Compositional memory matters for early molecular systems

Significance

Transient compartmentalization is thought to be essential for preventing parasites (short nonfunctional RNA sequences) from invading replicases (self-replicating RNA), but how compartmentalization behaviors specifically affect replicase-parasite dynamics remains unclear. Here, by combining experiments and modeling, we show that the extent of compositional memory—the degree to which molecular compositions are inherited within compartments across rounds of compartmentalization—alters replicase-parasite dynamics and therefore matters for early molecular systems.

Abstract

The error catastrophe refers to the proliferation of nonfunctional molecules in conditions where molecular replication has low accuracy, which is likely to correspond to conditions present at the Origin of Life. This error catastrophe can be avoided thanks to transient compartmentalization, provided that the compartments are sufficiently tight to prevent molecular leakage. Typically, transient compartmentalization models assume that the content of the compartments is completely pooled periodically, resulting in the complete loss of the compositional memory of the compartment. Furthermore, previous models that include the possibility of ecological interactions between molecular parasites and replicators within compartments generally do not study the effect of transient compartmentalization on their coevolution. To address both issues, we develop a framework that accounts for the coevolution of molecular replicators and parasites, along with specific compartmentalization dynamics that are transient yet partial, allowing compositional memory to accumulate from one round of compartmentalization to the next. We benchmark our model with a serial dilution experiment that displays complex oscillatory dynamics among four well-characterized RNA replicators. We also perform experiments to quantify the level of mixing in compartments when stronger stirring tends to homogenize their composition. We then model stirring-induced mixing and show how stirring alters the dynamics of compartmentalized replicators. We conclude that compositional memory arising from transient compartmentalization plays a major role in the dynamics of early molecular systems.

While chemical systems are often studied in well-mixed conditions with only a few species, it seems more likely that life emerged in a heterogeneous medium with complex composition (1–3). Such a medium would then contain regions in which diffusion is limited, which we call compartments. Compartmentalization, even when only partial and transient, could solve essential issues for the Origin of Life (3–12): It can keep important molecules nearby in a small volume, which could enhance reactivity (13, 14), it can limit side-reactions and parasite amplification, thus preventing the error catastrophe first identified by Eigen (15–18), and it could lead to new forms of autocatalysis and ecological interactions, which simply do not exist in well-mixed systems (19). Moreover, compartmentalization could facilitate compositional inheritance (20–23), which is a primitive form of heredity that does not require information-carrying polymers nor enzymes.

Recently, a scenario for the emergence of an RNA world in three stages has been proposed (24), in which the last two stages involve a form of compartmentalization. In stage 1, network autocatalysis occurs using oligomer substrates to form catalysts (25). In stage 2, autocatalysis coexists with templated ligation, and limited diffusion acts as a primitive form of compartmentalization (4, 7), thereby preventing the error catastrophe. Then, in stage 3, template-based replication takes over thanks to enhanced catalytic properties, smaller building blocks are used, and more efficient forms of compartmentalization appear via coacervates and lipid vesicles. The transition from stage 2 to stage 3, leading to the dominance of template-based replication, has been studied before by several authors (26, 27); the transition is stochastic and becomes less likely in well-mixed systems, suggesting that concentration fluctuations also matter for life to appear (28, 29).

This scenario supports the view that transient compartmentalization is crucial for the maintenance and the transfer of molecular information (4, 30–34). In practice, transient compartmentalization dynamics involve a periodic contact with the external environment and a certain level of spatial inhomogeneity. Some experiments confirmed the theoretical prediction that transient compartmentalization can limit parasite amplification (30–37). When performed over long timescales, experiments on compartmentalized RNA-based systems showed that RNA replicases coevolved with their parasites and formed increasingly complex molecular ecosystems (34, 38).

One question raised by these experiments (34, 38) is the role of compartmentalization during replicase-parasite coevolution (31–33). A related issue in these experiments was the lack of quantification of the composition of individual compartments and of the degree of mixing between compartments.

To address these challenges, we first revisit and extend a previous model of transient compartmentalization, which assume that compartment contents are fully mixed during periodic pooling (31). Our extension explicitly accounts for the coevolution of multiple replicases and parasites within compartments. We then introduce a model where compartments partially inherit their composition from one round to the next, linking this compositional memory to the average degree of stirring applied to the compartment population. Stirring induces fusion and mixing among compartments, thereby modulating the inheritance of molecular compositions.

Our model successfully reproduces the four-species oscillatory dynamics observed in a serial dilution experiment using previously characterized RNA replicators, which replicate via protein translation (34). These replicators include self-replicating RNA molecules (replicases) and parasitic sequences that exploit replicases for their own replication. Although the experimental system is based on protein translation, the theoretical model is more general and describes other replicator-parasite systems such as ribozymes and autocatalytic networks. We demonstrate that when parasites dominate the system—reducing the average number of molecules per compartment to near unity—sufficient mixing is essential to spatially isolate parasites from replicases and purify the system. Without adequate mixing, parasites persist even at low densities, leading to complete washout through successive dilutions. This behavior echoes the role of dispersal in other replicase-parasite models (39): parasites cannot proliferate if they fail to encounter replicases (e.g., due to limited diffusion).

We performed competition experiments using compartmentalized RNA-replicases and parasites, and we confirmed our theoretical expectation that stirring generally improves the survival of replicases. Weak stirring, conversely, preserves compositional memory, promoting parasite fixation. Using fluorescent dyes, we quantified the degree of mixing between compartments as a function of stirring strength by analyzing fluorescence distributions across droplets. Collectively, our findings reveal that compartmentalization alone is insufficient to prevent parasite invasion. Instead, controlled mixing is critical for enabling early molecular replicators to evolve into robust, self-sustaining systems.

Model and Observables

We model the serial transfer experiment with 4 species introduced in Fig. 1 A and B, where we specify the mechanisms of replication of RNA-replicases and RNA-parasites. Transient compartmentalization dynamics of RNA molecules has been theoretically modeled in previous work (31–34, 37) as a cycle consisting of four steps shown in Fig. 1C: i) compartmentalization: from an initial pool of replicases and parasites, compartments are formed by seeding them with molecules according to a Poisson distribution (40); ii) maturation: each compartment evolves depending on its initial composition during a maturation time $T_{\mathit{mat}}$ set by the environment. This time is assumed to be long enough for each compartment to reach its carrying capacity; iii) pooling: after maturation, compartments fuse and release their contents in a common pool, and the fractions of each subpopulations are updated; iv) dilution: the content of the pool is diluted homogeneously with a factor $d$, without affecting the fractions of each subpopulations. The process can start again.

Fig. 1.

(A) Representation of two possible evolution models of the RNA replication system containing two replicases and two parasites. (B) Illustration of the deterministic assumptions. (C) Transient compartmentalization with pooling (homogeneous case). (D) Illustration of the typical oscillatory dynamics obtained with the deterministic approach with second-order replication rates and mutations. Here, we show the results for model B, we used the rates ${\alpha }_{0,0}=0.5{\text{h}}^{-1}$, ${\alpha }_{1,1}=2.0{\text{h}}^{-1}$, ${\gamma }_{0,0}=7.0{\text{h}}^{-1}$, ${\gamma }_{1,1}=20.0{\text{h}}^{-1}$, ${\mu }_0=1.0\times {10}^{-2}{\text{h}}^{-1}$, ${\nu }_0=1.0\times {10}^{-2}{\text{h}}^{-1}$, and ${\nu }_1=1.0\times {10}^{-2}{\text{h}}^{-1}$. The carrying capacities are $K=K_r=K_p=28$ on the Left panel and $K=K_r=K_p=32$ on the Right panel, and the dilution factor is $d=10$. Limit cycles are shown in red (which is a single point for a steady state). We observe damped oscillations reaching a steady state on the Left panel and sustained oscillations (limit cycles) on the Right panel.

Maturation.

In the maturation step, we consider two different models, defined in Fig. 1B depending on how parasites are created. Both models are similar to that of (31) except that they account for the possible coevolution of replicases and parasites through mutations. Model A considers that mutant parasites (resp. replicases) evolve only from wild type parasites (resp. replicases) due to mutations. The dynamical equations (41) for the populations of the various species are given in Eq. 5.

In model B, parasites can evolve from replicases. In practice, this can happen due to deletions in the sequence of replicases [by recombination for instance (34)], which yield smaller sequences unable to replicate by themselves. The dynamical equation for the populations of the various species in this model is given by Eq. 6. Both models are based on experimental observations obtained by tracking evolutionary lineages of replicase and parasite RNAs (34).

Pooling.

After the maturation step, the contents of all compartments are pooled (they fuse and all their contents are mixed in a common pool). Therefore, compartments do not inherit from their content before pooling (compositional memory is erased). The new fraction of replicases $k$ in the pool is expressed in terms of the value at the beginning of the round by the following equation:

$$\begin{array}{c}\hfill {\overline{x}}_k(\mathit{x},\mathit{z},\lambda )=\frac{{\sum }_{\mathit{m},\mathit{y}}{\overline{m}}_k(\mathit{m},\mathit{y})P_{\lambda }(\mathit{m},\mathit{y}|\mathit{x},\mathit{z})}{{\sum }_{\mathit{m},\mathit{y}}\overline{n}(\mathit{m},\mathit{y})P_{\lambda }(\mathit{m},\mathit{y}|\mathit{x},\mathit{z})},\end{array}$$ [1]

where $\mathit{x}={\{x_s\}}_{0\le s\le N_m},\mathit{z}={\{z_s\}}_{0\le s\le N_y}$ are vectors describing the fractions of each mutant replicases (where $N_m$ is the total number of mutant replicases) and parasites (where $N_y$ is the number of mutant parasites), respectively. Further, $\lambda$ is the parameter of the Poisson distribution, and $P_{\lambda }(\mathit{m},\mathit{y}|\mathit{x},\mathit{z})$ is the multinomial probability to have a vector $\mathit{m}$ for the numbers of each replicase and a vector $\mathit{y}$ for the numbers of each parasite starting with a pool with fractions $\mathit{x}$ of replicases and fractions $\mathit{z}$ of parasites. Then ${\overline{m}}_k(\mathit{m},\mathit{y})$ (resp. $\overline{n}(\mathit{m},\mathit{y})$) is the final number of replicases after maturation of type $k$ (resp. final number of molecules after maturation) in the compartment given that it was initially filled with $n$ molecules among which $\mathit{m}$ were replicases and $\mathit{y}$ parasites. In principle, ${\overline{m}}_k(\mathit{m},\mathit{y})$, ${\overline{y}}_k(\mathit{m},\mathit{y})$ should be random variables but they become fully determined by $\mathit{m}$ and $\mathit{y}$ in our deterministic approach (see below). We can set the time of maturation to be $T$, or fix the final number in each compartment as a constant value $K+\sum _{1\le s\le N_m}{m}_s+\sum _{1\le s\le N_y}{y}_s$. In the following, we assume that $T$ is large enough for both conditions to be equivalent.

Results

Deterministic Approach.

There are two sources of stochasticity in the dynamics we consider: the first arises from inoculation (40), i.e. the preparation of the initial composition of compartments by sampling molecules from the pool. The second arises during the maturation phase, in which each compartment evolves stochastically through mutation or replication events.

In the deterministic approach, it is assumed that the maturation of a compartment is fully determined by its initial composition, which means that the only remaining source of stochasticity in the system is the inoculation step. This has the advantage of providing simple dynamics that are independent of the kinetic parameters of the maturation step, such as the replication rates. The only parameters controlling the evolution of the system are then the carrying capacity (assuming resources are homogeneously distributed among compartments), the dilution factor, the maturation time and mutation rates for mutations. The carrying capacity arises from the limitation of available resources in one compartment and constrains the total number of molecules maximally produced in that compartment.

This approach relies on specific assumptions about species interactions. In SI Appendix, sections 5–8, we explore how varying these assumptions—such as the potential for parasites to evolve and acquire generality (i.e., the ability to replicate with all replicases)—affects the system dynamics. For Fig. 1D, 2B, and 3, we adopt the following key assumptions:

Fig. 2.

(A) Schematic representation of the long-term replication experiment. 1) RNAs were replicated in water-in-oil droplets at 37 ^°C for 5 h. 2) Droplets were five-fold diluted with a fresh oil phase and the translation system. 3) The droplet population was stirred, allowing fusion and division, thereby redistributing RNAs among droplets. (B) Effect of stirring on the dynamics, we used the same deterministic rules and parameters but for different values of the stirring parameter $s$ (s $=$ 0.5 and s $=$ 0.99). The solid lines represent the simulations with stirring while the dotted lines represent the perfectly homogeneous case. Simulations with stirring were performed with $50,000$ compartments, the carrying capacities are $K_p=K_r=30$ and dilution factor $d=10$. Oscillations are damped in this panel because $K/d$ is small enough, (as illustrated in Fig. 1D and detailed in SI Appendix, section 1.F). (C) The RNA replication system containing two replicases (HL1-228 in yellow and HL2-228 in red) and two parasites (PL2-228 in green and PL3-228 in blue). These RNAs replicate by utilizing polymerases translated from the replicases. The widths of the colored arrows represent relative replication efficiencies (34). (D) We can reproduce experimental dynamics of four RNA species (2 replicases and 2 parasites), represented by solid circles connected by thin dashed lines. The solid lines show simulations performed using the same dilution factor $d=5$ and the experimentally determined replication rates of the individual species (34). Each species has a different carrying capacity, determined by its length. Other parameters are detailed in SI Appendix, section 2.D.

Fig. 3.

(A) Total fraction of replicases in the system, $x_{\mathit{tot}}$. Theoretical expectations (solid lines) are shown for different values of the ratio of carrying capacity to dilution factor $K/d$. Experimental data (solid symbols) are plotted against stirring strength (2.0, 5.2, 16, and 30 krpm), converted into the corresponding stirring parameters $s$. For visualization, the data points are plotted at the expected $s\pm 0.01$ to 0.02. The experiments were performed at different dilution factors, $d_{\text{exp}}$, as indicated in the panel. The same trends are observed in theory and experiments. Error bars indicate the mean $\pm$ one SD ($n=3$ to 4). (B) Effect of stirring on the compartment composition. In purple, we show the trajectories of individual compartments (for the number of mutant parasites $y_1$), while in solid blue lines we show the average over all compartments. As stirring gets stronger, the contents of compartments homogenize, and the compartments end up either full of mutant parasites or containing mostly no mutant parasites. In this limit, all compositional memory is lost.

Aggressive parasites.

In compartments containing both parasites and their associated replicases (i.e., replicases capable of replicating these parasites), parasites dominate and replicate until they reach their carrying capacity, which may differ for replicases and parasites depending on sequence length (31).

Strict specificity.

Parasites can replicate only with replicases of the same type, reflecting experimental observations of parasite-replicase ecosystems (38). Consequently, in compartments containing parasites of type $k$ and replicases of type $j\ne k$, only replicases replicate.

Beneficial mutations.

Mutated replicases or parasites exhibit significantly higher replication rates than their wild-type counterparts, with no back-mutations. For example, in compartments containing only replicases, those with the highest fitness (i.e., replication rate) dominate. Under these assumptions, each initial condition maps to a unique final composition.

Further, we assume that the maturation time is long enough for every compartment to reach its carrying capacity provided it contains replicases initially. The lengths of dominant parasite RNAs that emerged from replicases in previous evolution experiments were approximately $4$ times smaller than that of replicases (34). If one assumes that the carrying capacity originates from a limit in terms of the total number of nucleotides, then this factor four represents the ratio between the carrying capacity of the parasites and of the replicases as explained in SI Appendix, section 11. This effect is not taken into account in most simulations except for the comparison to experimental data in Fig. 2D. This effect mostly increases the amplitude of replicase-parasite oscillations and prevents them from reaching steady states. In this case, the average fraction of replicases tends to be lower, and the isolation of parasites in distinct compartments becomes less likely. Yet the effects of dilution and stirring are similar to the case where all species have the same carrying capacity (SI Appendix, section 11), and the general conclusions we draw are not modified.

Using this method, we obtain oscillatory dynamics between the different species, as can be seen in Fig. 1D. We see in Fig. 1D that oscillations manifest as cycles on the plane $(x,\lambda )$. When both $\lambda$ and $x$ are small, parasites become isolated, leading to an increase in the replicase fraction $x$. This subsequently raises the average number of individuals per compartment $\lambda$. At higher $\lambda$, parasites will more likely find replicases and start invading the system, thereby decreasing $x$. The resulting reduction in replicases leads to fewer replication events and thus a decrease in $\lambda$, and the cycle is ready to start again. The mechanism underlying these oscillations is detailed in SI Appendix, section 1.F. In contrast with the theoretical study of two-species dynamics (31), we explore the effects of mutations and four-species interactions.

Stochastic vs. Deterministic Approaches.

The deterministic approach ignores the stochasticity of mutations, but fortunately, their effect can be incorporated by computing the average numbers of mutants of each species appearing during one cycle (averaged over the initial compositions of the compartment), as explained in SI Appendix, section 1.C. In that way, the deterministic method can account for different values of the replication and mutation rates. To assess if this improved deterministic approach fully captures the dynamics that arise from stochastic chemical reactions, we compare it to Gillespie simulations (Direct Stochastic Simulation Algorithm) (41) in SI Appendix, Figs. S4 and S5. These simulations include the full stochasticity of all reactions and therefore account for all sources of stochasticity in the system. We find that the deterministic approach provides an accurate description in the regime where mutations are not too rare. Whenever mutation rates are small enough that mutations do not necessarily occur within one cycle, the dynamics predicted by the deterministic approach differ from those obtained with Gillespie simulations, yet it still predicts the correct average fractions of each species over cycles. In this limit, mutations become rare events that drastically modify the system. Depending on the conditions (typically the ratio $K/d$), we observe oscillations (either stable or unstable) or a stable node (31). In particular, some conditions lead to limit cycles, and the complexity of orbits increases with four species as compared to two species (SI Appendix, section 1.F). These complex dynamics stabilize the four species system (SI Appendix, section 1.F), because the threshold between stable and unstable spirals is shifted toward larger $K/d$ with four species.

This formalism can be extended to include more complex interactions between replicases and parasites, as observed experimentally (34). In one extension, we have distinguished generalist-parasites, which are able to replicate with all types of replicases, and specialist-parasites (42), which are able to replicate with only one type of replicases but more efficiently as explained in SI Appendix, sections 5A and 5B. We have also studied the role of various parameters such as the carrying capacities of replicases and parasites, mutation rates, and maturation time in SI Appendix, sections 9–12.

Stirring and Compositional Memory.

Despite these modeling efforts, experiments with the compartmentalized RNA systems still do not exactly match the assumptions of the models outlined above, because in practice, the solution is stirred periodically. This stirring causes some material exchange between compartments (sketched in Fig. 2A) due to fusion/fission events (34, 38) or other transfer mechanisms (43, 44), which are not described explicitly in the model. As a result, the composition of a given compartment before stirring is correlated with its composition after stirring in contrast to the assumptions made in the pooling step. We refer to this phenomenon as compositional memory, a feature which is central in some origin of life scenarios (20, 21, 45). This compositional memory is expected to be relevant irrespective of whether compartments are made of droplets, coacervates, or vesicles (46). The effect of mixing from one round of compartmentalization to the next has previously been assessed numerically using simulations (37), but that work did not model the complex ecological interactions between molecular species precisely.

To address the issue of stirring, we now consider the experimental protocol described in Fig. 2A. After an initial compartmentalization step, each subsequent cycle consists of a maturation step as before, followed by a dilution step at the compartment level, in which randomly chosen mature compartments are removed and replaced by empty ones to increase the total available volume. We also assume that new droplets are enriched (for instance by enclosing resources present in the environment). Then, the pooling step is replaced by a stirring step, in which the solution is stirred so as to induce fusions and divisions of compartments. If this stirring was strong enough to completely erase the influence of the former content of compartments on their new composition, we would recover equivalent dynamics as with pooling, as illustrated in Fig. 2B.

The stronger the stirring, the weaker the compositional memory should be. In the limit of strong stirring, we expect to recover the result of complete pooling, meaning that the system has no memory. To include compositional memory, we consider that the composition of compartments is updated to a new composition determined by an interpolation between their former compositions and the average composition of the whole population. This is a mean field assumption because every compartment is then interacting with the average composition of the total population. This approach does not allow us to study spatially localized effects or the fusion/fission dynamics of individual compartments, but it is sufficient to capture the average effect of stirring on droplet content. In practice, we find that the new composition of a compartment depends on its former composition because stirring is often not strong enough to fully homogenize the system. We quantify the average level of stirring-induced mixing by a parameter $s\in [0,1]$ ($s=0$ means that compartments do not interact with each other at all resulting in the absence of population scale dynamics, while $s=1$ is the complete pooling limit), and we denote with a hat the quantities computed after the stirring step. For instance, ${\widehat{n}}^{(i)}$ (the total number of individuals in compartment $i$ after dilution) is drawn according to a Poisson distribution of parameter:

$$\begin{array}{c}\hfill {\widehat{\lambda }}^{(i)}=(1-s){\overline{n}}^{(i)}+s\frac{〈\overline{n}〉}{d},\end{array}$$ [2]

where $〈\overline{n}〉$ is the average number of individuals after maturation and before dilution, and ${\overline{n}}^{(i)}$ is the number of individuals in compartment $i$ after maturation if it has not been removed during dilution, and $0$ otherwise. We deduce from this expression that $s=d/(d+1)$ corresponds to the level of stirring where the old composition and the average composition after dilution contribute equally to the definition of ${\widehat{\lambda }}^{(i)}$. Thus, when $s>d/(d+1)$, we expect to recover the perfectly homogeneous case. For instance, when $d=10$, the corresponding value is $s\simeq 0.91$. From Eq. 2, we also deduce that the SD of ${\widehat{\lambda }}^{(i)}$ scales as $1-s$ and we recover that the compartments are perfectly homogeneous for $s=1$.

Then, the new numbers ${\widehat{m}}_0^{(i)}$ (and similar for other species) are drawn according to a multinomial distribution with parameters

$$\begin{array}{cc}\hfill {\widehat{x}}_k^{(i)}& =\frac{(1-s){\overline{m_k}}^{(i)}+s\frac{〈\overline{m_k}〉}{d}}{(1-s){\overline{n}}^{(i)}+s\frac{〈\overline{n}〉}{d}},\hfill \\ \hfill {\widehat{y}}_k^{(i)}& =\frac{(1-s){\overline{y_k}}^{(i)}+s\frac{〈\overline{y_k}〉}{d}}{(1-s){\overline{n}}^{(i)}+s\frac{〈\overline{n}〉}{d}}.\hfill \end{array}$$ [3]

We keep the notations ${\overline{m}}_k,{\overline{y}}_k,\overline{n}$ for the numbers of each species in one compartment after maturation and before dilution, ${\overline{n}}^{(i)}$ is the number of individuals in compartment $i$. If $s=1$, we recover the dynamics discussed in the previous sections (Fig. 2B) and the distribution is homogeneous (no memory). Whenever $s<1$, we keep track of inhomogeneities (memory effect).

Note that this procedure implies following the trajectory of every compartment individually; this assumes a finite (fixed and large) number of compartments, in contrast to the previous assumption of an infinite number of compartments. For $s\to 1$ (strong stirring), the system loses its compositional memory, and we recover the results for the homogeneous case (perfect pooling). On the other hand, if $s\to 0$ (weak stirring), all the compartments are conserved over time and the system does not evolve (except for mutations).

To examine whether our theoretical model could reproduce experimental observations, we performed a serial dilution experiment using a previously constructed compartmentalized RNA replication system (34). This system comprises single-stranded RNA molecules functioning as replicases and parasites, together with a reconstituted Escherichia coli translation system, all encapsulated in water-in-oil droplets. Each replicase encodes the catalytic subunit of an RNA-dependent RNA polymerase, which becomes active within the translation system. Both replicases and parasites compete for polymerases translated from replicases and for nucleotide triphosphates (NTPs). We initiated the serial-dilution experiment with two replicases (HL1-228 and HL2-228, 2041 nt and 2042 nt, respectively) and two parasites (PL2-228 and PL3-228, both 514 nt), all previously characterized (34). Their replication network is detailed in Fig. 2C. Each round consisted of 1) RNA replication at 37 ^°C for 5 h, 2) five-fold dilution of droplets with freshly prepared droplets, and 3) stirring (16 krpm, 1 min) to induce droplet fusion and division (Fig. 2A). Step (3) not only replenished the translation system in individual droplets but also redistributed RNA molecules across the new droplet population, thereby representing the partial pooling of replicases and parasites. We found that the four RNA species exhibited sustained replication with oscillatory dynamics for at least 22 rounds. The dynamics were consistent with those observed previously using the same four RNA species together with another weakly replicating RNA (34). We then compared these dynamics with those generated by our theoretical model, based on the four RNA species with their experimentally determined replication rates (34) (Fig. 2D). We confirmed that the model $(s=0.79)$ largely reproduced the experimental dynamics. Details of the calculations are given in SI Appendix, section 2.D.

Stirring and replicases fraction.

Let us now study how stirring affects the degree by which molecular information is maintained. This can be assessed by following the total fraction of replicases $x_{\mathit{tot}}=x_0+x_1$ as a function of $s$ for a given set of parameters. As stirring becomes weaker, the increased heterogeneity of compartment composition decreases the probability for parasites to be isolated, resulting in a lower fraction of replicases. We find that the dependence of $x_{\mathit{tot}}$ as a function of $s$ is governed by the ratio of the carrying capacity to the dilution factor $K/d$ and that it displays different curvatures in the regimes $K/d\gg 1$ and $K/d<1$ (Fig. 3A).

To test whether these results could capture the experimental behavior, we subjected a replicase (HL2-228) and a parasite (PL2-228) in the RNA replication system (Fig. 2C) to a competition assay. The assay was performed as a single-round transfer experiment (Fig. 2A) with varying stirring strengths and dilution factors $d_{\text{exp}}$. First, we prepared droplets containing HL2-228 and PL2-228. Next, the droplets were diluted at different ratios (2- to 100-fold) with new droplets containing neither replicases nor parasites. We then induced droplet fusion and division by stirring at various speeds (2 to 30 krpm) for 1 min using a homogenizer. The different stirring strengths barely affected the droplet size distributions (SI Appendix, Fig. S14), and the average number of HL2-228 or PL2-228 molecules per droplet was likely less than one under all tested conditions. We did not vary other parameters that determine the carrying capacity, such as NTP concentrations. After incubating the droplets at 37 ^°C for 5 h to allow RNA replication, we measured the concentrations of replicases and parasites, and calculated $x_{\mathit{tot}}$. We found that $x_{\mathit{tot}}$ increased with both stirring strength and dilution factor (Fig. 3A), consistent with the theoretical prediction. The correspondence between experimental stirring strengths in krpm and stirring parameters $s$ was assessed from experiments detailed in the following section. These results further validate the theoretical model and the importance of accounting for stirring in the model, as it matters for the coevolution of replicases and parasites.

To further investigate how varying stirring strengths influence replicase-parasite competition, we used our model to calculate the number of mutant parasites per compartment as a function of $s$ (Fig. 3B), which is difficult to assess experimentally. Interestingly, we find that the number of mutant parasites remains heterogeneous under moderate stirring, whereas strong stirring homogenizes compartments, which end up either full or almost empty of mutant parasites. In this limit, all compositional memory is lost. These results can explain how the fraction of replicases increases under stronger stirring, as parasites are more frequently isolated.

Quantification of the level of stirring in experiments.

In the experiments, the degree of stirring mainly depends on the applied stirring strength (in rotations per minute or rpm). To quantify the extent of content mixing between droplets as a function of stirring strengths, we prepared two droplet populations containing either green or red fluorescent dyes—fluorescein isothiocyanate (FITC)-labeled dextran or tetramethylrhodamine isothiocyanate (TRITC)-labeled dextran)—hereafter referred to as G and R, respectively. The two droplet populations were mixed, stirred at varying strengths, and subjected to a two-round transfer experiment (Fig. 4A). The degree of content mixing was assessed from measurements of the fluorescence intensities of G and R in individual droplets after rounds 1 and 2 (SI Appendix, Fig. S13). To model this experiment, we rescale the fluorescence intensities by the average fluorescence per compartment for each color, and we call the corresponding distributions of green (resp. red) molecules ${\widehat{G}}_k$ (resp. ${\widehat{R}}_k$). In addition, we define ${\widehat{N}}_k={\widehat{G}}_k+{\widehat{R}}_k$.

Fig. 4.

Influence of the stirring parameter $s$ on the distribution of the fraction of one type of individuals, in a population with only two types of individuals. (A) Experimental procedure to assess the effect of stirring with two fluorescent dyes. After preparing the system from droplets containing either FITC- or TRITC-labeled dextran ($G$ or $R$) together with the translation buffer, as described in Materials and Methods, the system was stirred for $1$ min, followed by a 5 h incubation (maturation) phase. (B) Evolution of the experimental distributions (dotted lines, after round 1) for different stirring strengths and comparison with theoretical distributions (solid lines) starting from a perfectly bimodal distribution of ${\widehat{G}}_k/{\widehat{N}}_k$ for different values of the stirring parameter $s$. We observe that the distribution of $|{\widehat{G}}_k/{\widehat{N}}_k-1/2|$ shifts toward $0$ as stirring strengths increases, meaning that the distribution of ${\widehat{G}}_k/{\widehat{N}}_k$ shifts toward a monomodal, with a peak close to 0.5. We used $n_C=80$ (estimated from data, see SI appendix, section 2) interacting compartments. Experimental stirring was performed for 1 min at four different strengths ($2.0,5.2,16,30$ krpm). For each stirring strength, we evaluated the stirring parameter $s$ yielding a theoretical distribution corresponding the experimental distribution. (C) Effect of stirring on the probability distribution function of the fraction ${\widehat{G}}_k/{\widehat{N}}_k$ (probability that ${\widehat{G}}_k/{\widehat{N}}_k$ is between $u$ and $u+du$) with $n_C=80$ interacting compartments, starting from a perfectly bimodal distribution with peaks at ${\widehat{G}}_k/{\widehat{N}}_k=0$ and ${\widehat{G}}_k/{\widehat{N}}_k=1$. The distribution shifts toward a monomodal distribution as $s$ increases.

This rescaling assumes that during typical fusion/division events, droplets exchange a number of fluorescent molecules of the order of the average fluorescence per compartment. We call $\widehat{x}$ the fraction of fluorescence intensity due to green fluorescence in one compartment, which serves as a proxy for the fraction of G in the compartment, and we study the cumulative distribution function of $\widehat{x}$ (the probability of having $\widehat{x}$ lower than a given value $x$). In practice, we employ $|{\widehat{G}}_k/{\widehat{N}}_k-1/2|$ to correct for experimental deviations toward either ${\widehat{G}}_k$or ${\widehat{R}}_k$, introduced during microscopic observation. Values close to $0$ indicate well-mixed compartments containing approximately equal amounts of G and R, whereas values close to $0.5$ indicate compartments that contain either G or R almost exclusively. We found that the fraction of droplets with similar amounts of G and R gradually increased with stronger stirring and over two experimental rounds (Fig. 4B and SI Appendix, Fig. S13).

To model the observed shift in the distribution, we start with a bimodal distribution of ${\widehat{G}}_k/{\widehat{N}}_k$, assuming that compartments initially contain either only G or no G, and use the experimental distribution of ${\widehat{N}}_k$. We then apply a theoretical transformation to describe the effect of stirring on these distributions, yielding the final distribution conditioned on the initial state (see Materials and Methods, SI Appendix, section B for details). This transformation depends primarily on the stirring parameter $s$ and an effective number of compartments $n_C$. We calibrate $n_C$ by assuming that the maximal stirring condition (30 krpm) corresponds to $s\to 1$, as detailed in SI Appendix, section 2B. Additionally, we confirm in SI Appendix, section 2 that the transformed distribution approaches a fixed point as $s\to 1$, consistent with the distribution of $N_k$, where $s=1$ represents maximal stirring. The impact of this transformation on the probability density of ${\widehat{G}}_k/{\widehat{N}}_k$ is shown in Fig. 4C. With increasing stirring strength, the distribution transitions from a bimodal to a centered unimodal form.

In Fig. 4B we compare the result of this transformation, starting from an initially bimodal distribution, expressed as the cumulative distribution of $|{\widehat{G}}_k/{\widehat{N}}_k-1/2|$ (solid lines for different values of $s$), with the experimental distributions after round $1$ (dotted lines), and obtain a good agreement between them. In practice, increasing the number of effective compartments $n_c$ renders the distributions sharper (due to central limit theorem). Here, we use an effective number of compartments of $n_C=80$ (estimated from data as explained in SI Appendix, section 2), and by calibrating $s=1$ to $30$ krpm, we associate a value of $s$ with each stirring strength. We observe a good agreement between our theory and experimental data, with both showing a shift of the distribution toward a more monomodal form. Accordingly, our parameter $s$ captures the effect of stirring on droplet contents. We also recover that weak stirring (less than $5.2$ krpm) corresponds to small values of the stirring parameter $s\le 0.4$, whereas stronger stirring at $16$ krpm and $30$ krpm correspond to $s\ge 0.5$. We confirmed the robustness of these estimations by applying the same transformation and stirring parameters $s$ to experimental distributions after round 2, which are successfully reproduced by our model (SI Appendix, Fig. S11). The correspondence between stirring strengths and stirring parameters $s$ has been used to plot the data for $x_{\mathit{tot}}$ against $s$ in Fig. 3A.

Survival and Predominance.

Our theoretical description of stirring can now be used to study how this process impacts the dilution and maturation in the replicase-parasite dynamics. We study various networks of interactions between replicases and parasites. To this end, we build coexistence diagrams representing the expected fractions of each coexisting species in SI Appendix, section 4. In SI Appendix, Figs. S15–S18, we build these diagrams for models A and B. In both cases, a higher dilution factor favors replicases by isolating parasites more efficiently. In model B, a longer maturation time also favors parasites due to more frequent mutations from replicases to parasites. In SI Appendix, Figs. S20–S23, we examine how these coexistence diagrams depend on stirring. As detailed above, stronger stirring typically favors replicases, although it can also lead to extinction if $K/d$ is sufficiently small due to the finite number of compartments. In SI Appendix, Figs. S25 and S26, we observe that generalist parasites frequently take over specialist parasites. In SI Appendix, Fig. S27, we explore the case where mutant replicases acquire a better resistance to parasites but do not improve their replication rate. In SI Appendix, Figs. S28 and S29, parasites acquire generality but their replicability decreases; nevertheless, generalist parasites remain favored.

We observe that weaker stirring tends to favor parasites, as concluded in ref. 37 and detailed above in the present work. We recover the complete pooling limit for $s\to 1$. If the system has a strong compositional memory, dilution only weakly affects the system, whereas mutations occur even if the compartment is not diluted. With a weaker stirring, parasites are favored (as predicted in ref. 37), because parasites would less likely be isolated from replicases.

When the number of compartments is small enough, we may observe extinction of the whole population in the regime of strong dilution, provided that stirring is also strong enough. Under weak dilution, the heterogeneity in the number of individuals per compartment allows some compartments to contain more than one individual even if $K/d$ is smaller than $1$.

Discussion

In this work, we investigated the role of transient compartmentalization on the coevolution of parasites and replicases. The theoretical framework we have built is independent of the nature of the compartments, which may be for instance membrane-less aggregates, coacervates, or droplets. We find that the capacity of compartments to inherit from their past compositions (compositional memory) matters for the dynamics of these early molecular systems in agreement with the idea of compositional genomes or compositional memory (20–23).

We note that our experimental RNA replication system relied on a modern translation machinery, and replication occurred using encoded proteins. Such a complex replication scheme is unlikely to have existed in early stages of the origins of life. Unfortunately, more primitive RNA-based systems, such as ribozymes capable of sustained coreplication of replicases and parasites, are not yet available. The development of such systems would enable a more direct test of our theoretical model in a prebiotically relevant context. Nevertheless, the experimentally observed replicase-parasite dynamics were generally consistent with theoretical predictions based on simpler RNA replicators (Fig. 2D and 3A), suggesting that underlying compartmental effects may be broadly applicable. In addition, our findings may provide insights into the development of evolvable artificial cell systems in the context of bottom–up synthetic biology (47, 48). Genomic replication has been reconstituted through multiple mechanisms in diverse compartment types, including water-in-oil droplets, liposomes, and liquid-liquid phase-separated droplets, and compartmentalization has been shown to suppress parasite amplification (49–51). Although long-term coevolution of replicases and parasites has so far been demonstrated only in specific systems (34, 38), similar dynamics may be achievable in other platforms if compartment mixing is appropriately controlled.

Compartments allow the coevolution of replicases and parasites by preventing a complete takeover by parasites. It results in complex dynamics, with oscillations between four species that we can predict theoretically and that significantly differ from the behavior expected in bulk conditions. Dilution and carrying capacity play a central role in the ability of replicases to be maintained in the population during evolution. Transient compartmentalization dynamics are only possible if the system undergoes mixing steps, where parasites may be spatially isolated from replicases, resulting in their inability to replicate. Compositional memory is observed experimentally, and its effect on the predominance of parasites has been assessed. We also quantified the relation between the strength of stirring and the mixing of compartments, thanks to a combination of experiments and modeling. In future work, it will be important to further investigate this issue by systematically tracking the composition of individual compartments in experiments with compartmentalized RNAs. We hope that our work will guide further experiments in this field, as more complex and specific interactions between RNA molecules, as well as between RNA and proteins, will need to be included in future extensions of this work.

Materials and Methods

In this section, we detail experimental conditions and provide theoretical descriptions regarding the evolution of molecular replicators undergoing transient compartmentalization dynamics.

RNA Molecules.

All RNA molecules (HL1-, HL2-, PL2-, and PL3-228) were prepared by in vitro transcription. First, the DNA template for each RNA was amplified by PCR using the corresponding plasmid obtained previously (34). The resulting DNA products were digested with Sma I (Takara), transcribed in vitro using T7 RNA polymerase (Takara), and treated with DNase I (Takara). The transcribed RNAs were purified using the RNeasy Mini Kit (Qiagen).

Compartmentalized RNA Replication System.

The RNA replication system was prepared by mixing an arbitrary set of RNA molecules in 10 $\mathrm{\mu }$L of the Escherichia coli reconstituted translation system (52), as used previously (34). To encapsulate the replication system in water-in-oil droplets, the 10 $\mathrm{\mu }$L mixture was stirred in 1,000 $\mathrm{\mu }$L of buffer-saturated oil at various speeds for 1 min on ice using a homogenizer (Polytron PT-1300D, Kinematica). The buffer-saturated oil was prepared as follows. An oil phase composed of 95% mineral oil (Sigma-Aldrich), 2% Span 80 (Nacalai Tesque), and 3% Tween 80 (Fujifilm) was mixed with 1/20 vol. of the translation system buffer, lacking NTPs, tRNA, and translation proteins but containing an increased concentration of dithiothreitol (36 mM). After vigorous mixing, the oil mixture was incubated at 37 ^°C for 10 min, centrifuged at 20,000 g for 5 min at room temperature, and the upper oil phase was collected as buffer-saturated oil.

Long-Term RNA Replication Experiment.

The RNA replication system was prepared with 10 nM each of HL1-, HL2-, PL2-, and PL3-228. The mixture was emulsified in buffer-saturated oil at 16 krpm for 1 min to generate water-in-oil droplets. The droplets were incubated at 37 ^°C for 5 h to induce RNA replication coupled with protein translation, completing one replication round. For the next round, 200 $\mathrm{\mu }$L of the droplet suspension was transferred to a new mixture containing 800 $\mathrm{\mu }$L of buffer-saturated oil and 10 $\mathrm{\mu }$L of the translation system, and homogenized as above. The new droplets were then incubated again at 37 ^°C for 5 h. These cycles were repeated for 22 rounds. RNA concentrations were determined at the end of each cycle (after $5$h incubation) by intercalating dye-based RT-qPCR using One Step TB Green PrimeScript PLUS RT-PCR Kit (Takara) and sequence-specific primers (SI Appendix, Table S3). The primers were designed to specifically amplify each RNA species, and sequence mismatches with nontarget RNAs prevent cross-amplification (SI Appendix, Table S4). RNA concentrations were quantified using dilution series of each RNA.

Competition Assay.

The RNA replication system containing 0.05 nM HL2-228 and 0.15 nM PL2-228 was used to prepare water-in-oil droplets via homogenization at 16 krpm for 1 min as described above. To vary the dilution conditions, 10, 40, 200, or 500 $\mathrm{\mu }$L of the droplet suspension was mixed with 990, 960, 800, or 500 $\mathrm{\mu }$L of buffer-saturated oil and 9.9, 9.6, 8, or 5 $\mathrm{\mu }$L of the translation system, respectively. The mixtures were then stirred for 1 min at 2, 5.2, 16, or 30 krpm using the same homogenizer. After incubation at 37 ^°C for 5 h, RNA concentrations were measured by RT-qPCR as described above using sequence-specific primers (SI Appendix, Table S3).

Evaluation of Content Mixing in Droplet Populations.

Fluorescein isothiocyanate (FITC)-labeled dextran (FITC-dex, G) and tetramethylrhodamine isothiocyanate (TRITC)-labeled dextran (TRITC-dex, R), with average molecular weights of 10 kDa and 155 kDa, respectively, were purchased from Sigma-Aldrich. Alexa Fluor 647-labeled dextran (Alexa-dex; 10 kDa) was purchased from Invitrogen. Using these fluorescent dextrans, two types of water-in-oil droplets were prepared: droplets 1 containing 3 wt% FITC-dex and droplets 2 containing 3 wt% TRITC-dex, as described above but using a translation buffer lacking protein components. To examine droplet fusion induced by mechanical agitation, 100 $\mathrm{\mu }$L each of droplets 1 and 2 were mixed with 800 $\mathrm{\mu }$L of buffer-saturated oil and 10 $\mathrm{\mu }$L of the translation system buffer containing 0.01 wt% Alexa-dex. The droplet mixtures were then stirred for 1 min at 2, 5.2, 16, or 30 krpm, as described above. The droplet populations were then incubated at 37 ^°C for 5 h, completing the first round of the experiment. For the second round, new droplet populations were prepared in the same manner using 200 $\mathrm{\mu }$L of each postincubated droplet suspension in place of droplets 1 and 2, followed by incubation at 37 ^°C for another 5 h. At 5 h of rounds 1 and 2, aliquots were collected for microscopic observation. Fluorescence imaging was performed using an FV3000 confocal laser-scanning microscope (Evident) equipped with a 100$\times$ oil-immersion objective at room temperature. FITC, TRITC, and Alexa Fluor 647 were excited at 488 nm and 561 nm, and 640 nm, respectively. Images were analyzed using ImageJ (NIH). In merged triple-channel images, signals with circularity $\ge$0.7 were identified as individual droplets, and the mean intensities of each fluorophore were recorded for every droplet. Background intensities were calculated as the mean fluorescence of droplet-free regions. Misidentified droplets (based on transmitted light images) were manually excluded. To focus on “active” droplets containing either FITC or TRITC fluorophore, droplets with both FITC and TRITC intensities below 105% of their respective background intensities were omitted as unreliable detections. The ratio of FITC to TRITC intensity was then calculated for each remaining droplet after background subtraction. Droplet size was estimated by assuming circular geometry.

Perfectly Homogeneous Limit.

Let us consider a pool of replicators, with a fraction $x_k$ of replicases and $z_k$ a fraction of parasites. In the inoculation step, compartments are created with a total population drawn from a Poisson distribution of parameter $\lambda$, and a number for each species drawn from a multinomial distribution of parameters $x_k$ and $z_k$. Therefore, the probability to have a compartment with $m_k$ replicases of type $k$ for $0\le k\le K$ (modeled by a vector $\mathit{m}$) and $y_l$ parasites of type $l$ for $0\le l\le L$ (modeled by a vector $\mathit{y}$) is

$$\begin{array}{cc}\hfill {\mathbb{P}}_{\lambda ,\mathit{x},\mathit{z}}\left(\mathit{m},\mathit{y}\right)& =e^{-\lambda }\frac{{\lambda }^n}{n!}{\mathbb{P}}_{\mathit{x},\mathit{z}}\left(\mathit{m},\mathit{y}\left|{\displaystyle \sum _k}{m}_k+{\displaystyle \sum _l}{y}_l=n\right.\right)\hfill \\ \hfill & =e^{-\lambda }\frac{{\lambda }^n}{n!}\frac{n!}{m_0!...m_K!y_0!...y_L!}{x}_0^{m_0}...x_K^{m_K}{z}_0^{y_0}...z_K^{y_K}\hfill \\ \hfill & =e^{-\lambda }\frac{{\lambda }^n{x}_0^{m_0}...x_K^{m_K}{z}_0^{y_0}...z_K^{y_K}}{m_0!...m_K!y_0!...y_L!},\hfill \end{array}$$ [4]

where $n=\sum _k{m}_k+\sum _l{y}_l$. This step is a significant source of stochasticity and heterogeneity among compartments and the only one considered in the deterministic approach.

Maturation (model A).

A simple model consists in assuming that the sequence of the replicases mutates with time, in which case we can index the mutations by an integer $k$ (similar mutants are grouped together). We also suppose that new types of parasites can appear. The fitness of replicases corresponding to mutant $k$ replicating with replicase $j$ is ${\alpha }_{j,k}$, the fitness of parasites of type $l$ replicating with mutant $j$ is ${\gamma }_{j,l}$. With ${\mathit{e}}_i$ the vector of $0$ except at entry $i$, we can write the master equation for the probability of having $\mathit{m},\mathit{y}$ at time $t$ in a given compartment in Eq. 5 (SI Appendix, Fig. S1).

$$\begin{array}{cc}\hfill \frac{d{P}_t\left(\mathit{m},\mathit{y}\right)}{\mathit{dt}}=& {\displaystyle \sum _{i,j}}\left(m_i{\alpha }_{i,j}(m_j-1)P_t\left(\mathit{m}-{\mathit{e}}_j,\mathit{y}\right)+m_i{\gamma }_{i,j}(y_j-1)P_t\left(\mathit{m},\mathit{y}-{\mathit{e}}_j\right)\right)\hfill \\ \hfill & +{\displaystyle \sum _i}\left({\mu }_i(m_i+1)P_t\left(\mathit{m}-{\mathit{e}}_{i+1}+{\mathit{e}}_i,\mathit{y}\right)+{\nu }_i(y_i+1)P_t\left(\mathit{m},\mathit{y}-{\mathit{e}}_{i+1}+{\mathit{e}}_i\right)\right)\hfill \\ \hfill & -{\displaystyle \sum _{i,j}}\left(m_i{\alpha }_{i,j}{m}_j+m_i{\gamma }_{i,j}{y}_j+{\mu }_i{m}_i+{\nu }_i{y}_i\right)P_t\left(\mathit{m},\mathit{y}\right).\hfill \end{array}$$ [5]

The system evolves according to this master equation until the population reaches the carrying capacity of the compartment. The carrying capacity can be thought of as the total number of molecules or as the total number of nucleotides (it depends on the molecular lengths in this case) that can be produced. The latter effect is explored in SI Appendix, section 11. The processes that can occur during maturation are detailed in SI Appendix, Fig. S1. The model is similar to that of ref. 31 except that it accounts for the possible coevolution of replicases and parasites through mutations.

Maturation (model B).

In this model, we assume that parasites appear from deletions in the sequence of replicases (by recombination for instance (34)), yielding smaller sequences unable to replicate by themselves. The notations remain the same in this model. The mutations rates of the evolution equations of the maturation phase are different, and we obtain the master equation in Eq. 6 (SI Appendix, Fig. S2). The replication dynamics are the same in both models and we typically have that $m_0(t)=m_0/(1-{\alpha }_{0,0}{m}_{0}t)$ during this phase. In the following, we will study the differences that result from the choice of the model.

$$\begin{array}{cc}\hfill \frac{d{P}_t\left(\mathit{m},\mathit{y}\right)}{\mathit{dt}}=& {\displaystyle \sum _{i,j}}\left(m_i{\alpha }_{i,j}(m_j-1)P_t\left(\mathit{m}-{\mathit{e}}_j,\mathit{y}\right)+m_i{\gamma }_{i,j}(y_j-1)P_t\left(\mathit{m},\mathit{y}-{\mathit{e}}_j\right)\right)\hfill \\ \hfill & +{\displaystyle \sum _i}\left({\mu }_i(m_i+1)P_t\left(\mathit{m}-{\mathit{e}}_{i+1}+{\mathit{e}}_i,\mathit{y}\right)+{\nu }_i(m_i+1)P_t\left(\mathit{m}+{\mathit{e}}_i,\mathit{y}-{\mathit{e}}_i\right)\right)\hfill \\ \hfill & -{\displaystyle \sum _{i,j}}{m}_i\left({\alpha }_{i,j}{m}_j+{\gamma }_{i,j}{y}_j+{\mu }_i+{\nu }_i\right)P_t\left(\mathit{m},\mathit{y}\right).\hfill \end{array}$$ [6]

Modeling the Dye Experiment.

Let us call the initial distribution of fraction $x$ of G in the droplets $p_{\mathit{ini}}(x)$. After stirring, we write the cumulative probability of having a fraction $\widehat{x}$ lower than $x$ in a compartment $k$, using the mean field approach described above. We denote $G_k$ the rescaled amount of G in compartment $k$, $N_k$ the total rescaled fluorescence in compartment $k$, $n_c$ the number of interacting compartments and $N$ the total population of the system. We also assume that instead of taking the fraction of G as a single random variable, we have the following rules for mixing

$$\begin{array}{cc}\hfill {\widehat{N}}_k& =(1-s)N_k+s{\displaystyle \sum _{l\ne k}}\frac{N_l}{n_c-1},\hfill \\ \hfill {\widehat{G}}_k& =(1-s)G_k+s{\displaystyle \sum _{l\ne k}}\frac{G_l}{n_c-1}.\hfill \end{array}$$ [7]

Naturally, this transformation conserves the numbers of molecules of each type because $〈{\widehat{N}}_k〉=〈N_k〉:=n$ (with SD ${\sigma }_N$) and $〈{\widehat{G}}_k〉=〈G_k〉:=g$ (with SD ${\sigma }_G$). Note also that the random variables $N_k$ and $G_k$ are correlated due to the fact that $N_k=G_k+R_k$. The cumulative probability distribution of the updated fraction after one round of mixing is

$$\begin{array}{c}\hfill \mathbb{P}\left(\frac{{\widehat{G}}_k}{{\widehat{N}}_k}<x\right)=\mathbb{P}\left((1-s)(G_k-x{N}_k)<Y\right),\end{array}$$ [8]

with $Y=\frac{s}{n_c-1}\sum _{l\ne k}(x{N}_l-G_l)$.

Now, we can approximate the sums $\sum _{l\ne k}{G}_l$ and $\sum _{l\ne k}{N}_l$ as normally distributed, using the central limit theorem in the limit of a large number of compartments. Therefore, accounting for the fact that $G_l$ and $N_l$ are correlated we obtain

$$\begin{array}{c}\hfill Y\sim \mathcal{N}\left(s(xn-g),\frac{s^2}{n_c-1}\left(x^2{\sigma }_N^2+{\sigma }_G^2-2x\text{Cov}(N_l,G_l)\right)\right).\end{array}$$ [9]

For which we can use the measured values of SDs and covariances (from experimental distributions). If we assume that the distribution is initially bimodal in $G_k$ (initially the compartments are either full of G, or contain no G), we can also compute the statistics of $G_k$ as a function of the experimental statistics of $N_k$ : $g=n/2$, ${\sigma }_G=\sqrt{{\sigma }_N^2/2+n^2/4}$ and $\text{Cov}(G_k,N_k)={\sigma }_N^2/2+n^2/4$. Now, we can write the cumulative distribution probability, using the law of total probability:

$$\begin{array}{cc}\hfill \mathbb{P}(\frac{{\widehat{G}}_k}{{\widehat{N}}_k}<x)& =\mathbb{P}((1-s)(G_k-x{N}_k)<Y\left|N_k=\nu \right.)\hfill \\ \hfill & ={\int }_{-\infty }^{+\infty }dy{p}_Y(y)\sum _{\nu =0}^{\infty }{p}_N(\nu )\mathbb{P}\left(G_k<x\nu +\frac{y}{1-s}\left|N_k=\nu \right.\right).\hfill \end{array}$$ [10]

This formula depends on the initial cumulative probability, SDs, and covariance of the distributions of $N_k$ and $G_k$. In Fig. 4B, we used initial experimental distributions to obtain these values and compute the theoretical cumulative distribution after stirring for different values of $s$ (shown in solid lines). In the particular case where we start from a perfectly bimodal distribution for the conditional probability, we have $p_{G,ini}(\gamma |N_k=\nu )=({\delta }_{\gamma ,0}+{\delta }_{\gamma ,\nu })/2$, and the experimental distribution for $N$. Similarly we express the cumulative distribution function of $\left|{\widehat{G}}_k/{\widehat{N}}_k-1/2\right|$ in SI Appendix, section 2. A closed expression for the cumulative distribution of the fraction of $\left|{\widehat{G}}_k/{\widehat{N}}_k-1/2\right|$ follows after one round of stirring that is compared with the experimental data in Fig. 4B.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Main study data are included in the article and/or SI Appendix. All study data and codes used for simulations are made available in a GitHub repository (53).