# Homochirality from reversible reactions

Shiling Liang

[ToC]

## The motivations

### Reversible v.s. irreversible

Why is reversibility important? The reversibility sets thermodynamic constraints on the kinetic parameters of a chemical reaction system. In the absence of reversibility, one can not define the thermodynamic quantities such as energy difference. By defining thermodynamic quantities, one can further quantify how far is the system of interest away from equilibrium.

### What is special about second-order autocatalysis

If the system is chemostated to keep the substrate concentration constant, and has multiple species replicate themselves by quadratic autoctalysis, the reaction equation reads

$A_0\stackrel{\infty}{\rightleftharpoons} A\stackrel{k_f^{(i)}X_i}{\rightarrow} X_i \ \Rightarrow\ \frac{d}{dt}x_i = x_i^2 k_f^{(i)}a.$ Then the growth is hyperbolic $x_i(t)=\frac{x_i(0)}{a k_{f}^{(i)}x_i(0)^2-1},$ which means the population will grow to infinite in a finite time under unlimited resources, and the first species that reaches the threshold win the selection. Therefore one can not define “fitness” in this case.

On the other hand, for only one replicator case, if we introduce the reverse quadratic autocatalysis and put the system in a flow reactor, the reaction $A_0\stackrel{\infty}{\rightleftharpoons} A\underset{k_b X^2}{\stackrel{k_f X^2}{\rightleftharpoons}} X_i\stackrel{\phi}{\to}\empty\\ \Rightarrow \frac{d}{dt}x = k_f(1-x)x^2-k_bx^3-\phi x\,$ is one of the simplest chemical reaction models to show non-linear bistability, having two stable fixed points at $x=0$ and $x=\left(1+\sqrt{1-(4R_{eq}/A_0^2+1)\phi}\right)/(2+2R_{eq})$, where $R_{eq}=k_b/k_f$.

## Homochirality in the CSRT

Here we consider the homochirality in a continuous stirring reaction tank (CSTR). The basic mechanism comes from the nonlinear bistable nature of the second-order autocatalysis. Schlögl model as a standard model for chemical bi-stability has been intensively investigated. And the second order autocatalysis models have also been adopted to explain the homochirality of the Soai reaction. In 2004, Saito proposed a model of irreversible second-order autocatalysis with recyclings to show complete homochirality. Further models were proposed such noise-induced homochirality. However, missing some reverse reactions is a long-standing issue. Here we bring the reverse reaction back and show that homochirality can still be achieved under the reversible condition.

### The reactions and phase space

Here we study a system in the CSRT, the non-equilibrium is maintained by the in- and out- flow. Two chiral molecules can catalyze the reaction of the transformation from the substrate molecule $A$ to the corresponding chiral molecule. The reactions read \begin{aligned} &A + 2X\underset{k_b}{\stackrel{k_f}{\rightleftharpoons}} 3X,\\ &X\stackrel{\phi}{\to} \emptyset,\quad A_0\stackrel{\phi}{\to} A\stackrel{\phi}{\to} \emptyset. \end{aligned} where $X = D\text{ or }L$. The corresponding time-evolution equations read

\begin{equation} \begin{aligned} \frac{d}{dt}[D]&= \phi\left([A_0]^2[D]^2(k_f(1-[D]-[L])-k_b[D])-\phi[D]\right)\\ \frac{d}{dt}[L]&= \phi\left([A_0]^2[L]^2(k_f(1-[D]-[L])-k_b[L])-\phi [L]\right). \end{aligned} \end{equation}

From this, we can write down the dimensionless ode for the chemical reactions. Let $l = \frac{[L]}{[A_0]}$ and $d=\frac{[D]}{[A_0]}$, and also dimensionlesslize the parameters by letting $\eta=\frac{k_f[A_0]^2}{\phi}$ and $R_{eq}=\frac{k_b}{k_f}$, we have

\begin{equation} \begin{aligned} \frac{d}{dt}d&= \phi\left(\eta l^2((1-d-l)-R_{eq}d)- d\right)\\ \frac{d}{dt}l&= \phi\left(\eta d^2((1-d-l)-R_{eq}l)- l\right). \end{aligned} \end{equation}

The stationary state is controlled by two parameters: $\eta$ and $R_{eq}$, where $\eta$ characterizes the ratio of the forward reaction rate to the flow rate. $R_{eq}=\exp(-\Delta E/T)$ is determined by the temperature in the reactor and the energy difference between the achiral molecule and chiral molecule. Overall, $\eta\to 0$ means the system goes to equilibrium and $\eta\to \infty$ indicates that the out-of-equilibrium steady-state of the system is dominated by the flow, gives $l=d=0$.

Between the reaction-dominate regime and the flow-dominated regime, we can see rich behavior of the system including spontaneous symmetry breaking. Fig 1. Here the x and y axis are the normalized concentration of two chiral molecules.

With the increasing of $\eta$, one can see there is firstly a transition from extinction mode to a bi-stable mode, and then the two non-linear nullclines meet and an unstable racemic fixed point emerges. With further increasing of $\eta$, the racemic state turns to be stable which indicate that the reversible reaction start to play a role, adn the system is gradually approaching a equilibrium state.

The area of basin of attractions Fig 2. The are of basin of attractions.

### The bifurcation and stability

In the parameter space spanned by $\eta$ and $R_{eq}$, we can find the regimes in which the spontaneous symmetry breaking of chirality can happen.

In the system, there are three characterized timescales: the timescale of the forward reaction, of backward reaction and of the flow. The last one $\tau= 1/\phi$ characterizes the time for the injected chemicals staying in the reactor.

The all extinction regime is when the flow rate is much larger than the reactions so that the injected chemical do not have enough time to initialize the self-replication. $\eta<\eta_1^* = 4+4R_{eq}\Rightarrow \phi>4(k_f+k_b)[A_0]^2.$ Denoting this critical point as $\eta_1^*$. When the flux decreases, we can see a bistable state. It is bistable because the initializing of one chiral reaction take use of most substrate chemical $A$ and the other chiral molecule can not initialize the replication. With further decreasing of flux, there will be an unstable racemic fixed point at $\eta_2^* = 8+4R_{eq}.$ This unstable racemic fixed point will turn to a stable once $\eta$ reaches $\eta_3^* = \frac{4(1+R_{eq})^2}{R_{eq}}=\frac{4}{R_{eq}}+8+4R_{eq}.$ Only in the last regime we can see stable racemic fixed point. This regime only exist if we include the reversible autocatalytic reaction, as we can see the critical point to this regimes goes to infinity when $R_{eq}$ goes to zero, i.e. ther reaction turns to irreversible.  Fig 3a. The four regimes of the quadratic autocatalytic reaction in the CSRT.Fig 3b. The bifurcation diagram of the probability of one type of chiral molecule.

### How far away from equilibrium?

One benefit of introducing the reversible reactions is that now we can characterize how far is the system away from equilibrium. Here the energy difference between the achiral molecule and chiral molecule is $\Delta E= - k_BT\ln R_{eq}$, one can find the energy injected into the system in the steady-state: $J_E =\Delta E \phi ([A_0]-[A])=-k_BT\log[R_{eq}]\phi [A_0] (d+l).$

Here the energy does not only depends the flux strength, but also on the concentration of two chiral molecules. Which can be easily understood as the energy needed of complete recycling. When the flux strength is fixed, we can see multiple stable states and thus the energy flux also shows multi-branches. An interesting observation is that for the tri-stable regime, the racemic state needs higher energy injection than the homochiral state. Fig 4. The injected energy.

### Equivalence between the CSRT and irreversible recycling reactions

When the external reservoir only contains the achiral molecules, the in- / out- flows in the CSRT play the same role as an uncatalyzed recycling reaction. Let us consider the following reaction in the CSRT \begin{aligned} &A + 2X\underset{k_-}{\stackrel{k_+}{\rightleftharpoons}} 3X,\\ &X\stackrel{\phi}{\to} \empty\quad A_0\stackrel{\phi}{\to} A\stackrel{\phi}{\to} \empty \end{aligned}\Rightarrow \begin{aligned} \frac{d}{dt}[A]&=-F([X],[A])-\phi([A]-[A_0])\\ \frac{d}{dt}[X]&=F([X],[A])-\phi [X]. \end{aligned}

In steady-state, we have $[A]+[X]=[A_0]$, therefore the above equations can be reduced to a s differential equation:

$\frac{d}{dt}[X]=F([X],[A_0]-[X])-\phi [X].$

Alternatively, like extensively discussed in various literatures, an irreversible recycling is explicitly introduced \begin{aligned} A + 2X&\underset{k_-}{\stackrel{k_+}{\rightleftharpoons}} 3X\\ X &\stackrel{k_r}{\to}A \end{aligned}\Rightarrow\begin{aligned} \frac{d}{dt}[A]&=-F([X],[A])+k_r[X]\\ \frac{d}{dt}[X]&=F([X],[A])-k_r [X]. \end{aligned}

As the reaction system in this case does not exchange particles with the environment and the reactions insides do not change the total numbers of particles, we have $[C_{tot}]=[X]+[A]$. This constraint can also reduce the above equations to one $\frac{d}{dt}[X]=F([X],[C_0]-[X])-k_r[X].$ Here we can see the resulting differential equation of the recycling case is equivalent to the CSTR case by letting $\phi = k_r$ and $[C_0]=[A_0]$.

## Homochirality driven by temperature gradient

### Entropy roles

Although the quadratic autocatalytic reaction is considered as a reversible reaction, the recycling mechanism done out of the reactor, or by direct recycling process, is still including an irreversible one. One way to recycling is using a high temperature as a recycling force. For a net reaction of the form $A\leftrightharpoons X$, a high-temperature recycling mechanism can mostly achieve $[X]/[A]=1$, which is still not strong enough for the appearance of symmetry breaking in the cold compartment.

One way to achieve a stronger recycling is with the help of entropy. In this case we consider another type of reaction, that a chiral molecule is made up by two achiral building blocks $A$ $2A + 2X\underset{k_b}{\stackrel{k_f}{\rightleftharpoons}} 3X.$ Where $X=L\ \mathrm{or}\ D$. We can assume that the reactions in the warm compartment is fast enough to maintain at thermodynamic equilibrium and give the equilibrium distribution $[L_0]$ and $[D_0]$. With the diffusive transport between two boxes, the total mass are equal in two boxes so that $[A]+2[D]+2[L]=[A_0]+2[D_0]+2[L_0]=[C]$. In the warm box, the equilibrium distribution is $x_0=\frac{R_{eq}^w+8 [C]\alpha-\sqrt{R_{eq}^w(R_{eq}^w+16[C]\alpha)}}{32 [C]\alpha}$ where $x_0 = [L_0]/[C]$ or $[D_0]/[C]$, $R_{eq}$ and $\alpha$ are defined by $\frac{1}{\alpha} R_{eq}=k_b/k_f$. Therefore $R_{eq}^w=e^{-\Delta \mu/T_2}$ is the Boltzmann factor from the chemical energy difference between a chiral dimer $X$ and two building blocks $A$. The $\alpha$ here has the unit of inverse of concentration, thus can be understood as the inverse of the local concentration in the dimer state, and $\Delta S=S_m-S_d = - k_B\log([C]\alpha)$ is the entropy change of converting two monomers to a dimer. When the temperature of warm box goes to infinity and $[C]\alpha$ goes to zero, $x_0$ will goes to zero which means the chiral molecules are fully recycled to the achiral building blocks. Fig 5. The normalized concentration of a chiral species at equilibrium.

### Symmetry breaking of chiral dimers

With the transport between two boxes \begin{aligned} X\underset{\phi}{\stackrel{\phi}{\rightleftharpoons}} X_0\quad A\underset{\phi}{\stackrel{\phi}{\rightleftharpoons}} A_0 \end{aligned} and assuming $T_w\to \infty$, we focus on the reactions in the cold box \begin{equation}\begin{aligned} \frac{d}{dt}[L]&= [L]^2(k_f([C]-2[D]-2[L])^2-k_b[L])-\phi([L]-[L_0])\\ \frac{d}{dt}[D]&= [D]^2(k_f([C]-2[D]-2[L])^2-k_b[D])-\phi([D]-[D_0]) \end{aligned} \end{equation}

This reaction equation can be written in a dimensionless form by letting $\eta = \frac{k_f[C]^3}{\phi}$ and use the normalized concentrations \begin{equation}\begin{aligned} \frac{d}{dt}l&= \phi\left(4\eta l^4+\eta\left(8d+\frac{1}{[C]\alpha}R_{eq}-4\right)l^3+(1-2d)^2\eta l^2-l+l_0\right)\\ \frac{d}{dt}d&= \phi\left(4\eta d^4+\eta\left(8l+\frac{1}{[C]\alpha}R_{eq}-4\right)d^3+(1-2l)^2\eta d^2-d+d_0\right)\\ \end{aligned} \end{equation}

Again, the $\eta$ here characterize the ratio between the forward chemical reaction and the diffusion rate. In the following animations, we can see the chiral dimer formation mechanism can exhibit various modes under different parameters. The basin of attractions are colored using the enantiomeric excesses ($ee = (D-L)/(L+D)$) at time $t=t_{max}$ from corresponding point. The top-left panel shows a very similar behavior as of the $A\leftrightharpoons X$ reaction in the CSRT, because here $C \alpha$ is small enough so that the chiral dimers in the warm box are almost fully recycled. In the bottom-right panel, we see the system always stays in a stable racemic state due to insufficient recycling. Between these two regimes, richer dynamics are shown in the top-right and bottom-left panels, which worth further studying.    