## Steen Rasmussen, Mark A. Bedau, Liaohai Chen, David Deamer, David C. Krakauer, Norman H. Packard, and Peter F. Stadler

Print publication date: 2008

Print ISBN-13: 9780262182683

Published to MIT Press Scholarship Online: August 2013

DOI: 10.7551/mitpress/9780262182683.001.0001

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# Models of Protocell Replication

Chapter:
(p.213) 10 Models of Protocell Replication
Source:
Protocells
Publisher:
The MIT Press
DOI:10.7551/mitpress/9780262182683.003.0010

# Abstract and Keywords

This chapter attempts to model protocell production under the general perspective of autopoiesis. An example of an autopoietic system is a living cell that is capable of self-generation and self-reproduction using energy derived from its environment. This chapter investigates how, and under what conditions, an autopoietic protocell can emerge. It focuses on the lattice artificial chemistry (LAC), computational system that is used to model autopoietic protocells.

# 10.1 Introduction

The transition to cellular life was marked by the emergence of a chemical coupling between simple autocatalytic processes (perhaps a primitive form of metabolism) and a container. The emergence of cells allowed the propagation of information with multiple selective advantages, from having the reactants closer in space to allowing division of labor. It also provided the conditions for escaping from molecular parasites, which are known to destroy cooperative dynamics in hyperbolic replicators (see chapters 13 and 14).

Generally cell biomass increases, usually doubling, before the cell cycle ends with cell division. The cell cycle can be defined as the orderly duplication of intracellular components, including the cell genome (DNA), followed by division of the cell into two cells (figure 10.1). Current cells have a number of sophisticated molecular mechanisms to control cell cycle dynamics, including checkpoints and regulatory functions. Unicellular life forms replicate by simple splitting of the cell into two cells, a process known as binary fission. The process takes place provided that external resources allow the template-based copy of genetic information and the building of all required molecular components necessary for the build-up of two daughter cells.

Although lipid aggregates, micelles, and vesicles are known to spontaneously form under a wide range of conditions, the problem of cell growth and replication is far from trivial. Several problems emerge when modeling an effective catalytic cooperation between components leading to a coherent replication cycle. In this chapter, we address some early and recent steps toward understanding possible paths toward simple replicating protocells. They all involve the problem of generating a spatially structured pattern generated through some dynamical process that leads to partial or complete reproduction of the whole structure.

The whole cycle of cell reproduction is strongly constrained by two main factors: the kinetics associated with reactions among different chemical components and the interactions between these components and the container. Early models (to be (p.214)

Figure 10.1 Two main events are required for a protocell to complete a life cycle. The first is somehow controlled by the interaction of chemical reactions and membrane growth (G) with an external environment, likely to be subject to noisy fluctuations (here indicated as x) given the small scale at which the cell lives. The second deals with replication (R). Splitting can be associated with external fluctuations (E) or somehow controlled by the cell dynamics, or both.

reviewed here) explored these two features separately. Only some attempts (such as Rashevsky, 1960) explored how toy models of metabolism could effectively couple them, and the conditions under which the compartment enclosing metabolites would experience growth and eventually splitting. From a different perspective, simple models of nanoscale organization of lipid aggregates can trigger fission events that are connected to the problem of how self-organized systems can spontaneously experience some type of fission.

The growth of protocell-like systems (or parts of them) is strongly tied to a compromise between forces driving the system toward growth and forces that trigger destabilization. Growth is easily achieved by accretion of material that forms aggregates or through membrane expansion under increasing osmotic pressures. These processes can end up in some stable structure (a large aggregate) or reach some rupture threshold leading to breaking the cell container. Under certain conditions (to be discussed here) growth is followed by some type of instability that triggers the formation of smaller subsystems. At both the mesoscopic and microscopic scales, a physical implementation of the rules dealing with compartment dynamics must be considered.

Here we explore some basic results dealing with the kinetic behavior of spatially extended, replicating systems. Our review includes (a) basic replicator dynamics (and how particular reaction processes lead to specific dynamical patterns of growth), (b) spatially extended, reaction-diffusion systems with no membrane leading to self-replicating spots, (c) physical models of amphiphile aggregates involving some type of spontaneous or induced fission mechanisms, and (d) models of self-replicating cells including both a closed membrane and a simple metabolic core.

# (p.215) 10.2 Replicators and Producers

Primitive cellular life forms presumably involved the emergence of a catalytically coupled set of chemical reactions. In its simplest form, it might have included a vesicle or micelle coupled to a minimal metabolism. Such metabolism might have been favored by special, membrane-bound molecules acting as primitive enzymes. The so-called chemoton (Gánti, 1975; see also chapter 22) was suggested as a simple approach to this picture, where the three basic components of cellular organization—namely, metabolism, container, and information—would be tightly coupled. Metabolism would provide the building blocks for a population of replicating molecules, properly encapsulated by the membrane.

An important distinction exists between replicating and reproducing entities. Reproducers, in contrast to replicators, involve copy and development whereas replicators can be understood as informational molecules that can exist in several forms (A, B, C) and, when replicating, the new structures resemble the old ones. In the prebiotic scenario or chemical evolution context, natural selection is essentially the dynamics of replicators and thus dynamics and competition between replicators are key features to explore. In this sense, replicators can be units of selection (Szathmáry, 1997) since they have

1. 1. multiplication—entities should give rise to more entities of the same kind;

2. 2. heredity—A type entities produce A type entities, B type entities produce B type entities, and so on;

3. 3. variability—heredity is not exact; occasionally A type objects give rise to A′ type objects (it may be that A′ = B).

# 10.3 Self-Replication Spots

This type of cell-like, spatially distributed system belongs to the large class of so-called reaction-diffusion models. The best studied examples of the two types of reaction-diffusion systems are the Meinhardt system (Gierer and Meinhardt, 1972) and the diffusive Gray-Scott system (Pearson, 1993), respectively. The complex interplay between activator and inhibitor or substrate chemical, aided by the reaction and diffusion components, creates most startling spatiotemporal (Turing) patterns, such as spots, stripes, traveling waves, or spot replication. The Turing patterns are characterized by the active role that diffusion plays in destabilizing the homogeneous steady state of the system. They emerge spontaneously as the system is driven into a state in which it is unstable toward the growth of finite-wavelength stationary perturbations. Interestingly enough, the replication characteristic is a particularity of the diffusive Gray-Scott model alone, which makes it the ideal model for developmental research. (p.216) In this case, cell-like localized structures grow, deform, and replicate themselves until they occupy the entire space.

The Turing patterns from the work of Pearson (1993) on the diffusive Gray-Scott model were confirmed experimentally by Lee and coworkers (Lee et al., 1993), including spot replication (Lee et al., 1994). Theoretically, extensive work exists in the literature on the dynamics of this model concerning the “spot replication” in one, two, and three dimensions (Muratov and Osipov, 2000). The model was originally introduced in Gray and Scott (1985) as an isothermal system with chemical feedback in a continuously fed, well-stirred tank reactor, where the last property implied the lack of diffusion. The analysis of the system revealed stationary states, sustained oscillations and even chaotic behavior. The model considers the chemical reactions

$Display mathematics$
$Display mathematics$
which describe the autocatalytic growth of an activator, V, on the continuously fed substrate, U, and the decay of the former in the inert product, P, subsequently removed from the system. A major development was performed by Pearson (1993) who introduced the role of space by relaxing the constraint of a well-stirred tank and studied the system in two dimensions. In two dimensions, the concentrations of the two chemical components, u(x, y; t) and v(x, y; t) are given by
$Display mathematics$
$Display mathematics$

where Du and Dv are the diffusion coefficients, F is the dimensionless flow rate (the inverse of the residence time), and k is decay constant of the activator, V. The original study involved fixed diffusion coefficients, Du = 2 〉 10−5 and Dv = 10−5, with F and k being the control parameters. As a typical Turing pattern, the system has a steady state stable with respect to homogeneous temporal oscillations, which becomes unstable toward standing, space-periodic perturbations when diffusion is taken into account (see Mazin et al., 1996, for a detailed linear analysis of the Gray-Scott model).

For the numerical study of the partial differential equations, we used the conditions initially employed in Pearson (1993), consisting in a system size of RR, with R = 2.5 discretized through x →(x0, x1, x2, …, xN) and y →(y0, y1, y2, yN) with N = 256. The numerical integration of the partial differential equations was (p.217)

Figure 10.2 Replicating spots in a color map representation of the V concentration as a temporal process from (a) to (d). The values of the parameters are: F = 0.04, k = 0.0655, N = 128, R = 1.25.

performed by means of forward Euler integration, with a time step of τ ≅ 0.9 and spatial resolution h = R/N, and using the standard five-point approximation for the 2D Laplacian with periodic boundary conditions (figure 10.2). More precisely, the concentrations $(ui,jn+1,vi,jn+1)$ at the moment (n + 1)τ at the mesh position (i, j) are given by
$Display mathematics$
$Display mathematics$

with the Laplacian defined by

$Display mathematics$

The system was initialized with (ui;vi) = (1; 0) with the exception of a small central square of initial conditions (ui;vi) = (0.5; 0.25) perturbed with a 1% random noise. The works existent in the literature illustrate examples of patterns following a color map on the U-concentration, with the red color representing the (uR; vR) steady state and the blue one a value in the vicinity of the (uB; vB) state, for example, (u; v) = (0.3; 0.25).

(p.218) Spatial structures derived from spatially extended replicator dynamics are a first step toward localization of chemicals in a confined domain. Such a spatial localization is necessary to favor chemical reactions and selection processes. Although these kinds of spatial patterns have been found to have relevant, high-order selection properties, they are far from the real compartments defining cellular life. To gain insight into a more realistic scenario, we next consider two approaches largely based on considering the dynamics of membranes or aggregates.

# 10.4 Nanoscale Replicating Aggregates

If we consider aggregates of amphiphilic molecules with small numbers of components, we are actually looking at the nanoscale level. Here self-replication implies both growth and fission of amphiphile assemblies. Growth is understood as the outcome of an autocatalytic process: Amphiphiles are products of a chemical reaction that is driven by the presence of parent amphiphiles. As new monomers form, they are incorporated in the henceforth growing assembly. When the assembly reaches a critical size, it undergoes a fission process that results in two daughter assemblies. Here a microscopic approach to the underlying physics is needed.

Models that are used to study the dynamics of amphiphile aggregation are deduced from the technique of molecular dynamics (MD), in which forces derived from an assumed potential determine the motion of individual particles and hence the trajectory of the whole system in phase space.

Newton’s Second Law is hereby applied to calculate the trajectory from the position ri, mass mi and the potential Ui of each individual particle:

$Display mathematics$
(10.1)

The potential is assumed to be the superposition of pair-wise potentials for all particle pairs within a certain interaction range:

$Display mathematics$

Force field methods are well-established in the realms of molecular modeling, where the positions and forces of every single atom are calculated in each time step of a numerical integrator for equation 10.1. Unfortunately, the computational expense of fully atomistic molecular dynamics is considerably high when modeling the previously mentioned aggregation and fission processes.

Under the assumption that degrees of freedom of these lower scales do not affect amphiphile aggregation, coarse-graining techniques can de developed to reduce the (p.219) unmanageable complexity of atomic interactions. In these coarse-grained approaches, amphiphile molecules are usually represented by two or more hydrophilic and hydrophobic particles or “beads” that represent sections of the molecule and are connected by elastic springs. Water particles are either modeled as structureless particles or completely excluded from the model, but are handled implicitly by adding their effects to the amphiphile interactions. Depending on the presence of water particles, models are called either explicit or implicit.

Usually, these particles are not meant to represent single molecules, but rather exemplary particles or “lumps” of fluid. Thus, the coarse-grained models assume an underlying medium with which the simulated particles can exchange energy due to friction and thermal noise. For example, Langevin equations can be used to achieve this energy flow:

$Display mathematics$

where vi, is the particle velocity, η, the friction coefficient and ξ, a Gaussian-distributed random variable describing heat effects (Brownian motion). More sophisticated “thermostats” are used in the technique called dissipative particle dynamics (DPD) to ensure momentum conservation in the system.

In addition, particle interactions are usually modeled by a much simpler potential function than those used in fully atomistic MD. Most common is the so-called softcore potential

$Display mathematics$

where rij is the particle distance, rc, the cutoff distance and aij, the repulsion strength, that depends on the hydrophilic/hydrophobic character of the interacting particles. This potential induces a solely repulsive force that decreases linearly with distance, until at rij = rc both potential and force are 0. In implicit models, the repulsion of water particles that drives hydrophobic interaction is mimicked by an additional adhesive potential between hydrophobic particles. These simple potential functions further reduce the computational complexity because they do not obey the singularities found in more realistic descriptions (like, e.g., the Lennard-Jones potential common in atomistic MD).

## 10.4.1 Simulations of Vesicle Fission

In early 2003, Yamamoto and Hyodo simulated the fission process of vesicles. They used DPD—described in chapters 18 and 19—to observe the deformation and eventual fission of a two-component vesicle. Their simulation consists of water and (p.220)

Figure 10.3 Fission of vesicles with diverse amphiphiles. Reprinted with permission from Yamamoto and Hyodo (2003).

amphiphile particles, whereby amphiphiles are composed of one hydrophilic head and three hydrophobic tail particles, which are covalently bonded by elastic springs. Interactions between these particles follow a softcore repulsion potential, a common approach in DPD simulations. Repulsion parameters are chosen so that the forces correspond to cylindrical amphiphiles that form bilayers.

The simulations are initialized with a bilayer structure that bends to form a vesicle consisting of 3,024 amphiphiles that encapsulate 5,389 water particles (figure 10.3). After the initial vesicle has formed, 35% of the amphiphiles are randomly exchanged for a second type of amphiphile. The second amphiphile possesses the same repulsion (p.221) parameters as the former, but mutual repulsion of the monomers is supposed to be high (0.6 times stronger then the repulsion of identical monomers). Yamamoto and Hyodo (2003) observed that the amphiphiles drift within the bilayer and rearrange into separate phases to reduce the high mutual repulsion at domain edges. Depending on the ratio with which monomers are exchanged in the outer/inner layer of the membrane, the vesicle deforms. If the outer/inner ratio is greater than 3.0, a budding process is initiated, in which the exchanged amphiphiles form bulges on the membrane with the neck at the domain edge. Finally, these buds pinch off as a microvesicle from the parent vesicle.

Strictly speaking, their simulations do not model the spontaneous fission required for self-replication in the previously mentioned meaning: Their fission process is the result of an artificial exchange of amphiphiles after the initial vesicle is formed. However, their simulation sketches a possible way to an ongoing growth and fission process of vesicles. The system could be exposed to an ongoing influx of amphiphile monomers that get incorporated in the growing membrane. Successively added monomers rearrange within the membrane to form domains of identical amphiphiles. Once domains reach a critical size, the high potential at domain boundaries triggers the fission process.

Although this rough outline looks reasonable at first glance, some difficulties arise in the details of this scenario: Amphiphiles will enter only the outer membrane of the vesicle, hence increasing its surface tension. This might propagate the fission in the first place, but it is not clear how this will affect the fission process in the long term. The work of Yamamoto and Hyodo confirms that vesicle fission depends on both the number and rates of different amphiphiles in the outer and inner membrane. Thus, flipflop motion of amphiphiles from the outer to the inner membrane would be necessary to increase the surface tension needed for vesicle fission. It is not clear, however, which mechanism establishes flipflop motion against the energy gradient based on membrane curvature.

Noguchi and Takasu (2002) propose a fission process for vesicles that is mediated by a nanoparticle. They use an implicit model of the Langevin type to simulate budding, fission, and fusion of vesicles. Starting from a random initial condition with 1,000 amphiphiles, vesicles form spontaneously (figure 10.4). After vesicle formation, Noguchi and Takasu place what they call a nanoparticle at the center of mass of the vesicle: In their model, the nanoparticle represents a protein or colloid that acts as an initiator for morphological change, for example, vesicle fission. The nanoparticle is adhesive to the hydrophilic head groups of the amphiphiles, and therefore sticks to the inner membrane of the vesicle, where it introduces budding. The stronger the adhesion of the nanoparticle, the stronger will be the change of local curvature where the nanoparticle adheres to the membrane. For strong adhesion coefficients, the surface tension is induced but is so high that vesicle fission is energetically favored.

(p.222)

Figure 10.4 Vesicle fission in the presence of nanoparticle. Reprinted with permission from Noguchi and Takasu (2002).

While the nanoparticle might provide reliable vesicle fission, one now must find a mechanism for the nanoparticle to replicate. Otherwise, only the daughter vesicle that holds the nanoparticle is able to reproduce further.

In the previous examples we have not considered metabolism as an explicit part of the aggregation dynamics. Moreover, the importance of fluctuations (which might be huge) at this level has also been neglected. Both aspects are relevant in understanding the possible behavior exhibited by simple protocells. In the next section, we consider both a model membrane (at a large scale) and a toy metabolism together.

# 10.5 Mesoscopic, Mechanochemical Approximations

The objective of this section is to analyze the behavior of membranes from a mechanical point of view, with respect to the possible deformations that the membrane can experience and that can finally give rise to the division of the membrane into two structural daughters, which can grow and again return to divide themselves if the initial conditions can be regenerated in each daughter structure.

A first approximation (first applied by Rashevsky, 1960) to cell replication considers a minimal model of membrane physics, defined in terms of the average behavior of a continuous, closed membrane involving some sort of simple internal metabolism. Rashevsky presented the basic conditions required (under some constraints) to obtain replication that we summarize here. This is a mesoscopic approach that allows the incorporation of other components, such as chemical reactions able to create instabilities and symmetry breaking. The two membrane division mechanisms we consider are spontaneous and induced division.

## (p.223) 10.5.1 Spontaneous Division

Different models of vesicle shape transformations have been widely studied. Essentially, the shape of the vesicles is analyzed in a continuum model under two variants (Seifert, Berndl, and Lipowsky, 1991). First, the spontaneous-curvature model (Helfrich, 1973) analyzes changes in vesicle shape minimizing the bending energy for a given area and volume. Second, the bilayer-coupling model (Svetina and Zeks, 1983) assumes that the two monolayers do not exchange lipids between them. This model imposes the minimization of the bending energy for a given area, volume, and ΔA, where ΔA is the difference between the areas of the external and internal layers. Both models lead to the same shape equations (Svetina and Zeks, 1989).

Here we summarize the theoretical calculation Rashevsky (1960) suggested to determine the critical radius beyond which the cellular membrane becomes unstable and increases the probability of a spontaneous division. This model focuses not explicitly on the spontaneous shape transformations, but on the energetic stability as a function of membrane size and, implicitly, on the conditions favorable to cellular division. We consider a simple structure formed by a spherical membrane in whose interior metabolic reactions take place. When a substance is produced or consumed by a metabolic system, the forces that act on each element of volume and of surface (membrane) are directed outwards or inwards. In this situation, it is possible to calculate the variation of energy of the system resulting only from the work of these forces. The spontaneous division takes place only when the work these forces make in the process of driving the system from the initial configuration to complete division into halves is positive, which means the change in energy is negative.

We assumed, as a first approximation, that the structure is formed of a spherical membrane and that the metabolic reactions can take place only within this membrane because of the presence of a number of Ω catalytic particles intervening in the consumption and production of metabolites.

In the simplest case, where the rate of reaction is constant and the particles distribution is uniform, we can calculate the energy balance. Basically, it is necessary to consider three aspects: the variation of energy associated to the increase of surface, the work due to the pressure that different substances exert on the membrane when crossing it, and finally the work the diffusion forces make on the particles present in the volume limited by the membrane. Rashevsky proposed a method of calculation based on assuming that the net variation of the system’s energy is independent of the precise way the division occurs. The work made by the forces dividing the spherical membrane of radius r0 into two equal spheres of radius r1 is equal to the work needed to expand the initial sphere to infinite radius and later to contract it into a sphere of radius r1 (multiplying by 2 because there are two spheres).

Regarding energy associated to the increase of surface, if we suppose a spherical initial configuration of radius ro, and assume that the volume remains constant (p.224) throughout the entire division process (the sum of the volumes of two resulting halves is equal to the initial volume), we find the following relation between the final radius r1 and ro:

$Display mathematics$

with the total increase of surface being

$Display mathematics$

Assuming that γ is the coefficient of superficial tension of the membrane, the division of the membrane implies an increase in the superficial energy by

$Display mathematics$

independently of the mechanism used to obtain this division. From this point of view, the division of the membrane would not take place spontaneously because ΔWs 〉 0, unless other factors taking part in this process compensate this increase of energy.

The work on the membrane is a result of pressure of osmotic origin, produced by the different concentrations at either side of the membrane. Assuming a uniform distribution of particles inside and outside the membrane, the force that acts on each point of the spherical membrane is

$Display mathematics$

where ci and ce are the concentrations inside and outside the membrane, M is the molecular weight, T is the temperature, and R is the ideal gas constant (assuming that the substances are at low concentration in the dilution). This force can be expressed as

$Display mathematics$

where q is the rate of metabolic reaction and h is the membrane permeability (see Rashevsky, 1960).

In the first theoretical approach, from a purely physical point of view and without consideration of real cellular membranes composition, Rashevsky (1960) assumed that permeability, h, is a function of the thickness of the membrane, d, and that increasing the radius of the membrane decreases its thickness in agreement with the following relation: (p.225)

$Display mathematics$

The work made by this force to expand the radius from ro to infinity is

$Display mathematics$

In the calculus of the resulting spheres of radius r1, it suffices to replace q by 0.5·q and ro by r1 (r1 = 0.8·ro) in the previous expression. The resulting work to contract the sphere from infinity to r1 is

$Display mathematics$

and the net work of all the processes (multiplying by 2 the work of contraction because there are two resulting spheres) is

$Display mathematics$

Turning to the work done on catalytic particles present in the volume enclosed by the membrane, the existence of concentration gradients created by the metabolic reactions in the volume enclosed by the membrane implies the existence of diffusion forces acting on these particles. In particular, if we have n particles per volume unit (assuming a uniform distribution), and each one of these particles occupies a volume v, the force that acts per volume unit is

$Display mathematics$

with

$Display mathematics$

where δ is the density of the solvent. Vm is the volume of one molecule, and Mo is the molecuilar weight of the solvent (see Rashevsky, 1960, chapter VIII).

If we increase the volume by an amount dV, increasing the radius r0 by an amount dr0, it is possible to calculate the work made by the forces of diffusion:

$Display mathematics$
(p.226) where Ω is the number of catalytic particles, qo is the reaction rate per catalytic particle, and Di is the inner diffusion coefficient. If we expand the volume to infinity, we obtain
$Display mathematics$

Similarly to the previous point, the work of expansion of two halves can be calculated with the same expression, replacing Ω with 0.5·Ω and ro with r1 = 0.8·ro and multiplying by 2.

The final balance of work will consist of three terms, the first associated with modification of the total membrane surface, the second associated with the work done by the diffusion forces on the membrane, and finally the third associated with work done by the diffusion forces on the particles present in the inner volume.

$Display mathematics$

where we have changed

$Display mathematics$

From this final expression we can obtain a criterion of spontaneity in the cellular division:

• If ΔW 〉 0, spontaneous division is not possible.

• If ΔW 〈 0, spontaneous division is possible.

The critical value of the radius ro will be the one causing ΔW = 0.

We can analyze the values of this expression based on some of its most significant parameters. It is important to emphasize that significant variations in the permeability and the diffusion coefficient do not imply significant variations in the value of the critical radius, having the same order of magnitude (see figure 10.5). This implies that the assumption of the variability of permeability with membrane thickness is not of fundamental importance. The values of ro are in agreement with the experimental values of the radius of the actual cells.

## 10.5.2 Induced Division

It has been seen that the membrane can divide spontaneously when it reaches a critical radius value ro, as it is favorable from an energetic point of view. But the fact that this can occur does not guarantee that this division is unavoidable. Rashevsky (1960) (p.227)

Figure 10.5 Range of values of the radius ro versus the permeability h satisfying ΔW = 0. For values of ro leading to ΔW 〈 0 (or rro), spontaneous division can take place. For values of ro leading to ΔW 〉 0 (rro), the configuration of the membrane remains stable.

and other authors have suggested that division of the membrane can be induced by different factors and more likely by a joint operation of several of them simultaneously. One of the factors that play an important role is the time-varying nature of the osmotic pressures. From a mechanical point of view, the action of time-varying osmotic pressures by itself can give rise to membrane division.

The origins of these variable osmotic pressures can be very diverse. For example, metabolic reactions can generate skew distributions of concentrations of metabolites. Thus, they generate an asymmetric distribution of osmotic pressures on the membrane as they diffuse outwards. Another example is given by metabolic reactions associated to localized metabolic centers (for example, molecules or clusters of molecules with catalytic properties) that can move in the membrane-enclosed volume.

We have simulated the situation generated by two localized metabolic centers (each metabolic center corresponds to a molecule acting as a catalyst) in the volume limited by the membrane (figure 10.6). We suppose that inward flux of substances exists. Additionally, in the presence of these metabolic centers, particular reactions take place in which the substances are partially consumed and new substances are generated on the surface of the metabolic centers. The latter ultimately spread outwards. The variation of the concentrations in time resulting from diffusion is given by the diffusion equations: (p.228)

Figure 10.6 Different steps of membrane division. At each point of the membrane, the balance between pressures exerted by the incoming consumed substance, the outgoing produced substance and the surface tension are calculated. At the same time the metabolic centers are moving because of the electrostatic repulsion force. The model allows cell membranes to grow (with certain additional metabolic reactions that are able to produce membrane building blocks), and the volume of the cell can grow, too. The pictures show the concentration of the substance produced inside the volume limited by the membrane. Dashed line corresponds to the membrane boundary. For more details about the simulations, see Macia and Solé (2005).

(p.229)
$Display mathematics$

where ci, and ce are the concentrations inside and outside the membrane, and Di and De are the inner and outer diffusion coefficients, respectively. In the membrane, the continuity of the boundary conditions must occur, supposing that the flow that arrives at the membrane leaves it (there is no accumulation of matter in the membrane):

$Display mathematics$

where h is the permeability of the membrane and η is the normal direction to the membrane.

The substances generate a certain pressure when they cross the membrane inwards and outwards. In a first approach, we can suppose that the pressure acting on each point of the membrane is the sum of the pressures generated by each substance independently. Also, we can suppose that the pressure generated by each substance is proportional to the difference in its concentrations on each side of the membrane. Thus, the pressure that acts on each point of the membrane can be calculated as

$Display mathematics$

where s is the substance crossing the membrane, PS is the pressure associated to the surface tension, and Ks is the proportionality coefficient that can be approximated by the following expression:

$Display mathematics$

with Ms being the molecular weight of the substance s.

If, under these conditions, the metabolic centers are displaced by any cause, variability in the osmotic pressures acting on each point of the membrane would be generated. For example, the molecules that form these metabolic centers might have an electrical charge and produce a repulsion phenomenon. We have simulated these conditions in order to understand one division cycle. The simulations have allowed us to verify that simply the variation in the osmotic pressures leads to division of the membrane. In this example, we have considered two metabolic centers that consume the input substance. The products resulting from the metabolic reactions are the same for both metabolic centers.

# (p.230) 10.6 Conclusion

Modeling protocell replication dynamics requires a physics-inspired consideration of the membrane (container) together with an appropriate consideration of the kinetics of chemical reactions. So far, only a few models have been able to provide a full, reproducible set of steps leading to spontaneous replication. Most of these models incorporate the essential, bare bones of the underlying physics and consider some sort of active mechanism that can create instabilities in the container. Such deformations can result from energetic constraints, segregation of diverse amphiphiles, or active mechanisms. The latter can be associated with metabolic components that force the system to move out of equilibrium and split. These are, of course, the most relevant ones within the context of building artificial protocells, but they can be helped by considering other scenarios where splitting events are triggered by purely physical mechanisms.

In this chapter, we reviewed previous efforts in this direction. The limitations imposed by each approximation are obvious but also informative. The next steps toward the synthesis of artificial self-replicating cells require active processes that can regulate membrane dynamics in such a way that reaction kinetics dominates membrane growth. When thresholds of membrane stability are reached, splitting should be possible and the process can begin again. Steps taken in this direction reveal that such a scenario is feasible (Macia and Solé, 2005).

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