Background and Motivations

In research concerning the origins of life, many efforts focus on understanding of the very first steps which led to the emergence of organic compounds and metabolic pathways in pre-biotic chemical solutions. A less investigated issue regards a crucial evolutionary stage, i.e., the spontaneous compartmentalization of both early biochemical reactions and related metabolites into cell-like structures. This is a crucial aspect, as high concentration levels of metabolites are needed to get a reasonable probability for the occurrence of biochemical reactions.

A widely accepted hypothesis assigns to semi-permeable lipid vesicles (liposomes) the role of hosting the very first metabolic pathways, thus acting as precursors of the living cells. A strong argument supporting this hypothesis comes from the well known behavior exhibited by (amphipathic) lipids when they are put in water solutions. In this setting, lipids aggregate to form liposomes, a spontaneous process driven by hydrophobic forces [1]. Even though the mechanism of vesicle formation is well assessed, a critical issue that needs to be clarified concerns the mechanisms by which components of metabolic pathways can reach concentration levels that can be much higher inside the vesicles than outside.

Simple liposomes cannot actively modify their content because they are not equipped with the structures (e.g. trans-membrane channels) needed for managing the exchange of solutes with the surrounding environment. Thus, the anomalous concentration levels, found in liposomes, of the components of the early metabolic pathways must have occurred as a spontaneous process, i.e. as the result of a chemical and physical self-organization. Actually, trans-membrane channels are also a manifestation of self-organization, but here we refer to a form of self-organization that is, in some sense, simpler or, in other words, more homogeneous in space and time, with respect to the complex structures emerging in the cell dynamics.

To address this issue, Luisi et al. [2–4] performed an elegant series of experiments consisting of the direct observation of the encapsulation of solutes inside lipid vesicles (liposomes) forming in an aqueous environment. These experiments produced surprising and intriguing outcomes regarding the spontaneous self-organization of vesicles containing an unexpectedly high amount of encapsulated molecules. In particular, the authors found that, when lipid surfaces close up in a solute-containing solution to form vesicles, the entrapment frequency does not follow the expected Poisson distribution, but tends to assume a power-law behavior, characterized by many empty vesicles (no or very few trapped molecules), and a long decreasing tail with extremely crowded vesicles. This effect has been denoted as "superconcentration" [2–4] and, in the following, will be referred to as "anomalous solute crowding".

In this paper we propose a modeling approach for gaining some insights into the physical mechanisms underlying these experimental results. In particular we show that describing the system in the framework of the theory of renewal processes [5], and applying two simple hypotheses, the experimental behavior observed by Luisi et al. [2–4] is straightforwardly reproduced.

In the next subsection we will provide a more detailed description of both the experimental setting and the "anomalous solute crowding" phenomenon reported in [2–4], underlining that the emergence of a power-law non-Poisson distribution involve a strong cooperative mechanism. We then proceed in the "Methods" section by reviewing the connection between renewal point processes and self-organized (cooperative) systems. In particular, the subsection "Mathematical aspects of renewal theory" is devoted to a brief sketch of the basic mathematical aspects of renewal point processes that will be used in the next sections. In the "Results" section we introduce our model and we discuss how the proposed model can help in shedding some light on the observations of Luisi et al. [2]. In the last section "Conclusions" we discuss the results, also suggesting new possible experimental validation of the model predictions.

The experimental evidences

As sketched in the previous subsection, the experimental setting used by Luisi et al. [2] regards the formation of lipid vesicles in water solution containing a macromolecular solute. In particular, vesicles composed of 1-palmitoyl-2-oleoyl-snglycero-3-phosphati-dylcholine (POPC) or POPC/cholesterol were allowed to form in an aqueous solution containing soluble macromolecules such as ferritin. In this setting the liposomes, while forming, can trap the solute macromolecules. Once vesicles are formed, the encapsulated particles cannot flow out, due to the impermeability of the lipid membranes with respect to the water-soluble macromolecules. For the same reason, such macromolecules cannot cross liposome membranes from the outside. Thus, the amount of solutes present inside the vesicles after they have closed must necessarily have flown inside before the closure of the liposome membrane. Thus, the analysis of the liposome contents can be used to obtain realistic insights into the physics of solute encapsulation. In order to detect the molecular contents of each vesicle at the single molecule level, Luisi et al. used cryo-TEM, which is a very reliable method for imaging lipid vesicles. Cryo-TEM overcomes many limitations of the TEM instrumental technique, such as the presence of drying artifacts. Both ferritin molecules and ribosomes can easily be detected by cryo-TEM and counted individually and, thus, were used as solutes in the experiments. Unexpectedly, Luisi et al. noticed that lipid vesicles can spontaneously capture a very high number of macromolecular solutes, even when forming in diluted solutions. Thus, as a result of this unexpected behavior, a certain number of solute-rich vesicles, i.e., with anomalously high concentration levels of the solute, showed up in the chemical solution, and the overall distribution of solute particles trapped into liposomes was found to decay as an inverse power-law function (see Figure 1).

From a statistical point of view, the simplest assumption is that of a sequence of independent random trapping events of the solute molecules into the liposomes. Under this assumption, the entrapment of solute particles into liposomes of volume V (into a well stirred solution containing a solute S with concentration C₀) can be considered equivalent to a series of independent random samplings of the S particles. It is then expected that, on average, each vesicle will contain Nµ = NA V C₀ molecules, (with NA being the Avogadro number). Hence, under the assumption of independent trapping events and in absence of any other cooperative mechanism, the amount N of the solute S in the liposomes can be described by a Poisson distribution.

On the contrary, as shown in Figure 1, the direct observation of liposomes after solute entrapment revealed that the majority of vesicles (> 80%) were empty, while about 0.1% − 1% of vesicles contained a very high number of solutes (N > 20, up to ca. 300), in clear contrast to the prediction of the Poisson law and in agreement with a power-law decay. This means that lipid vesicles can spontaneously capture a very high number of macromolecular solutes, thus producing an intra-vesicular concentration of particles that is significantly higher with respect to the surrounding aqueous solution. Noticeably, this "anomalous solute crowding" occurs also when lipid vesicles form in diluted solutions. This finding overcomes the problem arising in prebiotic chemistry of explaining how, despite the low environmental concentration of solutes, proto-cells might have enclosed in their aqueous core enough molecules for their metabolic needs.

The power-law non-Poisson profile of the solute distribution is a signature of cooperative dynamics. As the system is clearly characterized by critical closure events marking the final step of vesicle-forming cooperative dynamics, the use of stochastic point processes to describe this system seems to be a natural choice.

In the next section we discuss the renewal character of the point process describing the dynamics of ferritin trapping into forming lipid vesicles, trying to give a qualitative interpretation in terms of cooperation among different components of the aqueous solution: water molecules, lipids, macromolecules of solute (e.g., ferritin). We also sketch the general connection between complex cooperative systems and renewal events.

Renewal condition and cooperation

In the last few years renewal processes are becoming increasingly popular for describing important aspects of the behavior of complex systems [5–12]. These systems are characterized by the presence of cooperative or self-organized structures that emerge from the strong non-linearities of the underlying microscopic dynamics. Cooperative structures are coherent and meta-stable, i.e., they have relatively long lifetimes, during which the emerging structure maintains essentially unaltered some basic features that make the structure itself "recognizable", until a critical (shorttime) transition event occurs at some random time [13–15]. After that, a fast drop of the structure features (memory and spatial topology) occurs. Consequently, the renewal process can be identified as a birth-death process of cooperation, thus defining the sequence of transition events determining the decay of a coherent, long-lived structure and, possibly, the quasi-concurrent emergence of a new one.

A renewal process is rigorously defined as a sequence of events randomly occurring in time, without any dependence from the previous events or from external forces, and it is described as a sequence of event occurrence times {t _n } (with n ∈ N ⁺), enjoying the property that the Waiting Times (WTs) between two subsequent events τ _n = t _n − t_n−1 are mutually statistically independent random variables.

In this work, the application and consequent interpretation of the renewal assumption is somewhat different with respect to the above described cases and closer to the definition of renewal process that can be found in the first pages of Cox's book [5]. Here the renewal process is again associated with a cooperative behavior, but the emergence of the self-organized structure is not associated with a transition event dynamically connecting one meta-stable structure to another. In the case under study, the cooperative interactions among lipids are given by the hydrophobic forces that give rise to a spheroidal self-organized structure, i.e., the liposome. The cooperative behavior extends also to the solute molecules. Each solute molecule interacts with the lipids during liposome formation and possibly with other solute molecules, in such a way that the number of solute molecules trapped in the liposomes are distributed according to an inverse power-law.

Notice that if there was no cooperation at all time and space scales, i.e., if there were no lipid-lipid interactions (liposome formation) neither ferritin-lipide and/or ferritin-ferritin interactions, totally independent events could be assumed. In this very simplified case, for any arbitrary chosen spatial volume, the statistical distribution of the number of both ferritin molecules and lipids in a given should be a Poisson one. As this is not the case, cooperation must emerge from some non-linear interactions among molecules.

Firstly, the "total independence" assumption is not compatible with liposome formation, which requires cooperation among lipid molecules, that is actually triggered by lipid-water interactions, i.e., by hydrophobic forces. Consequently, a first level of cooperation must be assumed and it is quite evident in the spontaneous emergence of structures such as liposomes.

However, the above observation is not sufficient to justify the anomalous solute crowding, which requires at least another cooperative force: a lipid-ferritin strong interaction or ferritin-ferritin (e.g., clustering) or both of them must be assumed.

Alternatively, the power-law distribution could be the consequence of an anomalous diffusion process of ferritin molecules inside water coupled with some clustering mechanism, which could depend only on ferritin itself or, on the contrary, on both ferritin and lipids. Note, however, that this hypothesis would again imply some kind of cooperation, but beetween ferritin and water, so that self-organization would yet emerge as a consequence of cooperative forces.

Here we limit to the first assumption, i.e., the lipid-ferritin interaction, and we show that this assumption is sufficient to derive the observed power-law decay in the overall distribution of trapped solute molecules.

It is important to underline that the model here proposed describes the average behavior of the system, without any explicit hypothesis on the chemical and physical microscopic mechanisms, but only resuming general principles on self-organized complex systems.

In the next subsection some mathematical definitions and results regarding renewal point processes are briefly reviewed. These results are exploited in the next section, where the proposed trapping model is introduced and discussed.

Mathematical aspects of renewal theory

Here we briefly recall the basic mathematical concepts of renewal theory. The critical closing event separates the time axis into a pre-event time period with an evolving number of ferritin molecules N (s), s < τ and a post-event period with a stationary, equilibrium, condition: N (s) = N _∞ , s > τ .

being Ψ(0) = P r{τ > 0} = 1.

so that, comparing with Eq. (5), it results that the event rate is constant in time: r(t) = r _p [5]. This is in agreement with the well-known property that, in a Poisson process, the mean number of events in a given time interval [t, t+Δt] is proportional to the length of the time interval itself: $〈N_p\left[t,t+\mathrm{\Delta }t\right]〉=r_p\mathrm{\Delta }t.$

Model assumptions

The lipid-solute interaction, discussed in the previous Section, is encoded in the capacity of the liposome to trap solute molecules, i.e., in the flux rate of molecules entering into the forming liposome through its open borders. Given the reasonable assumption that the probability of trapping is proportional to the area of the open surface of the liposome (permeability), the flux rate of solute turns out to be proportional to the closing rate r(τ ) of the liposome borders, being τ the time elapsed since the beginning of the liposome formation. The ferritin-lipid interaction hypothesis implies that the closing rate and, consequently, the ferritin flux rate, is affected by the passage of ferritin itself.

It is worth noting that the assumption of a ferritin-lipid interaction affecting the liposome closing rate is not incompatible with a possible anomalous diffusion of ferritin and/or lipid molecules inside water coupled with some clustering mechanism. However, this assumption needs a experimental validation and, consequently, the setup of new experimental work and will be the subject of future investigations.

We also note that, if the motion of solute particles in the aqueous solution was completely random, the net flux rate of solute on the liposome open borders would be zero. As a consequence, an asymmetric mechanism driving the flux rate must be assumed in order to get an average net flux of solute inside the liposome.

The asymmetric mechanism should involve a interaction ferritin-ferritin (clustering) and/or ferritin-lipid that changes depending on the position of the ferritin (inside or outside the forming vesicle). This asymmetry must determine a probability, for a ferritin molecule, of moving from the external environment into the vesicle greater than the opposite direction. Between these two hypotheses, the ferritin-lipid interaction might be more realistic. Indeed, soluble proteins such as ferritin do not clusterize spontaneously because of the composition of their surface which makes the interactions with water molecules thermodynamically more favourable than the protein-protein interaction.

In summary, our model is grounded on two hypotheses:

Then, the applicability of the renewal condition is exactly the same as in the Cox book [5] by substituting the failure events of some electronic devices, described there, with our closing events of some liposomes. In fact, under the assumptions (i) and (ii), the statistical ensemble of liposomes becomes a set of statistically independent realizations, so that both closing events and the associated closing times τ elapsed from the beginning of the experiment are mutually independent random variables and linear averaging can be applied to define mean quantities.

In the framework of renewal point processes, this model seems to be the minimal one explaining the emerging power-law behavior.

In fact, if we do not make this minimal assumption, the dynamics of lipids and ferritin would be independent and the flux of ferritin molecules across the liposome borders would be totally random, thus giving rise to a Poisson distribution of trapped ferritin molecules. Note that the renewal condition is associated with the emergence of an asymptotic self-organized structure (closed liposome + trapped ferritin molecules) following the occurrence of the closing event, i.e., the crucial event after which no flux of ferritin across the liposome surface is no longer possible. The renewal condition is then a natural hypothesis for the closing event, as the final state given by the asymptotic structure is well defined, but there's no way to know the exact time evolution that brought the system dynamics towards that particular asymptotic, stationary state.

In the next subsection we will show how, starting from these two very basic phenomenological hypotheses, it is possible to derive the observed power-law behavior.

Derivation of the model

Given the assumptions and hypotheses stated above, we will deal with the problem of finding the form for the event rate r(t) which is consistent with the experimentally observed power-law behavior for the distribution of ferritin molecules trapped inside the liposomes.

being N (t) the number of ferritin molecules trapped inside the liposome at time t. This is almost a natural choice, as we require a rate function that must slow down when N (t) increases, while the unit in the denominator avoid singularities at t = 0. r₀ is a dimensional constant representing the closing rate of the liposomes in absence of ferritin. N (t) is not a deterministic function, monotonically increasing with time, rather it is a fluctuating random variable with an average tendency to increase. In other words, the semi-permeability hypothesis imposes that the average number of ferritin molecules $〈N〉$(t) is a increasing function of time t. The rate r _c (t), given in Eq. (9), is also a random variable, as it depends on the random variable N (t). Essentially, r _c (t) is the closing rate associated with the stochastic dynamics of a single liposome interacting with the surroundings solute molecules (ferritin and lipids), while, under the renewal assumption, the average rate $〈r_c〉$(t) describes the average behavior of the total ensemble of independent systems. Each system is composed by a liposome approaching the closing event and by the surrounding ferritin molecules randomly passing through its surface.

with $〈\mathrm{\Delta }N〉\left(t\right)=0$ by definition. In summary, the average closing rate $〈r_c〉$ of a given liposome depends on $〈N\left(t\right)〉$, i.e., on the average number of solute particles present in the vesicle at time t. We note that, for the above derivation of Eq. (14) it is sufficient to assume a negligible fluctuation ΔN (t) for large times t, without resorting to the assumption of a Poisson process for N (t). On the contrary, the semi-permeability hypothesis is a fundamental one, as it allows to derive a simple linear expression for N (t). This hypothesis simply states that, while liposomes are forming, the flux of solutes per unit of time from the environment towards the inside of the vesicles (λ _in ) is greater than the flux in the opposite direction (λ _out ). It is reasonable to assume that the relative slowdown of the outward flux can be due to some interactions between the solute molecules and the lipids composing the inner face of the vesicle. Thus, there is a net flux of solutes per unit of time (λ = λ _in − λ _out ) directed towards the inner part of the forming vesicle. As a consequence, making an additional, but reasonable, linear assumption, we get:

Note that both Eqs. (8) and (16) become a purely inverse power-law distributions in the limit of large times: t >> 1/λ = T .

The derivation of the expression for P (N ) and the relationship given in Eq. (19), relating the exponent of P (N ) with the parameters of the closing rate function (r _c )(t) represent the main result of this work. As a preliminary validation, we made a fit with data taken from the paper of Luisi et al. [2]. Fitting Eq. (18) with the data, we obtain the best agreement for µ = 2.3, as shown in Figure 2.