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Parikh vectors
4BUW_1 7RXF_1 2RLB_1 Letter Amino acid
1 2 1 W Tryptophan
11 12 6 A Alanine
5 3 6 C Cysteine
17 9 15 E Glutamic acid
14 7 5 I Isoleucine
13 3 5 Y Tyrosine
12 4 7 N Asparagine
7 8 7 D Aspartic acid
11 5 8 F Phenylalanine
8 8 3 P Proline
25 13 13 G Glycine
8 7 4 M Methionine
19 3 14 S Serine
11 1 10 K Lycine
9 4 7 T Threonine
12 13 11 V Valine
14 9 9 R Arginine
9 10 9 Q Glutamine
19 3 3 H Histidine
15 7 11 L Leucine 

4BUW_1|Chains A, B|TANKYRASE-2|HOMO SAPIENS (9606)
>7RXF_1|Chains A, B|Steroid Delta-isomerase|Pseudomonas putida (303)
>2RLB_1|Chains A, B|Cation-dependent mannose-6-phosphate receptor|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 \(w=f(c)\)
4BUW , Knot 112 240 0.85 40 176 232 
MHHHHHHSSGVDLGTENLYFQSMLNTSGSGTILIDLSPDDKEFQSVEEEMQSTVREHRDGGHAGGIFNRYNILKIQKVCNKKLWERYTHRRKEVSEENHNHANERMLFHGSPFVNAIIHKGFDERHAYIGGMFGAGIYFAENSSKSNQYVYGIGGGTGCPVHKDRSCYICHRQLLFCRVTLGKSFLQFSAMKMAHSPPGHHSVTGRPSVNGLALAEYVIYRGEQAYPEYLITYQIMRPEG
7RXF , Knot 65 131 0.80 40 109 125 
MNLPTAQEVQGLMARYIELVDVGDIEAIVQMYADDATVEDPFGQPPIHGREQIAAFFRQGLGGGKVRACLTGPVRASHNGCGAMPFRVEMVWNGQPCALDVIDVMRFDEHGRIQTMQAYWSEVNLSVREPQ
2RLB , Knot 76 154 0.82 40 120 150 
TEEKTCDLVGEKGKESEKELALLKRLTPLFQKSFESTVGQSPDMYSYVFRVCREAGQHSSGAGLVQIQKSNGKETVVGRFNETQIFQGSNWIMLIYKGGDEYDNHCGREQRRAVVMISCNRHTLADNFNPVSEERGKVQDCFYLFEMDSSLACS

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 \(c\).

\(|P_{f(4BUW_1)}(2) \setminus P_{f(7RXF_1)}(2)|=126\), \(|P_{f(7RXF_1)}(2) \setminus P_{f(4BUW_1)}(2)|=59\). 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 1:100000000110110001010011000101011101010000100100010001000001101111100001101001000011000000000100000001000111010111011100110000101111111110110000000001011111010110000000100001110010110011010110110011100010101010111110011001001010011000110101
Pair \(Z_2\) Length of longest common subsequence
4BUW_1,7RXF_1 185 3
4BUW_1,2RLB_1 168 4
7RXF_1,2RLB_1 159 4

Newick tree

 
[
	4BUW_1:91.11,
	[
		2RLB_1:79.5,7RXF_1:79.5
	]:11.61
]

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 \((0.85)(0.8)(\frac{371 }{\log_{20} 371}-\frac{131}{\log_{20}131})=73.0\)
Status Protein1 Protein2 d d1/2
Query variables 4BUW_1 7RXF_1 97 73.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]