Assembly Theory Provides A Measure of Specified Complexity That Quantifies but does Not Explain Selection and Evolution

We have proposed that copy number operates as a specification. But what does copy number specify? Let’s consider two examples from different domains.

First, take the case of written human artifacts. Imagine an archaeologist happening upon a heretofore undiscovered ancient manuscript (given a pool of alphabetical letters any non-trivial manuscript of moderate length will have a high assembly index). At first the copy number of such an artifact will be one – it is unique in the world. But as the manuscript is carefully extracted, preserved, and interpreted, copies will begin to proliferate. There will be copies sent to museums around the world. Copies in academic papers interpreting the text. Copies in anthologies collecting such texts. Over time we would measure n_i ≫ 1 reflecting the value ascribed to the manuscript by society.

Contrast this with a random collection of words with assembly index commensurate to the ancient manuscript. These two objects may have similar a_i (and therefore be equivalently improbable); however, the copy number of the random collection of words is unlikely to exceed one. Why? Because such a text carries no value to the society in question. In this context, the long-term copy number acts as a reasonable – if blunt – external specification for ‘value ascribed to a written text by a society’.

Now consider a biological case, that of proteins: long chains of sequentially arranged amino acids that can fold into complex functional three-dimensional shapes. Given that cells contain machinery to transcribe sequences of DNA into multiple copies of identical proteins it is common for proteins to exhibit high copy number both within an individual cell and across the biosphere.

But now, imagine a scenario in which random mutation results in a new pair of start and stop codons appearing around a heretofore untranscribed string of DNA. This will result in a new protein. At first the protein will have copy number 1, but as the machinery of the cell repeatedly transcribes the novel DNA strand n_i will increase. If, however, the protein serves no function for the cell (or, worse, is detrimental to the cell) we can expect that future mutations, selection effects, or even error-detection mechanisms will winnow out the production and replication of this strand of random protein. Thus reducing n_i back down to zero.

Here again we see that the long-term behavior of n_i serves as a reasonable proxy for ‘value’ or ‘function’. In the case of proteins in the context of existing cellular infrastructure, the presence and persistence of high copy number serves as an external specification that points towards functionality.

We note, however, that there are issues with the use of copy number as a specification. Consider, for example, the presence of stochastic error in the copying process of an object. These errors may prove superficial – the copy still retains functional value commensurate to that of the original – however, without specifying a tolerance range for such variations AT may fail to categorize these superficial variants as members of the same class of object. This would lower the assembly measure for the object (by spreading n_i across multiple O_i) and potentially result in a false negative.

In addition, the normalization of n_i by N_T – the total number of objects in the ensemble – makes normalization of assembly somewhat ambiguous. The actual value of A will depend on what is and is not included in the ensemble making it challenging to speak of a rigorous absolute A_min threshold for rejecting an undirected process.

Nonetheless, such limitations are not fatal to the core structure of the theory. As we have seen in both the artifact and biological cases long-term copy number can serve as a workable specification to call out that the random sequence in question – as equally unlikely as any other random sequence of equivalent a_i – plays some sort of externally specified role.

In the framework of SC the probability under consideration must be computed for some hypothesized chance process. If this probability is low, and the event in question is highly specified, then we have good reason to rule out the chance process as the event’s cause. We have proposed that assembly index operates as a probability measure – but what chance process is it associated with?

This is where the physical motivation behind a_i proves valuable. Consider an object with assembly index a_i that is an ordered linear chain of initial building blocks. We assume there are β unique building block elements in the initial pool (for words in the English language β = 26, for proteins constructed of amino acids β = 20). An undirected construction process is one in which the choice of objects to combine at each assembly step is random (we assume a uniform distribution). We can estimate a bound on the probability of a given object being produced by such a process by estimating a bound on the number of possible product objects with index a_i. Doing this is not entirely straightforward, however, as it entails characterizing the minimal-length construction trajectory for objects to determine their a_i. Nonetheless we argue in the Appendix that there are at least β^a_i objects with index a_i. Thus the probability of a given object being generated by an undirected process is bounded above by p(O_i) ≲ β^−a_i = e^{−a_i ln β}.

Thus, we see how e^−a_i operates, heuristically, as a measure of the probability of an undirected process successfully identifying an object of interest in combinatorial construction space. Therefore, when understood as a measure of SC, assembly allows us to conclude that an undirected construction process is unlikely to explain the origin of an object with high a_i and high n_i.

Nonetheless, there are some noteworthy issues with a_i. For example, a highly ordered crystalline structure that occurs naturally by the laws of physics can readily have a high copy number and large a_i. If the crystal is simply the repeated presence of a single subunit, then a_i ∼ log₂ ℓ where ℓ is a measure of the size of the crystal and can grow quite large. Clearly one need not invoke a directed process to explain the existence of the crystal. Alternative measures of complexity are able to handle such edge cases and this limitation of AT is noted by some of its critics [see Zenil (13)].

Dembski and Ewert (9) emphasize that SC measures must avoid such false positives by carefully considering the space of possible causes when constructing their probability measures. A well-formed SC argument would note that the probability of a repeated crystalline structure is high on the known laws of physics and avoid concluding that a directed process must be at work. Though not articulated in Sharma et al. (1), this nuance is actually described in Figure 4 of Marshall et al. (11), reproduced here in Figure 2, which provides a technique for determining whether to infer a directed process from a given measure of a_i. The authors argue for analyzing the combination of a_i and object size ℓ. Recall that log₂ ℓ ≤ a_i < ℓ. If a_i is ‘too close’ to its lower bound, log₂ ℓ, then objects can be deemed too simple and therefore possibly attainable by an undirected process. Perhaps future iterations of AT will formally include such guards by conjoining all three of (a_i, n_j, ℓ) into A.

Figure 2.

Illustrative rejection regions for a_i and ℓ. Though not referenced in Sharma et al. (1), early work by Marshall et al. (11) (their Figure 4 governed by CC BY 4.0; reproduced here with slight aesthetic modifications) illustrates biosignature interpretations for different observed combinations of a_i and ℓ. The yellow area where a_i is too close to log₂ ℓ is rejected as being too simple – objects in this region can be explained by appeals to random chance and so a directed/biotic selection process cannot be securely inferred. On the other hand, objects in the red region where a_i is too close to ℓ are rejected for being ‘so complex that even living systems might have been unlikely to create them’ Marshall et al. (11); we discuss this in Section ‘Conclusion’. The green Goldilocks zone sits in-between and marks where a directed process can be inferred. Note that while the outer envelope is rigorously defined, these internal subregions are not; they are provided for illustrative purposes only.

Before proceeding, we should note that there are several alternative measures of complexity that predate AT. Perhaps most comparable is the work of Böttcher (14, 15) who develops a local measure of molecular complexity C_m derived from the information content of the degrees of freedom on a per-atom basis. C_m has many advantageous qualities including its computability and its linear nature. Under Böttcher’s schema, the molecular complexity of an ensemble of objects is largely given by the sum of the complexities of each member. a_i by comparison can behave in unintuitive ways when we consider ensembles of objects with overlapping assembly histories (see, for example, the complexity in Section ‘Computing a bound on N (a_i) for proteins’ of the Appendix). Moreover, Böttcher pairs C_m with a measure of the sequence information of biomolecules C_i to develop a model-free schema for understanding biosignatures across a wide spectrum of contexts – much like AT. Indeed, the parallels are striking: both Böttcher and the AT authors apply their measures to the problems of drug discovery, the detection of life in an astrobiological context, the use of experimental methods to estimate complexity [nuclear magnetic resonance (NMR) spectroscopy and mass spectrometry (MS), respectively], and questions of life’s origins. A deeper comparison of the two is warranted.

To further illustrate how assembly operates as a form of SC we shall work through a concrete example and recast the assembly Eq. (3) in terms of the common form equation for CSC (1).

We consider the case of some protein O with assembly index a observed with copy number n ≥ 1 in an ensemble of N ≥ n total proteins. We explore concrete scenarios and estimate values for these quantities later.

The assembly for this protein, in the context of this ensemble, is: (4) A=ean−1N A = {e^a}{{n - 1} \over N}

To recast this example in terms of SC we need a probability measure for how (im)probable it is for O to be discovered by an undirected process. As discussed in Section ‘Assembly index as measure of complexity’ this probability is bounded by p ≲ β^−a where β = 20 is the size of the pool of amino acid building blocks available for proteins. Since e^{a ln β} = β^a, we can express p in terms of assembly: (5) p<n−1ANlnβ p < {\left( {{{n - 1} \over {AN}}} \right)^{\ln \beta }}

Plugging Eq. (5) into Eq. (1), we find (6) SC=−log2rpν>−log2n−1ANlnβrνSC>lnβlog2ANn−1νr1/lnβ \matrix{ {SC = - {{\log }_2}r{p \over \nu } > - {{\log }_2}\left[ {{{\left( {{{n - 1} \over {AN}}} \right)}^{\ln \beta }}{r \over \nu }} \right]} \hfill \cr {SC > \ln \beta {{\log }_2}\left[ {{{AN} \over {n - 1}}{{\left( {{\nu \over r}} \right)}^{1/\ln \beta }}} \right]} \hfill \cr }

To ensure this is a measure of CSC we must choose a specification function ν and a normalization r such that νχ=∑x∈χνx<r \nu \left( \chi \right) = \sum\nolimits_{x \in \chi } {\nu \left( x \right) < r} . Recall that χ is the space of observations. Since AT’s specification is a function of copy number we shall take χ = {n|1 ≤ n ≤ N} – i.e. we consider the range of outcomes for observing up to N copies of O in the ensemble.

If we set (7) ν=n−1Nlnβ \nu = {\left( {{{n - 1} \over N}} \right)^{{\rm{ln\;}}\beta }} and (8) r=N r = N then it follows that (9) νχ=∑n=1Nn−1Nlnβ<N=r \nu \left( \chi \right) = \sum\limits_{n = 1}^N {{{\left( {{{n - 1} \over N}} \right)}^{\ln \beta }} < N = r} because ln β > 0 and n−1N<1 {{n - 1} \over N} < 1 . Thus these choices of ν and r give us ν(χ) < r.

Plugging Eqs (7) and (8) into Eq. (6), we find (10) SC>lnβlog2AN1/lnβ SC > \ln \beta {\log _2}{A \over {{N^{1/\ln \beta }}}} and we see that this normalized form of A serves as a lower bound on a CSC measure. Because of the conservation of CSC it follows that the probability of observing O is: (11) Pr≤2−SC≤NAlnβ Pr \le {2^{ - SC}} \le {N \over {{A^{\ln \beta }}}}

To expose the parameters in our model we can expand A: (12) Pr≤N1+lnββa(n−1)lnβ Pr \le {{{N^{1 + \ln \beta }}} \over {{\beta ^a}{{(n - 1)}^{\ln \beta }}}}

These probability equations reflect the features and limitations of assembly we have been discussing. We see that proteins with larger assembly index a and copy number n are, indeed, less probable to attain by undirected processes. However, the assembly (and the resulting probability) is poorly normalized and can be modified substantially by considering different sized ensembles N.

To put some numbers to this we note that proteins routinely have ℓ ≫ 100 amino acids. Since a < ℓ, we will assume an assembly index of a = 100 to probe proteins of moderate complexity. We consider two scenarios.

First, we consider observing n ∼ 10 copies of O within the proteome of a single E. coli cell. Such a cell typically has N ∼ 10⁶ proteins (16). This yields A ∼ 10³⁸ and Pr ≲ 10⁻¹⁰⁹. This low probability correctly rules out the chance occurrence of n copies of O within E. coli: O is only present in multiple copies because its amino acid sequence is specified (i.e. selected) by the DNA code for O and translated by the machinery in the cell.

Second, we consider a scenario more relevant to evolutionary timescales. Now O is a protein present in the majority of bacteria. There are currently ∼10³⁰ bacteria on earth (17) so we assume at least n ∼ 10³⁰ copies of O exist. We draw these n copies from the reservoir of all bacterial proteins that have existed over the course of evolution. This number is difficult to estimate but we’ll assume ∼10⁴⁰ bacteria have ever existed, each with ∼10⁶ proteins giving us N ∼ 10⁴⁶. With these numbers A ∼ 10²⁶ and Pr ≲ 10⁻³⁶. Again, we see a low probability correctly rule out undirected processes. Some sort of selection mechanism must have been at work to generate O.

Of course these numbers are somewhat contrived – the results are sensitive to the choice of ensemble, N, and the choice of a = 100 is somewhat arbitrary.

In particular, if a = 100 is too high then our probability bound is too low and we are at risk of incorrectly ruling out the efficacy of an undirected process. However, there is reason to believe that a is not substantially lower than the length of the protein in question and there are many proteins with length ℓ ≫ 100 – several lines of evidence point in this direction, particularly the incompressibility and high entropy content of proteins (18, 19).

In fact, the study of protein compressibility is closely related to the notion of assembly index. Some state-of-the-art approaches have been found that can compress protein sequences substantially: by looking for long-range correlations at the genome-scale of a species Adjeroh and Nan (20) are able to compress protein coding regions by 50%–80% depending on the species. While impressive from a data-storage point of view this does little on its own to paint a convincing picture that proteins are, in fact, broadly constructed recursively from fundamental building blocks at a higher level of abstraction than individual amino acids. Perhaps more sophisticated complexity analyses such as Block Decomposition Method (BDM) (21) will show that proteins are recursively constructed from simpler transformations of building blocks – this would provide a tighter bound on a_i. Or perhaps the presence of minor or synonymous mutations is confounding compression algorithms, and so analyzing three-dimensional protein structure instead of sequences will be necessary to improve compressibility (22). However, absent such an analysis our current knowledge of the field indicates that a_i will likely scale more-or-less linearly with ℓ and that the scaling factor will be modest (i.e. ∼O(0.1)).

Thus our estimates, though contrived, may nonetheless serve as plausible upper limits. This illustrates how assembly operates as a measure of SC to argue that an undirected process is unlikely to yield high-assembly objects. Indeed, in the case of proteins it is unlikely that random sequences will yield novel complex proteins [see, e.g., Tian and Best (23)]; instead a more complex search process is typically invoked to explain the discovery of novel proteins (24, 25). Assembly correctly points to the need for such a process.

The authors of AT refer to such mechanisms as selection: processes that can winnow combinatorial space along an object’s construction trajectory. Since their goal is to establish a broad framework for the construction of complex objects, AT’s authors choose to focus on the generic properties of selection by exploring abstract models in lieu of concrete physical systems. For example, to explore the dynamics of object discovery under selection, Sharma et al. (1) present a toy model represented by the equation: (13) dNa+1dt=kd(Na)α {{d{N_{a + 1}}} \over {dt}} = {k_d}{({N_a})^\alpha } where N_a is the number of unique objects at assembly index a, k_d represents the rate of discovery, and α represents the degree of selection. When α = 1 there is no selection (all objects at prior assembly index are available to construct subsequent objects) and the combinatorial space of possibilities explodes exponentially. However, if α < 1 then a selection process is at work reducing the number of objects from which to generate future objects. In principle, this alleviates the combinatorial explosion at high a and allows a selection process to construct high a_i objects within limited available resources.

But what does this model describe in practice? Solutions to Eq. (13) are provided in the supplementary information of Sharma et al. (1) and take the form N_a ∝ t^x where x is a function of a and α and always satisfies x ≥ 1 (its detailed behavior is unimportant here). This means the number of unique objects at assembly index a grows unbounded with time, even for small a. This is an unphysical result: the proposed solutions do not account for the combinatorial reality that even for large numbers of building blocks the possible number of unique objects at a given assembly index must be finite. Thus, it is unclear how to apply these solutions to realistic physical systems. Other exemplar models in Sharma et al. (1) are similarly limited. For example, the authors explore the construction of ‘linear chains defined as integers which are equivalent to linear polymers constructed from a single monomeric unit’. For such linear monomers the only distinguishing feature is their length; however, objects of identical length that have different construction histories are treated as distinct objects (e.g. there are two chains of length 4:1→1+12→2+24 4:1\buildrel {1 + 1} \over \longrightarrow 2\buildrel {2 + 2} \over \longrightarrow 4 and 1→1+12→2+13→3+14 1\buildrel {1 + 1} \over \longrightarrow 2\buildrel {2 + 1} \over \longrightarrow 3\buildrel {3 + 1} \over \longrightarrow 4 ).This is a confusing choice that obscures the relationship between actual objects (e.g. ‘4’) and the various trajectories that could have constructed the objects.

These linear monomers are subsequently used to explore the effects of directed selection vs undirected construction. A simulation is performed in which an undirected process constructs chains by drawing two chains from the assembly pool at random, attaching them together, noting the product, then adding it to the pool. This is compared to a directed process in which, at each step, the longest chain in the pool is chosen and combined with a randomly chosen chain from the pool. The authors then note that the products of the directed process are substantially longer than the products of the undirected process. Since assembly index simply scales with the log of length for these monomeric chains, the directed process therefore generates more complex objects. But both the model and this result are trivial in light of the sorts of real-life physical problems that selection must be invoked to solve.

These various toy models illustrate a key limitation of selection as construed in AT. Selection is conceived to operate via object concatenation and is parameterized with α as such. However, the real-life construction of objects (whether by the hands of human workers or the blind forces of evolution) does not only operate via concatenation. Consider, again, the example of proteins: it is well known that enzymes exhibiting promiscuous behavior can be optimized to catalyze related molecules via a process of stochastic exploration of navigable fitness landscapes via mutation (30). Modeling such a change-by-modification process is awkward in AT given its change-by-concatenation perspective of selection.

Finally, Sharma et al. (1) further characterize the selection process by stipulating that the timescale for producing copies, τ_P, and the timescale for discovering new objects, τ_D, must be commensurate. If τ_P ≫ τ_D then many unique objects with low copy number will be discovered and proliferate (these have low A); and if τ_D ≫ τ_P then there will be many copies of objects with low assembly index (these will also have low A). Expressing these constraints (α < 1, τ_D ∼ τ_P) provides helpful language, however, neither these constraints nor the simple models explored in Sharma et al. (1) describe a concrete set of mechanisms for implementing an actual viable material selection process.

Indeed for a material selection process to actually exist, function, and succeed a few things need to be true:

• There must be a mechanism for creating copies of objects such that earlier elements in an assembly trajectory are available for combination into future elements.

• There must be a mechanism for generating variations of objects – envisioned in AT as recombining prior elements in new ways.

• There must be a mechanism for filtering the objects at each step in the trajectory in a way that collapses combinatorial space.

Readers will recognize these as the three Lewontin conditions for evolution to hold (31). These are necessary but not sufficient. A viable selection process must also be shown to actually succeed in discovering high-a_i objects within the available probabilistic resources (time, number of trials, material, etc.) and in spite of any confounding factors (e.g. decay rates, polluting cross-linkages, etc.). AT does not attempt to concretely address any of these thorny issues for any actual physical system of interest – at best, AT narrowly focuses on the degree of challenge posed solely by the information dynamics of the system in question.

Nonetheless, AT’s senior authors argue that a material selection process must exist by dint of the existence of objects with high assembly. Lee Croning makes precisely this argument in the context of abiogenesis, arguing that 'existence is the proof' (32). Such a conviction, however, neither demonstrates that a material process capable of the necessary selection exists nor does it shed light on how it might operate. In fact, Marshall et al. (26) explicitly casts doubt, noting that high-a_i objects are ‘those so complex they require a vast amount of encoded information to produce their structures, which means that their abiotic formation is very unlikely’.

More pointedly, Sharma et al. (1) note that some known prebiotic synthesis reactions, ‘such as the formose reaction, end up producing tar, which is composed of a large number of molecules with too low a copy number to be individually identifiable’. This is one of many paradoxes associated with origin of life research (33) and it is not clear how a prebiotic selection process might emerge that is capable of selecting subcomponents of the combinatorially complex tar and moving the ensemble towards a high-copy number target such as life (34, 35). Indeed, it is increasingly apparent that origin of life experiments that manage to make progress typically do so only because of human intervention (36).

This brings us to the crux of the issue. In pointing to selection, AT draws attention to a core problem underpinning the formation of complex objects: that of information. Consider the toy model in Eq. (13), the parameterization of selection with a simple scalar α is, on its own, somewhat misleading. The challenge to solve is not merely identifying a low-α process capable of selecting arbitrary subset, Naα N_a^\alpha of objects. The challenge is to identify a directed process that can select particular subsets of objects that result in the eventual construction of relevant high-a_i objects. This is what it means for a directed selection process to tame the combinatorial explosion of possibilities: there must be some non-random bias that drives the exclusion of some possibilities and the inclusion of others.

Indeed we can estimate the information content of such a process (37). As described in Shannon (3), information can be computed in terms of probability via: (14) I=−log2P I = - {\rm{lo}}{{\rm{g}}_2}P so at each step in which a viable subset Naα N_a^\alpha is selected the selection process must utilize (15) I=−log2NaαNa=1−αlog2Na I = - {\rm{lo}}{{\rm{g}}_2}{{N_a^\alpha } \over {{N_a}}} = \left( {1 - \alpha } \right){\log _2}{N_a} bits of information to propagate the system forward. Where does this information come from? And how is it marshaled by a viable physical process to do the relevant work? Assembly theory does not (indeed, cannot) answer these questions; it merely helps us ask them in a more quantitative frame.

It is striking to compare this conclusion with what senior authors Cronin and Walker have argued in the past. In Cronin and Walker (38), both advocate for the need to ‘[focus] on information’ to make headway in the origin-of-life field. And in Walker and Davies (39) and Walker and Davies (40), Walker eloquently argues that the origin-of-life represents a phase shift in which information transitions from playing a passive role to taking on a more active role as a top-down causal force [drawing on Ellis (41)]. Walker describe this as ‘the hard problem’ of life. However, AT, as an adjudicator of undirected processes, can at best point us to where this ‘hard problem’ is at work; it does not make progress explaining how it works.

This can be seen in the shape of Cronin and Walker’s latest research program. Walker (4) discusses Cronin’s development of a ‘chemputer’ (42) – an impressive ‘grad-student-in-a-box’ programmatically driven chemical synthesis machine. According to Walker (4):

Lee right now has several chemputers running in his lab with primordial-soup chemistries in them, which he is evolving by programmable iteration over possible environments… Because we have assembly theory, we can quantify the assembly of the starting molecules and track how assembled things become over time to look for evidence of the emergence of evolution and selection within the boundary conditions of the experiment… Because the chemputer is automated, we hope to scale this up to search large volumes of chemical space all at once… [with] a cloud of potentially millions of chemical reactors.

Notably, while AT can help define success criteria for these experiments, it offers little insight into how to conduct them. Instead, the experimental approach most resembles a brute force search [albeit one in which each search attempt is heavily information laden – see Šiaučiulis et al. (43) for a sense of how much information is encoded in the protocols]. Ironically, AT itself casts doubt that such a random-sampling approach will succeed; indeed we can hazard a prediction: To the extent that the chemputer cluster can be directed towards finding a solution (perhaps by an AI-driven algorithm iterating on protocols to eventually generate high assembly objects) it will be because of a target-oriented design – an externally imposed selection – that has no material analogue in the prebiotic context of the early earth.