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Parikh vectors
7QOF_1 8HAN_1 1IQC_1 Letter Amino acid
30 7 22 E Glutamic acid
29 13 22 K Lycine
10 0 3 W Tryptophan
20 3 7 Y Tyrosine
34 18 24 A Alanine
38 12 20 L Leucine
28 4 15 F Phenylalanine
25 5 18 S Serine
22 7 15 I Isoleucine
19 2 9 M Methionine
41 6 16 V Valine
35 1 21 N Asparagine
27 4 17 D Aspartic acid
43 7 24 G Glycine
8 2 5 H Histidine
26 10 20 T Threonine
27 18 11 R Arginine
2 2 4 C Cysteine
21 8 14 Q Glutamine
19 6 21 P Proline 

7QOF_1|Chains A, B, C, D, E, F, G, H, I|Major capsid protein gp32|Bacteroides phage crAss001 (2301731)
>8HAN_1|Chains A, E|Histone H3.1|Homo sapiens (9606)
>1IQC_1|Chains A, B, C, D|di-heme peroxidase|Nitrosomonas europaea (915)
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 \(w=f(c)\)
7QOF , Knot 207 504 0.85 40 247 473 
MAGKLGKFQMLGFQHWKGLTSDNHLGAIFQQAPQKATNLMVQLLAFYRGKSLDTFLNSFPTREFEDDNEYYWDVIGSSRRNIPLVEARDENGVVVAANAANVGVGTSPFYLVFPEDWFADGEVIVGNLNQVYPFRILGDARMEGTNAVYKVELMGGNTQGVPAERLQQGERFSIEFAPVEKELSRKVGDVRFTSPVSMRNEWTTIRIQHKVAGNKLNKKLAMGIPMVRNLESGKQVKDTANMWMHYVDWEVELQFDEYKNNAMAWGTSNRNLNGEYMNFGKSGNAIKTGAGIFEQTEVANTMYYNTFSLKLLEDALYELSASKLAMDDRLFVIKTGERGAIQFHKEVLKTVSGWTTFVLDNNSTRVVEKVQSRLHSNALSAGFQFVEYKAPNGVRVRLDVDPFYDDPVRNKILHPMGGVAFSYRYDIWYIGTMDQPNIFKCKIKGDNEYRGYQWGIRNPFTGQKGNPYMSFDEDSAVIHRMATLGVCVLDPTRTMSLIPAILQG
8HAN , Knot 65 135 0.78 38 104 127 
ARTKQTARKSTGGKAPRKQLATKAARKSAPATGGVKKPHRYRPGTVALREIRRYQKSTELLIRKLPFQRLVREIAQDFKTDLRFQSSAVMALQEACEAYLVGLFEDTNLCAIHAKRVTIMPKDIQLARRIRGERA
1IQC , Knot 138 308 0.85 40 200 295 
ANEPIQPIKAVTPENADMAELGKMLFFDPRLSKSGFISCNSCHNLSMGGTDNITTSIGHKWQQGPINAPTVLNSSMNLAQFWDGRAKDLKEQAAGPIANPKEMASTHEIAEKVVASMPQYRERFKKVFGSDEVTIDRITTAIAQFEETLVTPGSKFDKWLEGDKNALNQDELEGYNLFKGSGCVQCHNGPAVGGSSYQKMGVFKPYETKNPAAGRMDVTGNEADRNVFKVPTLRNIELTYPYFHDGGAATLEQAVETMGRIQLNREFNKDEVSKIVAFLKTLTGDQPDFKLPILPPSNNDTPRSQPYE

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 \(c\).

\(|P_{f(7QOF_1)}(2) \setminus P_{f(8HAN_1)}(2)|=174\), \(|P_{f(8HAN_1)}(2) \setminus P_{f(7QOF_1)}(2)|=31\). 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 1:111011010111100101100000111110011001001110111100100100110011000100000001011100000111101000011111101101111001101111001110101111010010110111010101001100101111000111100100100101011110001000110101001101000100101000111001000111111110010010010001011100101010101000000111110000010100101100101100111110000110010000101011001100101001110001111001001110100011001011001110000001100100010001101110110001101101010101100011000110111111100000110110100101100010100000100111001101001010101000011100110111011010001011111101
Pair \(Z_2\) Length of longest common subsequence
7QOF_1,8HAN_1 205 3
7QOF_1,1IQC_1 165 3
8HAN_1,1IQC_1 178 3

Newick tree

 
[
	8HAN_1:10.08,
	[
		7QOF_1:82.5,1IQC_1:82.5
	]:17.58
]

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 \((0.85)(0.8)(\frac{639 }{\log_{20} 639}-\frac{135}{\log_{20}135})=145.\)
Status Protein1 Protein2 d d1/2
Query variables 7QOF_1 8HAN_1 188 117.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]