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Parikh vectors
5CBW_1 1MLY_1 9OAS_1 Letter Amino acid
7 23 10 P Proline
3 9 10 Y Tyrosine
9 10 9 N Asparagine
4 10 5 C Cysteine
3 21 15 Q Glutamine
8 25 8 I Isoleucine
11 18 29 S Serine
7 51 12 A Alanine
9 21 11 D Aspartic acid
6 26 11 E Glutamic acid
22 33 13 G Glycine
14 11 13 K Lycine
11 22 17 T Threonine
5 23 5 R Arginine
5 14 2 H Histidine
10 38 14 L Leucine
4 22 1 M Methionine
11 17 9 F Phenylalanine
1 8 2 W Tryptophan
15 27 14 V Valine 

5CBW_1|Chain A|Peptidyl-prolyl cis-trans isomerase F, mitochondrial|Homo sapiens (9606)
>1MLY_1|Chains A, B|7,8-diamino-pelargonic acid aminotransferase|Escherichia coli (562)
>9OAS_1|Chains A[auth F], D[auth K], H[auth L], K[auth R]|G001-14 Fab light chain|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 \(w=f(c)\)
5CBW , Knot 78 165 0.80 40 120 159 
MGNPLVYLDVDANGKPLGRVVLELKADVVPKTAENFRALCTGEKGFGYKGSTFHRVIPSFMCQAGDFTNHNGTGGKSIYGSRFPDENFTLKHVGPGVLSMANAGPNTNGSQFFICTIKTDWLDGKHVVFGHVIEGMDVVKKIESFGSKSGRTSKKIVITDCGQLS
1MLY , Knot 184 429 0.86 40 240 412 
MTTDDLAFDQRHILHPYTSMTSPLPVYPVVSAEGCELILSDGRRLVDGMSSWWAAIHGYNHPQLNAAMKSQIDAMSHVMFGGITHAPAIELCRKLVAMTPQPLECVFLADSGSVAVEVAMKMALQYWQAKGEARQRFLTFRNGYHGDTFGAMSVCDPDNSMHSLWKGYLPENLFAPAPQSRMDGEWDERDMVGFARLMAAHRHEIAAVIIEPIVQGAGGMRMYHPEWLKRIRKICDREGILLIADEIATGFGRTGKLFACEHAEIAPDILCLGKALTGGTMTLSATLTTREVAETISNGEAGCFMHGPTFMGNPLACAAANASLAILESGDWQQQVADIEVQLREQLAPARDAEMVADVRVLGAIGVVETTHPVNMAALQKFFVEQGVWIRPFGKLIYLMPPYIILPQQLQRLTAAVNRAVQDETFFCQ
9OAS , Knot 97 210 0.82 40 153 201 
DIQMTQSPSSLSASVGDRVTITCQASQDINNYLNWYQQKPGKAPKLLIYDASNLETGVPSRFSGSGSGTDFTFTISSLQPEDIATYYCQQYEFFGGGTKVEIKRTVAAPSVFIFPPSDEQLKSGTASVVCLLNNFYPREAKVQWKVDNALQSGNSQESVTEQDSKDSTYSLSSTLTLSKADYEKHKVYACEVTHQGLSSPVTKSFNRGEC

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 \(c\).

\(|P_{f(5CBW_1)}(2) \setminus P_{f(1MLY_1)}(2)|=36\), \(|P_{f(1MLY_1)}(2) \setminus P_{f(5CBW_1)}(2)|=156\). 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 1:110111010101010111011101010111001001011001001110010010011101100110100001011001010011000101001111110110111000100111001000110100111101101101100100110001000001110001010
Pair \(Z_2\) Length of longest common subsequence
5CBW_1,1MLY_1 192 5
5CBW_1,9OAS_1 153 3
1MLY_1,9OAS_1 193 3

Newick tree

 
[
	1MLY_1:10.98,
	[
		5CBW_1:76.5,9OAS_1:76.5
	]:25.48
]

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 \((0.85)(0.8)(\frac{594 }{\log_{20} 594}-\frac{165}{\log_{20}165})=123.\)
Status Protein1 Protein2 d d1/2
Query variables 5CBW_1 1MLY_1 158 107.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]