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Assembly theory is a framework for quantifying selection, evolution, and complexity. It, therefore, spans various scientific disciplines, including physics, chemistry, biology, and information theory. Assembly theory is rooted in the assembly of an object from a set of basic building units, forming an initial assembly pool and from subunits that entered the assembly pool in previous assembly steps. Hence, the object is defined not as a set of point particles but by the history of its assembly, where the assembly index is the smallest number of steps required to assemble the object.
1. Introduction
Assembly theory was formulated in 2017[1], introducing the concept of assembly index (initially called "pathway complexity") of an object as the smallest number of steps required to assemble this object from a set of basic building units, forming an initial assembly pool and from subunits that entered the assembly pool in previous assembly steps. The assembly index is, therefore, a measure of the complexity of the object, which is computable[5], unlike Kolmogorov complexity, for example, and captures the structural information about the object, unlike Shannon entropy. The theoretical background for the theory was researched[5] based on directed multigraphs showing that the assembly index of an object is computable for all finite objects.
Consider two binary strings C = [01010101] and D = [00010111] and the initial assembly pool containing two bits 0 and 1. Both strings have the same length N = 8 and the same Shannon entropy H(C) = H(D) = log2(2) = 1. However, the assembly index of the first string is a(C) = 3 (In step 1, assemble "01" and put it into the assembly pool, in step 2 assemble "01" with "01" taken from the assembly pool and put "0101" into the assembly pool, and in step 3 assemble "0101" assembled in the second step with "0101" taken from the assembly pool), while the assembly index of the second string is a(D) = 6, since only the substring "01" can be reused from the assembly pool[8].
Lower bound on the assembly index - OEIS A003313 (red); log2(𝑁) (red, dash–dot),
conjectured upper bound on the bitstring assembly index (green) and 𝑁−1 (green, dash–dot).
Basic building units depend on a particular application of the assembly theory. In chemistry, it found applications in drug discovery[3]. Furthermore, the theoretical value of the assembly index of a molecule, where the initial assembly pool contains chemical bonds, can be experimentally confirmed using tandem mass spectrometry, nuclear magnetic resonance, or infrared spectroscopy[4][7]. Therefore, the assembly index is the universal threshold between abiotic and biotic molecules and a robust and simple biosignature[1][2] to distinguish random, abiotic objects from biologically or technologically assembled ones, as only biotic samples can have a molecular assembly index above 15. The more complex a given object, the less likely an identical copy can exist without some information-driven mechanism that generates that object[6].