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Parikh vectors
4USX_1 3DXC_1 3MQE_1 Letter Amino acid
37 15 32 V Valine
0 6 12 C Cysteine
56 6 37 G Glycine
8 10 18 H Histidine
12 3 34 I Isoleucine
40 5 37 T Threonine
1 2 6 W Tryptophan
56 17 25 A Alanine
5 6 26 R Arginine
7 4 35 K Lycine
2 7 36 F Phenylalanine
7 3 27 Y Tyrosine
31 4 30 N Asparagine
17 7 39 P Proline
13 3 25 D Aspartic acid
15 8 29 Q Glutamine
6 7 38 E Glutamic acid
16 12 55 L Leucine
7 3 13 M Methionine
35 12 33 S Serine 

4USX_1|Chains A, B, C|TRIMERIC AUTOTRANSPORTER ADHESIN|BURKHOLDERIA PSEUDOMALLEI K96243 (272560)
>3DXC_1|Chains A, C|Amyloid beta A4 protein-binding family B member 1|Homo sapiens (9606)
>3MQE_1|Chains A, B, C, D|Prostaglandin G/H synthase 2|Mus musculus (10090)
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 \(w=f(c)\)
4USX , Knot 134 371 0.71 38 145 306 
MASMTGGQQMGRGSMAPTTPNGGKPSQDLTLVGAASGPVALHNVAPGTASTDAVNVGQLGAVTTGLGGGAAIDPKTGAVTAPSYTVYNADGTTSNVGNVGAAIDAINSTGIKYFHANSTKPDSQALGADSVAIGPNAVANNAGDVALGSGAVTSQAGGTLSETINGVTYSFAGTTPIGTVSVGAPGVERTITNVAAGRIGQSSTDAINGSQLYGTNQSIEALTDKMNSLGNTVANTLGSGASYNPQTGAVNGPANSGGVVTPTVIQEAANKWVSANPSTYVAPVATGTNGMAVGSGAVSTGQNSVALGTNASDGGRSNVVSVGAPGAERQVTNVAAGTQATDAVNLGQMNGANSSSVDKLAAALEHHHHHH
3DXC , Knot 70 140 0.82 40 112 133 
APKNELVQKFQVYYLGNVPVAKPVGVDVINGALESVLSSSSREQWTPSHVSVAPATLTILHQQTEAVLGECRVRFLSFLAVGRDVHTFAFIMAAGPASFCCHMFWCEPNAASLSEAVQAACMLRYQKCLDARSQHHHHHH
3MQE , Knot 231 587 0.83 40 282 552 
ANPCCSNPCQNRGECMSTGFDQYKCDCTRTGFYGENCTTPEFLTRIKLLLKPTPNTVHYILTHFKGVWNIVNNIPFLRSLIMKYVLTSRSYLIDSPPTYNVHYGYKSWEAFSNLSYYTRALPPVADDCPTPMGVKGNKELPDSKEVLEKVLLRREFIPDPQGSNMMFAFFAQHFTHQFFKTDHKRGPGFTRGLGHGVDLNHIYGETLDRQHKLRLFKDGKLKYQVIGGEVYPPTVKDTQVEMIYPPHIPENLQFAVGQEVFGLVPGLMMYATIWLREHNRVCDILKQEHPEWGDEQLFQTSRLILIGETIKIVIEDYVQHLSGYHFKLKFDPELLFNQQFQYQNRIASEFNTLYHWHPLLPDTFNIEDQEYSFKQFLYNNSILLEHGLTQFVESFTRQIAGRVAGGRNVPIAVQAVAKASIDQSREMKYQSLNEYRKRFSLKPYTSFEELTGEKEMAAELKALYSDIDVMELYPALLVEKPRPDAIFGETMVELGAPFSLKGLMGNPICSPQYWKPSTFGGEVGFKIINTASIQSLICNNVKGCPFTSFNVQDPQPTKTATINASASHSRLDDINPTVLIKRRSTEL

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 \(c\).

\(|P_{f(4USX_1)}(2) \setminus P_{f(3DXC_1)}(2)|=90\), \(|P_{f(3DXC_1)}(2) \setminus P_{f(4USX_1)}(2)|=57\). 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 1:11010110011010111001011010001011111011111001111010001101101111001111111101001110110001001010000110111110110001100101000010001111001111101110011011110111000111010001011000111001110101111110001001111011000001101001010000101100010011001100110110001001110111001111010110011001101010001111101001111101110010001111001001100011011111100010011110010011011010110000100111110000000
Pair \(Z_2\) Length of longest common subsequence
4USX_1,3DXC_1 147 6
4USX_1,3MQE_1 193 3
3DXC_1,3MQE_1 230 4

Newick tree

 
[
	3MQE_1:11.99,
	[
		4USX_1:73.5,3DXC_1:73.5
	]:41.49
]

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 \((0.85)(0.8)(\frac{511 }{\log_{20} 511}-\frac{140}{\log_{20}140})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 4USX_1 3DXC_1 121 86.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]