Necessary and sufficient conditions for protocell growth

Abstract

We consider a generic protocell model consisting of any conservative chemical reaction network embedded within a membrane. The membrane results from the self-assembly of a membrane precursor and is semi-permeable to some nutrients. Nutrients are metabolized into all other species including the membrane precursor, and the membrane grows in area and the protocell in volume. Faithful replication through cell growth and division requires a doubling of both cell volume and surface area every division time (thus leading to a periodic surface area-to-volume ratio) and also requires periodic concentrations of the cell constituents. Building upon these basic considerations, we prove necessary and sufficient conditions pertaining to the chemical reaction network for such a regime to be met. A simple necessary condition is that every moiety must be fed. A stronger necessary condition implies that every siphon must be either fed, or connected to species outside the siphon through a pass reaction capable of transferring net positive mass into the siphon. And in the case of nutrient uptake through passive diffusion and of constant surface area-to-volume ratio, a sufficient condition for the existence of a fixed point is that every siphon be fed. These necessary and sufficient conditions hold for any chemical reaction kinetics, membrane parameters or nutrient flux diffusion constants.

Appendices

Appendix 1: Analytical treatment of a simple example protocell with nutrient influx by active transport

Consider the same simple CRN example endowed with mass-action kinetics as described in Sect. 6, but where the first reaction is assumed unidirectional and only occurs across the membrane:

$$\begin{aligned}&A+B\rightarrow C\\&\quad 2B\rightleftarrows C \end{aligned}$$

A is the primary nutrient present in the outside growth medium, $c_{\text {out},A}>0$ and is absent from the protocell, C is the transformed nutrient under activation by B. B and C are the only species present inside. C also plays the role of membrane precursor with incorporation kinetic coefficient $K_{\text {me}}$ per unit area. We shall assume a constant surface area-to-volume ratio $\rho _{0}$ so that the surfacic kinetic coefficient of the first surfacic reaction can be transformed in a volumic kinetic coefficient, $k_{1}^{+}$. As in Sect. 6, the kinetic coefficients for the second reaction are $k_{2}^{\pm }$.

The protocell ODE system is given by:

$$\begin{aligned} \dot{c_{B}}= & {} -k_{1}^{+}c_{\text {out},A}c_{B}-2k_{2}^{+}{c_{B}}^{2}+2k_{2}^{-}c_{C}-\mu _{\text {inst}}c_{B} \end{aligned}$$

(52)

$$\begin{aligned} \dot{c_{C}}= & {} k_{1}^{+}c_{\text {out},A}c_{B}+k_{2}^{+}{c_{B}}^{2}-k_{2}^{-}c_{C}-\rho _{0}K_{\text {me}}c_{C}-\mu _{\text {inst}}c_{C} \end{aligned}$$

(53)

with the growth rate $\mu _{\text {inst}}$ given by:

$$\begin{aligned} \mu _{\text {inst}}=\frac{K_{\text {me}}c_{C}}{N_{\text {me}}} \end{aligned}$$

(54)

Looking for a fixed point, we set all time derivatives to zero. Adding Eqs. 52 and 53 with time derivatives set to zero results in the same equation as Eq. 32 which we restate here:

$$\begin{aligned} k_{2}^{+}{c_{B}}^{2}+\mu _{\text {inst}}(c_{B}+c_{C})=(k_{2}^{-}-\rho _{0}K_{\text {me}})c_{C} \end{aligned}$$

(55)

Replacing the growth rate $\mu _{\text {inst}}$ by its expression given in Eq. 54 gives a quadratic equation that uniquely determines $c_{B}$ as a function of $c_{C}$ provided $k_{2}^{-}>\rho _{0}K_{\text {me}}$ and $c_{C}<N_{\text {me}}(\frac{k_{2}^{-}}{K_{\text {me}}}-\rho _{0})$:

$$\begin{aligned} c_{B}=\phi _{1}(c_{C})=\frac{-K_{\text {me}}c_{C}+\sqrt{(K_{\text {me}}c_{C})^{2}+4k_{2}^{+}c_{C}N_{\text {me}}((k_{2}^{-}-\rho _{0}K_{\text {me}})N_{\text {me}}-K_{\text {me}}c_{C})}}{2k_{2}^{+}N_{\text {me}}} \end{aligned}$$

(56)

It can be verified that $\phi _{1}(c_{C})=0$ for $c_{C}=0$ and for $c_{C}=N_{\text {me}}(\frac{k_{2}^{-}}{K_{\text {me}}}-\rho _{0})$, that it is strictly positive for $c_{C}$ in between those two zeros, and that its derivative is infinite at $c_{C}=0$.

Adding Eqs. 52, 53 multiplied by two (thus obtaining the time derivative of the system density) and setting all time derivatives to zero, we obtain:

$$\begin{aligned} k_{1}^{+}c_{\text {out},A}c_{B}-2\rho _{0}K_{\text {me}}c_{C}-\mu _{\text {inst}}(c_{B}+2c_{C})=0 \end{aligned}$$

(57)

Replacing the growth rate $\mu _{\text {inst}}$ by its expression given in Eq. 54 and rearranging terms gives the following expression for $c_{B}$ as a function of $c_{C}$:

$$\begin{aligned} c_{B}=\phi _{2}(c_{C})=\frac{2K_{\text {me}}c_{C}(\rho _{0}N_{\text {me}}+c_{C})}{k_{1}^{+}c_{\text {out},A}N_{\text {me}}-K_{\text {me}}c_{C}} \end{aligned}$$

(58)

It can be verified that $\phi _{2}(c_{C})=0$ for $c_{C}=0$, that $\phi _{2}(c_{C})\rightarrow +\infty$ with $c_{C}\rightarrow \frac{k_{1}^{+}c_{\text {out},A}N_{\text {me}}}{K_{\text {me}}}$, that it takes strictly positive values for $c_{C}$ in between, and that its derivative is finite at $c_{C}=0$.

Because at $c_{C}=0$, the derivative of $\phi _{1}$ is infinite and that of $\phi _{2}$ is finite, $\phi _{1}(c_{C})>\phi _{2}(c_{C})$ in the neighbourhood of $c_{C}=0$. Besides, $\phi _{2}(c_{C})>\phi _{1}(c_{C})$ in the neighbourhood of some strictly positive $c_{C}$ value because of the following considerations:

• 1. If $k_{1}^{+}c_{\text {out},A}>k_{2}^{-}-\rho _{0}K_{\text {me}}$, then $\phi _{2}(c_{C})>\phi _{1}(c_{C})=0$ at $c_{C}=N_{\text {me}}(\frac{k_{2}^{-}}{K_{\text {me}}}-\rho _{0})$.

• 2. Else if $k_{1}^{+}c_{\text {out},A}\le k_{2}^{-}-\rho _{0}K_{\text {me}}$, then $\phi _{2}(c_{C})\rightarrow +\infty$ and $\phi _{1}(c_{C})\rightarrow \phi _{1}(N_{\text {me}}(\frac{k_{2}^{-}}{K_{\text {me}}}-\rho _{0}))$ (which is finite) when $c_{C}\rightarrow N_{\text {me}}(\frac{k_{2}^{-}}{K_{\text {me}}}-\rho _{0})$.

This implies that $c_{B}=\phi _{1}(c_{C})$ and $c_{B}=\phi _{1}(c_{C})$ intersect for some strictly positive set of concentrations $({c_{B}}_{0},{c_{C}}_{0})$.

We have thus proved the existence of a fixed point provided $k_{2}^{-}>\rho _{0}K_{\text {me}}$. This simple example illustrates the fact that in the case of active transport, the protocell may or may not exhibit a fixed point (other than the degenerate zero concentration vector corresponding to an empty protocell) depending on kinetic and membrane parameters.

Appendix 2: Analytical treatment of a mechanistic, coarse-grained whole-cell model inspired by Molenaar et al. (2009)

This appendix proves analytically that the whole-cell model described in Sect. 6.2 exhibits a stationary growth state for any parameter set.

1.1 ODE system

The ODE system governing the cell dynamics is composed of:

• 1. Three differential equations for metabolites:

  $$\begin{aligned} \dot{c_{S}}= & {} \rho _{0}\alpha \phi _{\text {input},S}-\nu _{P}-\mu c_{S} \end{aligned}$$

  (59)
  $$\begin{aligned} \dot{c_{P}}= & {} \nu _{P}-\nu _{L}-n_{1}\nu _{1}-n_{2}\nu _{2}-n_{R}\nu _{R}-n_{T}\nu _{T}-\mu c_{P} \end{aligned}$$

  (60)
  $$\begin{aligned} \dot{c_{L}}= & {} \nu _{L}-\rho _{0}F_{\text {output},L}-\mu c_{L} \end{aligned}$$

  (61)

• 2. Four differential equations for proteins:

  $$\begin{aligned} \dot{c_{i}}=\nu _{i}-\mu c_{i} \end{aligned}$$

  (62)

  where $i=1$, 2 or R, and:

  $$\begin{aligned} \dot{c_{T}}=\nu _{T}-\rho _{0}F_{\text {output},T}-\mu c_{T} \end{aligned}$$

  (63)

• 3. One differential equation for $\alpha$ (Eq. 48)

1.2 Expressing all concentrations as functions of $\alpha$ and of the normalized precursor concentration $p=c_{P}$/$K_{\text {m}_{R}}$

Looking for a stationary state, we set all time derivatives to zero. Combining Eqs. 41 and 62, both taken for $i=R$, gives:

$$\begin{aligned} \mu =\frac{k_{{\text {cat}}_{R}}n_{R}p}{1+n_{\text {tot}}p} \end{aligned}$$

(64)

where $p=c_{P}$/$K_{{\text {m}}_{R}}$ is the normalized precursor concentration. Combining Eqs. 41 and 62 for $j=1$, 2 and R together gives:

$$\begin{aligned} c_{i}=\frac{n_{i}}{n_{R}}c_{R} \end{aligned}$$

(65)

where $i=1$ or 2.

Combining Eqs. 41, 62 (taken for $i=R$) and 63 (taken for $i=R$ and T) gives:

$$\begin{aligned} c_{R}=\frac{n_{R}}{n_{T}}c_{T}\left( 1+\rho _{0}\frac{K_{T}}{\mu }\right) \end{aligned}$$

(66)

Equation 48 for $\alpha$ gives (excluding the degenerate case $\alpha =1$ for which $\mu =0$):

$$\begin{aligned} \alpha =\frac{\frac{F_{\text {output},T}}{N_{T}}}{\frac{F_{\text {output},L}}{N_{L}}+\frac{F_{\text {output},T}}{N_{T}}} \end{aligned}$$

(67)

or equivalently:

$$\begin{aligned} 1-\alpha =\frac{\frac{F_{\text {output},L}}{N_{L}}}{\frac{F_{\text {output},L}}{N_{L}}+\frac{F_{\text {output},T}}{N_{T}}} \end{aligned}$$

(68)

We are now in a position to express all concentrations (except $c_{S}$) as a function of two unknowns, $\alpha$ and the normalized precursor concentration p. For simplicity, we shall keep $\mu$ in these expressions, knowing than it is a simple function of p given by Eq. 64. Combining the above equation with Eq. 46, gives:

$$\begin{aligned} \mu =\frac{F_{\text {output},L}}{N_{L}} \end{aligned}$$

(69)

and:

$$\begin{aligned} \frac{\alpha }{1-\alpha }\mu =\frac{F_{\text {output},T}}{N_{T}} \end{aligned}$$

(70)

From which $c_{L}$ and $c_{T}$ are extracted:

$$\begin{aligned} c_{L}= & {} \mu \frac{N_{L}}{K_{L}} \end{aligned}$$

(71)

$$\begin{aligned} c_{T}= & {} \mu \frac{\alpha }{1-\alpha }\frac{N_{T}}{K_{T}} \end{aligned}$$

(72)

Feeding Eq. 72 in Eq. 66, and feeding the result in Eq. 65, gives:

$$\begin{aligned} c_{i}=\frac{\alpha }{1-\alpha }\frac{n_{i}}{n_{T}}N_{T} \left( \rho _{0}+\frac{\mu }{K_{T}}\right) \end{aligned}$$

(73)

where $i=1$, 2 or R. Finally, we recall that, by definition, $c_{P}=p\times K_{{\text {m}}_{R}}$.

1.3 Relating $\alpha$ to p

Setting the time derivative of $c_{L}$ (given by Eq. 61) to zero, using Eq. 40, and rearranging terms gives:

$$\begin{aligned} \frac{k_{{\text {cat}}_{2}}c_{P}c_{2}}{K_{{\text {m}}_{2}}+c_{P}}=K_{L}\left( \rho _{0}+\frac{\mu }{K_{L}}\right) c_{L} \end{aligned}$$

(74)

Defining $\theta =K_{{\text {m}}_{R}}$/$K_{{\text {m}}_{2}}$, using Eqs. 71 (for $c_{L}$), 73 (for $c_{2}$) and 64 (for $\mu$), and rearranging terms gives:

$$\begin{aligned} \frac{1-\alpha }{\alpha }=\frac{\frac{k_{{\text {cat}}_{2}}\theta }{1+\theta p}}{\frac{k_{{\text {cat}}_{R}}n_{R}}{1+n_{\text {tot}}p}}\frac{n_{2}}{n_{T}}\frac{N_{T}}{N_{L}}\frac{\rho _{0}+\frac{\mu }{K_{T}}}{\rho _{0}+\frac{\mu }{K_{L}}} \end{aligned}$$

(75)

Equation 75 defines a monotonic univocal relation between $\alpha$ and p, such that $0<\alpha (p=0)=\alpha _{0}, \,\alpha (p=+\infty )=\alpha _{\infty }<1$. $\alpha _{0}$ may be lower or greater than $\alpha _{\infty }$ depending on the specific choice of parameters.

1.4 Determining two independent relations of the kind $c_{S}=\phi _{1,2}(p)$

Setting the time derivative of $c_{P}$ (Eq. 60) to zero, dividing by $c_{R}$ and rearranging terms, gives:

$$\begin{aligned} \frac{k_{{\text {cat}}_{1}}c_{S}}{K_{{\text {m}}_{1}}+ c_{S}}\frac{n_{1}}{n_{T}}= & {} \frac{k_{{\text {cat}}_{2}}\theta p}{1+\theta p}\frac{n_{2}}{n_{T}}+\mu \frac{({n_{1}}^{2}+{n_{2}}^{2}+{n_{R}}^{2} +{n_{T}}^{2})n_{R}}{n_{T}}\nonumber \\&+ \,\mu \frac{ p K_{{\text {m}}_{R}}}{N_{T} \left( \rho _{0}+\frac{\mu }{K_{T}}\right) }\frac{1-\alpha }{\alpha } \end{aligned}$$

(76)

Combined with Eq. 75, this defines a function $c_{S}=\phi _{1}(p)$ over the interval $[0;p_{\text {max},1}[$, where $p_{\text {max},1}$ is the value for which the right-hand side (RHS) of Equation 76 verifies $RHS(p_{\text {max},1})=k_{{\text {cat}}_{1}}n_{1}$/$n_{T}$. This function is such that $\phi _{1}(p=0)=0$ (and this is the only zero of $\phi _{1}$), and $\phi _{1}(p)\rightarrow +\infty$ if $p\rightarrow p_{\text {max},1}$.

Setting the sum of the time derivatives of $c_{S}$, $c_{P}$ and $c_{L}$ (sum of Eqs. 59, 60 and 61) to zero and rearranging terms gives:

$$\begin{aligned} \rho _{0}\alpha \phi _{\text {input},S}= & {} \mu c_{S}+\mu \frac{({n_{1}}^{2}+{n_{2}}^{2}+{n_{R}}^{2}+{n_{T}}^{2})}{n_{T}} N_{T}(\rho _{0}+\frac{\mu }{K_{T}})\frac{\alpha }{1-\alpha }\nonumber \\&+\,\mu p K_{{\text {m}}_{R}} + \mu \left( \rho _{0}+\frac{\mu }{K_{L}}\right) N_{L} \end{aligned}$$

(77)

Combined with Eq. 75, this defines a function $c_{S}=\phi _{2}(p)$. Considering in turn the two different cases for nutrient influx:

• 1. Facilitated diffusion: $\phi _{\text {input},S}$ is the function of $c_{\text {out},S}$ and of $c_{S}$ given by Eq. 50. If $p=0$, then the RHS of Eq. 77 is null, and the LHS is also null iff $c_{S}=c_{\text {out},S}$. We thus have $\phi _{2}(p=0)=c_{\text {out},S}$. If $p\rightarrow +\infty$, then $RHS\rightarrow +\infty$ while the LHS cannot exceed a maximum value $\rho _{0}\alpha _{\infty }\mathcal {D}c_{\text {out},S}$ which is reached if $c_{S}=0$. Therefore, there must exist $p_{\text {max},2}$ such that $\phi _{2}(p_{\text {max},2})=0$.

• 2. Active transport, $\phi _{\text {input},S}$ is only a function of $c_{\text {out},S}$ and is thus a constant with respect to the internal state variables. It can be verified that $\phi _{2}(p)$ is a decreasing function of p such that $\phi _{2}(p=0)=+\infty$ and $\phi _{2}(p=p_{\text {max},2})=0$ where $p_{\text {max},2}$ is defined by letting $c_{S}=0$ in Eq. 77.

In any of the two above cases, we have $\phi _{2}(p=0)>\phi _{1}(p=0)$ and $\phi _{2}(p=p_{\text {max}})<\phi _{1}(p=p_{\text {max}})$ where $p_{\text {max}}=\text {min}\{p_{\text {max},1},p_{\text {max},2}\}$. This implies that $\phi _{1}(p)$ and $\phi _{2}(p)$ thus intersect for a finite $p_{\text {st}}$ which fully defines a stationary growth state.

In the second above case, it is to be expected that if active transport had been more rigorously modeled by taking into account the activator introduced in Sect. 7.2 (equivalent to an energy currency, which would be consumed or down-converted by the active transport process), a stationary growth state might only exist provided the rate of synthesis of this energy currency is sufficient. This would be a situation similar to that described by the simple example of Appendix 1.