Dynamics of Informazyme Systems

Evolution of Protocell-embedded Molecular Computation

Lead partner: ALife Group, Dublin City University

Dynamics of Informazyme Systems³

Assumptions

In this section we present a theoretical analysis of some generic dynamics which are possible in a limited class of informazyme systems. The analysis is based on the following assumptions:

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.

These assumptions all represent significant simplifications of any practical replicator chemistry; so that the dynamic possibilities of real replicator systems would presumably be wider than discussed here. Nonetheless, as will be demonstrated, this simplified replicator framework is sufficient to support the core phenomena of heredity and evolution in protocell populations; and has the key advantages of being tractable for useful theoretical analysis, and for simulation over evolutionary timescales with practical computational resources. Note in particular that this approach deliberately preserves what has already been mentioned as a key evolutionary constraint of synthetic proto-cells, as compared to modern living cells; namely that they lack a translational sub-system that can allow de-coupling of the space of genetic (informational) variation from the space of phenotypic (enzymatic) variation. The immediate challenge is to demonstrate that some meaningful protocell level evolution can be demonstrated despite this significantly impoverished hereditary mechanism. A longer term goal (beyond the scope of the immediate investigation reported here) is to explore precisely the evolutionary emergence of translation.

General Dynamic Equation

Let $c_{ij}$ denote the rate with which species

, acting as replicase, will replicate species

. Given our assumption of all-or-nothing recognition, and equal replication rates for all cases where recognition takes place, we can say, without loss of generality, that every $c_{ij}$ (“replication co-efficient”) will have a normalised value of either

(no recognition) or

(recognition).⁵ Under a continuous approximation, we will denote the concentration of any species

in the reactor by $x_i \in [0,1]$ (with $\displaystyle \sum _i x_i = 1$ ). Applying mass-action kinetics, the replication rate of each species will be:

∑ r(xi) = xixjcji j

The total replication rate for the reactor will be:

∑ ∑ ∑ R = p(xk) = xkxjcjk k k j

Under the condition of fixed molecular population, this must equal the total dilution outflow rate, and the dilution rate for each individual species will be:

d(xi) = xiR = x ∑ ∑ x x c i k j k j jk

Neglecting mutation (in the first instance) the dynamics of the reactor is then governed by the following system of ordinary differential equations (ODE) in each x_i

˙xi = r(xi∑) - d(xi) ∑ ∑ = xi( xjcji) - xi( xkxjcjk) j k j

Under the conditions specified, we are therefore dealing with various cases of a catalytic reaction network, in the sense of Stadler et al. (1993). To actually examine the concrete dynamics of any given reactor, we must specify the total number of species,

, and the corresponding $n \times n$ matrix of replication co-efficients $c_{ij}$ . Thus, even with the strong simplification of $c_{ij} \in \{0, 1\}$ , for any given

there are $2^{n^2}$ distinct possible systems. Exhaustive analysis rapidly becomes infeasible. We shall limit our detailed analysis to considering all possible cases for $n\leq 2$ . This provides a repertoire of “core” behaviours. We shall augment this with more qualitative discussions of how these core behaviours may be generalised or combined for systems with n>2

and/or perturbed by replication error (molecular level mutation).⁶

Terminology

Although we are discussing molecular level evolution, it will be convenient to use the following, ecologically based, terminology. Consider two distinct molecular species i, j

where $i \ne j$ :

If $c_{ii}=1$ we say this is a self-replicase; otherwise it is self-inert.
If both $c_{ij} = 1$ and $c_{ji} = 1$ we will say that and are mutualists relative to each other.
Where $c_{ij}=1$ but $c_{ji}=0$ 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 $c_{ji} = 0$ we may also say that is inert to replication by (and vice versa); if both $c_{ji} = 0$ and $c_{ij}=0$ we say they are mutually inert.

“Self-”systems

For “self”-systems we have n=1

; that is, there is just one species with concentration x_1

, one co-efficient, $c_{11}$ , and two possible systems:

$c_{11}=0$ : 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 ( $\dot {x}_1=0$ , ).
$c_{11}=1$ : 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 $\dot {x}_1=0$ . (The concept of mutation necessarily requires , so it is not meaningful in systems restricted to .)

Pairwise Systems

For pairwise systems we have n=2

; that is, there are two species with concentrations x_1

and

; and their interactions are represented by the matrix:

$\left (\begin {array}{cc} c_{11} & c_{12}\\ c_{21} & c_{22} \end {array}\right )$

As $c_{ij} \in \{0, 1\}$ , there are exactly 2^4 = 16

possible such pairwise interaction matrices. However, some of these are equivalent under a relabelling of x_1

and

; as a result there are just 10 dynamically distinct classes of pairwise system. All of these will be considered in turn. In each case, we shall initially neglect mutation, so that the reactor operates under the constraint x_1 + x_2 = 1

, and the (approximate) dynamics is fully characterised by the single differential equation:

2 2 2 x˙1 = x 1c11 + x1x2c21 - x1(x1c11 + x1x2(c12 + c21) + x2c22) = x21c11 + x1(1 - x1)c21 2 2 - x1(x 1c11 + x1(1 - x1)(c12 + c21) + (1 - x1) c22)

We shall also be interested in the overall replication rate, given by:

R = x21c11 + x1(1 - x1)(c12 + c21) + (1 - x1)2c22

For each distinct class we shall present the relevant pairwise interaction matrix or pair of equivalent matrices, the resulting simplified expression for $\dot {x}_1$ , optionally a graph of $\dot {x}_1$ , and a brief discussion of the resulting dynamic behaviour including example trajectories as appropriate.⁷ Finally we shall return to re-consider the general case ( n>2

) as potentially approximated by a simple superposition of pairwise systems ( n=2

), all instantiated in the same reactor at the same time.

Class 0

Pairwise interaction(s):

( ) 0 0 0 0

Differential equation:

˙x = 0 1

As there are no reactions at all, the reactor is completely inert. There is no turnover, and the rate of change of both species concentrations is, trivially, 0. In other words, the reactor remains exactly as it is initialised. In fact this particular result holds for arbitrary numbers of species (assuming they are all individually and mutually inert, i.e. all $c_{ij}=0$ ), and regardless of whether mutation is in effect (since there is no replication happening in the first place). This is a degenerate case of limited practical interest; however, we shall see that there are circumstances in which this generic situation (of a completely inert molecular population) is a possible fixed point of the long term (evolutionary) reactor dynamics.

Class 1

Pairwise interaction(s):

( ) 1 1 1 1

Differential equation:

˙x1 = 0

The two molecular species are (facultative) mutualists. By symmetry alone, the rate of change of both species’ concentrations is again 0, regardless of current state. However, in contrast to class 0, there is continuing turnover of the reactor contents at the maximum possible rate ( R=1.0

); accordingly, this would predict, under stochastic conditions, that there would be slow random drift in the relative concentrations, with eventual fixation of one or the other. However, because the intrinsic dynamic here is essentially neutral, then mutation, even at a low level, would significantly modify this behaviour, effectively introducing negative frequency dependent selection. In the simplest case, with just two possible species, each having an equal rate of mutation to the other, this would actively stabilise the state x_1=x_2=0.5

against drift.⁸ There would still be stochastic fluctuation around this state, with the level of fluctuation inversely related to the mutation rate (and, of course, the population size). With a somewhat larger number of mutationally related species, still all being pairwise facultative mutualists, this result may generalise to a stable equilibrium distribution across all species in the set (though with continuing replication/turnover at the maximum rate). Such a family of species would be somewhat analogous to the concept of a quasi-species in (externally catalysed) replicator systems (Eigen et al., 1989). If all species have equal inbound and outbound mutational pathways and rates, within this set, then this simplifies to a uniform distribution (i.e., all species having concentration 1/n

, where

is the number of species). This analysis is subject to the reactor size being large enough, relative to the number of distinct species in the set, that the equilibrium number of molecules of each species is still significant—i.e., so that all species can be simultaneously represented. In particular, we will require all $x_i \gg 1/M$ where

is the fixed total number of molecules in the reactor. With an approximately uniform distribution over

species, this means we require $n \ll M$ . With a combinatorially large family of mutant species, as is typical of modular replicators, this may not, in fact, be the case. In summary then, this form of interaction, under mutational conditions, tends in itself to lead to diversification into a potentially extensive family of mutualists (a quasi-species). If this mutationally accessible family of common mutualists is small then there may be stable co-existence; otherwise, because (by hypothesis) there is no “master” sequence with some significant selective advantage, there may be continuing ongoing drift of the quasi-species distribution (denoted say, by the instantaneous consensus sequence) with only a varying subset of the family being represented in the reactor at any given time.

Class 2

Pairwise interaction(s):

( 1 0 ) ( 0 0 ) 0 0 0 1

Differential equation:

2 ˙x1 = x1(1 - x1) > 0 ∀x1 > 0 (see figure 1)

Figure 1:

Class 2 Differential Equation

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 $\dot {x}_1 > 0 \; \forall {} x_1 > 0$ (where x_1

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, R(t)

) also rises to the maximum possible ( 1.0

Figure 2:

Class 2 Trajectory: Growth of self-replicase [ x_1(0)=0.05

]

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 1.0

due to the ongoing mutational load. To analyse this, let x_1

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:

x˙1 = x21(1 - V) - x31 2 = x1(1 - x1 - V)

Setting $\dot {x_1}=0$ , and neglecting the case of x_1=0

, there is a single fixed point at:

x1 = 1 - V

For

we have $\dot {x}_1 > 0$ and for x_1 > 1-V

we have $\dot {x}_1 < 0$ ; accordingly, this is a stable fixed point, and the self-replicase stably maintains itself at the concentration 1-V

. This will hold as long as the ODE approximation is valid (i.e., even for large

, provided the absolute number of molecules of x_1

is sufficient to mask stochastic fluctuations).

Class 3

Pairwise interaction(s):

( ) ( ) 0 1 0 0 0 0 1 0

Differential equation:

2 ˙x1 = - x 1(1 - x1) < 0 ∀x1 > 0 (see figure 3)

Figure 3:

Class 3 Differential Equation

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 ( x_1

) 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, x_2

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 x_2

, and thus ultimately becomes completely inert (effectively as already discussed above under the case of class 0 interactions).

Figure 4:

Class 3 Trajectory: Decay of Obligate Host [ x_1(0)=0.95

]

Class 4

Pairwise interaction(s):

( 1 1 ) ( 0 0 ) 0 0 1 1

Differential equation:

˙x1 = 0

One species ( x_1

) is a self-replicase, and the other is not; which, so far, is similar to class 2. However, in the class 4 case, the self-replicase will also act as a replicase for the other molecule, so there is a host-parasite relationship as in class 3. In this case we say the self-replicase is a facultative host for the obligate parasite. The situation is not as good for the host as in class 2, as some of its replicase activity is “wasted” on the parasite; nor yet as bad as class 3, because at least some of its replicase activity is still spent on replicating itself. In the ODE approximation these effects exactly balance out, and, as shown above, $\dot {x}_1 = 0 \; \forall {} x_1 \in [0, 1]$ . Superficially, this is formally similar to class 0 and class 1, but the practical dynamic is again quite different from both of those previous cases. Although $\dot {x}_1 = 0$ at all concentrations, the total replication rate now varies directly with x_1

R = x1

so if

drifts, then total replication rate will also “drift” in sympathy. It is under the assumption of non-zero mutation rate that this case becomes most distinct. The behaviour will be somewhat similar to that of class 1, in that the “master” sequence will tend to diversify, through mutation, across the available family, or quasi-species, of class 4 mutants; with some equilibrium distribution over this set (if the mutational pathways and rates are uniform over the set, the equilibrium concentration of all species will be equal at 1/n

). But unlike the case with class 1, this diversification now means that the overall replication rate also declines; with uniform distribution in the concentrations, this would ultimately be to an equilibrium rate of only 1/n

. If, as discussed for class 1, this family of mutants is actually combinatorially large, then we may have $n \simeq M$ (or indeed greater), with the “equilibrium” concentration representing less than a single molecule of functional replicase. In other words, under mutation, this class is more likely to share the decline toward complete inertness described for class 3. This decline may be relatively very slow (if the the mutation rate is low); but would still be essentially monotonic.

Class 5

Pairwise interaction(s):

( ) ( ) 0 1 1 0 0 1 1 0

Differential equation:

˙x1 = x21(1 - x1) + (1 - x1)2x1 = x (1 - x )(x + (1 - x )) 1 1 1 1 = x1(1 - x1) > 0 ∀x1 > 0 (see figure 5)

Figure 5:

Class 5 Differential Equation

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 x_1>0

(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 $t\simeq 10$ for class 5, compared to $t \simeq 30$ for class 2. Formally, the growth law for the class 5 case is strictly logistic.

Figure 6:

Class 5 Trajectory: Logistic Takeover [ x_1(0)=0.05

]

The overall rate of replication for this case is simply R=x_1

, as it was in class 3; but as x_1

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:

x˙1 = x1(1 - x1 - V)

where

is the per-molecule mutation rate (to class 5 mutants). Again, this has a stable fixed point at x_1=1-V

. Indeed, this result would hold for any mix of mutants that are all in class 2 or class 5 relative to the “original” self-replicase (and are otherwise inert).

Class 6

Pairwise interaction(s):

( ) ( ) 1 0 1 1 1 1 0 1

Differential equation:

˙x = - x (1 - x )2 1 1 1 < 0 ∀x1 > 0 (see figure 7)

Figure 7:

Class 6 Differential Equation

Both species are self-replicases; but one ( x_2

) is a (facultative) parasite of the other ( x_1

), which acts as a (facultative) host. As with class 3, we have $\dot {x}_1 < 0 \; \forall {} x_1 >0$ , so x_1

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, x_2

, 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 x_2

initially has essentially arbitrarily low concentration. Figure 8 shows an example trajectory for x_2(0)=0.05

. While there is an initial period of slow growth, the parasite does inevitably achieve a “critical” concentration (say $x_2 \simeq 0.2$ at $t \simeq 16$ in this example) after which the displacement is then completed very rapidly.

Figure 8:

Class 6 Trajectory: Selective displacement by facultative parasite [ x_1(0)=0.95

]

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

Class 7

Pairwise interaction(s):

( 1 1 ) ( 0 1 ) 1 0 1 1

Differential equation:

2 ˙x1 = x1(1 - x1) > 0 ∀x1 > 0 (see figure 9)

Figure 9:

Class 7 Differential Equation

Figure 10:

Class 7 Trajectory: Growth of self-replicase [ x_1(0)=0.05

]

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 x_1(0)=0.05

). 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 x_1

) the differential equation becomes:

x˙1 = x1(x2- 2x1 + (1 - V)) 1

with a stable fixed point at $x_1 = 1 - \sqrt {V}$ . Thus, the self-replicase will still reliably grow to dominate the reactor, but must carry a significantly larger ongoing mutational load than in the cases of class 2 or class 5. This may be qualitatively understood as arising from the fact that the mutants benefit from a degree of amplification by the self-replicase, which is not the case for the other two classes. We can reasonably infer that, in the case of a mix of mutants of class 2, 5 and 7, this growth and dominance of the self-replicase will still be observed, with a steady state concentration somewhere between $1-\sqrt {V}$ and 1-V

Class 8

Pairwise interaction(s):

( 0 1 ) 1 0

Differential equation:

x˙1 = x1(1 - x1)(2x1 - 1) > 0 ∀x1 < 0.5 < 0 ∀x1 > 0.5 (see figure 11)

Figure 11:

Class 8 Differential Equation

Neither species is a self-replicase, but each can act as a replicase (obligate host) for the other (obligate parasite). Setting $\dot {x_1}=0$ , and neglecting the case of x_1=0

, there is a single fixed point at:

x1 = 0.5

Of course, we also therefore have x_2=1-x_1=0.5

in this state (which would follow from the symmetry of the situation in any case). As noted above, for x_1 < 0.5

we have $\dot {x}_1 > 0$ and for x_1 > 0.5

we have $\dot {x}_1 < 0$ . Accordingly this is a stable fixed point. The reactor will relax back into it, even under essentially arbitrarily large perturbation. Figure 12 shows an example trajectory, initialised from (x_1, x_2) = (0.05, 0.95)

. While not particularly significant to the investigation here, this case is equivalent to the canonical situation of single stranded RNA replication where each complementary strand effectively serves as a catalyst for the production of the other. More generally, this case is formally equivalent to a two-component hypercycle (Eigen and Schuster, 1977). It can clearly be generalised to a hypercycle containing an arbitrary number of components (say

); and in that case, the stable fixed point will be with equal concentrations ( 1/n

) across all components (assuming, as usual, that the absolute number of molecules is still sufficient to mask statistical fluctuation, i.e., $n \ll M$ ). However, this situation changes significantly once mutation is introduced. In this case, a hypercycle is always vulnerable to collapse. There are several scenarios here; but, in general, any mutant species which is able to exploit any one of the components of the hypercycle as a replicase, without passing this support on to the next component of the hypercycle, is a potential trigger for such degeneration.¹⁰

Figure 12:

Class 8 Trajectory: Stable Co-existence/Hypercycle [ x_1(0)=0.05

]

Class 9

Pairwise interaction(s):

( 1 0 ) 0 1

Differential equation:

x˙1 = x1(x1 - 1)(2x1 - 1) < 0 ∀x1 < 0.5 > 0 ∀x1 > 0.5 (see figure 13)

Figure 13:

Class 9 Differential Equation

Both molecular species are independent self-replicases. In a certain sense this is exactly complementary to class 8. The expression for $\dot {x}_1$ is precisely the negation of that in class 8. Accordingly, the dynamics has exactly the same fixed points: the two states of x_1=0

where one or the other species is no longer present, and the state x_1=x_2=0.5

. 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 (x_1, x_2) = (0.51, 0.49)

. 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.

Figure 14:

Class 9 Trajectory: Survival of the Common [ (x_1(0), x_2(0))=(0.51, 0.49)

]

This result generalises directly to the

species case. There is an unstable fixed point with $x_i = 1/n \; \forall i$ ; and

stable fixed points with a single x_i = 1

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 $\sqrt {V}$ (but generally much closer to the minimum).

Protocell Inheritance Mechanism

We now discuss more carefully the operation of a molecular population of informazymes as the “informational subsystem” (inheritance mechanism) of a protocell, in the light of the theoretical analysis above. Consider a protocell containing a population of informazyme molecules which is “self-sustaining”. By that we mean that the molecules in some set of informazyme species are replicated sufficiently consistently that the relative concentrations of these species remain essentially constant, as the protocell grows. Under independent assortment of molecules to offspring, both daughter cells will then inherit populations with this same pattern of informazyme species concentrations; and thus the potential for indefinitely extended inheritance through a protocell lineage is established. The simplest possibility for this is an informazyme population dominated by a single self-replicase species. Assuming the initial existence of such an established self-replicase species, this will generate a spectrum of molecular-level mutants, on an ongoing basis. We now summarise the potential impact of each pairwise class of possible mutant species which may arise:

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 $\sqrt {V}$ (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.

In summary then, given that we wish to formulate a protocell model which can support evolution of computation, this theoretical analysis suggests that we should devise an informazyme sub-system that:

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.

Notes

³A preliminary version of the analysis in this section is presented in (Kelly et al., 2008).

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 $c_{ij}$ vary continuously i.e., $\in [0,1]$ .

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.

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Artificial cell evolution and functionality: Evolution of Protocell-embedded Molecular Computation

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Evolution of Protocell-embedded Molecular Computation

Dynamics of Informazyme Systems³

Assumptions

General Dynamic Equation

Terminology

“Self-”systems

Pairwise Systems

Class 0

Class 1

Class 2

Class 3

Class 4

Class 5

Class 6

Class 7

Class 8

Class 9

Protocell Inheritance Mechanism

Notes

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Artificial cell evolution and functionality: Evolution of Protocell-embedded Molecular Computation

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Evolution of Protocell-embedded Molecular Computation

Dynamics of Informazyme Systems3

Assumptions

General Dynamic Equation

Terminology

“Self-”systems

Pairwise Systems

Class 0

Class 1

Class 2

Class 3

Class 4

Class 5

Class 6

Class 7

Class 8

Class 9

Protocell Inheritance Mechanism

Notes

Dynamics of Informazyme Systems³