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ENVIRONMENT: a computational platform to stochastically simulate reacting and selfreproducing lipid compartments
This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2010 Phys. Biol. 7 036002 (http://iopscience.iop.org/1478-3975/7/3/036002) View the table of contents for this issue, or go to the journal homepage for more
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IOP PUBLISHING
PHYSICAL BIOLOGY
doi:10.1088/1478-3975/7/3/036002
Phys. Biol. 7 (2010) 036002 (13pp)
ENVIRONMENT: a computational platform to stochastically simulate reacting and self-reproducing lipid compartments Fabio Mavelli1,4 and Kepa Ruiz-Mirazo2,3 1 2 3
Chemistry Department, University of Bari, Italy Biophysics Research Unit (CSIC—UPV/EHU), University of The Basque Country, Spain Department of Logic and Philosophy of Science, University of The Basque Country, Spain
E-mail:
[email protected] and
[email protected]
Received 11 February 2010 Accepted for publication 14 July 2010 Published 11 August 2010 Online at stacks.iop.org/PhysBio/7/036002 Abstract ‘ENVIRONMENT’ is a computational platform that has been developed in the last few years with the aim to simulate stochastically the dynamics and stability of chemically reacting protocellular systems. Here we present and describe some of its main features, showing how the stochastic kinetics approach can be applied to study the time evolution of reaction networks in heterogeneous conditions, particularly when supramolecular lipid structures (micelles, vesicles, etc) coexist with aqueous domains. These conditions are of special relevance to understand the origins of cellular, self-reproducing compartments, in the context of prebiotic chemistry and evolution. We contrast our simulation results with real lab experiments, with the aim to bring together theoretical and experimental research on protocell and minimal artificial cell systems. S Online supplementary data available from stacks.iop.org/PhysBio/7/036002/mmedia
In this context, we have developed a computational platform called ENVIRONMENT [9] suitable for studying the stochastic time evolution of reacting lipid compartments. This software is an improvement of a previous program that simulated the stochastic time evolution of homogeneous, fixedvolume, chemically reacting systems [10], extending it to more general conditions in which a collection of similar such systems interact and change in time. A preliminary description of this computational tool was included in the Proceedings of BIOCOMP ’08 [11], but here a more complete and well-grounded account of it will be given. The overall simulation approach we take can be applied, in principle, to many different situations and type of systems (full-fledged living cells included), although our initial objective has been to provide an instrument for analysis of bottom-up and semisynthetic attempts to construct relatively simple artificial protocells [7, 12].
1. Introduction Over the last two decades synthetic and semi-synthetic lipid compartments have been extensively used as in vitro models to investigate the reactive and interactive behavior of cells, including their possible ancient precursors and their artificial implementations [1–8]. Nevertheless, despite the great advances that these approaches have brought about in our understanding of the properties and dynamics of those supramolecular structures (e.g. liposome populations), as well as in biomedical applications (e.g. drug delivery), it is not always easy to interpret what is happening in reality, especially when such complex colloidal systems comprise chemical reactions, together with growth and division or reproduction processes. 4
Author to whom any correspondence should be addressed.
1478-3975/10/036002+13$30.00
1
© 2010 IOP Publishing Ltd
Printed in the UK
Phys. Biol. 7 (2010) 036002
F Mavelli and K Ruiz-Mirazo
In particular, the main motivation for our approach is elucidating the role of randomness [13] in the time behavior of chemically reacting and self-reproducing lipid compartments, such as vesicles or micro emulsions. In fact, in compartmentalized reacting systems (CRS) where the molecular population of the reactants is very low, random fluctuations due to the stochastic nature of reacting events (intrinsic stochasticity) can bring an open system towards unexpected time evolutions [10]. Additionally, this effect can be enlarged by the spreading of different initial concentrations of biological molecules encapsulated in lipid compartments, depending on the experimental preparation procedure (extrinsic stochasticity). In this context, ENVIRONMENT has been designed to study the stochastic time evolution of compartmentalized lipid reacting systems by means of a general and widely accepted Monte Carlo procedure, the Gillespie algorithm [14, 15], and mimicking as closely as possible experimental conditions and preparation methods. As it stands, the general project we have started is divided in two main lines of research. The first consists in modeling and simulating the structural properties and dynamic behavior of lipid vesicle populations [11, 13, 16], comparing them directly with real experimental data (for instance, those coming from Szostak’s [17–20] or Luisi’s [21–25] labs). This gives us the opportunity to test our approach and our simplifying assumptions (e.g. the adopted propensity probability laws), and it also allows estimating dynamic and structural parameters, by fitting experimental data. The second line of research explores hypothetical protocell models that keep a relatively low degree of molecular complexity. In particular, we have introduced and studied the ‘minimal lipid-peptide cell’, a prebiotic cell model where lipid vesicle dynamics is coupled with the condensation of oligo-peptides that could eventually form solute transport channels through the membrane [26, 27]. More recently, we have also started analyzing the feasibility of other more complex schemes, like the ‘Ribocell’ [28]: a minimal cell model that considers the possibility of two hypothetical ribozymes being enclosed within the protocell (one catalyzing its own replication, and the other the synthesis of amphiphilic molecules from lipid precursors) [29, 30]. Nevertheless, in this paper we will focus attention on the first research line, with the aim of demonstrating the main computational features and potential of our platform and, more in particular, on how it can be applied to simulate an equilibrium (or near to equilibrium) vesicle population in an aqueous solution, under different experimental conditions. In this way, by contrasting the results of our simulations with real experimental data, we hope to convince readers that this is a helpful and flexible computational tool, which can be used to study more complex scenarios.
of the stochastic time behavior of a CRS as easy as possible. A CRS is defined as a collection of different chemically reacting domains that can exchange molecules by means of molecular fluxes. Each of these domains, individually, is assumed to be physically and chemically homogeneous, with concentration gradients only present at domain boundaries. Therefore, with this approach one can represent very diverse types of system and dynamic processes: from a population of vesicles that exchange molecules with the external solution (further details in section 3) to a process of molecular diffusion (through a collection of adjacent equal solution volumes that exchange molecules only with the neighboring domain elements—see the supplementary materials for further details at stacks.iop.org/PhysBio/7/036002/mmedia). The time state of a CRS decomposable in D reaction domains can then be described by the molecular matrix of integers M: ⎞ ⎛ x1,1 x1,2 · · · x1,D ⎜ x2,1 x2,2 · · · x2,D ⎟ ⎟ ⎜ M=⎜ . .. .. ⎟ , ⎝ .. . . ⎠ xN,1 xN,2 · · · xN,D which stores the molecular numbers xi ,d of the N different species Xi (i = 1, 2, . . . N) in each domain. The stochastic time evolution of the CRS is described by the reaction diffusion master equation (RDME), which describes the time evolution of the probability of each possible state of the system [31–33]. The RDME is governed by the total propensity density function W(M), where W (M)dt gives the probability that in the next time interval [t, t + dt] an event occurs that changes the state of the system, i.e. the molecular matrix M. Along with M, the state of the system is also defined by the volume, the interface area and the connection map of all domains. Rather than trying to solve directly the RDME, Gillespie proposed a stochastic simulation algorithm (SSA), which works with the propensity density function W (M), for homogeneous well-stirred chemically reacting systems [14, 15]. As an extension of this method, diverse inhomogeneous stochastic simulation algorithms (ISSAs), like the one we use, are being developed. Since all these algorithms are based on the propensity density function, W (M), a general expression for it will be derived next, followed by a description of the recent improvements of both SSAs and ISSAs. Further details on the ENVIRONMENT architecture can be found in the supplementary materials at stacks.iop.org/PhysBio/7/036002/mmedia. 2.1. The total propensity density function W (M) can be defined as the sum of the propensity density functions of any single event that can occur in a CRS, be it either a reacting process or a flux: W (M) = W P (M) + W J (M) =
D R d=1 r=1
2. The platform ENVIRONMENT +
The platform ‘ENVIRONMENT’ is a collection of object oriented C++ classes that cooperate to make the simulation
N D
D
n=1 d=1 d =1 d=d
2
J wd→d (xn,d ),
P wd,r (M × ed )
Phys. Biol. 7 (2010) 036002
F Mavelli and K Ruiz-Mirazo
2.2. Monte Carlo algorithms
where WP (M) is the summation of the propensity density P function wd,r (M × ed ) of each possible reacting event (r = 1, 2, . . . , R) in all domains (d = 1, 2, . . . , D), plus the summation WJ (M) of the propensity density function J wd→d (xn,d ) of any molecular flux between two adjacent domains d and d , ed being the vector (δ i ,d )Nx1 (i = 1, 2, . . . , N). j Of course, wd→d (xn,d ) will be null for unconnected domains according to the time-dependent connection map. Writing down the stoichiometry of the rth reacting process in the dth domain as follows,
Due to the high dimension of the state space, neither analytical nor direct numerical solutions of the RDME are possible for minimally interesting systems. Therefore, different Monte Carlo procedures have been proposed in the past for individual realizations of the Markov processes described by the RDME. In the mid-1970s, Gillespie introduced the SSA direct method (DM) that produces detailed realizations of the stochastic evolution of homogeneous well-stirred chemically reacting systems [14, 15]. Gillespie proved that this method gives an exact simulation of the chemical master equation (CME), which is governed only by the propensity density function describing reactions in a single domain. Over the years different improvements of the original procedure were proposed, with the aim to reduce its computational cost. Diverse computational programs implementing these new features were also presented and are now available for the scientific community (see, as a review, [34]; and, as an example: [36]). These amendments can be grouped into two categories: optimizations and approximations. Within the first group we can mention: the next reaction method (NRM) [37], the optimized direct method (ODM) [38], the sorting direct method (SDM) [39] and the logarithmic direct method (LDM) [40], among others. All these methods are exact simulation procedures of the CME. Alternatively, approximated methods sacrifice the precision of the DM to be faster. They can also be divided in two groups: time-leaping methods [41–43], which assume that many reactions can occur in the same time interval without significantly changing the propensity function WP , and system-partitioning methods or hybrid methods [44, 46], which approximate analytically or numerically fast reactions and stochastically simulate the slow processes. In order to adapt Gillespie’s SSA to spatially inhomogeneous problems, a mesoscopic approach is commonly used by discretizing the reacting system into subvolumes, each of which is assumed to be homogeneous (so that intra-subvolume reactions can be treated as in the homogeneous case). A first example of a simulation of the RDME was done in 1979 [47], when Gillespie’s DM was used to simulate a 1D system decomposed into 21 subvolumes. More recently, efforts have been focused in speeding up the ISSA. The next subvolume method (NSM) was presented [48] as a natural development of the NRM since it adopts the same priority queue structures [37]. For instance, MesoRD [49] is a fairly used implementation of this algorithm (see [50] as an example). A tau-leaping method [51] and a multinomial simulation algorithm [52] have also been presented, with the aim to circumvent the fact that the diffusive transfer between subvolumes dominates the computational cost. Finally, hybrid methods have also been proposed to treat diffusion deterministically, while reactions are handled stochastically by the SSA [53–55]. In this rapidly evolving field, many different programs have been developed that implemented methods to stochastically simulate inhomogeneous chemical and biological systems. It is out of the scope of the present paper to review all these software tools and the interested reader will have to turn to various review papers, included
krd
a1,r X1 + a2,r X2 + · · · + aN,r XN −−→ b1,r X1 + b2,r X2 + · · · + bN,r XN , P (M × ed ) of a reacting the propensity density function wd,r event becomes [10] N
xj,d P d wd,r (M × ed ) = cr aj,r j
=
N krd xj,d (xj,d N ( a )−1 j d j j,r
− 1) . . . (xj,d + 1 − aj,r ),
where d = Vd NA is the dth domain scale factor (i.e. the domain volume times Avogadro’s number), crd is the absolute density probability coefficient and krd is the kinetic constant (i.e. the reaction rate per unit volume and concentration) of the rth reaction in the dth domain. The previous formula simply represents a generalization of the stochastic kinetic theory basic assumption [31, 32], adapting it to the general stoichiometric notation we just introduced [10]. In fact, crd dt gives the probability that a particular arrangement of molecules (in agreement with the stoichiometry of the rth reaction) will react in the next dt time interval. So the combinatorial factors take into account all the possible nonequivalent molecular combinations, to return the rth reacting event propensity density function for each reacting domain. Moreover, it is possible to show that the absolute density probability coefficient crd can be expressed in terms of the macroscopic kinetic constant krd provided that the deterministic and the stochastic approaches must be convergent, in average, at the thermodynamic limit (i.e. for very large populations of reacting species) [10, 34]. J The propensity density function wd→d (xn,d ) for a flux that moves one Xn molecule from the d to the d domain can be expressed in different ways, depending on the physical nature of the two domains. For instance, in the case of a molecular random walk between two equivalent domains (e.g. volume elements of the same aqueous phase), it can be expressed as follows: Dn xn,d J wd→d , (1) (xn,d ) = Ad,d ld,d NA Vd where Ad,d is the interface area between the two adjacent domains, Vd is the d domain volume, Dn is the diffusion coefficient of the nth species and ld,d is the distance between the centre of mass of the two volume domains [35]. For equal cylindrical domains, Ad ,d ld ,d = Vd and the previous formula simply becomes Dn J wd→d xn,d . (xn,d ) = NA ld2 3
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3.1. Geometrical properties and stability of vesicles
in [36, 56, 57]. In any case, we would like to stress that ENVIRONMENT was designed for the simulation of lipid CRS in order to study chemically reacting vesicle solutions, but with the possibility of extending this approach also to aggregates of different morphology. To our knowledge, no other programs have been developed that are directly suitable for this kind of reacting systems. The way in which a lipid vesicle in the aqueous solution is represented and implemented will be discussed in the next section. Here we just want to remark that our program was optimized to handle lipid CRS that can change their volume, their total number and also their connections during the simulations. In particular, we are interested in studying selfreproducing vesicles, as in vitro models of minima cells. Therefore, depending on the internal metabolic network under consideration, they may divide, increasing the total aggregate population, or burst due to an osmotic crisis, staying in the systems as flat bilayers. Both these events change the number of reacting lipid compartments and the number of connections in the global system. At the moment, only exact algorithms, in particular the DM and the NSM, are implemented in the program to perform stochastic simulations. Moreover, to circumvent the problem of different reaction time scales (i.e. stiff reactions), the slow-scale procedure [58] or the pseudoequilibrium approximation [44] can also be used for particular reversible processes. Special attention has been paid to the W updating method: after each iteration (i.e. after any reacting or diffusive event) in order to recalculate it, only the terms that have really changed their contribution to the overall summation are taken into account [38].
The geometrical properties of the lipid membrane, i.e. its volume Vμ and its surface Sμ , are defined in terms of its molecular composition, by the summations: Sμ = 0.5
N
αi xi,μ
Vμ =
i=1
N
νi xi,μ ,
i=1
where α i and ν i are the head area and the hydrophobic tail volume of the ith species, respectively. It should be stressed that the membrane scale factor μ = NA Vμ changes as soon as the membrane composition is changed. The bilayer thickness λμ can then be estimated by the ratio λμ = Vμ /Sμ . In addition, the volume of the core VC is constrained to counterbalance the osmotic pressure and, for this reason, at any iteration it is rescaled according to the following formula:
N N xi,E xi,C , VC = VE (2) i=1
i=1
where VE is the constant volume of the external environment. Equation (2) can be obtained by simply assuming that the T = xi,E /VE is overall external osmolite concentration C E always equal to the internal one CCT = xi,C /VC thanks to an instantaneous flux of water across the lipid membrane. This is in agreement with the high permeability to water molecules that is exhibited by lipid membranes (an assumption that will be tested with specific simulations, in any case, in the next section). As a consequence of this, the membrane surface Sμ and the aqueous core volume Vc follow two different dynamics, and this may bring the vesicles toward unstable conditions. The stability of the vesicle membrane can then be monitored by means of the reduced surface φ: Sμ Sμ φ= ∅ = , (3) 3 SVC 36π V 2
3. In silico vesicle model
C
In this section our in silico model for a reacting lipid vesicle will be presented and discussed. In agreement with the CRS assumptions, a single vesicle in the aqueous solution will be decomposed into three homogeneous reacting domains: the external aqueous environment (comparatively very large, approx. constant volume), the lipid bilayer (or vesicle membrane) and the encapsulated internal water pool (the vesicle core), the latter ones having a variable volume and surface. The membrane surface can be defined as the interface area between the vesicle aqueous core and the external environment, neglecting the difference between the inner and the outer membrane layers due to the bilayer thickness. Both aqueous domains can exchange water, amphiphiles and molecular substrates through the membrane. Moreover, the vesicle membrane can also open up due to an osmotic pressure shock, becoming a flat bilayer and changing the connectivity of the internal and the external water domains. Therefore, the main problem with the simulations of these reacting systems is that both the volume and the connections between different compartments may change in time. Now, we briefly review the main features and assumptions adopted in modeling the dynamics of vesicular aggregates through the platform ENVIRONMENT.
that is, the ratio of the actual membrane surface Sμ , divided by the spherical area that would perfectly wrap the actual core volume. Assuming that, for a given size, the spherical shape (φ = 1) represents the minimum energy state, swollen (φ < 1) and deflated (φ > 1) vesicles are in high energetic conditions, due to the elastic and the bending tension, respectively. Therefore, there is only a small range of φ values, around 1, in which vesicles are stable: √ 3 (4) (1 − ε) φ 2 (1 + η) , with ε and η being the osmotic and dividing tolerance coefficients, respectively. In fact, vesicles in hypotonic solutions can swell, stretching the membrane until they reach a critical state: φ = 1 − ε. The osmotic tolerance ε can be experimentally determined by measuring the maximum difference in the osmolite aqueous concentration (between the internal core and the external environment) that vesicles can bear. For oleic acid and POPC vesicles, ε was found equal to 0.21 and 0.59, respectively [11, 17]. At the critical bursting point, vesicles are assumed simply to break down, releasing all their internal content into the environment and remaining in solution as flat bilayers. A bilayer sealing process is not considered at the moment, but will be a future improvement. 4
Phys. Biol. 7 (2010) 036002
F Mavelli and K Ruiz-Mirazo
In turn, deflated vesicles are supposed to be able to divide in order to minimize the bending energy. The dividing condition is reached when they can form two equal volume spherical daughters (φ = 21/3 ). So η introduces a tolerance that is linked to the relative flexibility exhibited by any membrane. After splitting all the content of the mother vesicle is randomly distributed between the twin daughters. In our simulations, both tolerances can be fixed or, alternatively, be calculated as linear functions of the membrane composition: ε=
N i=1
εi χi,μ
η=
N
ηi χi,μ ,
3.3. Membrane transport processes The transport of molecules from the external environment to the vesicle core through the lipid membrane can be stochastically simulated by explicitly calculating both the incoming and outgoing flux probabilities. In order to do so, one should estimate the propensity density function for both fluxes, according to equation (1) and setting ld,d = λμ and Ad,d = Sμ or defining the propensity density function related to solute exchange as the balance of both fluxes: p T r (xn ) = w J (xn,E ) − w J (xn,C ) E→C
(5)
i=1
where χ i ,μ are the molecular fractions of the different lipids, while εi and ηi are the osmotic and dividing tolerances of the pure membranes, respectively.
(9)
where Dn is the diffusion coefficient (dm2 s−1 mole−1 ) and (Cn,E − Cn,C ) is the difference between the external and the internal concentrations (mole dm−3 ) of the nth solute. pT r dt gives the probability that one solute molecule crosses the membrane in the time interval [t,t + dt[ in the opposite sense of the concentration gradient. The relationship
3.2. Lipid fluxes A vesicle membrane is in constant dynamic equilibrium with its aqueous surroundings: i.e. it exchanges hydrophobic molecules (including its own amphiphilic components) both with the external environment and the internal core. To simulate all these fluxes, we define the probability for molecular uptake,
1−φ J wE/C→μ = kin Sμ exp (6) Ci,E/C , φ
Dn = Pn λμ NA
(10)
links the molecular diffusion coefficient to the macroscopic permeability. The expression for pT r implies that a variation in the membrane composition can affect the transport propensity probability, since both the bilayer thickness λμ and the solute diffusion coefficient Dn depend on the nature of the lipid bilayer. Whereas λμ can be estimated as discussed before (λμ = Vμ /Sμ ), the dependence of the diffusion coefficient on the membrane composition is a more complex problem [19]. In a first approximation, for a binary mixture, the solute diffusion coefficient could be assumed as linearly dependent on the membrane composition, according to the formula
to be proportional to the molecular concentration in the aqueous solutions Ci,E/C = xi,E/C /(NA VE/C ) and to the surface area of the membrane Sμ . The exponential factor additionally introduced increases and decreases the propensity to uptake molecules for deflated and swollen vesicles, respectively (as experimentally observed for oleic acid vesicles [17]). The propensity probability for molecular release was, in turn, assumed to be proportional to the number of molecules of that type present in the membrane xi ,μ : J = kout xi,μ . wμ→E/C
C→E
|(Cn,E − Cn,C )| , = Dn Sμ λμ
DnMix = Dn,A + (Dn,B − Dn,A )χBs ,
(11)
where Dn,A and Dn,B are the diffusion coefficients of the nth solute across the pure membrane of amphiphiles (A and B, respectively), and χBs is the surface fraction of amphiphile B. With ENVIRONMENT it is then possible to simulate transport processes across a binary-mixed membrane, with a composition that changes in time [59].
(7)
Both these two density probability laws will be validated by matching the experimental time behavior of vesicle solutions (see section 4.3). In the case of a membrane composed only of a single type of lipid molecule L, the vesicle surface can be simply expressed as Sμ = α L xL,μ /2 and the aqueous lipid concentration in equilibrium with the bilayer can be obtained (making use of equations (6) and (7) and of the fact that the uptake and release propensity probabilities must be equal at equilibrium):
kout 2 1−φ Eq CL,E/C = exp − (8) kin αL φ
4. Stochastic simulation results In this section, the outcomes of different stochastic simulations will be presented and discussed, in order to test the performance of our platform and reproduce the experimentally observed time behavior of some vesicle populations. In particular, section 4.1 is devoted to study molecular diffusion along a constant section tube and molecular transport across a vesicle lipid membrane, showing the flexibility of our software platform (which can handle these two, so different systems). Furthermore, the approximation of an instantaneous flux of water across the lipid bilayer (equation (2)), will be contrasted with the case of the water transport, dealt deterministically, assuming finite water permeability. Section 4.2 is, instead, devoted to study a population of 50 vesicles. The main aim is to check the coherence between the assumed propensity density probability
As the reader may have noticed, in our approach any aggregate size is equally probable since no factor including the vesicle size is present in the previous equation. This is in agreement with what is observed experimentally: in fact, vesicle solutions extruded at different sizes remain stable for very long time periods [21]. 5
Phys. Biol. 7 (2010) 036002
F Mavelli and K Ruiz-Mirazo -3
1.2
x 10
1
CX,C / M
0.8
0.6
0.4
Myristoleic Acid 1:1 Oleic/Myristoleic 4:1 Oleic/Myristoleic Oleic Acid
0.2
0
0
5
10
15
20 time /s
25
30
35
40
Figure 2. Molecular transport of ribose across 50 nm vesicle membranes made of a myristoleic/oleic acid binary mixture: internal solute concentrations against time. Myristoleic acid (31.0 × 10−8 cm s−1 ) and oleic acid (11.0 × 10−8 cm s−1 ) permeabilities are taken from [19] and directly used for the two limit cases. CRib,E = 1.0 ×10−3 M and the final number of ribose molecules when t tends to infinite is xRib = CRib,E Vc NA = 315. This determines an average displacement from √ the deterministic concentration time course about σCRib,c = xRib / (NA VC ) = 6.0 × 10−5 M.
Figure 1. One-dimensional molecular flux of dextrose S: comparison between stochastic simulations (points) and deterministic curves (lines). Dextrose concentration profiles (S) at different times are reported against the tube length r. The tube is closed on the left end (r = 0) and opened on the other one (r = 2.0 × 10−14 dm) and it has been decomposed into 20 compartments of equal volume Vd = 1.0 × 10−15 dm3 . At the beginning all the tube compartments are empty except for the first one (on the left), where the concentration is set equal to C0 = 1.0 × 10−3 M, which corresponds to about 6.0 × 105 dextrose molecules. Since the number of diffusing molecules is large enough the stochastic outcomes converge with the deterministic curves and random fluctuations appear negligible.
compartments can be different, depending only on the balance of the propensity probabilities of the incoming and the outgoing fluxes. The good agreement obtained supports our approach, at least as far as simulating diffusion processes through a tube driven by a concentration gradient. Next, in figure 2 (points), the simulated transport processes of ribose across a 50 nm radius vesicle membrane (made of a myristoleic/oleic acid binary mixture, at different composition ratios) are reported. These data are compared with the deterministic curves (lines)
3PRib t , (13) CRib,C = CRib,E 1 − exp − RC
for the lipid uptake and release (equations (6) and (7)), respectively, and the average simulated behavior of a vesicle population at equilibrium. Finally, in section 4.3 a comparison between experimental data reported in the literature and stochastic simulation results will be shown for ‘vesicle competition’ experiments. 4.1. Molecular diffusion and membrane transport
which are obtained for a spherical vesicle Sμ /VC = 3/RC where no ribose molecules are present at the beginning but, instead, we introduce an osmolite concentration much larger than CRib,E (see appendix B). For the limit cases of pure lipid vesicles, permeability values were taken from the literature [19], while the diffusion coefficients of the mixed membranes have been calculated as previously described—by equation (11). In this case the agreement between the stochastic simulations and the deterministic curves is worse than before. But this is probably due to the small absolute number of solute molecules that determines a random displacement from the deterministic time courses of the concentration (around 6%). We would like to underline that this feature of the program allowed us to investigate the hypothetical transformation of (prebiotically more plausible) fatty acid vesicles into more stable liposomes (made of doubletailed phospholipids) [59], taking into account the abrupt change expected in the membrane permeability.
Figure 1 shows the stochastic simulation results (points) for the diffusion process of S dextrose molecules along a water tube, closed at one end and with a constant section A. These stochastic outcomes are compared to the deterministic curves (lines):
1 Vd π DS t − 2 −r 2 exp , (12) C(r, t) = C0 A NA 4DS t obtained by analytically solving Fick’s second law [35]. The tube was described as a collection of homogenous equal volume compartments Vd that can exchange molecules by means of two pairs of incoming and outgoing fluxes, toward the left and right neighboring compartments, except for the ends (since the tube was assumed to be closed on the left and open on the right—see figure S2 for a sketch, available at stacks.iop.org/PhysBio/7/036002/mmedia). The flux propensity density probability was calculated according to equation (1). So the concentration in neighboring 6
Phys. Biol. 7 (2010) 036002
F Mavelli and K Ruiz-Mirazo
5
6
-5
x 10
14
VC / nm3
14
-5
x 10
14
x 10
One Vesicle
5
L,C
C Eq
12
12
12
10
10
10
8
8
8
6
6
6
4
4
4
L,C
4
2 -5
-4
-3
-2
-1
0
1
CL,C mol/dm3
3
2
0.5 0.4 CRib,C / M
-5
x 10
0.3 0.2 0.1 0 -5
φ = 0.79 -4
-3
-2 -1 log10(time/s)
0
1
0
2
1
φ = 1.00 2
3
0
1 2 time / sec
φ = 1.36 3
0
1
2
3
Figure 4. Oleic acid concentration in the vesicle aqueous core against time for a 3 mM oleic acid solution of 50: (a) swollen (φ = 0.79); (b) isotonic (φ = 1.00) and (c) deflated (φ = 1.36) vesicles. The simulation starts from an ideally monodisperse 50 nm radius vesicle population. Equilibrium values (black lines) are calculated by equation (8) using the parameters reported in table 1: (a) 51.1 μM, (b), 66.7 μM and (c) 86.7 μM. Light gray lines are the internal concentration of free lipid molecules in a single vesicle, while the dark gray lines are averages (calculated over the vesicle population).
Figure 3. Vesicle internal core shrinkage and growth after the addition of ribose XRib in the external environment of a 50 nm oleic acid vesicle: comparison between stochastic outcomes (points) and numerical deterministic analysis (gray lines). The volume and ribose concentration of the vesicle aqueous core are reported in the upper and the lower plot, respectively. The external ribose concentration after the addition is assumed to remain constant. The permeability values used for the oleic acid vesicle membrane to water PW = 8.4 × 10−3 cm s−1 and to ribose PX = 11.0 × 10−8 cm s−1 were taken from [19].
Table 1. Simulation parameters for oleic acid and POPC vesicles: kin and kout are the kinetic constants for the release and uptake Eq processes; CL,E/C (φ = 1) is the lipid aqueous concentration at equilibrium in the isotonic solution; α is the lipid head area; ε is the Eq osmotic tolerance. CL,E/C has been calculated by equation (8).
In order to test the assumption of an instantaneous flux of water across the lipid membrane (equation (2)), we simulated the shrinkage and subsequent growth of a vesicle as a result of the addition of a large amount of ribose to the external environment. Figure 3 shows the comparison of the stochastic outcomes (points) with the deterministic curves (gray lines) that reproduce the experimental results reported by Sacerdote and Szostak [19]. The instantaneous addition of ribose to a solution of lipid vesicles is a method that has been used in order to determine experimentally the membrane permeability both to water and to diverse molecular compounds, by following the vesicle volume change through a spectrofluorometric assay [19]. In fact, starting from an isotonic condition, the abrupt addition of molecules outside the vesicle membrane will determine a flux of water from the vesicle core toward the external environment (so as to decrease the internal volume and, thus, increase the vesicle internal concentration). This process will be stopped when a new isotonic condition is reached (typically, in a few milliseconds). Therefore, the transport of the solute molecules from the outside, driven by the concentration gradient, takes place and this process will increase the internal osmolite concentration—and, then, the core volume to its original value. This phenomenon is well reproduced by the deterministic analysis (see appendix C), in agreement with what has been observed experimentally. The comparison with the stochastic outcomes shown in figure 3 also validates the assumption of an instantaneous flux of water adopted by ENVIRONMENT, given the high value of the water permeability. Nevertheless, the limit of this approximation appears when describing the time behavior of the vesicle core
Parameters
Oleic acid
POPC
kin /s−1 M−1 nm−2 kout /s−1 Eq CL,E/C (φ = 1)/M α nm−2 ε
7.6 × 103 7.6 × 10−2 6.67 × 10−5 0.3 0.21
7.6 × 103 7.6 × 10−7 2.86 × 10−10 0.7 0.59
volume in the case of very abrupt variations of the external osmolite concentration. 4.2. Membrane solution lipid exchange In this section we present and discuss simulations of a population of oleic acid vesicles at equilibrium using the parameters reported in table 1. We start testing the lipid uptake and release processes by simulating a 3 mM oleic acid volume solution containing 50 (a) swollen (φ = 0.79), (b) isotonic (φ = 1.00) and (c) deflated (φ = 1.36) monodispersed vesicles of 50 nm (initial radius). In all the studied cases, as reported in figure 4, the internal lipid concentration CL,C , averaged over the vesicle population, fluctuates around the equilibrium concentration values calculated by equation (8) (dark gray lines). Light gray lines reflect the time course of the internal lipid concentration in a 7
Phys. Biol. 7 (2010) 036002
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(a)
1.15
1.1
μ
μ
μ
μ
S end+Δ S *e-kGt Ol.Ac. Isotonic Ves. Ol.Ac. Swollen Ves. POPC Isotonic Ves. POPC Swollen Ves.
μ
S /S 0
μ
1.05
1
0.95
0.9
0.85 -1.5
-1
-0.5
0 0.5 log10(time / s)
1
1.5
2
Figure 6. Simulation outcomes of isotonic versus swollen vesicle competition experiments: the averaged relative surfaces of swollen (stars) and isotonic (circles) oleic acid vesicles versus swollen (triangles) and isotonic (points) POPC vesicles are reported against time, including error bars. Averages were carried out over an initially monodispersed population of 50 vesicles. The line and dash gray curves are the time trends calculated with functions in the legend, by setting kD = 0.08 s−1 and kG = 0.09 s−1 , as reported experimentally [17].
(b)
to the number of lipids xLμ in the membrane and the radius 1 1 R is proportional to the surface squared root Sμ 2 ∼ xL,μ 2 for a spherical vesicle, the observed linear relationship is what would be expected between a random variable and its standard deviations due to random noise. Moreover, the χ 2 statistics also confirm that the simulated average size distribution at equilibrium is a Gaussian distribution. Figure 5. Stochastic simulations of a 3.0 mM oleic acid solution containing 50 vesicles with a defined initial radius R0 : (a) in the graph above, the broadening in the size distribution is reflected by the membrane surface standard deviation σ S ; (b) in the graph below, the surface standard deviations σ S are averaged for each simulation over the last 0.5 s time range of the run, and the values obtained are reported against each initial radius R0 .
4.3. Vesicle competition experiments Having tested the behaviour of the platform through simulating lipid vesicles at equilibrium, we turn now to study more complex, far from equilibrium experimental conditions, checking the performance and validity of the platform against recent data on vesicle population ‘competitive dynamics’, as provided by various labs.
single vesicle: as the random displacement from the average expected values increases, the number of reacting molecules decreases. For instance, in the isotonic case, CL,Eq = 66.7 μM corresponds to an average number of 21 lipid molecules. In order to make sure that the observed deviations are simulated really as random fluctuations, we follow the standard deviation σ S of the vesicle surface area distribution against time. Starting from a population of 50 isotonic (φ = 1.0) spherical vesicles (with the same initial radius R0 ), the broadening of the membrane surface distribution is followed in time, simulating an ideal vesicle extrusion process. In figure 5(a) all the studied cases are reported: σ S starts from 0.0 and after about 10 s is already fluctuating around an average value σ S , which increases linearly with R0 (see figure 5(b)). Since the vesicle surface area Sμ = α L xL,μ /2 is proportional
4.3.1. Isotonic versus swollen vesicles. The competition between isotonic and swollen vesicles for the uptake of free lipid molecules in solution was reported a few years ago by Szostak and coworkers, for oleic acid vesicles [17]. This phenomenon has been ascribed to the elastic tension that increases the rate of the monomer uptake in the membranes of swollen vesicles. As a consequence of this, the surface of those vesicles increases, diminishing that of the isotonic ones. This behaviour is well grasped using our approach, as shown in figure 6. In fact, the relative average surface time course of both swollen (stars) and isotonic (circles) oleic acid vesicles obtained by stochasic simulations exhibits two exponential time trends that match pretty well those experimentally observed and are best fitted by using two 8
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slightly different phenomenological constants: kD = 0.09 s−1 and kG = 0.08 s−1 for the surface decay and growth, respectively [17]. The agreement between our simulated data and the experimental evidence allowed us to validate the probability laws (6) and (7), as well as to estimate the best kout and kin constant values (table 1). Furthermore, we could also simulate correctly the behavior of POPC vesicles (reported in figure 6) which, after mixing, do not exhibit the same competitive dynamics, even after several hours [17]. This was obtained by fixing kin and downscaling the value of kout (by, aproximately, 5 orders of magnitude), in agreement with the lower CAC value expected for POPC molecules, see table 1.
(a)
4.3.2. Oleic acid versus POPC vesicles. As a final case study, our approach is applied to the analysis of the competition dynamics between two populations of vesicles with very different mean sizes, initially made also of two different types of lipids, POPC and oleic acid, as reported by Cheng and Luisi [21]. When equilibrium is reached, after mixing the two extruded vesicle suspensions, only a single peak was observed (by dynamic light scattering analysis), which is located around the initial radius of the extruded POPC vesicles. In particular, we shall focus on the experiments where a population of 30 ± 8.9 nm POPC vesicles (1.0 mM POPC) is mixed with a 64.1 ± 14.4 nm oleic acid vesicle suspension (1.0 mM oleic acid, too). In spite of the fact that light scattering efficiency is proportional to the squared volume of the scatters, a single peak around 30 nm was recorded, after long waiting times [21]. Figure 7 shows how this behavior is also well reproduced by our simulations. The upper histogram of figure 7(a) portrays the initial size distribution of the simulated POPC and oleic acid vesicle populations, which mimic the experimental ones before mixing. The lower histogram shows the final size distribution, located around 30 nm, i.e. near the initial average radius of the POPC vesicles. The stochastic simulations make possible a better interpretation of this phenomenon: after mixing, POPC vesicles uptake free oleic acid molecules very fast, lowering the aqueous (free/monomer) oleic acid concentration (simulation results not shown). Oleic vesicles then release their components in order to re-establish the oleic acid equilibrium concentration in the environment. As a consequence of this, there is a net flux of oleic acid molecules from the oleic to the POPC vesicles. This is actually confirmed by the change in the membrane composition reported in figure 7(b), which goes from 1.0 at the beginning (for pure POPC vesicles— white bars) to the value 0.5 at the end (for mixed POPC/oleic acid vesicles—gray bars). This 1:1 composition corresponds to the aqueous concentration ratio of the two lipid solutions before mixing. Furthermore, figure 7(b) also shows that the total number of vesicles increases after mixing. In fact, our approach predicts that the mixed POPC/oleic acid vesicles enlarge their membrane surface, meeting the condition for splitting into two twin daughters, as experimentally observed [29, 30]. The new mixed vesicle size distribution reaches equilibrium in less than 1 min and this simulation result is in agreement with the optical density spectra reported
(b)
Figure 7. Competition between POPC and oleic acid vesicles of different extruded radii: (a) oleic acid and POPC vesicle size distributions in 1.0 mM solutions before mixing (upper plot) and after mixing (lower plot) and (b) variation in composition and number of initially pure POPC vesicles, after the mixing. The oleic acid molecules are completely absorbed by the POPC aggregates by means of free monomer uptake, which determines the size growth and the division of the POPC/oleic acid mixed vesicles.
by Cheng and Luisi [21], but also with the dynamic light scattering [23] and the turbidity [60] analysis of the time behavior of unilamellar phospholipid vesicles perturbed by an instantaneous addition of fresh fatty acid, i.e. the so-called matrix effect [23, 25, 60, 61].
5. Discussion and final remarks In this paper we have described the new software platform ‘ENVIRONMENT’, developed with the aim to simulate the time evolution of chemically reacting lipid compartments, based on the stochastic kinetics theory [31–33] and the wellaccepted Gillespie method [14, 15, 34, 36–47]. This method is an exact Monte Carlo procedure for sampling time trajectories in the phase space, in agreement with the chemical master equation [15], while in the thermodynamic limit (i.e. for high molecular populations of the reacting species), it can be seen 9
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06348-C02-01, from the Spanish Ministry of Science and Education (MICINN).
as a robust algorithm for integrating the ordinary differential equation set that gives the deterministic time evolution of the reacting system [62]. In our program we adapt the Gillespie DM in order to cope with compartmentalized lipid systems that are decomposed into a collection of homogeneous domains, in accordance with the mesoscopic approach to the simulations of the RDME. Moreover, the code has been optimized to simulate systems where lipid compartments can change in size and number during the run and it has been applied to reproduce the dynamic behaviour of lipid vesicles that are being experimentally used as protocell models, both in the presence and absence of chemical reaction networks. Here we focused on relatively simple case studies, in which the comparison with analytic-deterministic curves or experimental data could give more direct support to the platform, testing its underlying stochastic nature. More specifically, we have shown how our platform can carry out simulations of both a single vesicle and a vesicle population (which can also increase in number, due to division processes). By assuming a dependence of the lipid uptake probability on the reduced membrane surface φ (i.e. on the elastic energy of the lipid bilayer), we also reproduced the observed isotonic/swollen vesicle competition dynamics and could estimate the kinetic parameters for both oleic acid and POPC aggregates. Thereafter, we used these values to simulate the competition among POPC and oleic acid vesicles, obtaining both an equilibrium size distribution and a process time scale which are in agreement with the experimental data. Thus, ENVIRONMENT seems to be a suitable tool to study theoretically the dynamics of lipid vesicles. It can be used both to simulate reacting liposomes, as compartmentalized nano-sized reactors (since it takes into account the role of stochastic fluctuations), as well selfreproducing vesicular systems that mimic protocellular behavior. Furthermore, it can be applied to test different hypotheses on the initial solute distribution present in compartmentalized systems, according to different preparation methods. In fact, as it has been recently highlighted by various authors [63–65], there is increasing evidence that the observed experimental solute distributions in this type of systems cannot be explained by a simple random encapsulation hypothesis. Furthermore, in self-reproducing lipid vesicles that mimic cellular division processes, the random redistribution of internal molecular species between the two daughters represents another source of extrinsic stochasticity, which must be taken into consideration (as we have shown when analyzing the ribocell model [30]). In conclusion, we believe that current [26, 27, 29, 59] and future developments of this computational platform will demonstrate that it is a research tool with high potential and flexibility, which will hopefully help us gain a better understanding of the time behavior of complex, chemically reacting, lipid systems.
Appendix A. The software architecture The program ENVIRONMENT was coded in the objectoriented ANSI C++ language and it was designed to make future developments as simple as possible. Thus, the problem of simulating the stochastic time evolution of a compartmentalized chemically reacting system has been decomposed into a hierarchy of C++ classes. The class CReactor has been designed to correspond to a homogeneous chemically reacting domain. This class is a container of species and processes, as member objects. Species and processes are instances of classes themselves. The CSpecies class is designed to store all data regarding the population of a reacting chemical species (molecules, different size polymers, amphiphile aggregates, nucleic acid strands) and to implement methods for manipulating this information. The CProcess class, instead, is designed to simulate chemical reactions. Each instance of CProcess must be linked to instances of CSpecies classes involved in each specific reaction and belonging to the same CReactor object. CProcess must also contain all the parameters necessary to calculate the propensity density probability of the specific reacting event. Moreover, if the reaction takes place during the simulation run, the CProcess instance must communicate to all the linked CSpecies instances to change the species populations according to the specific stoichiometry. Therefore, any CReactor instance can manage the stochastic time evolution of a column of the matrix M driven by reacting events, according to Gillespie’s DM [14, 15]. In order to study CRS, two other classes were introduced: CPot and CSystem, respectively. CPot is merely the base class for both CSystem and CReactor classes, while CSystem is a container of different classes, all derived from CPot. This means that a CSystem instance can contain both CReactor and CSystem objects and the latter, in turn, other CReactor and CSystem objects (and so on). This approach makes our program suitable to study real cells, where many reacting compartments with different levels of encapsulation are actually present. Each CSystem object can contain also instances of the class CFlux, which was designed to simulate molecular fluxes between connected homogenous reacting domains. Figures S1 and S2 (available at stacks.iop.org/PhysBio/7/036002/mmedia) show the computational objects instanced during the simulation of a reacting vesicle and a one-dimensional diffusion process. Both these phenomena can be easily described in terms of CRS by the ENVIRONMENT platform. Finally, we should also remark that at the beginning of each program run there are no objects allocated in memory. All the instructions and settings can be read from a set of initialization files, one for each CPot instance, so the same executable can simulate a single reacting system or a system with a complex hierarchy of interconnected reacting domains. Furthermore, since the computer memory is
Acknowledgments KRM is a ‘Ram´on y Cajal’ Research Fellow. The authors collaborated under support from research grant FFI200810
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managed dynamically, the program is apt to study the evolution of populations of proto-cells, which may grow and reproduce, or collapse, during its running time. Anyone who would be interested in testing this code can write an email to the corresponding author to get an OS WIN32 executable with the input files necessary for reproducing all the simulation data shown in this paper.
Experimental and calculated values for myristoleic/oleic acid 50 nm radius mixed vesicles. The permeability values are relative to ribose. Parameters Permeability (cm s−1 ) ×108 Molecular diffusion coefficient (dm2 mole−1 s−1 ) ×10−8 Membrane thickness (nm) Surface head area (nm2 ) Molecular volume (nm3 )
Appendix B. Deterministic analysis of concentration time changes in the vesicle core due to an instantaneous addition of a small amount of solute Equation (13) describes the deterministic time course of the vesicle internal concentration CRib,C of ribose when a transport process takes place across the lipid membrane (due to the external addition of an amount of ribose, such that CRib,E >CRib,C after the addition). So the number of molecules that, in average, go into the vesicle, through the membrane, per unit of time is given by
4:1
Oleic acid
31.0a
19.9d
14.3d
11.0a
5.97b
4.31c
3.31c
2.65b
3.2e 0.3 0.48
3.6e – –
3.84e – –
4.0e 0.3 0.6
a
Reported by Sacerdote and Szostak [19]. Calculated by equation (10). c Calculated by equation (11). d Calculated by inverting equation (10). e Values calculated by means of Vμ /Sμ . b
Sμ DX dNX (CRib,E − CRib,C ). = dt λμ
Appendix C. Deterministic analysis of vesicle volume time changes due to an instantaneous addition of a solute large amount
Dividing both members of the previous equation by NA VC and remembering the relationship between the molecular diffusion coefficient and the permeability (equation (10)), one obtains
The instantaneous addition of a solute to the external environment of a lipid vesicle is a method that has been used by Sacerdote and Szostak [19] in order to experimentally determine both the membrane permeability to water and to different solute molecules. In a 50 nm radius extruded oleic acid vesicle suspension (in isotonic conditions—with an osmolite concentration CB ,E = C◦ B ,C = 0.5 M), a fresh amount of ribose is instantaneously added, giving a ribose initial concentration CRib,E = 0.5 M and C◦ Rib,C = 0.0 M). If the external aqueous solution is so large that the concentration CB ,E and CRib,E can be considered constant, then the following pair of differential equations must be solved:
Sμ dCRib,C PRib (CRib,E − CRib,C ). = dt VC If the external solution is so large that CRib,E can be considered constant and the amount of ribose added is so small that CRib,E is much lower than the overall osmolite concentration, the previous equation can be transformed as follows: −
Myristoleic acid 1:1
3 d C PRib C, = dt RC
where Sμ /VC = 3/RC since the core volume and the surface vesicle changes during the transport process have been considered negligible. This equation can then be easily integrated to give equation (13) if no molecules of X are present inside the vesicle at the beginning: C0 = CRib,E = 1.0 mM. In agreement with the data reported by Sacerdote and Szostak [19], the stochastic simulations have been performed in the presence of a total osmolite concentration of 0.5 M, while the ribose external concentration, immediately after the addition, was CRib,E = 1.0 mM. In that way, the increment in the total external concentration determines a decrease in the VC around 0.2%, as can be calculated by using equation (2):
N N VE 0.501 CET VC = x x = = 0.998. = i,E i,C T 0 0 0.5 CC VC VC i=1 i=1
⎧
xRib,C dxRib,C ⎪ ⎪ ⎪ ⎨ dt = NA PRib Sμ CRib,E − V N C A
⎪ + x dV x C B,C Rib,C ⎪ ⎪ − CB,E − CX,E , = vaq NA Paq Sμ ⎩ dt VC NA in order to get the time course of the average number of ribose molecules xRib,C inside the vesicle and the average vesicle core volume VC . The number of B molecules xB,C inside the vesicle is constantly equal to the initial value xB,C = NA C◦ B ,C (4π 503 /3) since no molecules of this kind are released from the vesicle core. The permeability values PW = 8.4 × 10−3 cm s−1 and PRib = 11.0 × 10−8 cm s−1 were taken from the literature [17] and the obtained volume time trend is congruent with those reported. The differential equation set was numerically solved with a method suitable to the solution of stiff problems [66].
In the following table, the permeability and diffusion coefficients of ribose, calculated by means of equation (11) for mixed membranes made by a blend of myristoleic acid and oleic acid, are reported, along with the values for the pure fatty acid vesicles. 11
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