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Parikh vectors
3QMO_1 1IBQ_1 4LCW_1 Letter Amino acid
33 25 16 V Valine
30 14 16 Q Glutamine
38 35 17 G Glycine
35 13 10 K Lycine
37 29 16 T Threonine
6 3 9 W Tryptophan
27 20 15 Y Tyrosine
27 1 14 R Arginine
38 11 21 E Glutamic acid
34 13 12 I Isoleucine
14 0 7 M Methionine
34 48 15 S Serine
39 11 14 P Proline
30 23 14 A Alanine
25 29 14 D Aspartic acid
13 2 4 C Cysteine
60 22 25 L Leucine
37 15 10 F Phenylalanine
29 8 9 N Asparagine
24 3 13 H Histidine 

3QMO_1|Chains A, B|Prostaglandin G/H synthase 2|Mus musculus (10090)
>1IBQ_1|Chains A, B|ASPERGILLOPEPSIN|Aspergillus phoenicis (5063)
>4LCW_1|Chains A, C|Major histocompatibility complex class I-related gene protein|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 \(w=f(c)\)
3QMO , Knot 242 610 0.84 40 287 572 
MLFRAVLLCAALGLSQAANHHHHHHPCCSNPCQNRGECMSTGFDQYKCDCTRTGFYGENCTTPEFLTRIKLLLKPTPNTVHYILTHFKGVWNIVNNIPFLRSLIMKYVLTSRSYLIDSPPTYNVHYGYKSWEAFSNLSYYTRALPPVADDCPTPMGVKGNKELPDSKEVLEKVLLRREFIPDPQGSNMMFAFFAQHFTHQFFKTDHKRGPGFTRGLGHGVDLNHIYGETLDRQHKLRLFKDGKLKYQVIGGEVYPPTVKDTQVEMIYPPHIPENLQFAVGQEVFGLVPGLMMYATIWLREHNRVCDILKQEHPEWGDEQLFQTSRLILIGETIKIVIEDYVQHLSGYHFKLKFDPELLFNQQFQYQNRIASEFNTLYHWHPLLPDTFNIEDQEYSFKQFLYNNSILLEHGLTQFVESFTRQIAGRVAGGRNVPIAVQAVAKASIDQSREMKYQSLNEYRKRFSLKPYTSFEELTGEKEMAAELKALYSDIDVMELYPALLVEKPRPDAIFGETMVELGAPFSLKGLMGNPICSPQYWKPSTFGGEVGFKIINTASIQSLICNNVKGCPFTSFNVQDPQPTKTATIAASASHSRLDDINPTVLIKRRSTEL
1IBQ , Knot 131 325 0.77 38 166 296 
SKGSAVTTPQNNDEEYLTPVTVGKSTLHLDFDTGSADLWVFSDELPSSEQTGHDLYTPSSSATKLSGYSWDISYGDGSSASGDVYRDTVTVGGVTTNKQAVEAASKISSEFVQDTANDGLLGLAFSSINTVQPKAQTTFFDTVKSQLDSPLFAVQLKHDAPGVYDFGYIDDSKYTGSITYTDADSSQGYWGFSTDGYSIGDGSSSSSGFSAIADTGTTLILLDDEIVSAYYEQVSGAQESYEAGGYVFSCSTDLPDFTVVIGDYKAVVPGKYINYAPVSTGSSTCYGGIQSNSGLGLSILGDVFLKSQYVVFNSEGPKLGFAAQA
4LCW , Knot 128 271 0.88 40 195 261 
MRTHSLRYFRLGVSDPIHGVPEFISVGYVDSHPITTYDSVTRQAEPRAPWMAENLAPDHWERYTQLLRGWQQMFKVELKRLQRHYNHSGSHTYQRMIGCELLEDGSTTGFLQYAYDGQDFLIFNKDTLSWLAVDNVAHTIKQAWEANQHELLYQKNWLEEECIAWLKRFLEYGKDTLQRTEPPLVRVNRKETFPGVTALFCKAHGFYPPEIYMTWMKNGEEIVQEIDYGDILPSGDGTYQAWASIELDPQSSNLYSCHVEHSGVHMVLQVP

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 \(c\).

\(|P_{f(3QMO_1)}(2) \setminus P_{f(1IBQ_1)}(2)|=150\), \(|P_{f(1IBQ_1)}(2) \setminus P_{f(3QMO_1)}(2)|=29\). 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 1:1110111101111100110000000100001000010010011000000000011010000010110010111010100100110010111011001111001110011000001100110001001000101100100000111111000101111010001100001100111000111010100111111100100011000000111100111011010010100100000101100101000111101011010000101101101100101111001111111111010111000001001100001011000110000111110010111000100101001010101011100010000011001001001011110010100000010011000011100110011001000111011110011111011101010000010000100000010101000100101000111010110001011010111110010101111001101111101011110110010010100111011101100101001100010101100101001010001011101000010010101110000001
Pair \(Z_2\) Length of longest common subsequence
3QMO_1,1IBQ_1 179 4
3QMO_1,4LCW_1 190 3
1IBQ_1,4LCW_1 193 3

Newick tree

 
[
	4LCW_1:97.74,
	[
		3QMO_1:89.5,1IBQ_1:89.5
	]:8.24
]

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 \((0.85)(0.8)(\frac{935 }{\log_{20} 935}-\frac{325}{\log_{20}325})=163.\)
Status Protein1 Protein2 d d1/2
Query variables 3QMO_1 1IBQ_1 208 156.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]