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Parikh vectors
6JOF_1 5CEP_1 8RSA_1 Letter Amino acid
20 12 4 R Arginine
21 21 3 G Glycine
5 7 4 M Methionine
3 12 3 F Phenylalanine
11 12 10 T Threonine
20 14 12 A Alanine
5 9 0 W Tryptophan
2 21 10 K Lycine
3 6 8 C Cysteine
14 21 5 E Glutamic acid
15 15 4 H Histidine
24 29 2 L Leucine
4 6 6 Y Tyrosine
23 23 9 V Valine
18 16 5 D Aspartic acid
7 9 7 Q Glutamine
9 19 3 I Isoleucine
15 15 4 P Proline
17 25 15 S Serine
2 8 10 N Asparagine 

6JOF_1|Chain A|tRNA (guanine-N(1)-)-methyltransferase|Mycobacterium tuberculosis H37Rv (83332)
>5CEP_1|Chain A|Mitogen-activated protein kinase kinase kinase 12|Homo sapiens (9606)
>8RSA_1|Chains A, B|RIBONUCLEASE A|Bos taurus (9913)
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 \(w=f(c)\)
6JOF , Knot 107 238 0.82 40 148 226 
MRIDIVTIFPACLDPLRQSLPGKAIESGLVDLNVHDLRRWTHDVHHSVDDAPYGGGPGMVMKAPVWGEALDEICSSETLLIVPTPAGVLFTQATAQRWTTESHLVFACGRYEGIDQRVVQDAARRMRVEEVSIGDYVLPGGESAAVVMVEAVLRLLAGVLGNPASHQDDSHSTGLDGLLEGPSYTRPASWRGLDVPEVLLSGDHARIAAWRREVSLQRTRERRPDLSHPDGSHHHHHH
5CEP , Knot 131 300 0.83 40 185 286 
MGSEDLWEVPFEEILDLQWVGSGAQGAVFLGRFHGEEVAVKKVRDLKETDIKHLRKLKHPNIITFKGVCTQAPCYCILMEFCAQGQLYEVLRAGRPVTPSLLVDWSMGIAGGMNYLHLHKIIHRDLKSPNMLITYDDVVKISDFGTSKELSDKSTKMSFAGTVAWMAPEVIRNEPVSEKVDIWSFGVVLWELLTGEIPYKDVDSSAIIWGVGSNSLHLPVPSSCPDGFKILLRQCWNSKPRNRPSFRQILLHLDIASADVLSTPQETYFKSQAEWREEVKLHFEKIKSEGTGNSHHHHHH
8RSA , Knot 64 124 0.83 38 107 121 
KETAAAKFERQHMDSSTSAASSSNYCNQMMKSRNLTKDRCKPVNTFVHESLADVQAVCSQKNVACKNGQTNCYQSYSTMSITDCRETGSSKYPNCAYKTTQANKHIIVACEGNPYVPVHFDASV

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 \(c\).

\(|P_{f(6JOF_1)}(2) \setminus P_{f(5CEP_1)}(2)|=79\), \(|P_{f(5CEP_1)}(2) \setminus P_{f(6JOF_1)}(2)|=116\). 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 1:1010110111101011000111011001110101001001000100010011011111111011111011001000001111101111110010100100000111101000110001100110010100101100111110011111101110111111101100000000011011101100001101011011011101001011110001010000000101001010000000
Pair \(Z_2\) Length of longest common subsequence
6JOF_1,5CEP_1 195 7
6JOF_1,8RSA_1 175 3
5CEP_1,8RSA_1 194 3

Newick tree

 
[
	5CEP_1:10.28,
	[
		6JOF_1:87.5,8RSA_1:87.5
	]:12.78
]

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 \((0.85)(0.8)(\frac{538 }{\log_{20} 538}-\frac{238}{\log_{20}238})=85.7\)
Status Protein1 Protein2 d d1/2
Query variables 6JOF_1 5CEP_1 110 97.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]