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Parikh vectors
1JUX_1 1KTV_1 6DXQ_1 Letter Amino acid
0 22 12 S Serine
2 44 15 T Threonine
0 3 4 W Tryptophan
0 40 25 D Aspartic acid
0 14 13 H Histidine
0 20 13 M Methionine
2 56 21 I Isoleucine
0 49 28 L Leucine
0 64 18 V Valine
0 42 18 R Arginine
0 24 12 Q Glutamine
0 59 25 G Glycine
0 42 11 K Lycine
0 38 25 P Proline
0 22 10 Y Tyrosine
2 50 33 A Alanine
2 1 7 C Cysteine
0 66 17 E Glutamic acid
0 13 14 N Asparagine
0 22 19 F Phenylalanine 

1JUX_1|Chain A|5'-D(*IP*TP*IP*TP*AP*CP*AP*C)-3'|
>1KTV_1|Chains A, B|ELONGATION FACTOR G|Thermus thermophilus (274)
>6DXQ_1|Chains A, B, C, D|4-oxalomesaconate hydratase|Sphingobium sp. SYK-6 (627192)
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 \(w=f(c)\)
1JUX , Knot 5 8 0.43 8 5 6 
ITITACAC
1KTV , Knot 264 691 0.83 40 259 627 
MAVKVEYDLKRLRNIGIAAHIDAGKTTTTERILYYTGRIHKIGEVHEGAATMDFMEQERERGITITAAVTTCFWKDHRINIIDTPGHVDFTIEVERSMRVLDGAIVVFDSSQGVEPQSETVWRQAEKYKVPRIAFANKMDKTGADLWLVIRTMQERLGARPVVMQLPIGREDTFSGIIDVLRMKAYTYGNDLGTDIREIPIPEEYLDQAREYHEKLVEVAADFDENIMLKYLEGEEPTEEELVAAIRKGTIDLKITPVFLGSALKNKGVQLLLDAVVDYLPSPLDIPPIKGTTPEGEVVEIHPDPNGPLAALAFKIMADPYVGRLTFIRVYSGTLTSGSYVYNTTKGRKERVARLLRMHANHREEVEELKAGDLGAVVGLKETITGDTLVGEDAPRVILESIEVPEPVIDVAIEPKTKADQEKLSQALARLAEEDPTFRVSTHPETGQTIISGMGELHLEIIVDRLKREFKVDANVGKPQVAYRETITKPVDVEGKFIRQTGGRGQYGHVKIKVEPLPRGSGFEFVNAIVGGVIPKEYIPAVQKGIEEAMQSGPLIGFPVVDIKVTLYDGSYHEVDSSEMAFKIAGSMAIKEAVQKGDPVILEPIMRVEVTTPEEYMGDVIGDLNARRGQILGMEPRGNAQVIRAFVPLAEMFGYATDLRSKTQGRGSFVMFFDHYQEVPKQVQEKLIKGQ
6DXQ , Knot 149 340 0.85 40 215 327 
MMIIDCHGHYTVLPKAHDEWREQQKAAFKAGQPAPPYPEISDDEIRETIEANQLRLIKERGADMTIFSPRASAMAPHVGDQSVAVPWAQACNNLIARVVDLFPETFAGVCMLPQSPEADMTSSIAELERCVNELGFIGCNLNPDPGGGHFKHPPLTDRFWYPFYEKMVELDVPAMIHVSGSCNPAMHATGAYYLAADTIAFMQLLQGNLFADFPTLRFIIPHGGGAVPYHWGRFRGLADMLKQPSLDTLLMNNVFFDTCVYHQPGINLLADVIDNKNILFGSEMVGAVRGIDPTTGHYFDDTKRYIDALDISDQERHAIFEGNTRRVFPRLDAKLKARGL

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 \(c\).

\(|P_{f(1JUX_1)}(2) \setminus P_{f(1KTV_1)}(2)|=2\), \(|P_{f(1KTV_1)}(2) \setminus P_{f(1JUX_1)}(2)|=256\). 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 1:10101010
Pair \(Z_2\) Length of longest common subsequence
1JUX_1,1KTV_1 258 5
1JUX_1,6DXQ_1 216 2
1KTV_1,6DXQ_1 156 4

Newick tree

 
[
	1JUX_1:12.77,
	[
		6DXQ_1:78,1KTV_1:78
	]:51.77
]

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 \((0.85)(0.8)(\frac{699 }{\log_{20} 699}-\frac{8}{\log_{20}8})=209.\)
Status Protein1 Protein2 d d1/2
Query variables 1JUX_1 1KTV_1 260 131
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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