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Parikh vectors
9BXC_1 8BBH_1 7XCA_1 Letter Amino acid
48 1 16 D Aspartic acid
44 0 16 I Isoleucine
50 2 20 L Leucine
42 0 10 Y Tyrosine
41 0 18 V Valine
47 2 24 A Alanine
42 0 16 N Asparagine
59 0 10 K Lycine
17 0 17 P Proline
35 0 27 T Threonine
3 1 6 W Tryptophan
28 0 9 R Arginine
49 1 10 E Glutamic acid
33 1 20 G Glycine
16 2 11 H Histidine
49 1 15 S Serine
6 2 3 C Cysteine
25 2 9 Q Glutamine
27 1 7 M Methionine
39 2 7 F Phenylalanine 

9BXC_1|Chains A, B|Ribonucleoside-diphosphate reductase subunit alpha|Bacillus subtilis (1423)
>8BBH_1|Chain A|Glucose-6-phosphate isomerase|Homo sapiens (9606)
>7XCA_1|Chains A, B|Capsid protein spike domain|Porcine astrovirus 1 (1239567)
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 \(w=f(c)\)
9BXC , Knot 271 700 0.84 40 286 659 
MSQNQVPKWIQLNNEIMIQKDGKFQFDKDKEAVHSYFVDYINQNTVFFHNLKEKLDYLVENQYYEEEFLSLYSFEDIKEVFKTAYAKKFRFPSFMSAFKFYNDYALKTNDKKKILERYEDRISIVALFFANGDTEKAKEYVNLMINQEYQPSTPTFLNAGRKRRGELVSCFLLEVNDSLNDISRAIDISMQLSKLGGGVSLNLSKLRAKGEAIKDVENATKGVVGVMKLLDNAFRYADQMGQRQGSGAAYLNIFHRDINDFLDTKKISADEDVRVKTLSIGVVIPDKFVELAREDKAAYVFYPHTIYKEYGQHMDEMDMNEMYDKFVDNPRVKKEKINPRKLLEKLAMLRSESGYPYIMFQDNVNKVHANNHISKVKFSNLCSEVLQASQVSSYTDYDEEDEIGLDISCNLGSLNILNVMEHKSIEKTVKLATDSLTHVSETTDIRNAPAVRRANKAMKSIGLGAMNLHGYLAQNGIAYESPEARDFANTFFMMVNFYSIQRSAEIAKEKGETFDQYEGSTYATGEYFDKYVSTDFSPKYEKIANLFEGMHIPTTEDWKKLKAFVAEHGMYHSYRLCIAPTGSISYVQSSTASVMPIMERIEERTYGNSKTYYPMPGLASNNWFFYKEAYDMDMFKVVDMIATIQQHIDQGISFTLFLKDTMTTRDLNRIDLYAHHRGIKTIYYARTKDTGQDSCLSCVV
8BBH , Knot 16 19 0.82 26 18 17 
XCFEQLLSGAHWMDQHFCA
7XCA , Knot 123 271 0.84 40 175 257 
MDIGINSDGETPVTFKAYRMMPEDTIYLRFKPDTLSVVSNFQPAKRPMLAKTYSGDTLTVGQGNNKTAIHTVVRISDPTWFSADWDPISTPQPIAEIYCKAGTTTVGDILAAYQVHGLGNHTTTAYVVRMTAGANPQVSAGIVTNKGTNDYDLKTANSNAGFSWNLGSGTWYLMMSFGDALGSLGTWRWTPNELSANYTIYNCEIIPCLLLANDDFHIVIPTKNALVPLVARERHLDQGRQVQIQPTEPPASEVEDVGKLAAALEHHHHHH

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 \(c\).

\(|P_{f(9BXC_1)}(2) \setminus P_{f(8BBH_1)}(2)|=275\), \(|P_{f(8BBH_1)}(2) \setminus P_{f(9BXC_1)}(2)|=7\). 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 1:1000011011010001110001010100000110001100100001110010001001100000000110100100100110010100101101101101000011000000011000000101111111010000100010111000001001011011000010110011101000100100110101010011111010100101010110010010011111101100110010011000101110101100010011000010100010100101111110011011000011011010010000100100101001000110010100001010011001111000010101110001001010001001010010001101001000000000001110100011010110110000100010110001001000001001111001001100111111010101100111000101001100111110100100010110001001000010001010010001000101000011011011011000010010111100110000010111010100100001011111001000001000000111111000111000100101101101110100010011010111000100001001010100011001001000001000010011
Pair \(Z_2\) Length of longest common subsequence
9BXC_1,8BBH_1 282 3
9BXC_1,7XCA_1 185 4
8BBH_1,7XCA_1 181 2

Newick tree

 
[
	9BXC_1:12.38,
	[
		7XCA_1:90.5,8BBH_1:90.5
	]:36.88
]

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 \((0.85)(0.8)(\frac{719 }{\log_{20} 719}-\frac{19}{\log_{20}19})=209.\)
Status Protein1 Protein2 d d1/2
Query variables 9BXC_1 8BBH_1 263 135
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]