{f(1SPH_1)}(2) \setminus P_{f(7RPG_1)}(2)|=25\), \(|P_{f(7RPG_1)}(2) \setminus P_{f(1SPH_1)}(2)|=114\). 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 1:1100010101001101011011100100001010100010010100111110111101101010101100001101100010001110
Pair \(Z_2\) Length of longest common subsequence
1SPH_1,7RPG_1 139 4
1SPH_1,5JFV_1 198 3
7RPG_1,5JFV_1 185 3

Newick tree

 
[
	5JFV_1:10.09,
	[
		1SPH_1:69.5,7RPG_1:69.5
	]:33.59
]

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 \((0.85)(0.8)(\frac{333 }{\log_{20} 333}-\frac{88}{\log_{20}88})=76.7\)
Status Protein1 Protein2 d d1/2
Query variables 1SPH_1 7RPG_1 96 63.5
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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