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Parikh vectors
3QYQ_1 1WKP_1 4KLZ_1 Letter Amino acid
19 14 18 R Arginine
6 7 6 Q Glutamine
7 13 5 P Proline
31 18 10 V Valine
2 2 0 W Tryptophan
4 7 8 Y Tyrosine
28 6 14 A Alanine
10 13 2 N Asparagine
14 7 14 D Aspartic acid
13 2 9 K Lycine
13 8 10 F Phenylalanine
5 0 2 C Cysteine
28 12 10 G Glycine
19 10 12 I Isoleucine
22 14 11 L Leucine
18 11 9 S Serine
25 9 15 E Glutamic acid
9 4 4 H Histidine
7 2 5 M Methionine
13 12 9 T Threonine 

3QYQ_1|Chains A, B, C, D|Deoxyribose-phosphate aldolase, putative|Toxoplasma gondii (508771)
>1WKP_1|Chains A, B, C, D|FLOWERING LOCUS T protein|Arabidopsis thaliana (3702)
>4KLZ_1|Chain A|GTP-binding protein Rit1|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 \(w=f(c)\)
3QYQ , Knot 126 293 0.81 40 172 272 
MGSSHHHHHHENLYFQGIYKQFTSRTLLNFFEVAALTDGETNESVAAVCKIAAKDPAIVGVSVRPAFVRFIRQELVKSAPEVAGIKVCAAVNFPEGTGTPDTVSLEAVGALKDGADEIECLIDWRRMNENVADGESRIRLLVSEVKKVVGPKTLKVVLSGGELQGGDIISRAAVAALEGGADFLQTSSGLGATHATMFTVHLISIALREYMVRENERIRVEGINREGAAVRCIGIKIEVGDVHMAETADFLMQMIFENGPRSIVRDKFRVGGGFNLLKELRDCYESWDSVGVS
1WKP , Knot 81 171 0.81 38 131 165 
GSHMSINIRDPLIVSRVVGDVLDPFNRSITLKVTYGQREVTNGLNLRPSQVQNKPRVEIGGEDLRNFYTLVMVDPDVPSPSNPHLREYLHWLVTDIPATTGTTFGNEIVSYENPSPTAGIHRVVFILFRQLGRQTVYAPGWRQNFNTREFAEIYNLGLPVAAVFYNSQRES
4KLZ , Knot 86 173 0.85 38 131 166 
GSSREYKLVMLGAGGVGKSAMTMQFISHRFPEDHDPTIEDAYKIRIRIDDEPANLDILDTAGQAEFTAMRDQYMRAGEGFIICYSITDRRSFHEVREFKQLIYRVRRTDDTPVVLVGNKSDLKQLRQVTKEEGLALAREFSCPFFETSAAYRYYIDDVFHALVREIRRKEKEA

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 \(c\).

\(|P_{f(3QYQ_1)}(2) \setminus P_{f(1WKP_1)}(2)|=101\), \(|P_{f(1WKP_1)}(2) \setminus P_{f(3QYQ_1)}(2)|=60\). 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 1:11000000000010101100010000110110111100100000111100111001111110101111011000110011011110101110110101010010101111100110010011010010001101000101110010011110010111011010110110011111101110110000111100101101011011100011000001010110001111001110101101011001011101110011001100010111110110010000001001110
Pair \(Z_2\) Length of longest common subsequence
3QYQ_1,1WKP_1 161 4
3QYQ_1,4KLZ_1 151 3
1WKP_1,4KLZ_1 154 3

Newick tree

 
[
	1WKP_1:79.82,
	[
		3QYQ_1:75.5,4KLZ_1:75.5
	]:4.32
]

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 \((0.85)(0.8)(\frac{464 }{\log_{20} 464}-\frac{171}{\log_{20}171})=86.1\)
Status Protein1 Protein2 d d1/2
Query variables 3QYQ_1 1WKP_1 105 82.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]