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Parikh vectors
1SQX_1 2AGK_1 1HNC_1 Letter Amino acid
33 13 13 S Serine
22 10 9 Q Glutamine
30 17 16 E Glutamic acid
8 1 7 M Methionine
15 6 3 H Histidine
17 11 9 N Asparagine
14 11 15 F Phenylalanine
17 6 12 Y Tyrosine
12 8 3 C Cysteine
32 22 10 G Glycine
17 12 12 I Isoleucine
45 30 27 L Leucine
14 24 18 K Lycine
44 11 10 A Alanine
25 10 11 R Arginine
24 23 15 D Aspartic acid
30 19 5 V Valine
18 4 12 P Proline
24 15 7 T Threonine
5 7 3 W Tryptophan 

1SQX_1|Chain A|Ubiquinol-cytochrome-c reductase complex core protein I, mitochondrial precursor|Bos taurus (9913)
>2AGK_1|Chain A|1-(5-phosphoribosyl)-5-[(5-phosphoribosylamino)methylideneamino] imidazole-4-carboxamide isomerase|Saccharomyces cerevisiae (4932)
>1HNC_1|Chains A, B, C, D|GLUTATHIONE S-TRANSFERASE|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 \(w=f(c)\)
1SQX , Knot 185 446 0.84 40 239 424 
TATYAQALQSVPETQVSQLDNGLRVASEQSSQPTCTVGVWIDAGSRYESEKNNGAGYFVEHLAFKGTKNRPGNALEKEVESMGAHLNAYSTREHTAYYIKALSKDLPKAVELLADIVQNCSLEDSQIEKERDVILQELQENDTSMRDVVFNYLHATAFQGTPLAQSVEGPSENVRKLSRADLTEYLSRHYKAPRMVLAAAGGLEHRQLLDLAQKHFSGLSGTYDEDAVPTLSPCRFTGSQICHREDGLPLAHVAIAVEGPGWAHPDNVALQVANAIIGHYDCTYGGGAHLSSPLASIAATNKLCQSFQTFNICYADTGLLGAHFVCDHMSIDDMMFVLQGQWMRLCTSATESEVLRGKNLLRNALVSHLDGTTPVCEDIGRSLLTYGRRIPLAEWESRIAEVDARVVREVCSKYFYDQCPAVAGFGPIEQLPDYNRIRSGMFWLRF
2AGK , Knot 115 260 0.82 40 168 249 
TKFIGCIDLHNGEVKQIVGGTLTSKKEDVPKTNFVSQHPSSYYAKLYKDRDVQGCHVIKLGPNNDDAAREALQESPQFLQVGGGINDTNCLEWLKWASKVIVTSWLFTKEGHFQLKRLERLTELCGKDRIVVDLSCRKTQDGRWIVAMNKWQTLTDLELNADTFRELRKYTNEFLIHAADVEGLCGGIDELLVSKLFEWTKDYDDLKIVYAGGAKSVDDLKLVDELSHGKVDLTFGSSLDIFGGNLVKFEDCCRWNEKQG
1HNC , Knot 101 217 0.83 40 153 209 
PMTLGYWNIRGLAHSIRLLLEYTDSSYEEKKYTMGDAPDYDRSQWLNEKFKLGLDFPNLPYLIDGTHKITQSNAILRYIARKHNLCGESEKEQIREDILENQFMDSRMQLAKLCYDPDFEKLKPEYLQALPEMLKLYSQFLGKQPWFLGDKITFVDFIAYDVLERNQVFEPSCLDAFPNLKDFISRFEGLEKISAYMKSSRFLPRPVFTKMAVFGNK

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 \(c\).

\(|P_{f(1SQX_1)}(2) \setminus P_{f(2AGK_1)}(2)|=125\), \(|P_{f(2AGK_1)}(2) \setminus P_{f(1SQX_1)}(2)|=54\). 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 1:01001011001100010010011011000000100011111011000000000111011001110100001101100010011101010000000100101100011011011101100001000010000011100100000010011100101011010111001011000100100101000100000110111111111000011011000101101000001110101001010010000011111011111011111010011101101111000000111101001110111000100010010100100111110110001010011111010110100010000110100110011100101001100011001100100111101000110101011001000010000111111111001100001001111101
Pair \(Z_2\) Length of longest common subsequence
1SQX_1,2AGK_1 179 3
1SQX_1,1HNC_1 198 4
2AGK_1,1HNC_1 169 3

Newick tree

 
[
	1SQX_1:97.43,
	[
		2AGK_1:84.5,1HNC_1:84.5
	]:12.93
]

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 \((0.85)(0.8)(\frac{706 }{\log_{20} 706}-\frac{260}{\log_{20}260})=124.\)
Status Protein1 Protein2 d d1/2
Query variables 1SQX_1 2AGK_1 159 125
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]