Effect of Migration in a Diffusion Model for Template Coexistence in Protocells

Abstract

The compartmentalization of distinct templates in protocells and the exchange of templates between them (migration) are key elements of a modern scenario for prebiotic evolution. Here we use the diffusion approximation of population genetics to study analytically the steady-state properties of such a prebiotic scenario. The coexistence of distinct template types inside a protocell is achieved by a selective pressure at the protocell level (group selection) favoring protocells with a mixed template composition. In the degenerate case, where the templates have the same replication rate, we find that a vanishingly small migration rate suffices to eliminate the segregation effect of random drift and so to promote coexistence. In the nondegenerate case, a small migration rate greatly boosts coexistence as compared with the situation where there is no migration. However, increase of the migration rate beyond a critical value leads to the complete dominance of the more efficient template type (homogeneous regime). In this case, we find a continuous phase transition separating the homogeneous and the coexistence regimes, with the order parameter vanishing linearly with the distance to the transition point.

Appendices

Appendix A: Local Fixation Probability for the M=0 Nonergodic Segregation Regime

As shown by Fontanari and Serva (2013), setting M=0 in Eq. (14) yields two possible steady-state solutions: the solution corresponding to the ergodic coexistence phase, which is a combination of two Delta functions and a regular function, ϕ(x)=A ₀ δ(x)+A ₁ δ(x−1)+Bϕ _r (x) with A ₀+A ₁+B=1, and the solution corresponding to the nonergodic segregation phase, which is a combination of two Delta functions, ϕ(x)=A ₀ δ(x)+A ₁ δ(x−1) with A ₀+A ₁=1. The nonergodic regime, which is our focus here, occurs for C<π ²+S ²/4. Note that in both regimes, A ₁ may be interpreted as the probability that the type 1 template fixates in a given protocell and a similar interpretation holds for A ₀ as well. However, the result $\bar{x} = A_{1}$, which we used to draw the curve for M=0 in Fig. 4, holds in the segregation regime only. In Fontanari and Serva (2013) we have calculated the dependence of the weight A ₁ on the initial probability density ϕ(x,0) for S=0 only, and in this appendix we generalize that calculation for S≥0.

We begin by rewriting Eq. (9) for M=0,

$$\begin{aligned} \frac{\partial}{\partial \tau} \phi (x,\tau ) =& \frac{\partial^2}{\partial x^2} \bigl[ x (1-x)\phi (x, \tau ) \bigr] + S \frac{\partial}{\partial x} \bigl[ x (1 - x ) \phi (x,\tau )\bigr] \\ &{} + \bigl[ C x (1-x )-\bar{C} (\tau ) \bigr] \phi (x, \tau ), \end{aligned}$$

(33)

and introducing the abbreviation $\langle f (x) \rangle_{\tau}= \int_{0}^{1} f (x) \phi ( x , \tau ) \,dx$ for the expected value of a regular function f(x) at time τ. Hence,

$$\begin{aligned} \frac{d}{d\tau} \bigl\langle f ( x ) \bigr\rangle _\tau = & \biggl\langle x ( 1- x ) \frac{\partial^2 f (x)}{\partial x^2}\biggr\rangle _\tau -S \biggl\langle x ( 1- x ) \frac{\partial f (x)}{\partial x} \biggr\rangle _\tau \\ &{} + C \bigl\langle x ( 1- x ) f ( x ) \bigr\rangle _\tau - \bar{C} ( \tau ) \bigl\langle f (x) \bigr\rangle _\tau \end{aligned}$$

(34)

with $\bar{C} ( \tau ) = C \langle x ( 1- x ) \rangle_{\tau}$. The idea is to choose a function f(x) such that the first three terms of the right-hand side of Eq. (34) cancel out. This choice depends on the value of the parameter Γ≡C−S ²/4, as discussed next. We note that Γ<π ² in the nonergodic regime.

Region 0<Γ<π ²

In this region we choose $f (x) = e^{S x/2} \sin ( \sqrt{\varGamma} x + \theta )$, where θ is an arbitrary constant. Then Eq. (34) rewrites

$$ \frac{d}{d\tau} \bigl\langle e^{S x/2} \sin ( \sqrt{\varGamma} x + \theta )\bigr\rangle _\tau = - \bar{C} ( \tau ) \bigl\langle e^{S x/2} \sin ( \sqrt{\varGamma} x + \theta ) \bigr\rangle _\tau, $$

(35)

which has the formal solution

$$ \frac{ \langle e^{S x/2} \sin ( \sqrt{\varGamma} x + \theta ) \rangle_\tau}{ \langle e^{S x/2} \sin ( \sqrt{\varGamma} x + \theta ) \rangle_0} = \exp \biggl[ - \int_0^\tau \bar{C} ( \eta ) d \eta \biggr] . $$

(36)

As the right-hand side of this equation does not depend on θ, neither does the ratio in its left-hand side. Hence, equating the ratios evaluated at θ=0 and $\theta = \pi/2 - \sqrt{\varGamma}/2$ yields

$$ \frac{ \langle e^{S x/2} \sin ( \sqrt{\varGamma} x ) \rangle_\tau}{ \langle e^{S x/2} \cos [ \sqrt{\varGamma} ( x - 1/2 ) ] \rangle_\tau} = \frac{ \langle e^{S x/2} \sin ( \sqrt{\varGamma} x ) \rangle_0}{ \langle e^{S x/2} \cos [ \sqrt{\varGamma} ( x - 1/2 ) ] \rangle_0} . $$

(37)

In the limit as τ→∞ we have

$$ \bigl\langle e^{S x/2} \sin ( \sqrt{\varGamma} x ) \bigr\rangle _\infty = A_1 e^{S/2} \sin ( \sqrt{\varGamma} ) $$

(38)

and

$$\begin{aligned} \bigl\langle e^{S x/2} \cos \bigl[ \sqrt{\varGamma} ( x - 1/2 ) \bigr]\bigr\rangle _\infty = & \bigl( A_0 + A_1 e^{S/2} \bigr) \cos (\sqrt{\varGamma}/2 ) \\ = & \bigl[ 1 + A_1 \bigl( e^{S/2} - 1 \bigr) \bigr] \cos (\sqrt{\varGamma}/2 ), \end{aligned}$$

(39)

which leads to

$$ A_1 = \frac{1}{1 + e^{S/2} ( \varXi_0 -1 )}, $$

(40)

where

$$ \varXi_0 = 2 \sin (\sqrt{\varGamma}/2 ) \frac { \langle e^{S x/2} \cos [ \sqrt{\varGamma} ( x - 1/2 ) ] \rangle_0}{ \langle e^{S x/2} \sin ( \sqrt{\varGamma} x ) \rangle_0} . $$

(41)

In the limit as Γ→π ², we have Ξ ₀→2 regardless of the initial probability density ϕ(x,0), and so $A_{1} \to A_{1}^{c} = 1/ (1 + e^{S/2} )$. In addition, for the initial probability density ϕ(x,0)=δ(x−1/2) used to calculate $\bar{x}$ at M=0 in Fig. 4, the dependence on Γ (and hence on C) disappears, and so $A_{1} = A_{1}^{c}$.

Region −S ²/4<Γ<0

In this region the choice f(x)=e ^Sx/2(e ^ux+θe ^−ux) with $u = \sqrt{-\varGamma}$ and θ arbitrary leads to the canceling of the first three terms of the right-hand side of Eq. (34), yielding

$$ \frac{ \langle e^{S x/2} ( e^{ux} + \theta e^{-ux} ) \rangle_\tau}{ \langle e^{S x/2} ( e^{ux} + \theta e^{-ux} ) \rangle_0} = \exp \biggl[ - \int_0^\tau\bar{C} ( \eta ) \,d \eta \biggr] . $$

(42)

The same argument used in the analysis of the Γ>0 region allows us to equate the ratio that appear in the left-hand side of this equation for θ=−1 and θ=0,

$$ \frac{ \langle e^{S x/2} \sinh ( u x ) \rangle_\tau}{ \langle e^{S x/2 +ux} \rangle_\tau} = \frac{ \langle e^{S x/2} \sinh ( u x ) \rangle_0}{ \langle e^{S x/2 + ux} \rangle_0} . $$

(43)

Finally, taking the limit as τ→∞ yields

$$ A_1 = \frac{1}{1 + e^{S/2} [ \varOmega_0\sinh ( u ) -e^u ]}, $$

(44)

where

$$ \varOmega_0 = \frac { \langle e^{S x/2 + ux } \rangle_0}{ \langle e^{S x/2} \sinh ( u x ) \rangle_0} . $$

(45)

By taking the limit as u→0 we can easily verify that A ₁ is continuous at the boundary of the two regions. In addition, in the limit as C→0, i.e., u→S/2, we recover the classical formula for the fixation of an allele with selective disadvantage S (Crow and Kimura 1970),

$$ A_1 = \frac{ \langle e^{Sx} \rangle_0 - 1}{e^S - 1 }. $$

(46)

Similarly to our finding in the analysis of the previous region, the initial probability density ϕ(x,0)=δ(x−1/2) yields A ₁=1/(1+e ^S/2) regardless of the value of C, as shown in Fig. 4.

Appendix B: Critical Line for the Limit as M→0

Setting M=0 in Eq. (25) yields

$$ z'_c + z^2_c - S z_c +C=0 $$

(47)

with the condition

$$ z_c ( 1 )=0 . $$

(48)

In the region C−S ²/4>0 its solution is

$$ z_c (x )= \frac{S}{2} -\gamma \tan ( \gamma x + \theta ), $$

(49)

where $\gamma= \sqrt{C- S^{2} /4}$, and θ=θ(γ,S) is fixed by condition (48) as

$$ \frac{S}{2} -\gamma \tan ( \gamma + \theta )=0 . $$

(50)

We note that the critical value C _c (S) is in the region C−S ²/4>0 (see Fig. 5). To evaluate Eq. (28), we use the equality

$$ \exp ( -Sx + y_c ) = \frac{\cos ( \gamma x + \theta )}{\cos (\theta )} \; \exp ( -S x/2 ), $$

(51)

which follows directly from the definition $y_{c} (x) = \int_{0}^{x} z_{c} ( \xi ) d \xi$ with z _c given by (49). Now the integrals in Eq. (28) can be readily evaluated, yielding

$$ \frac{S}{2} -\gamma \tan ( \theta )= \frac{ \gamma [ e^{-\frac{S}{2}}\sin ( \gamma+ \theta ) -\sin ( \theta ) ] - \frac{S}{2} [ e^{-\frac{S}{2}}\cos (\gamma+ \theta )-\cos ( \theta ) ] }{ e^{-\frac{S}{2}}\cos (\gamma+ \theta )} . $$

(52)

This equation can be further simplified using the equalities $\sin ( \gamma + \theta )= S/ ( 2\sqrt{C} )$ and $\cos (\gamma + \theta )=\gamma/\sqrt{C}$, which follow from Eq. (50). The final result is simply

$$ \gamma = \sqrt{C} e^{\frac{S}{2}}\cos ( \theta ) . $$

(53)

Finally, we rewrite Eq. (50) as

$$ \theta = \arctan \biggl( \frac{S}{2\gamma} \biggr) - \gamma $$

(54)

in order to make clear that Eq. (53) yields a relation C=C _c (S), which is the critical line M→0 depicted in Fig. 5.