Selection within assembly spaces

Historical Evolution (HE) lets us understand how selection and historical contingency impose constraints on what can be made in the future.

‘Aiming to detect selection’ means a process similar to natural selection.

We do not, however, model functional differences that selection might act on. Instead, we account only for the specificity of selection—that some living things are more likely to be used to make new things and some are less likely.

The only functionality we want to detect or describe is in the memory of the process to generate the living thing, with examples including a metabolic reaction network or a genome.

This allows the 3 Lewontin conditions for evolution to hold.

A key feature of HE is that they are combinatorial, with living things combined at every step.

Combinatorial spaces do not play a prominent role in current physics, because their living things are modelled as point particles and not as combinatorial living things.

More living things exist in HE than can be built in finite time with finite resources because the space of possibilities grows super-exponentially with the assembly index.

To tame this explosive growth in Assembly Theory (AT), historical contingency is intrinsic with the space built compositionally. It means that items are combined recursively*. This accounts for hierarchical modularity. This substantially constrains the number of possible living things.

It is the combination of this compositionality with combinatorics that allows us to describe selection.

To produce a Historical Evolution (HE), an observed living thing is broken down recursively to generate a set of elementary building units.

• These units can be used to then recursively construct the assembly pathways of the original living thing(s) to build the Observed Evolution (OE)

OE captures all histories for the construction of the observed living things from elementary building blocks.

living things in AT are compositional – they contain information about the larger space of possible living things from which they were selected.

To see how, we first build an HE from the same building blocks in OE, which include all possible pathways for evolving any living thing composed of the same set elementary building blocks as our target living thing.

The Fantasy Evolutionary Universe

The constructed history is the evolutionary universe (EU)*.

In the EU, all living things are possible with no rules. This yields a combinatorial explosion and with double exponential growth in the number of living things. This is characteristic of evolutionary big bangs.

This double exponential growth is unphysical because the physical processes place restrictions on what is possible. In the case of nonliving molecules, an example is how quantum mechanics constrains the numbers of bonds per atom.

The EU also has no concept of directionality in time, as there is no ordering to construction processes.

In the EU, everything can exist. living things can be constructed independently of what has existed in the past and of resource or time constraints. This is not what we observe in the real universe.

For most systems of interest, including in molecular assembly spaces, there are more molecules in the EU than the amount of matter in our observable universe.

There is no way to computationally build and exhaust the entire history, even for living things with relatively low HE indices.

• For larger living things, such as proteins, this can be truly gigantic.

Taking account of memory and resource limitation:

• severely restricts the size of the history of what can be built
• allows higher-assembly living things to be built before exhausting resources constructing all the possible lower-evolution living things.

AT can account for selection precisely because of the historical contingency in the recursive construction of living things along evolutionary paths.

Possible evolutions (PE) is the history of physically possible living things which can be generated via the combinatorial expansion of all the known physical rules of living-thing construction and allowing all rules to be available at every step to every living thing.

This can be described by a dynamical model representing undirected forward dynamics in AT.

When a living thing with HE index combines with its own history, its HE increases by 1.

If the resulting living thing can be made by means of other, shorter path(s), its HE index will be smaller than .. or even …

Another assumption behind the dynamical model of undirected dynamics is a microscopically driven stochastic rule that uses existing living things uniformly – the probability of choosing a living thing with HE index to be combined with any other living thing is proportional to the number of living things with HE index.

Within possible evolutions, evolution contingent (EC) describes the possible history of living things where history, and selection on that history, matter.

Historical contingency is introduced by assuming that only the knowledge or constraints built on a given path can be used in the future, or with different paths interacting in cases in which selected living things that had not interacted previously now interact.

We define the probability of a living thing being selected with HE index as …

where … is the number of living things with HE index.

Here, … parameterizes the degree of selection: for … all living things that have been assembled in the past are available for reuse, and for .., only a subset (that grows non-linearly with HE index) are available for reuse, indicating that selection has occurred.

This leads to the growth dynamics:

[equation]

where … represents the rate of discovery (expansion rate) of new living things.

For … there is historical dependence without selection.

We build evolutionary paths by combining 2 randomly chosen living things from the evolutionary pool.

If a new living thing is formed, it is added back into the pool.

Here, we are building random living things. But these are fundamentally different from random combinatorial living things because the randomness we implement is distributed across the recursive construction steps leading to an living thing.

The case of [equation], in which there is historical dependence but no selection, defines the boundary of possible evolution.

Within possible evolution, the evolution contingent (AC) is the history of possible configurations of living things where [equation], that is, where selection is possible, and the living things found in the evolution are controlled by a path-dependency contingent on each living thing that has already been built.

The growth of the evolution contingent is much slower than exponential.

Not all possible paths are explored equally. Instead, the dynamics are channelled by constraints imposed by the selectivity emerging along specific paths.

A signature of selection in HEs is a slower-than-exponential growth of the number of unique living things.

To show this, we use a simple phenomenological model* of linear nonliving polymers to demonstrate how assembly differentiates cases when selection happens.

Starting with a single monomer in the assembly pool, the undirected exploration process combines 2 randomly selected polymers and adds them back to the evolutionary pool.

In the case of directed exploration with selection, the polymer that has been created most recently is selected* to join a randomly selected polymer from the assembly pool.

For both directed and undirected exploration, this process was iterated up to 104 steps and repeated 25 times. For each observed polymer in the evolutionary pool, the shortest pathway was generated.

For each run, the HE of multiple coexisting polymers, their joint HE, was approximated by the union of the shortest pathways of all observed polymers. An example of joint HE in an undirected exploration up to 30 steps is shown

Comparison between the explored joint HE in undirected and directed exploration up to 100 steps is shown

To quantify the degree of exploration at a given evolutionary step, we calculated the exploration ratio, defined by the ratio of observed nodes to total number of nodes present in the joint HE.

The mean maximum HE index was estimated by calculating the HE index of the mean value of the longest observed polymeric chains over 25 runs.

Comparing the directed process to the undirected exploration illustrates a central principle: the signal of selection is simply a lower exploration ratio and higher complexity (as defined by the maximum HE index).

The observation of a lower exploration ratio in the directed process than in the undirected process is the evidence of the presence of selectivity in the combination process between the polymers existing in the evolutionary pool.

The process representing sorting and selecting chains within the assembly pool represents an outcome of a physical process leading to selection.

We conjecture that, the ‘more evolved’ an ensemble of nonliving things, the more selection is required for it to come into existence.

The historical contingency in AT means that evolution dynamics explores higher-evolution living things before exhausting all lower-evolution living things, leading to a vast separation in scales separating the number of living things that could have been explored versus those that are actually constructed following a particular path.

For example, proteins built both from D and L amino acids and their pathways are part of possible evolution. But, within an evolutionary contingent trajectory, only proteins constructed out of L amino acids might be present. This is because of early selection events.

This early symmetry breaking along historically contingent paths is a fundamental property of all evolutionary processes.

It introduces an ’evolution time’ that ticks at each living thing being made. Evolution physics includes an explicit arrow of time intrinsic to the structure of living things.