Chemical Schemes for Maintaining Different Compositions Across a Semi-permeable Membrane with Application to Proto-cells

Abstract

Osmotic pressure arising from a higher total chemical concentration inside proto-cells is thought to have played a role in the emergence and selection of self-replicating proto-cells. We present two chemical schemes through which different equilibrium compositions can coexist on each side of a semi-permeable membrane. The first scheme relies upon the concept of moieties and associated number of degrees of freedom. The second scheme relies upon the concept of siphons and of pass reaction capable of transferring matter from outside a siphon into it. Using simple example reaction networks, we show that both schemes are compatible with stationary proto-cell growth with up-concentration, but suffer from shortcomings. To alleviate these we propose a third scheme derived from the second one by having the pass reaction catalyzed by the membrane surface instead of occurring in bulk solution. This may have proven an intermediate step before having the pass reaction occurring only when the nutrient crosses the membrane. This suggests an evolutionary path for the emergence of active transport.

Introduction

A biological cell is a system that maintains itself out of equilibrium and in a different compositional state than the outside growth medium. Although modern evolved cells maintain a balance of osmotic pressure across both sides of the membrane through complex regulation mechanisms (Csonka 1989), a higher total chemical concentration inside the cell and associated positive osmotic pressure is thought to have played a role in contributing to the growth of early proto-cells (Chen et al. 2004).

It is not obvious how such a concentration difference may be maintained on two sides a of semi-permeable membrane. Serra and Villani (2013) have addressed this issue and proposed a scheme where a semi-permeable membrane separates two compartments of different sizes, with specific chemical reactions being catalyzed by the membrane surface. Diffusion of the permeating species can induce a transient higher concentration in the smaller compartment, and this up-concentration can be made stationary with a constant flow through the larger compartment.

The present article also addresses this issue, by presenting two chemical schemes that do not require any external flow to maintain such a concentration difference. We present chemical schemes where the concentrations of permeating chemical species are equal on both sides of a semi-permeable membrane, and yet where other concentrations are maintained at different equilibrium values on each side. And we evaluate to which extent such an up-concentration can be maintained inside a stationarily growing out-of-equilibrium proto-cell.

The first scheme relies upon the concept of moieties and number of degrees of freedom for a conservative Chemical Reaction Network (CRN). Moieties are linear combinations of concentrations that are left unchanged by the chemical reactions (Schuster and Höfer 1991). The number of linearly independent moieties is the number of degrees of freedom of a CRN (Wei 1962). Our first scheme exploits this by having only some degrees of freedom fixed by the equality of concentrations of permeating chemical species across the membrane, while leaving at least one degree of freedom free to vary on either side of the membrane.

The second scheme relies upon the concept of siphons. Siphons are subsets of species whose absence cannot be compensated by the chemical reactions (Angeli et al. 2007). If a CRN has a siphon that is shorter than the full set of species, there exist equilibria with all species in the siphon absent, distinct from equilibria with all species present. It is thus possible to envision situations where a membrane is semi-permeable to species outside this shorter siphon, and where only the permeating species are present on one side of the membrane while all species are present on the other side.

Assuming that one of the chemical species (membrane precursor) self-assembles in a structured membrane that is itself semi-permeable to other species (nutrient), and using simple example chemical reaction networks, we find that these two schemes are theoretically compatible with stationary proto-cell growth, while maintaining an up-concentration regime within some kinetic and membrane parameter range. However both schemes present shortcomings that may prevent practical implementation.

A shortcoming of the first scheme is that the membrane grows not only from the inside but also from the outside, which reduces the level of up-concentration. The second scheme relying upon siphons does not present such shortcoming because the membrane precursor is assumed to be part of the shorter siphon and is thus not present in the outside growth medium. However, this second scheme is sensitive to growth medium contamination that may result from any lysed cell.

To alleviate this, a third scheme is proposed, derived from the second one by having the key nutrient input reaction (denoted pass reaction) catalyzed by the membrane surface instead of occurring in bulk solution. A further optimized scheme consists in having this key nutrient input reaction only occurring upon crossing the membrane, which is active transport.

Definitions and Methods

In this text, a vector v is written in bold font. Its i ^th coordinate is denoted v _i . The i ^th coordinate of a vector carrying its own index v _k is denoted v _k,i .

Definitions Related to CRNs

Following standard practice in Chemical Reaction Network Theory (Érdi and Tóth 1989) a stoichiometry matrix S may be associated to any CRN. For the CRN in a closed system, the time derivative of the concentration vector c is given by $\dot {\textit {\textbf {c}}}=S\textit {\textbf {f}}$ where f is the rate vector. We denote $\mathcal {S}$ the full set of species participating in the CRN.

Moieties and Degrees of Freedom for a Conservative CRN

A CRN is conservative if each chemical species may be assigned a positive mass such that mass conservation be guaranteed for every chemical reaction. In algebraic terms, this is equivalent to having a strictly positive mass vector m such that m ^T S=0 (m _i is the molecular mass of species A _i ). There may exist more than one solution to mass conservation. The set {m | m _i >0, i=0,…,N−1 and m ^T S=0} is a pointed convex cone having p linearly independent generating vectors {b _k } _{k=0,…,p−1}. These are moieties and the associated positive linear combinations of concentrations $\left \{\boldsymbol {b}_{k}^{\mathrm {T}}\boldsymbol {c}\right \}_{k=0,\dots ,p-1}$ are left invariant by the chemical reactions. The support of a moiety b is the subset of chemical species A _i along which b has non-zero components. p is also the number of degrees of freedom (Wei 1962) in the sense that the equilibrium state of the CRN in a closed system is uniquely determined by the conserved quantities $\left \{\boldsymbol {b}_{k}^{\mathrm {T}}\boldsymbol {c}_{0}\right \}_{k=0,\dots ,p-1}$ where c ₀ is the initial concentration vector. Only conservative CRNs are considered in this work with all material fluxes made explicit in the proto-cell model.

Siphons and pass Reactions

A siphonZ is a subset of species such that for each species A _i in Z and every reaction where A _i appears as product, then at least one of the reactant species also belongs to Z. A siphon is minimal if it does not contain strictly any other siphon. Siphons are essentially the sets of chemical species whose absence cannot be compensated by the chemical reactions that take place inside the cell (Angeli et al. 2007). Building upon the concept of siphon, we here introduce the notion of pass reaction that will prove useful in the case of a proto-cell. Consider a conservative CRN having a siphon Z that is shorter than the full set of species $\mathcal {S}$. A pass reaction is a reaction involving species in both Z and $\mathcal {S}\setminus Z$ and resulting in a net positive mass transfer from $\mathcal {S}\setminus Z$ into Z.

Proto-cell Model

Figure 1 illustrates the proto-cell model that is used to assess the compatibility with stationary growth and up-concentration for the proposed schemes.

Fig. 1

Schematic of the proto-cell model. The nutrient A _nu flows across the membrane by passive diffusion (first and second schemes) or by active transport (third scheme). The membrane results from the self-assembly of the membrane precursor A _me that may be present on both sides (first scheme) or only inside (second and third scheme)

Assumptions

• 1. Chemistry represented by the same CRN inside and outside of the proto-cell.

• 2. Constant surface area-to-volume ratio denoted ρ ₀ (e.g. filamentous growth).

• 3. Self-assembly of a precursor $A_{\text {me}}\in \mathcal {S}$ in a structured membrane: its incorporation into the growing membrane is kinetically controlled, with coefficient K _me per unit area, or k _me=ρ ₀ K _me per unit volume. Denoting N _me the number of molecules per unit area in the self-assembled membrane, C _me=ρ ₀ N _me is the effective membrane concentration as if all self-assembled molecules were dissolved in the cell volume.

• 4. Semi-permeability of self-assembled membrane to a nutrient A _nu. The resulting flux is F _input,nu per unit area, or f _input,nu=ρ ₀ F _input,nu per unit volume.

• 5. Spatially homogeneous concentrations within each compartment.

• 6. Large outside growth medium volume: the outside concentration vector c _out remains constant, even in the presence of a growing proto-cell.

Ordinary Differential Equation (ODE) System

The time evolution of concentrations inside the proto-cell is determined by:

$$ \dot{\boldsymbol{c}}=\boldsymbol{S}\boldsymbol{f}+\boldsymbol{f}_{\text{input}}-\boldsymbol{f}_{\text{output}}-\lambda\boldsymbol{c} $$

(1)

where c is the concentration vector inside the proto-cell, S is the stoichiometry matrix, f=f(c) is the reaction rate vector, f _input is the nutrient flux vector with only non-zero component f _input,nu along A _nu, f _output is the membrane precursor incorporation flux vector with only non-zero component f _output,me along A _me and λ c represents the dilution factor with growth rate λ:

$$ \lambda=\frac{1}{\mathcal{V}}\frac{\mathrm{d}\mathcal{V}}{\mathrm{d}t}=\frac{1}{\mathcal{A}}\frac{\mathrm{d}\mathcal{A}}{\mathrm{d}t} $$

(2)

where $\mathcal {V}$ (resp. $\mathcal {A}$) is the proto-cell volume (resp. surface area). The latter equality results from Assumption 2.

From assumption 4, in the case where nutrient flows by diffusion with effective diffusion constant $\mathcal {D}$, we have:

$$ f_{\textrm{input,nu}}=\mathcal{D}(c_{\textrm{out,nu}}-c_{\text{nu}}) $$

(3)

From Assumption 3 where k _me was defined as an effective kinetic coefficient, we have:

$$ f_{\textrm{output,me}}=k_{\text{me}}c_{\text{me}} $$

(4)

λ is related to f _output,me and C _me as follows. The membrane surface area $\mathcal {A}$ increases as the result of A _me incorporation with two possible situations: (i) either incorporation originates only from the inside because A _me is absent from the outside growth medium or because there is some physical constraint preventing incorporation from the outside (e.g. membrane polarity), or (ii) it originates from both sides. In case (i), as A _me gets incorporated into the growing membrane with rate (per unit area) F _output,me=K _me c _me, the membrane area grows as:

$$ \lambda=\frac{1}{\mathcal{A}}\frac{\mathrm{d}\mathcal{A}}{\mathrm{d}t}=\frac{F_{\textrm{output,me}}}{N_{\text{me}}}=\frac{f_{\textrm{output,me}}}{C_{\text{me}}}=k_{\text{me}}\frac{c_{\text{me}}}{C_{\text{me}}} $$

(5)

In case (ii) there is an additional contribution arising from c _out,me and the above equation should be replaced by:

$$ \lambda = \frac{k_{\text{me}}c_{\text{me}}}{C_{\text{me}}} + \frac{k_{\text{me}}c_{\textrm{out,me}}}{C_{\text{me}}} $$

(6)

Equations 1–6 give the ODE system governing the time evolution of concentrations. The CRN is characterized by its stoichiometry matrix S and by the reaction kinetics represented by the dependence of the reaction flux vector on the concentration vector, f=f(c). The membrane is characterized by k _me and C _me accounting for precursor incorporation, as well as by $\mathcal {D}$ accouting for nutrient diffusion. And the environment is characterized by the outside concentration vector c _out.

Stationary Growth

A proto-cell is defined as compatible with stationary growth if the concentration vector trajectory (solution of the above ODE system) asymptotically converges towards constant strictly positive concentrations. With the assumption 2 of a constant surface area-to-volume ratio, this typically corresponds to a steady-state filamentous growth with the filament length increasing indefinitely at a constant rate. In such a situation, the various contributions in the above ODE system exactly compensate one another for each species: considering for example the case of the membrane precursor A _me, its disappearance rate through incorporation into the growing membrane as well as by dilution is exactly compensated by its rate of synthesis by the CRN (which results from the nutrient being metabolized into all species, including A _me). Such an asymptotic property assumes that the outside growth medium volume is infinitely large (assumption 6). In practical situations where this volume is finite, growth would eventually slow down and stop when most of the nutrient has been metabolized.

Minimum Conditions for Stationary Growth

We have previously proved necessary and sufficient conditions for proto-cell stationary growth (Bigan et al. 2014):

• 1. A necessary condition is that each moiety b must be fed. Otherwise, all species in the support of b go extinct by dilution.

• 2. A sufficient condition is that each siphon contains the support of a moiety (that is fed).

A stronger necessary condition is that for each siphon Z that is not directly fed, there must exist a pass reaction connecting species in $\mathcal {S}\setminus Z$ to species in Z. Otherwise, all species in Z go extinct by dilution.¹

Numerical Parameters and Methods

All schemes presented in this article rely upon simple example CRNs involving three species A, B and C participating in monomolecular or bimolecular reactions, with A chosen as nutrient, A _nu=A, and C as membrane precursor, A _me=C.

All reactions were endowed with mass-action kinetics (numerical values are given in figure captions). Nominal parameters were chosen for the proto-cell as : $\mathcal {D}=10\, \textrm{s}^{-1}$, c _out,nu=1×10⁻³ M, k _me=1×10⁻³ s⁻¹ and C _me=5×10⁻² M. The rationale for these numerical choices is given in Appendix A.

Proto-cell parameters were varied around their nominal values the following way. First, (k _me,C _me) were kept constant at their nominal values and the ODE system was integrated for varying $(\mathcal {D}, c_{\textrm {out,nu}})$ values. Second, $(\mathcal {D}, c_{\textrm {out,nu}})$ were kept constant at their nominal values and the ODE system was integrated for varying (k _me,C _me) values. In both cases each variable parameter was let span six orders of magnitude in 25 logarithmic increments, and each of the resulting 25×25 trajectories was integrated using two different choices of initial conditions: all concentrations either equal to 10⁻⁹ M, or to 1 M.

Such large spans are likely to include some numerical values that are not physically accessible. They have been chosen to include any practical situation and to illustrate the robustness of proposed schemes with respect to parameter variation.

ODEs were handled in vectorial form using the odeint native ODE solver from the scientific Python software package (Python version 2.7.3) with dynamical adjustment of the time step.

First Scheme: Exploiting the Number of Degrees of Freedom

General Description

Figure 2 illustrates the first scheme. Let us consider a CRN having p moieties (i.e. p degrees of freedom), the support of which overlap for k<p species, and let us assume the membrane is semi-permeable to these k species. At equilibrium, concentrations for these k species is equal on both sides. This fixes k out of the p degrees of freedom. There remains p−k additional degrees of freedom to let other concentrations vary. It is thus possible to sustain different compositions on each side of the membrane. The reason why the support of the p moieties should overlap for the k permeating species is for each moiety to be fed (necessary condition for proto-cell stationary growth).

Fig. 2

Schematic representation of the first scheme

Simple Example: At Equilibrium

Consider the following CRN consisting of the single bidirectional reaction:

$$A\rightleftarrows B+C $$

It has p=2 moieties: b ₀ with (b _0,A ,b _0,B ,b _0,C )=(1,1,0) and b ₁ with (b _1,A ,b _1,B , b _1,C )=(1,0,1). Their supports overlap for k=1 chemical species, A. Let us endow this CRN with mass-action kinetics with k ^± as forward and reverse kinetic coefficients, and assume this CRN operates on each side of a membrane that is semi-permeable to A. At equilibrium, concentration of A is equal on both sides, c _A =c _out,A . The only additional equilibrium constraint is k ⁺ c _A =k ⁻ c _B c _C , which leaves p−k=1 degree of freedom to let c _B and c _C vary on each side.

For a given c _A , at equilibrium the total metabolite pool concentration P=c _A +c _B +c _C is given by:

$$ P=c_{A} + c_{B} + \frac{k^{+}c_{A}}{k^{-}c_{B}} $$

(7)

which is minimized for:

$$ c_{B}=\sqrt{\frac{k^{+}}{k^{-}}c_{A}} $$

(8)

Assuming the concentration of B outside is so chosen to minimize P _out, any other choice of equilibrium inside concentrations results in an up-concentration, P>P _out.

It remains to be seen whether such an equilibrium up-concentration may hold for a stationarily growing proto-cell. This assessment is carried out below using the choice of outside concentrations given by Eq. 8.

Simple Example: Out-of-Equilibrium Proto-cell Assessment

Consider a proto-cell based on this example CRN with C chosen as membrane precursor, and A as nutrient. Choosing the nutrient as the species for which the support of the two moieties overlap ensures that the necessary condition (each moiety must be fed) is met. The sufficient condition is also met because it can be verified that the only two minimal siphons ({A,B} and {A,C}) coincide with the support of the two moieties so that the necessary (each moiety must be fed) and sufficient (each siphon contains the support of a moiety that is fed) conditions coincide for this particular example. This ensures the existence of a fixed point for the ODE system given by Eqs. 1–6, regardless of chosen kinetic or membrane parameters.

Numerical Results

All trajectories asymptotically converged towards constant positive concentrations corresponding to stationary proto-cell growth and this stationary state was found to be independent of the choice of initial conditions. The growth rate for nominal $(\mathcal {D}, c_{\text {out},A}, k_{\text {me}}, C_{\text {me}})$ was λ≈6.3×10⁻³ s⁻¹ (resp. λ≈1.9×10⁻² s⁻¹) with precursor incorporation only from the inside (resp. from both sides). As intuitively expected, it increased with increasing $\mathcal {D}$, c _out,A or k _me, and with decreasing C _me. Figure 3 gives the stationary up-concentration P/P _out contour plots for varying $(\mathcal {D}, c_{\text {out},A})$ (left) and (k _me,C _me) (right), with precursor incorporation only from the inside (top) or from both sides (bottom). It can be seen that there exists a limited parameter range leading to up-concentration, P/P _out>1, and that up-concentration is reduced with membrane precursor incorporation occurring from both sides. This is because the increased growth rate leads to decreased concentrations as a result of dilution (see Eq. 1).

Fig. 3

Contour plots of the stationary proto-cell up-concentration P/P _out for the first scheme with varying $(\mathcal {D}, c_{\text {out},A})$ (left) and (k _me,C _me) (right), with precursor incorporation only from the inside (top) or from both sides (bottom). Top and bottom plots share the same x-axis. Kinetic coefficients were chosen as k ⁺=1×10² s⁻¹ and $k^{-}=k^{+}(\mathrm {c}^{-{\ominus }-})^{-1}\exp (-6)\approx 0.25~\mathrm {M}^{-1}\cdot \mathrm {s}^{-1}$ (see Appendix 1)

Shortcoming

Even though the proto-cell up-concentration capability was assessed using the most favourable choice of c _out,B given by Eq. 8 for a given c _out,A (any other choice of c _out,B would result in a reduced up-concentration), the parameter range compatible with up-concentration is limited and may not correspond to any practical situation. Although this might be different for other CRNs than the particular one considered here, the main shortcoming of this scheme is that regardless of the chosen CRN, with growth from both sides the up-concentration capability is reduced. This shortcoming is alleviated with the second scheme presented in the following section.

Second Scheme: Exploiting Siphons

General Description

The main shortcoming of the previous scheme originated from the presence of the membrane precursor in the outside growth medium. It is often thought that the chemistry may somehow be different inside and outside a living cell because some chemical species (e.g. enzymes) are not present or quickly denaturated outside. The concept of siphons is instrumental in formalizing this notion of absent species.

Figure 4 illustrates the second scheme. If we wish the membrane precursor A _me to be possibly absent of the outside growth medium, then the CRN should have a siphon Z that is shorter than the full set of species $\mathcal {S}$ and that contains this precursor, A _me∈Z, so that equilibria without any A _me in the outside growth medium may exist. If we further wish two different equilibria to coexist on both sides of a semi-permeable membrane, one side (outside) without any species in Z being present, and the other side with all species in $\mathcal {S}$ being present, then the membrane should be semi-permeable to some nutrient A _nu outside Z, $A_{\text {nu}} \in \mathcal {S}\setminus Z$.

Fig. 4

Schematic representation of the second scheme. There exists a siphon Z shorter than the full set of species $\mathcal {S}$ that contains the membrane precursor A _me. The membrane is semi-permeable to some nutrient A _nu outside the siphon. There exists some pass reaction(s) capable of transferring mass from $\mathcal {S}\setminus Z$ into Z

One straightforward consequence is that Z is not fed, so that the sufficient condition for proto-cell stationary growth is no longer met. This does not exclude the possibility of a working proto-cell provided the stronger necessary condition (there must exist a pass reaction connecting species in $\mathcal {S}\setminus Z$ to species in Z) is met as we shall see in the simple example below, but the existence of a stationary growth regime may no longer be granted for any choice of kinetic or membrane parameters.

Simple Example: At Equilibrium

Consider the following CRN consisting of two bidirectional reactions:

$$\begin{array}{@{}rcl@{}} A+B\rightleftarrows C\\ 2B\rightleftarrows C \end{array} $$

The only possible mass assignment (up to a multiplying factor) is (m _A ,m _B ,m _C )=(1,1,2). The mass vector m is thus the only moiety. {B,C} is the only minimal siphon, and A+B→C is a pass reaction because the net mass transfer into Z is m _C −m _B =m _A >0. Let us endow this CRN with mass-action kinetics with $k_{1,2}^{\pm }$ as forward and reverse kinetic constants for the first and second reaction, and assume this CRN operates on each side of a membrane that is semi-permeable to A. At equilibrium, concentration of A is equal on both sides, c _A =c _out,A . Let us further assume that only A is present outside, c _out,B =c _out,C =0. This is possible because {B,C} is a siphon. Conversely, let as assume that at least some molecule of B or C is initially present inside, which leads to relaxation towards an inside equilibrium state with all species A, B and C being present.² This equilibrium state is uniquely determined by the concentration of A, c _A =c _out,A , because the CRN has only one degree of freedom.³ Detailed balance for the two reactions results in equilibrium concentrations given by:

$$ c_{B}=\left( \frac{k_{1}^{+}}{k_{1}^{-}}\right)\left( \frac{k_{2}^{-}}{k_{2}^{+}}\right)c_{\text{out},A} $$

(9)

and:

$$ c_{C}=\left( \frac{k_{1}^{+}}{k_{1}^{-}}\right)^{2}\left( \frac{k_{2}^{-}}{k_{2}^{+}}\right){c_{\text{out},A}}^{2} $$

(10)

At equilibrium, the total metabolite pool concentration or the system density is thus obviously higher inside than outside because the inside contains not only the same amount of A, but also some B and C that are both absent from the outside.

This equilibrium up-concentration, P/P _out>1, may not necessarily hold for a proto-cell because any stationary growth regime with positive concentrations is such that the nutrient concentration is lower inside than outside.

Simple Example: Out-of-Equilibrium Proto-cell Assessment

Consider a proto-cell based on such a CRN with A chosen as nutrient and C chosen as membrane precursor. The necessary condition (each moiety must be fed) is met because the support of the only moiety is the full set of species that is fed. But as highlighted in the general description above, the sufficient condition is not met because the minimal siphon {B,C} is not fed. This did not preclude proto-cell stationary growth in the investigated numerical parameter range. The stronger necessary condition for stationary proto-cell growth is also met because of the pass reaction A+B→C.

Numerical Results

All trajectories asymptotically converged towards constant positive concentrations corresponding to stationary proto-cell growth and the stationary point was found to be independent of the choice of initial conditions. The growth rate for nominal $(\mathcal {D}, c_{\text {out},A}, k_{\text {me}}, C_{\text {me}})$ was λ≈1.61×10⁻⁴ s⁻¹. As intuitively expected, it increased with increasing $\mathcal {D}$, c _out,A or k _me, and with decreasing C _me. Figure 5 gives the stationary up-concentration contour plots for varying $(\mathcal {D},c_{\text {out},A})$ (left) and (k _me,C _me) (right). It can be seen that nearly all $(\mathcal {D}, c_{\text {out},A})$ values resulted in up-concentration, except a narrow range in the lowest $\mathcal {D}$ and highest c _out,A corner, and that all (k _me,C _me) values resulted in up-concentration.

Fig. 5

Contour plots of the stationary proto-cell up-concentration P/P _out for the second scheme with varying $(\mathcal {D}, c_{\text {out},A})$ (left) and (k _me,C _me) (right). Kinetic coefficients were chosen as $k_{1}^{+}=k_{2}^{+}=1\times 10^{4}~\mathrm {M}^{-1}\cdot \mathrm {s}^{-1}$, $k_{1}^{-}=k_{1}^{+}\mathrm {c}^{-{\ominus }-}\exp (-6)\approx 25\, \mathrm{s}^{-1}$ and $k_{2}^{-}=k_{2}^{+}\mathrm {c}^{-{\ominus }-}\exp (-3)\approx 5\times 10^{2}~\mathrm {s}^{-1}$ (see Appendix 1)

This second scheme could appear well suited for up-concentration. Yet it suffers a major shortcoming.

Shortcoming

The shortcoming of this second scheme is the possible contamination of the outside growth medium. Any lysed cell releases some molecules of B and C in the outside growth medium. Any such molecule present in the outside growth medium suffices to bring the outside growth medium towards a new equilibrium with all A, B, and C present. We verified numerically that in such a case no up-concentration could be observed for any $(\mathcal {D}, c_{\text {out},A}, k_{\text {me}}, C_{\text {me}})$ parameter choice.

Third Scheme: Towards Active Transport

General Description

The shortcoming of the previous scheme arises from the sensitivity of the outside equilibrium state (supposed to only contain species in $\mathcal {S}\setminus Z$) to contamination by chemical species from the proto-cell (containing any species in $\mathcal {S}$). This is alleviated if Z is insulated from $\mathcal {S}\setminus Z$ by having the pass reaction catalyzed by the membrane surface. In such a case, transformation of species belonging to $\mathcal {S}\setminus Z$ into species belonging to Z (activated by contamination of the outside growth medium with some species belonging to Z) only occurs in the vicinity of the membrane and will only lead to a negligible displacement of the equilibrium state in the bulk outside growth medium, provided the outside growth medium volume is very large.

The ultimate insulation of Z from $\mathcal {S}\setminus Z$ is reached if the pass reaction is further confined and only occurs when species in $\mathcal {S}\setminus Z$ cross the membrane, which is active transport. A third species participating in the CRN should then be assigned a specific role, besides the membrane precursor and nutrient: the activator A _act (e.g. ATP), the consumption of which is required to import the primary nutrient A _pnu (e.g. glucose) from the outside growth medium inside the proto-cell. The corresponding reaction could be modeled as a unidirectional (or bidirectional if full consistency with thermodynamics is sought, Ederer and Gilles 2007) surfacic reaction occurring across the membrane:

$$A_{\text{pnu}}+A_{\text{act}}\rightarrow A_{\text{tnu}}+{\sum}_{A_{i}\in \mathcal{S}_{\text{byprod}}}\eta_{i}A_{i} $$

where A _tnu (e.g. glucophosphate) is the transformed nutrient under the action of A _act and where $A_{i}\in \mathcal {S}_{\text {byprod}}$ are additional potential byproduct species (e.g. ADP) that may be produced with stoichiometry coefficient η _i .

The original primary nutrient A _pnu is never present inside the proto-cell (unless it can also diffuse across the membrane) while A _tnu is the transformed nutrient inside the proto-cell. This nutrient input reaction is thus the only reaction in which A _pnu participates. The full set of species excluding the primary nutrient, $\mathcal {S}\setminus \{A_{\text {pnu}}\}$, is obviously a siphon Z. The nutrient input reaction is a pass reaction for this siphon Z but this siphon is not explicitly fed and the sufficient condition for proto-cell stationary growth is not met. But this does not preclude its possibility as shown below with a simple example.

Simple Example: At Equilibrium

Consider the same example as used to illustrate the second scheme but having the pass reaction A+B→C (as well as its reverse reaction C→A+B) only occurring across the membrane. The CRN operating on each side of a membrane (that is semi-permeable to A) only consists of $2B\rightleftarrows C$ which is the only reaction occurring in bulk solution. A plays the role of primary nutrient A _pnu, B that of activator A _act, and C plays both roles of transformed nutrient A _tnu and of membrane precursor A _me.

We shall denote $K_{1}^{\pm }$ (resp. $k_{1}^{\pm }=\rho _{0}K_{1}^{\pm }$) the forward and reverse kinetic coefficients per unit area (resp. per unit volume) of the pass reaction, and $k_{2}^{\pm }$ the forward and reverse kinetic coefficients (per unit volume) of the bulk reaction. Starting from an initial situation with only A present outside, c _out,B =c _out,C =0, and only B and C inside, c _A =0, this initial segregation will persist and the concentrations c _B and c _C will reach an equilibrium state with detailed balance for both the pass reaction (occurring across the membrane) and the bulk reaction, given by the same Eqs. 9 and 10 as for the second scheme. If for example $\frac {k_{1}^{+}}{k_{1}^{-}}>\frac {k_{2}^{+}}{k_{2}^{-}}$ (as is the case with reference chemical potentials chosen in Appendix 1), then c _B >c _out,A which results in up-concentration inside. It remains to be checked whether such an equilibrium up-concentration can hold for an out-of-equilibrium stationarily growing proto-cell.

Simple Example: Out-of-Equilibrium Proto-cell Assessment

Only species in Z={B,C} are present inside the proto-cell, c _A =0, while only species in $\mathcal {S}\setminus Z=\{A\}$ are present outside, c _out,B =c _out,C =0. The CRN only consists of the reaction $2B\rightleftarrows C$ with C being both the effective nutrient entering the proto-cell under the activation of B, and the membrane precursor. The ODE system to be considered only governs the time evolution of c _B and c _C with an effective nutrient input flux expression given by:

$$ f_{\textrm{\text{input},C}}=k_{1}^{+}c_{\text{out},A}c_{B}-k_{1}^{-}c_{C} $$

(11)

which replaces Eq. 3.

The active transport is a surfacic process with rate per unit area. In the above expression, we have converted rates per unit area into rates per unit volume owing to Assumption 2 of constant surface area-to-volume ratio.

The active transport is often thought of as a unidirectional process, similar to enzymatic reactions. However, thermodynamical constraints impose that any process be bidirectional although possibly strongly biased (Ederer and Gilles 2007; Noor et al. 2013). A buildup of C inside the proto-cell would ultimately lead to a decrease in its rate of entry. This is the reason why we have kept this active transport reaction bidirectional in Eq. 11.

Numerical Results

As $k_{1}^{+}$ plays a role analogous to that of the effective diffusion coefficient $\mathcal {D}$, it was varied the same way as $\mathcal {D}$ in the two previous schemes around a nominal $k_{1}^{+}=1\times 10^{4}~\mathrm {M}^{-1}\cdot \mathrm {s}^{-1}$ (nominal value used for bimolecular reactions as explained in Appendix 1, same value as for the second scheme) keeping the ratio $k_{1}^{+}/k_{1}^{-}$ constant. The nominal $k_{1}^{-}$ was determined as described in Appendix 1 (same value as for the second scheme).

All trajectories asymptotically converged towards constant positive concentrations corresponding to stationary proto-cell growth and the stationary point was found to be independent of the choice of initial conditions. The growth rate for nominal $(\mathcal {D}, c_{\text {out},A}, k_{\text {me}}, C_{\text {me}})$ was λ≈1.62×10⁻⁴ s⁻¹. As intuitively expected, it increased with increasing $\mathcal {D}$, c _out,A or k _me, and with decreasing C _me. Figure 6 gives the stationary up-concentration contour plots for varying $(k_{1}^{+},c_{\text {out},A})$ (left) and (k _me,C _me) (right). All ($k_{1}^{+}, c_{\text {out},A})$ and (k _me,C _me) values resulted in up-concentration. This confirms the compatibility of this third scheme with proto-cell stationary growth with up-concentration for this simple example CRN.

Fig. 6

Contour plots of the stationary proto-cell up-concentration P/P _out for the third scheme (active transport) with varying $(\mathcal {D}, c_{\text {out},A})$ (left) and (k _me,C _me) (right). Kinetic coefficients were kept the same as for the second scheme

Discussion

In this section we discuss a possible evolutionary path from the second scheme to the third one (active transport).

We stated above that growth medium contamination in the second scheme could be alleviated provided the pass reaction be catalyzed by the membrane surface. Assuming for simplicity that the pass reaction only takes place within a small volume $\mathcal {A}\delta$ from the membrane (where $\mathcal {A}$ is a and δ is an effective diffusion length), this is equivalent to reducing the pass reaction kinetics by a factor $\mathcal {A}\delta /\mathcal {V}=\rho _{0}\delta$ where $\mathcal {V}$ is the proto-cell volume and ρ ₀ is the constant surface area-to-volume ratio. Arbitrarily reducing the pass reaction kinetics (both forward and reverse coefficients) by three orders of magnitude, we have verified numerically that the up-concentration patterns shown in Fig. 5 did not change significantly. This confirms that a chemistry of the kind described in the second scheme and such that the pass reaction be catalyzed by the membrane surface is indeed capable of stationary proto-cell growth with up-concentration.

Would such a modified second scheme really be immune to growth medium contamination? Any lysed cell would release some molecules of B or C which could then trigger the pass reaction in the outer vicinity of the membrane (assuming a non-polar membrane with surface catalysis on both sides). Assuming for simplicity that the concentrations of B and C approximately take their equilibrium values (given by Eqs. 9 and 10) within a volume $\mathcal {A}\delta$ from the membrane, their diffusion in the bulk growth medium should eventually lead to a complete displacement of the growth medium composition. However, (i) with a large growth medium volume the dynamics of such a contamination could be significantly slower than the growth rate so that its effect could be negligible over a very large number of proto-cell generations, and/or (ii) this contamination could be minimized even with only a moderate flow of A through the growth medium.

Starting from such a situation, a step advantage could be gained by having surface catalysis only operate inside the proto-cell. This may be the case with membrane-bound enzymes. Their rate of synthesis could be significantly higher inside than outside as the result of the initial up-concentration granted in the starting point. The original non-polar surface catalysis mechanism could then be displaced by more favourable kinetics for the new polar surface catalysis, or even completely shut down assuming some evolution of the membrane itself. This could be advantageous in terms of up-concentration and of growth rate. The only source of growth medium contamination would then be lysed cells, which would remain negligible because without the pass reaction operating outside, the presence of some molecules of B or C would not trigger the transformation of A into additional molecules of B and C.

Having reached such a modified second scheme with polar surface catalysis (but with A still flowing across the membrane by passing diffusion), further evolution towards the third scheme (active transport) could have been driven by the fact that the kinetic coefficient $k_{1}^{+}$ (which controls nutrient influx in the third scheme and may result from enzymatic processes) may have more room for variation than the effective diffusion coefficient $\mathcal {D}$ (which controls nutrient influx in the second scheme and results from molecular characteristics of A _nu as well as structural properties of the membrane). A higher $k_{1}^{+}$ would result in higher growth rate and a preferentially selected proto-cell.

Conclusion

We have presented chemical schemes that are theoretically capable of sustaining different compositional equilibria on two sides of a semi-permeable membrane. The first scheme exploits the degrees of freedom of a CRN, only some of which are fixed by the equality of concentrations on both sides of the membrane for permeating species. The second scheme exploits the fact that a CRN having shorter siphons than the full set of species may exhibit different equilibria depending on whether species belonging to the siphon are initially present or not. Using simple examples we have confirmed that these two schemes are theoretically compatible with proto-cell stationary growth with up-concentration. However both schemes suffer from shortcomings that may make their operation impractical. For the first one, growth may occur from both sides of the membrane which reduces the up-concentration capability. For the second one, possible contamination of the growth medium by species belonging to the siphon leaking out of the proto-cell may result in a dramatic displacement of the outside growth medium compositional equilibrium and in the loss of up-concentration. This is solved if the pass reaction (which transforms the nutrient) is catalyzed by the membrane surface instead of occurring in bulk solution. The ultimate insulation between the outside and the inside consists in having such a pass reaction occurring only across the membrane, which is active transport (third scheme). This suggests that selection based on up-concentration and enhanced growth capability resulting from positive osmotic pressure may have played an evolutionary role in the emergence of active transport. Numerical results obtained using simple examples do not preclude the behavior of other more complex reaction networks, but simply prove the principle of the proposed schemes. Likewise, the different schemes have been presented in isolation. They are not exclusive, and might potentially be combined to result in new possibilities.

Appendix A: Rationale for the Choice of Numerical Parameters

The following numerical parameters were chosen to assess compatibility with stationary proto-cell growth with up-concentration:

• 1. Standard reference chemical potentials were chosen as $\frac {{\upmu }_{0,A}}{\mathcal {\mathrm {R}}T}=+3$, $\frac {{\upmu }_{0,B}}{\mathcal {\mathrm {R}}T}=0$ and $\frac {{\upmu }_{0,C}}{\mathcal {\mathrm {R}}T}=-3$. This results in standard reference chemical potential drops Δμ that are typical of biochemical reactions. Reactions were endowed with mass-action kinetics. Forward kinetic coefficients were chosen as k ⁺=1×10² s⁻¹ (resp. k ⁺=1×10⁴ M⁻¹⋅s⁻¹) for monomolecular (resp. bimolecular) forward reactions, consistent with typical k _cat and $\frac {k_{\text {cat}}}{K_{\mathrm {m}}}$ for biochemical reactions. To ensure consistency with thermodynamics (i.e. detailed balance and minimization of free energy for the CRN in a closed system), kinetic coefficients for reverse reactions were determined from:

  $$ k^{-}=k^{+}\left( \mathrm{c}^{-{\ominus}-}\right)^{\delta}\exp\left( -\frac{\Delta\upmu}{\mathcal{\mathrm{R}}T}\right) $$

  (12)

  where c^−⊖−=1 M is the standard concentration for which standard chemical potentials are defined, δ is the difference in molecularity between forward and reverse, Δμ>0 is the reference chemical potential drop across the reaction, $\mathcal {\mathrm {R}}$ is the ideal gas constant and T is the absolute temperature.

• 2. Nominal $(\mathcal {D}, c_{\textrm {out,nu}}, k_{\text {me}}, C_{\text {me}})$ values were selected as believed to have biological sense for E. coli chosen as an arbitrary reference organism. These values are $\mathcal {D}=10~\mathrm {s}^{-1}$, c _out,nu=1×10⁻³ M, k _me=1×10⁻³ s⁻¹ and C _me = 5 × 10⁻² M. The chosen c _out,nu is a typical glucose concentration used in E. coli growth medium. The rationale for the chosen C _me value is the following: the total dry weight in exponential growth phase is estimated to be 500 fg, out of which 8 % corresponds to the membrane (Neidhardt 1996). This gives a total dry weight of 40 fg for the membrane, corresponding to a density of 40 g⋅l⁻¹ for an estimated 1 μm³ total cell volume. The membrane is primarily constituted of phosphatidylethanolamine with molecular weight 744 g⋅mol⁻¹ (Huijbregts et al. 2000). This gives $C_{\text {me}}=\frac {40~\mathrm {g}\cdot \mathrm {l}^{-1}}{744~\mathrm {g}\cdot \text {mol}^{-1}}\approx 5\times 10^{-2}~\mathrm {M}$. As there is no equivalent direct way of estimating a biologically sensible k _me, we proceeded indirectly. The total metabolite concentration in E. coli is 300 mM (Neidhardt 1996). We arbitrarily assumed that membrane precursors constitute 10 % of this pool, i.e. c _me≈30 mM. Cell division time is typically 40 min corresponding to a growth rate of 4×10⁻⁴ s⁻¹. This is given for our model by Eq. 5, from which we extract: $k_{\text {me}}=\lambda \frac {C_{\text {me}}}{c_{\text {me}}}\approx 1\times 10^{-3}~\mathrm {s}^{-1}$. Last, $\mathcal {D}$ was chosen so that the asymptotic value of c _me approximately matches the above expected c _me.