Abstract

Synthetic or semi-synthetic minimal cells are those cell-like artificial compartments that are based on the encapsulation of molecules inside lipid vesicles (liposomes). Synthetic cells are currently used as primitive cell models and are very promising tools for future biotechnology. Despite the recent experimental advancements and sophistication reached in this field, the complete elucidation of many fundamental physical aspects still poses experimental and theoretical challenges. The interplay between solute capture and vesicle formation is one of the most intriguing ones. In a series of studies, we have reported that when vesicles spontaneously form in a dilute solution of proteins, ribosomes, or ribo-peptidic complexes, then, contrary to statistical predictions, it is possible to find a small fraction of liposomes (<1%) that contain a very large number of solutes, so that their local (intravesicular) concentrations largely exceed the expected value. More recently, we have demonstrated that this effect (spontaneous crowding) operates also on multimolecular mixtures, and can drive the synthesis of proteins inside vesicles, whereas the same reaction does not proceed at a measurable rate in the external bulk phase. Here we firstly introduce and discuss these already published observations. Then, we present a computational investigation of the encapsulation of transcription-translation (TX-TL) machinery inside vesicles, based on a minimal protein synthesis model and on different solute partition functions. Results show that experimental data are compatible with an entrapment model that follows a power law rather than a Gaussian distribution. The results are discussed from the viewpoint of origin of life, highlighting open questions and possible future research directions.

1 Introduction

The laboratory construction of cell-like systems of minimal complexity is a useful way to explore, by means of model systems, how primitive living cells originated from self-assembly and self-organization processes. Based on the encapsulation of molecules like DNA, RNA, and proteins inside lipid vesicles (liposomes), the research on semi-synthetic minimal cells (here, for brevity, synthetic cells) [19, 26, 27] has been considerably expanded in recent years, especially from the biochemical viewpoint. For example, several important case studies and insights have been reported when simple and complex biochemical reactions have been reconstituted inside vesicles [10, 20, 25].

The realization of transcription (TX) and translation (TL) reactions inside vesicles is one of the main targets of current research. TX-TL reactions bring about the synthesis of a protein from the corresponding DNA sequence, and therefore allow the study of synthetic cells endowed with a nontrivial reaction network. This helps to model some of the processes that could have occurred in primitive cells by using, however, modern molecules. The production of enzymes inside synthetic cells is essential for attempting the reconstruction of key patterns (for example, lipid biosynthesis and consequent vesicle growth and division [13]) or the replication dynamics of RNA [12]. Moreover, synthetic cells incorporating TX-TL machinery might be considered innovative tools for biotechnological applications [16, 37].

The typical experimental approach for assembling such compartmentalized systems is shown in Figure 1 and consists of two distinct steps.

Figure 1. 

Schematic representation of how (bio)chemical systems (e.g., Oparin reaction, enzyme systems, TX-TL) are encapsulated inside lipid vesicles following the strategy described in the pioneering work by Luisi [42] and Deamer [7]. First, vesicles are formed in an aqueous solution containing the solutes that need to be encapsulated, by simply adding membrane-forming compounds (lipids or fatty acids). These will self-assemble in closed membranous shells (vesicles), capturing solutes from the solution. A heterogeneous population of solute-filled vesicles is generally obtained. Note that vesicle formation and solute encapsulation occur simultaneously according to a bottom-up self-organization pattern. After vesicle formation, non-entrapped solutes are generally removed, degraded, or inhibited, so that the reaction can only occur inside vesicles. In the case of TX-TL reactions, the concentration of the messenger RNA or protein is typically monitored. In some cases, one or more reactants are added after the reactant encapsulation, letting them enter the vesicle by passive permeation or by the help of pores [25].

Figure 1. 

Schematic representation of how (bio)chemical systems (e.g., Oparin reaction, enzyme systems, TX-TL) are encapsulated inside lipid vesicles following the strategy described in the pioneering work by Luisi [42] and Deamer [7]. First, vesicles are formed in an aqueous solution containing the solutes that need to be encapsulated, by simply adding membrane-forming compounds (lipids or fatty acids). These will self-assemble in closed membranous shells (vesicles), capturing solutes from the solution. A heterogeneous population of solute-filled vesicles is generally obtained. Note that vesicle formation and solute encapsulation occur simultaneously according to a bottom-up self-organization pattern. After vesicle formation, non-entrapped solutes are generally removed, degraded, or inhibited, so that the reaction can only occur inside vesicles. In the case of TX-TL reactions, the concentration of the messenger RNA or protein is typically monitored. In some cases, one or more reactants are added after the reactant encapsulation, letting them enter the vesicle by passive permeation or by the help of pores [25].

In the first step, the lipid compartment is allowed to form by simply putting together the solutes that need to be entrapped inside the vesicle and the lipids that will form the vesicle membrane. Some simple vesicle preparation methods are particularly apt to model the formation of primitive cells, for example, thin-lipid-film hydration [2], ethanol injection [3], and micelle-to-vesicle transformation [4]. Note that in these preparation methods, vesicle formation and solute entrapment occur simultaneously.

In the second step, the so obtained system, which consists of a population of compartments of different size, is generally treated to remove (or inhibit) the external non-entrapped molecules, and the intra-vesicle reaction is followed in order to investigate the properties and the dynamics of the reaction (e.g., protein folding, yield, reaction rate, interaction with lipids, and confinement effects).

Whereas several intriguing results have been reported with respect to the biochemical characterization of protein-producing vesicles (reviewed in [34, 36]), only minor attention has been paid to some physical aspects related to the generation of synthetic cells from separated components. The mechanism underlying the entrapment of several different solutes inside artificial vesicles is indeed an intriguing phenomenon that can reveal how primitive cells emerged from a mixture of separated molecules.

Considering that the knowledge of how solutes get entrapped during the formation of vesicles is currently very limited, our group, coordinated by P. L. Luisi, recently started a detailed study of the formation of solute-filled lipid vesicles according to the simple methods cited above. The study consists in the analysis of the individual entrapment efficiency of just one solute type or of complex multimolecular mixtures. The idea is to determine the solute occupancy distribution inside vesicles and compare it with the theoretical expectations based on the statistics of random entrapment. In contrast to previous studies, which generally deal with the population-averaged entrapment efficiency (for exceptions, see [8, 17, 40]), we focused on the analysis of the individual vesicle content.

The population-averaged entrapment efficiency is simply defined as the ratio between the amount of entrapped solutes (summed over all vesicles) and the total amount of solute used to prepare vesicles; it does not take into account vesicle diversity [38]. Individual entrapment efficiency is evaluated by comparing the actual and the expected numbers of entrapped solute molecules inside each vesicle. In the absence of specific solute-lipid interactions (null hypothesis), the intra-vesicle solute(s) concentration should correspond to the bulk concentration of the solution employed to prepare vesicles.

Clearly, the determination of individual vesicle content requires non-averaging analytical techniques such as microscopy and flow cytometry. In collaboration with A. Fahr, we applied cryo-transmission electron microscopy for studying spontaneously formed vesicles prepared in the presence of macromolecules like ferritin [18], ribosomes [29], and ribo-peptidic complexes [33]. A detailed analysis has shown that, in contrast to the expectations, in these cases the intra-vesicle solute occupancy distribution does not follow a Poisson curve, but it is rather shaped as a power law, as shown in Figure 2a. The vast majority of vesicles typically encapsulates a limited number of solute molecules, which lies in the range of expected values. However, few vesicles (typically less than 1%) contain a very large number of solutes, so that their intra-vesicle concentration turned out to be up to one order of magnitude higher than the concentration of the solutes in the external solution (Figure 2b). The larger effect was observed in the case of small vesicles (diameter < 100 nm), as also independently reported by [17]. In the most interesting cases, a large part of the vesicle lumen (20–70 vol%) was filled by the solutes, so that the intra-vesicle microenvironment of these few superfilled vesicles corresponds, roughly speaking, to a crowded biological cytoplasm.

Figure 2. 

(a) Comparison between the theoretical (expected) and experimental (measured) solute occupancy distribution, namely, a Poisson bell-shaped curve and a power law. (b) Empty and ferritin-containing liposomes coexist in the same solution. The internal ferritin concentration of the vesicles on the bottom left is about 4 times higher than expected (30 versus 8 μM). Reproduced from [18] with permission from Wiley.

Figure 2. 

(a) Comparison between the theoretical (expected) and experimental (measured) solute occupancy distribution, namely, a Poisson bell-shaped curve and a power law. (b) Empty and ferritin-containing liposomes coexist in the same solution. The internal ferritin concentration of the vesicles on the bottom left is about 4 times higher than expected (30 versus 8 μM). Reproduced from [18] with permission from Wiley.

These results indicate that during the steps of vesicle formation, solutes can actually self-concentrate within the vesicles, and although this unexpected phenomenon occurred only in a small subpopulation of vesicles, it has a quite intriguing relevance in origin-of-life scenarios.

In fact, the superconcentration effect can explain how solute-filled primitive cells could have originated when lipids could self-assemble in closed compartments starting from dilute aqueous solution. The formation of solute-rich—and therefore chemically efficient—cell-like systems could derive from physical mechanisms where the role of lipids and lipid membrane is not limited to passive containment, but might actively contribute to the encapsulation and concentration of molecules. While the mechanisms underlying this behavior are still under investigation, an open question is whether or not this effect is indeed sufficient to trigger otherwise inefficient reactions by concentrating the reactants in the small vesicle volume.

We reasoned that if vesicles form in a dilute reaction mixture, it can happen that whereas the vast majority of vesicles entrap the expected (small) numbers of reactants, few vesicles could be indeed capable of encapsulating and concentrating the reaction mixture above a critical threshold, so that the reaction becomes efficient. Within these vesicles, the interaction among molecules that must recognize each other is much more efficient, and the reaction rate enhanced with respect to that in the external dilute environment. Consequently, this would provide a physical mechanism accounting for the origin of compartmentalized protometabolism in lipid vesicles.

1.1 Spontaneous Concentration of TX-TL Mixture inside Liposomes [35]

In order to test the hypothesis of spontaneous concentration of multimolecular mixtures inside liposomes, the encapsulation of the whole TX-TL machinery has been recently assessed [35]. Two different TX-TL kits, here taken as models of a biochemical network, have been used: (i) Escherichia coli cell extracts, and (ii) the PURE system [31, 32]. The use of the PURE system is particularly advantageous, because its chemical composition is exactly known and can be modified at will, and it is also susceptible of detailed mathematical modeling [6, 15, 21].

The PURE system is composed of ribosomes, tRNAs, RNA polymerase, translation factors, aminoacyl tRNA synthetases, and energy-regenerating enzymes, for a total of about 80 macromolecules and about two dozen low-molecular-weight compounds (nucleotides, amino acids, salts), and operates as shown in Figure 3. The PURE system reactions can be schematically grouped in four modules: transcription (TX), translation (TL), amino acylation of tRNAs (RS), and energy regeneration (EN). When a DNA sequence is added to the PURE system, this multimolecular complex machinery produces the corresponding protein.

Figure 3. 

The four main reactions of cell-free protein synthesis by the PURE system. The four reactions are marked with boxes and drawn schematically. Reproduced from [32], with permission from Elsevier.

Figure 3. 

The four main reactions of cell-free protein synthesis by the PURE system. The four reactions are marked with boxes and drawn schematically. Reproduced from [32], with permission from Elsevier.

The transcription-translation kit can be diluted to a level where protein synthesis takes place at a very low rate (so that the amount of produced protein is so low that it is indistinguishable from a negative control sample where no DNA is added). This system would simulate a primitive scenario where solutes were present in diluted form in a sea or fresh-water lagoon, but were not very reactive due to their low concentration. If, according to the superconcentration effect shown in Figure 2, the PURE system solutes can be concentrate inside liposomes, we expect that protein synthesis would take place only inside liposomes, whereas no observable reaction would take place in the external medium.

This experiment has been indeed carried out [35]. In order to follow the reaction, the enhanced green fluorescent protein (eGFP) was used, and, as shown in Figure 4, it was possible to observe that a few vesicles (1–2%) appear green-fluorescent, whereas the others and the background do not. Note that in this experiment, in contrast to what is shown in Figure 1, the non-entrapped solutes have not been removed. The background is not fluorescent, because the TX-TL reaction occurring in bulk proceeds at such a low rate that the eGFP yield lies below the detection limit, or is actually zero. Control experiments carried out in the absence of DNA (so that no protein can be synthesized at all) gave samples whose fluorescence was equal to the background fluorescence shown in Figure 4. By this experiment it was demonstrated that despite the initially low PURE system concentration, a complex network like the TX-TL coupled reactions spontaneously self-concentrates inside liposomes. By observing the behavior of such a model system it can be argued that it provides a rational physical explanation for the emergence of functional primitive cells.

Figure 4. 

Confocal microscopy images of POPC liposomes prepared by the ethanol injection method in the presence of a dilute PURE system solution (0.65×); vesicles' membrane stained with Nile red. Green-fluorescence channel (ex. 488 nm, em. 505–525 nm; left) reveals only a couple of fluorescent vesicles, those that synthesize eGFP inside; red-fluorescence channel (ex. 543 nm, em. 550–580 nm, right) shows all vesicles present in the sample. Reproduced from [35] with the permission of Wiley.

Figure 4. 

Confocal microscopy images of POPC liposomes prepared by the ethanol injection method in the presence of a dilute PURE system solution (0.65×); vesicles' membrane stained with Nile red. Green-fluorescence channel (ex. 488 nm, em. 505–525 nm; left) reveals only a couple of fluorescent vesicles, those that synthesize eGFP inside; red-fluorescence channel (ex. 543 nm, em. 550–580 nm, right) shows all vesicles present in the sample. Reproduced from [35] with the permission of Wiley.

These experimental observations have been qualitatively explained as follows [35]. During the lipid hydration step (done by the thin film or the injection method), most of the vesicles entrap the PURE system components in such a way that their intra-vesicle concentration is too low to give an experimentally detectable amount of eGFP. In the limiting cases, vesicles could miss one or more components, so that the reactions could not take place at all. On the other hand, the few fluorescent vesicles are those that (i) incorporated all PURE system components, and (ii) concentrated at least some of the PURE system components, so that the internal reaction could proceed much better than the reaction in the external phase. This second condition is necessary because without self-concentration, the amount of produced protein would be as low as in the external phase, and the vesicle fluorescence would be similar to the background. According to what has been recalled in the previous section about the average number of encapsulated molecules, these results strongly suggest that the statistics of solute entrapment significantly deviates from the expectations.

In these circumstances, therefore, the following question becomes central: Considering that previous data on macromolecular entrapment have shown that solute entrapment can be described as obeying a power law, what happens in the case of multicomponent mixtures (like the TX-TL one) encapsulated inside vesicles?

An answer to this question can be obtained by measuring the intra-vesicle concentration of all PURE system solutes, and determining the solute occupancy distributions for each of them. This is currently out of reach from the experimental viewpoint, due to the chemical complexity of the TX-TL kit. It should be recalled here that up to now the direct measurement of solute distributions has been achieved by using solutes that could be visualized well by electron microscopy, such as ferritin and ribosomes.

Another possibility to get insights into this phenomenology is using mathematical models. Deterministic and stochastic in silico modeling is an emerging topic in synthetic cell research [6, 11, 15, 21, 30, 39], and its use for investigating the pattern of primitive cells is well known [22, 23]. When based on realistic physico-chemical parameters, in silico models help and integrate quite well with the experimental approaches, as happens in conventional synthetic biology for the design and synthesis of genetic circuits [1, 9].

In this work we develop a mathematical model that predicts, under simplifying assumptions, the outcome of an experiment based on the entrapment of the PURE system inside lipid vesicles. The goal of the study is to evidence the role of stochastic phenomena in the entrapment process (cf. Figure 1) rather than in the reaction kinetics. By generating solute-filled vesicle populations in silico, we will show how different solute occupancy distributions shape the population of vesicles and generate different patterns of protein production.

2 Mathematical Model

This section focuses on the model used to simulate the eGFP expression in a vesicle population. First, we will introduce the set of ordinary differential equations (ODEs) that describes the time evolution of the protein production both in a bulk aqueous solution and/or in the vesicle aqueous core. Then we will outline the stochastic procedures to distribute the PURE system components among the vesicles, which will be based on two different probability density functions.

2.1 Protein Expression Kinetics

We have recently derived a simple TX-TL deterministic kinetic model for PURE system dynamics [24], based on a previously published work [39]. The PURE system's reactions are shown in Figure 3; the model includes four reactions and two degradations, referring only to the main processes occurring during the protein expression: (1) transcription, (2) RNA degradation, (3) translation, (4) degradation of TL catalysts, (5) aminoacyl tRNA charging, and (6) energy regeneration, as shown by Figure 5. A detailed discussion of the model and the derivation of the numerical parameters can be found in [24]; only a short summary is given here.

Figure 5. 

The TX-TL model is based on six chemical equations that correspond to the main reactions of the PURE system. See Section 2.1 for details.

Figure 5. 

The TX-TL model is based on six chemical equations that correspond to the main reactions of the PURE system. See Section 2.1 for details.

The model reported in Figure 5 is based on two main polymerization reactions, namely transcription (1) and translation (3), whose rates are given as polymerization events per unit of time. The transcription consists in the polymerization of nucleoside triphosphates (NTPs) to give polymerized nucleotides (nt); it is catalyzed by the species TXcat, and it requires the presence of DNA, which acts as a template. The reaction also releases pyrophosphate (not shown). Once synthesized, the polymerized nucleotides species nt can undergo pseudo-first-order spontaneous degradation. Next, the translation consists in the polymerization of aminoacyl tRNAs (ATs) in polymerized amino acids (a), with the release of free tRNA (T). This reaction is catalyzed by the species TLcat, and energy is required to drive the process (NTP is transformed into the exhausted species NXP). Like transcription, translation also requires a template molecule, nt in this case. Importantly, the TLcat species actually represents the set of catalysts that cooperate for the success of the reaction. The fictitious species TLcat is not stable and undergoes spontaneous decay, as reported by [39, 41]. Step 5 concerns the charging of tRNA (T) with free amino acids (A), with energy consumption, and it is catalyzed by the species RScat. Finally, the energy regeneration step (6) converts NXP into NTP, at the expense of the ultimate phosphate donor (creatine phosphate, CP), which is then converted to creatine (C).

The rate equations for the four steps (1-3-5-6) illustrated in Figure 5 are assumed to have the following expression:
formula
where ki represents the rate constant of the ith process (in s−1), [cati] the concentration of the catalyst, [Xj] the concentration of the jth substrate(s) or of the template (according to [39]), and Kij its cognate Michaelis-Menten constant. The two degradation processes (steps 2 and 4) have been instead simply modeled as pseudo-first-order processes (ri = ki,deg [Si]). Table 1 summarizes the 14 chemical species, the 6 rate constants, and the 10 Michaelis-Menten constants introduced by this model. All concentrations and Michaelis-Menten constants are expressed in μM, and consequently the rates ri are in μM/s. It is worthwhile to note that the species initially present as PURE system components are only nine (DNA, TXcat, TLcat, RScat, ENcat, A, T, NTP, CP); they are entered in bold in Table 1 along with their standard concentrations in the PURE system kit.
Table 1. 

List of chemical species and kinetic parameters of the TX-TL model [24].

Molecular speciesInitial conc. (μM)Rate constantMichaelis-Menten constant
SymbolValue (s−1)SymbolValue (μM)
DNA 0.013 kTX 1.99 KTX,NTP 80 
TXcat 0.1 knt,deg 5.8 × 10−5 KTX,DNA 5.0 × 10−3 
TLcat 2.064 kTL 0.27 KTL,AT 10 
RScat 0.16 kTL,deg 2.1 × 10−4 KTL,NTP 10 
ENcat 0.08 kRS 6.20 KTL,nt 340 
nt kEN 100 KRS,A 23 
a   KRS,T 0.7 
A 1000   KRS,NTP 200 
T 1.9   KEN,NXP 40 
AT   KEN,CP 200 
NTP 1500     
NXP     
CP 1000     
    
Molecular speciesInitial conc. (μM)Rate constantMichaelis-Menten constant
SymbolValue (s−1)SymbolValue (μM)
DNA 0.013 kTX 1.99 KTX,NTP 80 
TXcat 0.1 knt,deg 5.8 × 10−5 KTX,DNA 5.0 × 10−3 
TLcat 2.064 kTL 0.27 KTL,AT 10 
RScat 0.16 kTL,deg 2.1 × 10−4 KTL,NTP 10 
ENcat 0.08 kRS 6.20 KTL,nt 340 
nt kEN 100 KRS,A 23 
a   KRS,T 0.7 
A 1000   KRS,NTP 200 
T 1.9   KEN,NXP 40 
AT   KEN,CP 200 
NTP 1500     
NXP     
CP 1000     
    

Note: In bold, the nine initially present species in the PURE system.

The fictitious species nt and a are conveniently introduced to simplify the kinetic treatment, and the actual concentration of messenger RNA [mRNA] and protein [Protein] are obtained by dividing [nt] and [a] by 3L and L, respectively (where L is the protein length). Initiation and termination (TL) steps, as well as other details of TX-TL reactions, are not included in the model.

In the current model, the protein being produced by the PURE system is the enhanced green fluorescent protein (eGFP, L = 265), and therefore a maturation step should be added (the O2-mediated cyclooxidation of the fluorochrome tripeptide precursor). However, for generality, here we omitted the maturation step (which, for GFP, is kmat ∼ 0.003 s−1 [39]) and we use L = 300.

2.2 Stochastic Encapsulation Procedures

To the best of our knowledge, there is no quantitative theory that rationalizes the physics of solute entrapment inside lipid vesicles from a molecular perspective, also because the details of the entrapment dynamics are intimately linked to the mechanism of vesicle formation, which is in turn not so well understood.

It is possible to recognize that two basic mechanisms, or their combinations, are common to all vesicle preparation methods. These are (1) bending and self-closure of an open lipid bilayer disk to give a closed bilayer lipid vesicle, and (2) fission of a membrane “bud” from preexisting lipid bilayers (from a larger mother vesicle or from lipid layers stratified over a surface) [14]. In both cases, the solutes present in the vicinity of the curved membrane that is going to become a vesicle are later found in the vesicle lumen.

The simplest hypothesis for estimating the expected entrapment efficiency states that—in the absence of specific attractive or repulsive lipid-solute interactions—solute molecules are redistributed among vesicles randomly with a probability of entrapment proportional to the size of the internal aqueous volume. Under this assumption, it is possible to prove that the probability of finding N solute molecules inside a vesicle volume Vv is the Binomial function B(N, Vv/∑Vv), which converges to the Poisson probability P(N|〈N〉) in the case of a population of monodispersed compartments, that is, compartments of equal volume. If the average number of encapsulated molecules is not too low (viz., is greater than 25), the discrete Poisson probability can be approximated with the continuous Gaussian density function with the same average 〈N〉 and standard deviation equal to 〈N1/2. More recently, experimental evidence has been reported that large proteins can be entrapped in a vesicle aqueous core according to the power law density functions [18, 29]. Therefore the aim of this article is to compare the effects of these two different solute density probability functions on the time behavior of the synthesizing-protein monodispersed vesicle population. Moreover, since different species of solutes are present in solution, we will assume for all of them an ideal behavior, that is, we will consider the interactions between solute molecules negligible, and then we will suppose each type of solute molecules is entrapped independently of the other.

2.2.1 Gaussian Law Procedure

When vesicles form in a solution containing different solute species, assuming a pure random encapsulation, it is expected that the average number of entrapped molecules of the kth species at time zero, 〈Nk〉, is simply given by
formula
where NA is Avogadro's number, the bulk concentration of the kth solute, and 〈Vves〉 the average vesicle volume [5, 28]. This implies that the average initial concentration inside a vesicle, 〈Ck〉 = 〈Nk〉/(NAVves〉), corresponds to the bulk value . In the case of more than one solute, Equation 2 holds for each species under the hypothesis of independent encapsulation events [28], that is, assuming ideal behavior for all solute molecules. Therefore, a population of Nves monodispersed vesicles (diameter 2.0 μm) can be generated and filled with all nine PURE system components by drawing random numbers distributed according to a Gaussian density function with average 〈Nk〉 and standard deviation 〈Nk1/2. Negative values when drawn have been put equal to zero. We have used the so-defined Gaussian density function instead of the Poisson one because when 〈Nk〉 is not extremely small, these two distributions essentially coincide.

On the other hand, when 〈Nk〉 is less than 25 the solute can be distributed among a vesicle population Nves according to the Poisson probability distribution, first calculating the overall number of solute molecules contained in all the internal vesicle volumes , and then randomly distributing this number of molecules among vesicles by iteratively drawing random integer numbers in the range 1 to Nves and assigning the solute molecules to the drawn vesicles. This second algorithm is much more time-consuming, and it is adopted only when the average number of molecules to be distributed is less than 25. Moreover, these two procedures have been compared by generating vesicle populations (Nves = 5,000 vesicles) with solute bulk concentrations from 0.01 to 1,000 μM (average number of molecules > 25). Good agreement between the two methods was found (data not shown). Finally, the formula to transform integer numbers of molecules distributed into local vesicle initial concentrations is .

2.2.2 Power Law Procedure

In agreement with the findings reported with model systems [18, 29], the solute distribution inside vesicles does not necessarily obey the Gaussian density function, but it can also be shaped as a power law density function. The function used to model a power law distribution is
formula
where C is the solute concentration inside vesicles and α is the scaling parameter, which typically lies in the range 2 < α < 3 for such natural and man-made phenomena as, for instance, the populations of cities, the intensities of earthquakes, and the sizes of power outages. In this article, the value of α has been fixed at 2.1, whereas the normalization constant βk was set as follows:
formula
so that the average concentration 〈Ck〉 of the kth solute distributed according to Equation 3 becomes equal to value of the bulk concentration . Then, assuming ideal behavior for all the solute molecules, a population of Nves vesicles with solute concentrations randomly distributed according to the power law can be generated for each of the nine PURE system components by using an inversion procedure, consisting in (a) the generation of uniformly distributed random numbers r between 0 and 1, and (b) applying
formula

2.2.3 Gaussian Law versus Power Law Encapsulation

In the following a comparison between solute distributions calculated according to the power and the Gaussian density function is performed in order to get insights into the main differences produced by the vesicle population's initial conditions. Both distributions are built in order to correspond to the same mean bulk concentration and to have the same mean number of entrapped molecules, 〈Nk〉, as reported in Table 2 and discussed previously in Sections 2.2.1 and 2.2.2. Note that in the case of large vesicles (diameter 2 μm), which are the relevant ones in this study because they refer to the case reported in [35], the smallest 〈Nk〉 is 33 (for the DNA species), which does not pose problems on replacing the Poisson distribution with the Gaussian one. If, on the other hand, smaller vesicles were to be considered (diameter 0.2 μm), it would be correct to use the Poisson distribution, given the very low 〈Nk〉 values for most solutes.

Table 2. 

List of and 〈Nk〉 (contained in a vesicle with diameter 2 μm) of the nine PURE system solutes considered in the present model.

SpeciesNk
DNA 0.013 3.3 × 101 
TXcat 0.1 2.5 × 102 
TLcat 2.064 5.2 × 103 
RScat 0.16 4.0 × 102 
ENcat 0.08 2.0 × 102 
1000 2.5 × 106 
1.9 4.8 × 103 
NTP 1500 3.8 × 106 
CP 1000 2.5 × 106 
SpeciesNk
DNA 0.013 3.3 × 101 
TXcat 0.1 2.5 × 102 
TLcat 2.064 5.2 × 103 
RScat 0.16 4.0 × 102 
ENcat 0.08 2.0 × 102 
1000 2.5 × 106 
1.9 4.8 × 103 
NTP 1500 3.8 × 106 
CP 1000 2.5 × 106 

Figure 6 shows a comparison between the distribution of a solute when its expected average intra-vesicle concentration () is 1.9 or 0.013 μM, as in the case of the two PURE system components T and DNA respectively. As is well known, the Gaussian distribution is peaked and the most probable values are distributed around the peak located at the average value: 〈Ck〉 = 〈Nk〉/(NAVv). Moreover, the probability of finding values far from the peak region rapidly vanishes. In terms of solute entrapment, this means that most of the vesicles contain similar numbers of solute molecules, and that it is very improbable to find a vesicle with a very large number of solute molecules inside. In contrast, the power law distribution has its most probable value to the left of the population average (Figure 6). Accordingly, most of the vesicles will contain very few solute molecules. However, due to the long tail of the power law distribution, there is a non-negligible probability of finding vesicles with a very large number of entrapped solute molecules in comparison with the Gaussian curve, as confirmed by the very different values of the standard deviations of the two simulated distributions reported in the legend of Figure 6.

Figure 6. 

Comparison between Gaussian (red circles) and power law (blue circles) simulated solute occupancy distribution in the case of PURE system components: (a) T () and (b) DNA (). The distributions have been simulated for the entrapment of solutes in 5 × 103 monodispersed vesicles (diameter 2 μm). The power law distribution has been calculated by using α = 2.1, and the value of βk has been chosen so that the two distributions have the same mean (cf. Section 2.2.2). Statistical values of simulated data are reported in the legends.

Figure 6. 

Comparison between Gaussian (red circles) and power law (blue circles) simulated solute occupancy distribution in the case of PURE system components: (a) T () and (b) DNA (). The distributions have been simulated for the entrapment of solutes in 5 × 103 monodispersed vesicles (diameter 2 μm). The power law distribution has been calculated by using α = 2.1, and the value of βk has been chosen so that the two distributions have the same mean (cf. Section 2.2.2). Statistical values of simulated data are reported in the legends.

Indeed, this is the central message we would like to emphasize here. According to a power law distribution, events that are almost impossible according to the Gaussian distribution become possible (although not very probable) according to the power law.

3 Results

3.1 Protein Expression in Bulk Aqueous Solution

Figure 7 shows the profile of eGFP production as obtained by the kinetic model introduced in Section 2.1 by using the optimized kinetic parameters reported in Table 1. The kinetic profiles have been obtained for different values of the scaling factor F that multiplies the initial concentrations of the PURE system components listed in Table 1. When F = 1.0 (red curve in Figure 7), the initial concentrations are set to the standard values of the PURE system kit and the protein production exhibits a maximum rate (∼8 × 10−5 μM/s) at about 40 min and a final concentration of about 0.6 μM, in good agreement with the experimental observations. In all the others cases, the initial concentrations are multiplied by F in order to get insight into the sensitivity of the protein production to the set of the initial concentrations. All curves follow a sigmoidal profile and reach a plateau after 4–5 h with a protein yield that increases on increasing the concentrations of the PURE system components.

Figure 7. 

Time course of eGFP synthesis in bulk solution for different scaling factor F of the initial concentrations of the PURE system components. The curves are obtained by numerically solving the deterministic kinetic model described in Section 2.1, using the kinetic parameters and the initial concentrations reported in Table 1.

Figure 7. 

Time course of eGFP synthesis in bulk solution for different scaling factor F of the initial concentrations of the PURE system components. The curves are obtained by numerically solving the deterministic kinetic model described in Section 2.1, using the kinetic parameters and the initial concentrations reported in Table 1.

In Figure 8, the asymptotic concentration [eGFP] of the produced protein is plotted against the dilution factor F by varying F from 0.025 to 5 and estimating the protein concentration after 5 h with the model. The model predicts that the cell-free protein production in the standard conditions typically used in our experiments (∼0.6 μM when F = 1) could indeed be increased by a factor ∼4 if the PURE system components were 5-fold concentrated; however, the dependence appears not to be linear in the scanned range. Further details about the calculated kinetic profiles can be found in [24]; here we would like to remark that there is good correspondence between calculated and available experimental data in the range 0.5 ≤ F ≤ 1 [35]. In fact, the protein concentration value experimentally obtained by diluting the PURE system 1 : 1 from its standard concentration with a buffer solution (F = 0.5) [35] essentially agrees with the calculated data reported in Figure 8.

Figure 8. 

Protein yield against the scaling factor F. Here [eGFP] is estimated as the eGFP concentration value calculated after 5 h. When the PURE systems components are diluted 1 : 1 (F = 0.5) with a solution buffer, the eGFP concentration is around the experimental detection limit (0.2 μM).

Figure 8. 

Protein yield against the scaling factor F. Here [eGFP] is estimated as the eGFP concentration value calculated after 5 h. When the PURE systems components are diluted 1 : 1 (F = 0.5) with a solution buffer, the eGFP concentration is around the experimental detection limit (0.2 μM).

3.2 Protein Expression in Vesicle Population

3.2.1 The Role of Extrinsic Fluctuations

The data shown in Figures 7 and 8 constitute the background for understanding the effect of the concentration of PURE system components on the protein synthesis in a vesicle population. The PURE system produces protein as a function of the concentration of its components. Figure 8 shows the amount of produced protein when all PURE system components have been concentrated (F > 1) or diluted (F < 1) simultaneously, but what happens if the PURE system components are randomly distributed among vesicles?

When lipid vesicles spontaneously form in an aqueous medium containing different solutes, even in the absence of specific lipid-solute interaction, it is expected that the local solute composition within each single vesicle differs from the bulk average composition. In other words, the vesicle formation actually corresponds to a random “sampling” process. As a result of stochastic fluctuations, each vesicle will capture a unique set of solutes, which will produce—in turn—a unique protein production pattern, giving rise to a certain protein production distribution within a vesicle population.

These fluctuations, which will generate a population of vesicles with different compositions [38], are called extrinsic stochastic effects, and refer to the variance associated with the chemical composition of a reacting mixture. In contrast, intrinsic stochastic effects refer to the variance associated with the occurrence of individual chemical steps, given a reacting mixture of a certain composition. In this article, we are mainly interested in elucidating the role of extrinsic stochastic effects, and for this reason we use a deterministic kinetic model to predict the protein expression in order to be able to neglect the intrinsic stochastic effects. Therefore the time evolution of the vesicle population in relation to the protein production will be affected only by the different initial conditions of each lipid compartment.

In terms of our simplified model, this scenario corresponds to having, for each vesicle v of a population, a unique initial concentration vector , referring to local (intra-vesicle) concentrations:
formula
where (i = 1, 2, …, 9) are the local initial concentrations of the PURE system components randomly distributed according to a certain density function probability: Gaussian or power law. Next, we investigated how extrinsic stochastic effects, which are encoded by , tune the protein production within a vesicle population.

3.2.2 Time Course of Protein-Producing Vesicles

Four different populations of Nves vesicles are created by generating random initial concentration vectors , one for each vesicle v (v = 1, 2, …, Nves), according to the Gaussian and the power law solute distributions, and setting two different scaling factors: F = 1 and F = 0.5.

Under the hypothesis of ideal behavior of solutes, the joint probability of co-encapsulation of the PURE system components (nine chemical species in our model) can be simply modeled as the product of the probability of each solute's independent encapsulation. Therefore, the components of the random initial concentration vector are nine random variables, which are distributed according to Gaussian or power law density functions, as previously discussed. In the case F = 1, the nine random initial concentration values are generated by simulating the stochastic entrapment of the PURE system components at their standard concentration (Table 2), while when F = 0.5, that is, for the 1 : 1 diluted PURE system, concentrations are generated by halving the values of Table 2.

For the four different vesicle populations, the internal protein production of each vesicle is calculated according to the deterministic kinetic model discussed in Section 2.1. The value of is then estimated as the protein produced after 5 h and for each population we determine the fraction of vesicles capable of producing eGFP above the concentration threshold of 0.2 μM, which corresponds to the experimentally determined detection value [35]. Note that we always consider an Nves = 5,000 monodispersed population of spherical vesicles, with exactly the same diameter (2 μm) and constant volume Vv = πD3/6. The fixed value of the diameter stems from experimental data [35]. It is important to remark that the relaxation of this constraint on vesicle size simply increases the vesicle diversity; it does not add anything more to the concept of extrinsic stochastic effects discussed in this article.

The protein time courses and the distributions of the protein concentration for the four considered vesicle populations are shown in Figures 9 and 10, which refer, respectively, to Gaussian and power law distribution functions.

Figure 9. 

Simulated protein expression in a population of 5000 vesicles with solute distributed according to the Gaussian density function. On the left, time courses of the internal protein production of each single vesicle (blue curves) for different scaling factors: F = 1 (a) and F = 0.5 (c); on the right, statistics of the protein yield [eGFP] (histograms) for different scaling factors: F = 1.0 (b) and F = 0.5 (d). The red curves in plots (a) and (c) (lines) are protein time courses calculated for the initial concentration vector set equal to CB and 0.5CB respectively, while dashed red lines are time courses obtained by adding a random noise to the initial concentrations (see the main text for details). The vertical red dashed line in the (b) and (d) histograms indicates the eGFP experimental detection limit, while %N>0.2μM is the percentage of vesicles with a protein yield greater than 0.2 μM.

Figure 9. 

Simulated protein expression in a population of 5000 vesicles with solute distributed according to the Gaussian density function. On the left, time courses of the internal protein production of each single vesicle (blue curves) for different scaling factors: F = 1 (a) and F = 0.5 (c); on the right, statistics of the protein yield [eGFP] (histograms) for different scaling factors: F = 1.0 (b) and F = 0.5 (d). The red curves in plots (a) and (c) (lines) are protein time courses calculated for the initial concentration vector set equal to CB and 0.5CB respectively, while dashed red lines are time courses obtained by adding a random noise to the initial concentrations (see the main text for details). The vertical red dashed line in the (b) and (d) histograms indicates the eGFP experimental detection limit, while %N>0.2μM is the percentage of vesicles with a protein yield greater than 0.2 μM.

Figure 10. 

Simulated protein expression in a population of 5000 vesicles with solute distributed according to the power law density function. On the left, time courses of the internal protein production of each single vesicle (blue curves) for different scaling factors: F = 1 (a) and F = 0.5 (c); on the right, statistics of the protein yield [eGFP] (histograms) for different scaling factors: F = 1.0 (b) and F = 0.5 (d). The vertical red dashed line in the (b) and (d) histograms indicate the eGFP experimental detection limit, while %N>0.2μM is the percentage of vesicles with a protein yield greater than 0.2 μM.

Figure 10. 

Simulated protein expression in a population of 5000 vesicles with solute distributed according to the power law density function. On the left, time courses of the internal protein production of each single vesicle (blue curves) for different scaling factors: F = 1 (a) and F = 0.5 (c); on the right, statistics of the protein yield [eGFP] (histograms) for different scaling factors: F = 1.0 (b) and F = 0.5 (d). The vertical red dashed line in the (b) and (d) histograms indicate the eGFP experimental detection limit, while %N>0.2μM is the percentage of vesicles with a protein yield greater than 0.2 μM.

In the case of the Gaussian law at standard concentrations (F = 1), the vesicle population shows limited diversity. In fact, the protein-versus-time profiles form a rather narrow bundle around the average curve (red curve) generated, as a control curve, by using the standard initial concentrations . Almost all kinetic curves are between the two boundary curves (red dashed lines) obtained by integrating the kinetic differential equation set using as initial concentrations , where σk is estimated as random noise: σk = 〈Nk1/2/(NAVv). Most of the vesicles produce about 0.6 μM protein after 5 h (Figure 9b—note the bell-shaped distribution), and 100% of vesicles are recognized as functionally active, that is, above the threshold value ≈0.2 μM.

In agreement with Figure 8, when the PURE system components are diluted 1 : 1 with buffer (F = 0.5, Figure 9c), the protein production curves lie in a concentration range much lower than those obtained with the undiluted one (F = 1.0), in particular, around an average value of 0.1 μM (Figure 9d). All vesicles in the population would be recognized, therefore, as inactive, because the intra-vesicle protein concentration is below the detection level, that is, lower than 0.2 μM.

Completely different is the scenario when solutes are distributed according to the power law (with α = 2.1). If the PURE system is not diluted (F = 1; see Figure 10a), most of the kinetic curves refer to low protein production (below 0.2 μM), and only 5.4% are above the detection level. Moreover, a small fraction of vesicles (0.36%) are characterized by high levels of protein production, up to 0.75 μM. The distribution of protein concentrations after 5 h is not bell-shaped, and mirrors the solute occupancy power law distributions. When compared with Figure 9a,b, the vesicle population will be recognized as essentially ineffective in producing protein, even if a small fraction of vesicles produces enough protein to overcome the detection limit.

In the case of a diluted PURE system (F = 0.5, Figure 10c), the random population is again characterized by a large number of vesicles that produce low protein amounts, below 0.1 μM, and that are therefore classified as inactive. Interestingly, however, a few vesicles, despite the PURE system dilution, are able to produce protein quite well (up to 0.9 μM). The distribution of produced protein (Figure 10) reveals that about 0.8% of vesicles are active as if they were formed in the presence of an undiluted PURE system. Clearly, this is a consequence of the fact that the long tail of power law distributions (Figure 6) does not rule out the possibility of entrapping a large number of solutes, despite the expected low mean value. Note also that on moving from F = 1 to F = 0.5, a clear reduction of the number of fluorescent vesicles is predicted by the model (from 5.4% to 0.8%), in agreement with the observations [35].

This behavior is well evidenced by the plot in Figure 11, where the density functions p([eGFP]) of the produced eGFP, estimated by in silico simulations after 5 h, are compared for Gaussian and power law distributions of the PURE system components when F = 0.5. In the case of the power law (blue curve), p([eGFP]) exhibits a higher peak at lower concentration than in the case of the Gaussian law (red curve), but in both cases below the detection level around 0.2 μM. However, the estimated density function obtained with the power law solute distribution shows a long tail at concentration values higher than the detection threshold, and this corresponds to non-negligible probabilities of finding active compartments among the vesicle population.

Figure 11. 

Comparison of protein yield [eGFP] density functions calculated from the simulations outcomes of vesicle populations performed using the Gaussian and the power law initial distribution of PURE system components diluted 1 : 1 (F = 0.5). The [eGFP] density functions values are estimated as the ratio between the fractions of vesicles belonging to 10 logarithmically equally spaced concentration classes (from the minimum to the maximum [eGFP] value) and the size of the classes.

Figure 11. 

Comparison of protein yield [eGFP] density functions calculated from the simulations outcomes of vesicle populations performed using the Gaussian and the power law initial distribution of PURE system components diluted 1 : 1 (F = 0.5). The [eGFP] density functions values are estimated as the ratio between the fractions of vesicles belonging to 10 logarithmically equally spaced concentration classes (from the minimum to the maximum [eGFP] value) and the size of the classes.

4 Concluding Remarks

The encapsulation of solutes, and in particular macromolecules, inside lipid vesicles is a key physical process that has relevance to origin-of-life studies (modeling protocell formation) and synthetic biology (synthetic cell construction). Inspired by previous results on ferritin and ribosome encapsulation, we have recently demonstrated that this self-concentration phenomenon can be exploited to concentrate complex multimolecular mixtures like the TX-TL kit inside liposomes. Results indicated that in contrast to expectations, few liposomes were able to synthesize a protein even if they were prepared by using a diluted PURE system, demonstrating that the formation of liposomes brings about a local concentration of solutes and therefore triggers the onset of complex reactions.

In this article we have presented a mathematical model of TX-TL coupled reactions that can be used to predict the time course and the amount of protein production starting from any PURE system composition. By generating an in silico population of liposomes whose content is determined by two stochastic solute distributions, it has been possible to show that whereas the experimental results are not compatible with a simple Poisson encapsulation distribution function, a power law fits much better with the observations, so that a rational and quantitative explanation of data has been achieved.

Although rewarding in terms of modeling, our approach is still unable to explain the physical reasons underlying a power law solute distribution inside liposomes, and more experimental data are needed. In particular, it would be very interesting to investigate the nature of the pre-liposomal intermediates and their interaction with macromolecular solutes, possibly measuring whether or not the presence of solutes interferes with the rate of vesicle formation. This will allow a direct verification of a supposed mechanism focused on interplay between the mechanics of liposome closure and the rate of solute capture (encounter rate) [29].

Our approach emphasizes the role of stochastic events in synthetic cell research and therefore the importance of integrating stochastic simulations [6, 15] with experimental approaches, especially in the field of synthetic biology.

Finally, we would like to remark that the construction and the study of cellular models of minimal complexity is a fundamental tool for understanding the basic physico-chemical principles and constraints that regulated and guided the emergence of primitive cells. Since the same principles also apply to man-made synthetic cells, we are confident that in vitro synthetic biology approaches will be essential not only for exploring open questions in origin of life, but also for gaining knowledge (the synthetic, or constructive, approach) and developing new tools for the biotechnology arena.

Acknowledgments

We are grateful to Pier Luigi Luisi for his guidance in the field of synthetic cells and for useful comments on the manuscript. The experimental research on solute encapsulation was started by one of us (P.S.) within Luisi's group (Roma Tre University, Rome, Italy), and in collaboration with Prof. Alfred Fahr, Dr. Tereza Souza, Dr. Frank Steiniger, and Erica D'Aguanno of the Friedrich-Schiller University of Jena (Germany). The modeling work was started within the PRIN2008 (2008FY7RJ4) Synthetic Cells project and further expanded thanks to networking initiatives such as EU-COST Actions CM0703 (Systems Chemistry) and CM1304 (Emergence and Evolution of Complex Chemical Systems). We thank Margherita Caputo (University of Bari) and Francesca D'Angelo (Roma Tre University) for their involvement in the initial phase of the work.

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Author notes

Contact author.

∗∗

Department of Chemistry, University of Bari, Via E. Orabona 4, 70126 Bari, Italy. E-mail: fabio.mavelli@uniba.it

Department of Science, Roma Tre University, Viale G. Marconi 446, 00146 Rome, Italy. E-mail: pasquale.stano@uniroma3.it