Lead partner: ALife Group, Dublin City University
Dynamics of Informazyme Systems3
- A fixed size flow reactor containing a population of informazyme molecules. The reactor is buffered to maintain ready availability of free monomers. Dilution outflow is matched to the reaction rate to ensure a fixed maximum number of molecules (informazymes).
- In this first instance, the only enzymatic reaction considered is replication of a (recognised) template. Thus, an informazyme can act as a replicase for all, and only, those molecules which it can bind to, as determined by its “active” or “folded” conformation.
- We assume identity (as opposed to complementary) replication. This is analogous to, say, DNA (double-stranded) rather than RNA (single-stranded) replication.4
- We assume the reactor is well-stirred, i.e., every molecule has an equal rate of collision with every other molecule; and that, in any specific collision, each molecule has an equal rate of participating as replicase or template.
- Recognition is considered to be all-or-nothing. Once recognition takes place, replication rate is taken to be the same for any informazyme acting on any recognised template (regardless even of template length).
- Replication is error-prone—i.e., the produced molecule will not always match the template exactly in sequence. However, in general, we shall first analyse the case where such mutation is neglected; and then attempt to consider the effect of mutation as a perturbation of this “underlying”, mutation-free, dynamic.
- If we say this is a self-replicase; otherwise it is self-inert.
- If both and we will say that and are mutualists relative to each other.
- Where but we will say that, relative to each other, is a host and a parasite.
- A mutualist, host or parasite may be said to be facultative if it is also a self-replicase; otherwise it is obligate.
- If we may also say that is inert to replication by (and vice versa); if both and we say they are mutually inert.
- : The species is self-inert. Given that this is the only species present, then there are no reactions, and the system as a whole is completely inert (, ).
- : The species is a self-replicase. Given that this is the only species present, then although there is constant turnover at the maximum rate (), in the absence of mutation this just produces identical replacement molecules so we still have . (The concept of mutation necessarily requires , so it is not meaningful in systems restricted to .)
One molecule is a self-replicase and the other is completely inert. Accordingly, neglecting stochastic effects at very low absolute numbers of molecules, such a self-replicase will essentially deterministically invade and displace the inert species, even if starting from arbitrarily low initial concentration. This formally follows from the fact, noted above and in figure 1, that (where is taken as labelling the self-replicase). The growth of the self-replicase follows a classic “S-shaped” (though not technically logistic) curve, with a sharp, positive feedback, “switching” dynamic, as shown in figure 2. Note that, as the self-replicase takes over the reactor, the total replication rate (reactor flow rate, ) also rises to the maximum possible ().
In this case, even with relatively high mutation rates (but assuming that all mutants are still individually and mutually inert), the core dynamic of growth of the self-replicase to the maximum sustainable concentration is very robust; the only modification being that this maximum concentration is then somewhat less than due to the ongoing mutational load. To analyse this, let still denote the concentration of the self-replicase species. If we let the per-molecule mutation rate be denoted by , then the dynamics is represented by:
Neither species is a self-replicase, but one can act as a replicase for the other; so the latter is an obligate parasite of the former (which is classified as an “obligate host” in our terminology). The functional replicase () is then effectively a finite initial resource which is irreversibly “used up” and decays to zero concentration, given the initial presence of even an arbitrarily small concentration of the parasite. The overall replication rate may initially rise (as more “template” molecules, become available for replication), but then as the replicase concentration declines, the replication rate necessarily peaks and then enters a monotonic decline. As shown in figure 4, over a moderately extended period of decay, the reactor asymptotically approaches a state containing only species , and thus ultimately becomes completely inert (effectively as already discussed above under the case of class 0 interactions).
One species is a self-replicase and also a parasite of the other species; the latter is self-inert, and thus acts as an obligate host. As with class 2, the self-replicase has a positive rate of growth at all concentrations (see figure 5) and will essentially deterministically invade and displace the second species, even if starting from arbitrarily low initial concentration. The difference here is that, even if the self-replicase itself is initially in low concentration, it will still achieve an exponential replication rate (because it receives replication support from its obligate host, which is, by assumption here, in high concentration). Thus the displacement can be initiated more rapidly than for class 2 (see figure 6); in this specific example, from the same starting concentration of 0.05, the takeover is complete by time for class 5, compared to for class 2. Formally, the growth law for the class 5 case is strictly logistic.
The overall rate of replication for this case is simply , as it was in class 3; but as now grows rapidly to take over the reactor, the overall replication rate similarly rapidly grows to the maximum possible and there is then continuing turnover at this rate. As with class 2, even with relatively high mutation rates (but assuming that all mutants are still obligate hosts of the self-replicase), the core dynamic of growth of the self-replicase to the maximum sustainable concentration is very robust. The differential equation taking account of this (restricted) form of mutation is:
Both species are self-replicases; but one () is a (facultative) parasite of the other (), which acts as a (facultative) host. As with class 3, we have , so will inevitably decline. However, in this case, instead of the reactor asymptotically approaching a completely inert state, it is simply taken over by the other self-replicase species, , which will then continue to replicate at the maximum rate. This is a properly “selective”, quasi-deterministic, displacement of one self-replicase by another, because, at all relative concentrations, the latter achieves a higher replication rate. The displacement will take place even if initially has essentially arbitrarily low concentration. Figure 8 shows an example trajectory for . While there is an initial period of slow growth, the parasite does inevitably achieve a “critical” concentration (say at in this example) after which the displacement is then completed very rapidly.
Note that although we describe this as properly “selective”, or “Darwinian”, displacement, there is no intrinsic fitness difference between the two species here. That is, if examined in isolation from each other, both species show exactly the same dynamics. This is by deliberate assumption in the current analysis. It is only when incubated together that the asymmetric “host-parasite” interaction between them means that one is consistently favoured over the other, and can successfully invade from rarity. This case gives rise to a very distinctive, evolutionary dynamic at the molecular level. In essence, if there is continuing generation of mutants which are class 6 relative to a currently dominant self-replicase, we can predict the possibility of a potentially indefinitely extended series of displacement events. At least, this will be so, provided the rate at which these mutations arise is not too rapid, so that there is time for each displacement event to complete before another one is initiated. The details of this evolutionary dynamic will be determined by the pattern of mutational connectivity. As will be discussed subsequently, this case is of key significance to the investigation of protocell evolution (at least within the simplified model being considered here). This is because this situation allows a reactor to “switch” from being dominated by one informazyme species to being dominated by a different informazyme species. More generally, when the informazymes are contained within a protocell instead of a static reactor, the dominant informazyme species will be both heritable (at the level of protocell reproduction), and potentially coupled to some significant protocell level trait. A Class 6 “molecular evolution” displacement may “propagate upwards” in organisational terms, to manifest as a single “mutation event” at the protocell level, which can then be the target for protocell level selection. Indeed, this is the only case, under the specific protocell modelling framework analysed here, where the molecular level dynamics can reliably give rise to such a protocell level mutational event.9
One species is a self-replicase; the other is an obligate (self-inert) parasite of it. However, unlike class 4, the parasite also functions as a host for the self-replicase. The means that the (parasitic) replication service provided by the self-replicase to the self-inert species is offset by the (parasitic) replication service provided by the self-inert species back to the self-replicase. As with class 2 and class 5, the self-replicase will essentially deterministically invade and displace the other species, even if starting from arbitrarily low initial concentration. The duration of the transition is essentially intermediate between classes 2 and 5, as shown in figure 10 (compare with figures 2 and 6, all based on the same example initial state of ). Also as with classes 2 and 5, this behaviour generalises to relatively high mutation rates (assuming that the mutants are, similarly, both obligate parasites and obligate hosts of the self-replicase). Neglecting back-mutations (from the mutants back to species ) the differential equation becomes:
Neither species is a self-replicase, but each can act as a replicase (obligate host) for the other (obligate parasite). Setting , and neglecting the case of , there is a single fixed point at:
Both molecular species are independent self-replicases. In a certain sense this is exactly complementary to class 8. The expression for is precisely the negation of that in class 8. Accordingly, the dynamics has exactly the same fixed points: the two states of , where one or the other species is no longer present, and the state . However, the latter state is now unstable rather than stable. The effect is that any perturbation means the system rapidly collapses into a state where one species displaces the other. Figure 14 shows an example trajectory initialised from . In effect, there is positive frequency dependent selection, so that whichever species first achieves a higher concentration than the other benefits from a positive feedback effect which further amplifies its concentration until it takes over the complete reactor. The effect is well known, and is termed the survival of the common (Szathmáry and Maynard Smith, 1997). It is characteristic of any system of replicators undergoing hyperbolic rather than exponential growth.
This result generalises directly to the species case. There is an unstable fixed point with ; and stable fixed points with a single and all other concentrations equal to . That is, whichever species chances to achieve a greater concentration than the others will benefit from the positive feedback and rapidly displace all other species. In the case of continuing generation of class 9 mutants (at rate V per replication of the dominant species) there is still no possibility for any of these to invade the established dominant self-replicase. Instead, as with classes 2, 5 and 7, there will be an equilibrium mutant concentration. The equilibrium analysis is more complex is this case, and will not be considered in detail here. However, we can say that the load will be somewhat greater than class 2 or 5 (as the mutants can, in principle, achieve some limited degree of amplification through self-replication) but less than class 7 (as the degree of amplification will be much smaller than would be the case if supported by the dominant species); i.e., the equilibrium mutant concentration will be between a minimum of and a maximum of (but generally much closer to the minimum).
- Class 0, 3, 8: Not applicable (by definition, these classes arise only when neither species is a self-replicase).
- Class 2, 5, 7, 9: Such mutants are actively selected against relative to the established dominant. Continuing mutation will mean an equilibrium load of such mutants, at a concentration no higher than (where is the total rate of generating mutants of these classes, per replication of the dominant species). Prima facie then, these have no negative effect on the operation of the informazyme inheritance mechanism; and arguably have the positive benefit of continuously sampling the space of molecular mutants. However, it should also be noted that in focussing on the pairwise interactions between these mutants and the dominant species, we are neglecting the possible impact of their interactions among themselves. This is a significant limitation of the analysis attempted here.
- Class 6: Such a mutant will be actively selected for relative to the established dominant species, and will displace it, becoming a new dominant in turn. Indeed, the key significance of the “survival of the common” (class 9) dynamic is that, of all the dynamic behaviours analysed here, the only one that allows one established self-replicase to be reliably displaced by another is the class 6 facultative parasite. Provided the expected time between such mutations (given some average protocell size) is small compared to the protocell gestation time, this provides an effective molecular mechanism for protocell level mutation.
- Class 1: These are selectively neutral at the molecular level. Accordingly, if such mutations are possible, it is expected that the originally dominant species will progressively diversify across a family of mutually class 1 species which can stably co-exist. Prima facie, all of these might be considered as members of a single “quasi-species” which is still (collectively) self-replicating at a high rate, and is therefore still capable of being effectively transmitted to offspring protocells. However, this conclusion assumes that the number of distinct species constituting the quasi-species is not too large (otherwise there may not be enough molecules for each to be represented), and that they are all class 1 relative to each other (as well as to the original “master” species). There is no particular reason for these assumptions to hold. Accordingly, in the face of even modest rates of class 1 mutation, the informazyme population may degenerate into a relatively chaotic mix of many dissimilar species, each present in small absolute numbers, and with impaired overall reaction rate. Such a population would no longer serve to transmit coherent “information” to offspring protocells, i.e., it could not function as an effective inheritance mechanism at the protocell level.
- Class 4: This is similar to class 1, in that again these are selectively neutral at the molecular level, and the originally dominant species will progressively diversify across these. However, a key difference from class 1 is that in this case the overall replication rate will unconditionally decrease as the concentration of class 4 mutants increases.
- Supports self-replicase activity.
- Supports emergence of class 6 molecular mutants.
- May permit class 2, 5, 7, 9 molecular mutants.
- Either prevents class 1 and 4 molecular mutants or ensures that there is some effective mechanism for controlling their impact.
4The role of complementarity in early molecular evolution is a complex one (Szathmáry and Maynard Smith, 1997). However, for the purposes of the present study, this will not be a focus of investigation.
5In the simulation experiments described later, this normalisation of reaction rates will correspond to scaling time by dividing the number of bi-molecular collisions by the total number of molecules present.
6We also conjecture, though without any detailed discussion, that this repertoire of core behaviours represents a useful “idealisation” even of the much more general case where the vary continuously i.e., .
7Where a class consists of two interaction matrices which are equivalent under relabelling, we shall adopt the convention that all labels used in the discussion refer to the first given version of the matrix.
8This behaviour is analogous to the default selectional stabilisation of equal sex ratio in sexual species.
9This is, of course, in marked contrast to modern cells, with chromosomal organisation, strictly regulated copies of the “informational” molecules in a single cell (ploidy), and the use of a translational subsystem which de-couples information and function. In that case, almost arbitrary molecular level mutations can result in cell-level mutations/phenotypic effects. But synthesising artificial cells with that level of function will be much harder than the protocells considered here.
10It is known that, in principle, hypercycle organisation can be stabilised in a spatially distributed reactor, where finite diffusion rates allow spatial inhomogeneity to be generated and maintained (e.g., Hogeweg and Takeuchi, 2003). However, such spatial structure at the molecular level is deliberately eschewed in the current investigation, to allow a focus on protocells as the sole mechanism for such “spatial” containment or inhomogeneity.