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Parikh vectors
4PBH_1 3GCS_1 8ARU_1 Letter Amino acid
12 19 16 R Arginine
15 23 19 E Glutamic acid
24 16 30 G Glycine
14 19 13 K Lycine
16 10 7 M Methionine
53 24 29 A Alanine
10 14 16 N Asparagine
12 15 11 Q Glutamine
34 44 15 L Leucine
14 12 13 F Phenylalanine
23 22 19 V Valine
19 22 21 S Serine
4 5 12 W Tryptophan
7 15 12 Y Tyrosine
20 27 17 D Aspartic acid
2 3 6 C Cysteine
7 12 17 H Histidine
15 22 7 I Isoleucine
17 17 25 P Proline
16 19 18 T Threonine 

4PBH_1|Chain A|TRAP dicarboxylate transporter, DctP subunit, putative|Ruegeria pomeroyi (246200)
>3GCS_1|Chain A|Mitogen-activated protein kinase 14|Homo sapiens (9606)
>8ARU_1|Chain A|Formylglycine-generating enzyme|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 \(w=f(c)\)
4PBH , Knot 141 334 0.81 40 187 318 
MTISFKGLARGVACAALVLAALPAAAKEFRLGLITPSPHTWTKAAEAFGAELSEKSGGAHSVSVFPARQLGNEAQMLQQLQTGALDMAFMTVAEVSNRVPNMGAFYAPYLAGDINHAAAILRSDTARGMLAVLPQEAGVVGVGFGSAGMRQILSRGAVNSAADLSGLKLRITPFDPILDFYNALGAAPTPMPLPAVYDALANGQVDAIDMDVELINVLKCHEHADTILISNHMMFPMVGLISARVYAGMSDADKAMISELMAKHVDSTLDVYMVKEPEWTDALTKVGKTFKRVDQSFFGDAIAQWETIWADKAPSLPELRKTAADLQAENLYFQ
3GCS , Knot 155 360 0.84 40 220 344 
GSQERPTFYRQELNKTIWEVPERYQNLSPVGSGAYGSVCAAFDTKTGLRVAVKKLSRPFQSIIHAKRTYRELRLLKHMKHENVIGLLDVFTPARSLEEFNDVYLVTHLMGADLNNIVKSQKLTDDHVQFLIYQILRGLKYIHSADIIHRDLKPSNLAVNEDSELKILDFGLCRHTDDEMTGYVATRWYRAPEIMLNWMHYNQTVDIWSVGCIMAELLTGRTLFPGTDHIDQLKLILRLVGTPGAELLKKISSESARNYIQSLTQMPKMNFANVFIGANPLAVDLLEKMLVLDSDKRITAAQALAHAYFAQYHDPDDEPVADPYDQSLESRDLLIDEWKSLTYDEVISFVPPPLDQEEMES
8ARU , Knot 142 323 0.84 40 202 305 
ADLGSSMEFEANAPGPVPGERQLAHSKMVPIPAGVFTMGTDDPQIKQDGEAPARRVTIDAFYMDAYAVSNTEFEKFVNSTGYLTEAEKFGDSFVFEGMLSEQVKTNIQQAVAAAPWWLPVKGANWRHPEGPDSTILHRPDHPVLHVSWNDAVAYCTWAGKRLPTEAEWEYSCRGGLHNRLFPWGNKLQPKGQHYANIWQGEFPVTNTGEDGFQGTAPVDAFPPNGYGLYNIVGNAWEWTSDWWTVHHSVEETLNPKGPPSGKDRVKKGGSYMCHRSYCYRYRCAARSQNTPDSSASNLGFRCAADRLPTMDSGRGSHHHHHHH

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 \(c\).

\(|P_{f(4PBH_1)}(2) \setminus P_{f(3GCS_1)}(2)|=64\), \(|P_{f(3GCS_1)}(2) \setminus P_{f(4PBH_1)}(2)|=97\). 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 1:1010101110111011111111111100101111010100100110111101000011100101111001100101100100111011110110100011011110110111010011111000010111111100111111111011100110011100110101101010110111010011111101111111001110101011010101101100000100111000111111111010101110010011100111001000101011001010011001100100100011101110100111001101101000110101001010
Pair \(Z_2\) Length of longest common subsequence
4PBH_1,3GCS_1 161 4
4PBH_1,8ARU_1 165 3
3GCS_1,8ARU_1 194 3

Newick tree

 
[
	8ARU_1:93.00,
	[
		4PBH_1:80.5,3GCS_1:80.5
	]:12.50
]

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 \((0.85)(0.8)(\frac{694 }{\log_{20} 694}-\frac{334}{\log_{20}334})=99.0\)
Status Protein1 Protein2 d d1/2
Query variables 4PBH_1 3GCS_1 127 121
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]