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Parikh vectors
4CAE_1 5RCX_1 1CXW_1 Letter Amino acid
23 22 5 D Aspartic acid
37 27 1 L Leucine
20 22 5 F Phenylalanine
13 22 3 P Proline
24 58 2 S Serine
19 51 9 T Threonine
7 6 1 W Tryptophan
21 37 3 A Alanine
4 2 4 C Cysteine
7 2 0 H Histidine
26 17 4 Y Tyrosine
26 31 0 V Valine
13 46 8 G Glycine
29 20 0 I Isoleucine
7 2 1 M Methionine
24 8 4 E Glutamic acid
34 19 3 K Lycine
15 3 4 R Arginine
28 12 1 N Asparagine
7 12 2 Q Glutamine 

4CAE_1|Chains A, B, C|GLYCYLPEPTIDE N-TETRADECANOYLTRANSFERASE|PLASMODIUM VIVAX (5855)
>5RCX_1|Chain A|Endothiapepsin|Cryphonectria parasitica (5116)
>1CXW_1|Chain A|HUMAN MATRIX METALLOPROTEINASE 2|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 \(w=f(c)\)
4CAE , Knot 165 384 0.85 40 223 370 
DYKFWYTQPVPKINDEFNESVNEPFISDNKVEDVRKDEYKLPPGYSWYVCDVKDEKDRSEIYTLLTDNYVEDDDNIFRFNYSAEFLLWALTSPNYLKTWHIGVKYDASNKLIGFISAIPTDICIHKRTIKMAEVNFLCVHKTLRSKRLAPVLIKEITRRINLENIWQAIYTAGVYLPKPVSDARYYHRSINVKKLIEIGFSSLNSRLTMSRAIKLYRVEDTLNIKNMRLMKKKDVEGVHKLLGSYLEQFNLYAVFTKEEIAHWFLPIENVIYTYVNEENGKIKDMISFYSLPSQILGNDKYSTLNAAYSFYNVTTTATFKQLMQDAILLAKRNNFDVFNALEVMQNKSVFEDLKFGEGDGSLKYYLYNWKCASFAPAHVGIVLL
5RCX , Knot 170 419 0.81 40 190 380 
MSSPLKNALVTAMLAGGALSSPTKQHVGIPVNASPEVGPGKYSFKQVRNPNYKFNGPLSVKKTYLKYGVPIPAWLEDAVQNSTSGLAERSTGSATTTPIDSLDDAYITPVQIGTPAQTLNLDFDTGSSDLWVFSSETTASEVDGQTIYTPSKSTTAKLLSGATWSISYGDGSSSSGDVYTDTVSVGGLTVTGQAVESAKKVSSSFTEDSTIDGLLGLAFSTLNTVSPTQQKTFFDNAKASLDSPVFTADLGYHAPGTYNFGFIDTTAYTGSITYTAVSTKQGFWEWTSTGYAVGSGTFKSTSIDGIADTGTTLLYLPATVVSAYWAQVSGAKSSSSVGGYVFPCSATLPSFTFGVGSARIVIPGDYIDFGPISTGSSSCFGGIQSSAGIGINIFGDVALKAAFVVFNGATTPTLGFASK
1CXW , Knot 34 60 0.77 34 53 57 
TALFTMGGNAEGQPCKFPFRFQGTSYDSCTTEGRTDGYRWCGTTEDYDRDKKYGFCPETA

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 \(c\).

\(|P_{f(4CAE_1)}(2) \setminus P_{f(5RCX_1)}(2)|=92\), \(|P_{f(5RCX_1)}(2) \setminus P_{f(4CAE_1)}(2)|=59\). 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 1:000110001110100010001001110000100100000011110010100100000000100110000100000110100010111111001001001011100010001111101110010100001011010110100010000111111001000101001101100111011011001000000101001101110010001010011010010001010010110000101100111001001010111000011011111001100010000101001101001100111000000101100100100010100110011111000010110110110000110010110101010001001001011110111111
Pair \(Z_2\) Length of longest common subsequence
4CAE_1,5RCX_1 151 4
4CAE_1,1CXW_1 218 3
5RCX_1,1CXW_1 185 3

Newick tree

 
[
	1CXW_1:10.28,
	[
		4CAE_1:75.5,5RCX_1:75.5
	]:32.78
]

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 \((0.85)(0.8)(\frac{803 }{\log_{20} 803}-\frac{384}{\log_{20}384})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 4CAE_1 5RCX_1 138 136.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]