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Parikh vectors
1RDX_1 5YEZ_1 1RHY_1 Letter Amino acid
12 15 10 R Arginine
8 14 4 Q Glutamine
18 12 12 E Glutamic acid
4 4 12 H Histidine
24 13 15 I Isoleucine
15 9 5 P Proline
16 8 3 N Asparagine
19 12 13 D Aspartic acid
6 2 4 C Cysteine
27 10 16 G Glycine
26 4 10 K Lycine
9 7 6 M Methionine
10 11 8 F Phenylalanine
21 11 13 T Threonine
0 2 0 W Tryptophan
15 9 4 Y Tyrosine
29 10 9 V Valine
32 19 21 A Alanine
28 24 20 L Leucine
18 13 17 S Serine 

1RDX_1|Chains A, B|FRUCTOSE 1,6-BISPHOSPHATASE|Sus scrofa (9823)
>5YEZ_1|Chains A, B, C, D|Cell density-dependent motility repressor|Salmonella typhimurium (90371)
>1RHY_1|Chains A, B|Imidazole glycerol phosphate dehydratase|Filobasidiella neoformans (5207)
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 \(w=f(c)\)
1RDX , Knot 147 337 0.84 38 201 323 
TDQAAFDTNIVTLTRFVMEQGRKARGTGEMTQLLNSLCTAVKAISTAVRKAGIAHLYGIAGSTNVTGDQVKKLDVLSNDLVINVLKSSFATCVLVTEEDKNAIIVEPEKRGKYVVCFDPLDGSSNIDCLVSIGTIFGIYRKNSTDEPSEKDALQPGRNLVAAGYALYGSATMLVLAMVNGVNCFMLDPAIGEFILVDRNVKIKKKGSIYSINEGYAKEFDPAITEYIQRKKFPPDNSAPYGAAYVGSMVADVHRTLVYGGIFMYPANKKSPKGKLRLLYECNPMAYVMEKAGGLATTGKEAVLDIVPTDIHQRAPIILGSPEDVTELLEIYQKHAAK
5YEZ , Knot 100 209 0.85 40 154 201 
GAMDYTLRKIKIAAAHSLSLGLLPTIVKQMPTQFTYAVEAIDVDQAVDMLREGQSDFIFSYHDENLQQAPFDNIRLFESRLFPVCANNGRGEPRYTLEQPHFPLLNYSQNSYQGRLINRTLTRHAELSFSTFFVSSMSELLKQVAMDGCGIAWLPEYAIRQEITDGRLIVLDADELVIPIQAYAYRMNTRMSQVAETFWRDLRGLQAAL
1RHY , Knot 91 202 0.79 38 131 186 
MSERIASVERTTSETHISCTIDLDHIPGVTEQKINVSTGIGFLDHMFTALAKHGGMSLQLQCKGDLHIDDHHTAEDCALALGEAFKKALGERKGIKRYGYAYAPLDESLSRAVIDISSRPYFMCHLPFTREKVGDLSTEMVSHLLQSFAFAAGVTLHIDSIRGENNHHIAESAFKALALAIRMAISRTGGDDVPSTKGVLAL

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 \(c\).

\(|P_{f(1RDX_1)}(2) \setminus P_{f(5YEZ_1)}(2)|=107\), \(|P_{f(5YEZ_1)}(2) \setminus P_{f(1RDX_1)}(2)|=60\). 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 1:0001110001101001110010010101010011001001101100110011110101111000101001001011000111011000110011100000011110100010011010110100010011011011110000000010000110110011111011010101111111011001110111101111000101000101001001010010111000100001110001101110110111010001101111101100001010101100001110110011111001001110111001000111111010010011010000110
Pair \(Z_2\) Length of longest common subsequence
1RDX_1,5YEZ_1 167 4
1RDX_1,1RHY_1 166 3
5YEZ_1,1RHY_1 171 3

Newick tree

 
[
	5YEZ_1:85.00,
	[
		1RDX_1:83,1RHY_1:83
	]:2.00
]

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 \((0.85)(0.8)(\frac{546 }{\log_{20} 546}-\frac{209}{\log_{20}209})=96.7\)
Status Protein1 Protein2 d d1/2
Query variables 1RDX_1 5YEZ_1 122 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]