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Parikh vectors
4MZX_1 2ABQ_1 5IZR_1 Letter Amino acid
24 14 47 R Arginine
30 10 42 D Aspartic acid
51 27 112 L Leucine
14 4 15 M Methionine
35 20 68 S Serine
21 16 42 Q Glutamine
38 30 51 G Glycine
25 19 20 I Isoleucine
13 18 32 K Lycine
15 13 36 F Phenylalanine
19 14 46 T Threonine
40 29 41 V Valine
25 10 21 N Asparagine
6 1 13 C Cysteine
40 11 50 P Proline
7 0 13 W Tryptophan
75 29 58 A Alanine
23 30 64 E Glutamic acid
18 7 19 H Histidine
17 4 24 Y Tyrosine 

4MZX_1|Chain A|Benzoylformate decarboxlyase|Pseudomonas putida (303)
>2ABQ_1|Chains A, B|fructose 1-phosphate kinase|Bacillus halodurans (86665)
>5IZR_1|Chains A, B, C, D|Cytosolic phospholipase A2 delta|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 \(w=f(c)\)
4MZX , Knot 215 536 0.84 40 242 486 
MASVHGTTYELLRRQGIDTVFGNPGSNELPFLKDFPEDFRYILALQEACVVGIADGYAQASRKPAFINLHSAAGTGNAMGALSNAWNSHSPLIVTAGQQTRAMIGVEALLTNVDAANLPRPLVKWSYEPASAAEVPHAMSRAIHMASMAPQGPVYLSVPYDDWDKDADPQSHHLFDRHVSSSVRLNDQDLDILVKALNSASNPAIVLGPDVDAANANADCVMLAERLKAPVWVAPSAPRCPFPTRHPCFRGLMPAGIAAISQLLEGHDVVLVIGAPVFRYHQYDPGQYLKPGTRLISVTCDPLEAARAPMGDAIVADIGAMASALANLVEESSRQLPTAAPEPAKVDQDAGRLHPETVFDTLNDMAPENAIYLNESLSTTAQMWQRLNMRNPGSYYFCAAGGLGFALPAAIGVQLAEPERQVIAVIGDGSANYSISALWTAAQYNIPTIFVIMNNGTYGYLRWFAGVLEAENVPGLDVPGIDFRALAKGYGVQALKADNLEQLKGSLQEALSAKGPVLIEVSTVSPVKRSHHHHHH
2ABQ , Knot 129 306 0.80 38 170 287 
MIYTVTLNPSIDYIVQVENFQQGVVNRSERDRKQPGGKGINVSRVLKRLGHETKALGFLGGFTGAYVRNALEKEEIGLSFIEVEGDTRINVKIKGKQETELNGTAPLIKKEHVQALLEQLTELEKGDVLVLAGSVPQAMPQTIYRSMTQIAKERGAFVAVDTSGEALHEVLAAKPSFIKPNHHELSELVSKPIASIEDAIPHVQRLIGEGIESILVSFAGDGALFASAEGMFHVNVPSGEVRNSVGAGDSVVAGFLAALQEGKSLEDAVPFAVAAGSATAFSDGFCTREEVERLQQQLQRTIKKEG
5IZR , Knot 304 814 0.83 40 301 729 
GAMGSESLSPGGPPGHPYQGEASTCWQLTVRVLEARNLRWADLLSEADPYVILQLSTAPGMKFKTKTLTDTSHPVWNEAFRFLIQSQVKNVLELSIYDEDSVTEDDICFKVLYDISEVLPGKLLRKTFSQSPQGEEELDVEFLMEETSDRPENLITNKVIVARELSCLDVHLDSTGSTAVVADQDKLELELVLKGSYEDTQTSFLGTASAFRFHYMAALETELSGRLRSSRSNGWNGDNSAGYLTVPLRPLTIGKEVTMDVPAPNAPGVRLQLKAEGCPEELAVHLGFNLCAEEQAFLSRRKQVVAKALKQALQLDRDLQEDEVPVVGIMATGGGARAMTSLYGHLLALQKLGLLDCVTYFSGISGSTWTMAHLYGDPEWSQRDLEGPIRYAREHLAKSKLEVFSPERLASYRRELELRAEQGHPTTFVDLWALVLESMLHGQVMDQKLSGQRAALERGQNPLPLYLSLNVKENNLETLDFKEWVEFSPYEVGFLKYGAFVPPELFGSEFFMGRLMRRIPEPRICFLEAIWSNIFSLNLLDAWYDLTSSGESWKQHIKDKTRSLEKEPLTTSGTSSRLEASWLQPGTALAQAFKGFLTGRPLHQRSPNFLQGLQLHQDYCSHKDFSTWADYQLDSMPSQLTPKEPRLCLVDAAYFINTSSPSMFRPGRRLDLILSFDYSLSAPFEALQQTELYCRARGLPFPRVEPSPQDQHQPRECHLFSDPACPEAPILLHFPLVNASFKDHSAPGVQRSPAELQGGQVDLTGATCPYTLSNMTYKEEDFERLLRLSDYNVQTSQGAILQALRTALKHRTLE

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 \(c\).

\(|P_{f(4MZX_1)}(2) \setminus P_{f(2ABQ_1)}(2)|=121\), \(|P_{f(2ABQ_1)}(2) \setminus P_{f(4MZX_1)}(2)|=49\). 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 1:11010100001100011001110110001111001100100111100101111101010100011110100111010111110011000011110110000111110111001011011011101000110110110110011011011101110101100010001010000110001000101000010111011001001111111010110101001111001011111110110011100010101111111111001101001111111111000000110010110011010001101101111011110111110111011000000110111011010001101010011001001110011010001000101100101001100010111111111111111011010001111110101000101110110001101111100100101011111101001111011110101110101101101001001010100110101111101001011000000000
Pair \(Z_2\) Length of longest common subsequence
4MZX_1,2ABQ_1 170 4
4MZX_1,5IZR_1 143 4
2ABQ_1,5IZR_1 179 4

Newick tree

 
[
	2ABQ_1:91.93,
	[
		4MZX_1:71.5,5IZR_1:71.5
	]:20.43
]

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 \((0.85)(0.8)(\frac{842 }{\log_{20} 842}-\frac{306}{\log_{20}306})=145.\)
Status Protein1 Protein2 d d1/2
Query variables 4MZX_1 2ABQ_1 184 143.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]