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Parikh vectors
4MYT_1 2MNQ_1 1AHG_1 Letter Amino acid
13 1 21 N Asparagine
40 3 20 D Aspartic acid
65 6 27 E Glutamic acid
56 1 18 I Isoleucine
49 1 40 L Leucine
44 3 20 T Threonine
3 0 5 W Tryptophan
51 3 46 A Alanine
24 0 16 Q Glutamine
64 3 25 V Valine
42 0 22 R Arginine
22 0 20 F Phenylalanine
38 0 15 P Proline
1 0 5 C Cysteine
59 0 30 G Glycine
14 0 6 H Histidine
42 4 17 K Lycine
20 0 8 M Methionine
22 3 23 S Serine
22 0 12 Y Tyrosine 

4MYT_1|Chain A|Elongation factor G|Thermus thermophilus (274)
>2MNQ_1|Chain A|THYMOSIN ALPHA-1|Homo sapiens (9606)
>1AHG_1|Chains A, B|Aspartate aminotransferase|Escherichia coli (strain K12) (83333)
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 \(w=f(c)\)
4MYT , Knot 264 691 0.83 40 259 627 
MAVKVEYDLKRLRNIGIAAHIDAGKTTTTERILYYTGRIHKIGEVHEGAATMDFMEQERERGITITAAVTTCFWKDHRINIIDTPGHVDFTIEVERSMRVLDGAIVVFDSSQGVEPQSETVWRQAEKYKVPRIAFANKMDKTGADLWLVIRTMQERLGARPVVMQLPIGREDTFSGIIDVLRMKAYTYGNDLGTDIREIPIPEEYLDQAREYHEKLVEVAADFDENIMLKYLEGEEPTEEELVAAIRKGTIDLKITPVFLGSALKNKGVQLLLDAVVDYLPSPLDIPPIKGTTPEGEVVEIHPDPNGPLAALAFKIMADPYVGRLTFIRVYSGTLTSGSYVYNTTKGRKERVARLLRMHANHREEVEELKAGDLGAVVGLKETITGDTLVGEDAPRVILESIEVPEPVIDVAIEPKTKADQEKLSQALARLAEEDPTFRVSTHPETGQTIISGMGELHLEIIVDRLKREFKVDANVGKPQVAYRETITKPVDVEGKFIRQTGGRGQYGHVKIKVEPLPRGSGFEFVNAIVGGVIPKEYIPAVQKGIEEAMQSGPLIGFPVVDIKVTLYDGSYHEVDSSAMAFKIAGSMAIKEAVQKGDPVILEPIMRVEVTTPEEYMGDVIGDLNARRGQILGMEPRGNAQVIRAFVPLAEMFGYATDLRSKTQGRGSFVMFFDHYQEVPKQVQEKLIKGQ
2MNQ , Knot 18 29 0.69 22 27 27 
XSDAAVDTSSEITTKDLKEKKEVVEEAEN
1AHG , Knot 166 396 0.83 40 217 370 
MFENITAAPADPILGLADLFRADERPGKINLGIGLYYDETGKIPVLTSVKKAEQYLLENETTKLYLGIDGIPEFGRCTQELLFGKGSALINDKRARTAQTPGGSGALRVAADFLAKNTSVKRVWVSNPSWPNHKSVFNSAGLEVREYAYYDAENHTLDFDALINSLNEAQAGDVVLFHGCCHNPTGIDPTLEQWQTLAQLSVEKGWLPLFDFAYQGFARGLEEDAEGLRAFAAMHKELIVASSYSKNFGLYNERVGACTLVAADSETVDRAFSQMKAAIRANYSSPPAHGASVVATILSNDALRAIWEQELTDMRQRIQRMRQLFVNTLQEKGANRDFSFIIKQNGMFSFSGLTKEQVLRLREEFGVYAVASGRVNVAGMTPDNMAPLCEAIVAVL

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 \(c\).

\(|P_{f(4MYT_1)}(2) \setminus P_{f(2MNQ_1)}(2)|=235\), \(|P_{f(2MNQ_1)}(2) \setminus P_{f(4MYT_1)}(2)|=3\). 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 1:1110100010010011111010110000000110001010011010011101011000000110101110001100001011001101010101000101101111110000110100001100100001101111001000110111110010001110111101111000010111011010100010011001001111000100100000011011101000111001010010000111110010101010111110110001101110111001101101111010010101101010101111111101110101101011010010100100100000100001101101010000010010110111111100010100111001101110010110111011101000100001001110110001010100010010011011101010111001000101010110101100001001101010110001101001010101011101011011011111111000111100110011001111111110101010010000100011110111011100110010111101110101001000110111010100101111010101011011111101110100100000101011111000001100100011010
Pair \(Z_2\) Length of longest common subsequence
4MYT_1,2MNQ_1 238 4
4MYT_1,1AHG_1 136 5
2MNQ_1,1AHG_1 202 3

Newick tree

 
[
	2MNQ_1:12.24,
	[
		4MYT_1:68,1AHG_1:68
	]:53.24
]

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 \((0.85)(0.8)(\frac{720 }{\log_{20} 720}-\frac{29}{\log_{20}29})=205.\)
Status Protein1 Protein2 d d1/2
Query variables 4MYT_1 2MNQ_1 252 131
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]