Noise-induced symmetry breaking far from equilibrium and the emergence of biological homochirality

The origin of homochirality, the observed single-handedness of biological amino acids and sugars, has long been attributed to autocatalysis, a frequently assumed precursor for early life self-replication. However, the stability of homochiral states in deterministic autocatalytic systems relies on cross-inhibition of the two chiral states, an unlikely scenario for early life self-replicators. Here we present a theory for a stochastic individual-level model of autocatalytic prebiotic self-replicators that are maintained out of thermal equilibrium. Without chiral inhibition, the racemic state is the global attractor of the deterministic dynamics, but intrinsic multiplicative noise stabilizes the homochiral states. Moreover, we show that this noise-induced bistability is robust with respect to diffusion of molecules of opposite chirality, and systems of diffusively coupled autocatalytic chemical reactions synchronize their final homochiral states when the self-replication is the dominant production mechanism for the chiral molecules. We conclude that nonequilibrium autocatalysis is a viable mechanism for homochirality, without imposing additional nonlinearities such as chiral inhibition.

Figure 1

Ball-and-stick model of a generic $\alpha$-amino acid and its mirror image. $\alpha$-amino acids are organic compounds with a chiral carbon connected to an amino group ($-{\text{NH}}_2$), a carboxylic acid group ($-\text{COOH}$), a hydrogen atom ($-\text{H}$), and a side chain ($-\text{R}$) that varies depending on the particular amino acid. The $\mathrm{L}\text{or}\mathrm{D}$ chirality of amino acids is determined by the CORN rule: an amino acid is $\mathrm{L}-\mathrm{chiral}\left(\mathrm{D}-\mathrm{chiral}\right)$ if by wrapping your left hand (right hand) fingers around the direction of CORN ($-\text{CO}$, $-\text{R}$, and $-\text{N}$ groups in order) your thumb points toward the direction of the hydrogen atom.

Figure 2

(a) Phase portrait of Frank's model: the racemic state is an unstable fixed point (red dot), while the homochiral states are stable fixed points (green dots). (b) If chiral inhibition is replaced by linear decay reaction, the ratio of $\mathrm{D}$ and $\mathrm{L}$ molecules stays constant. (c) Adding even the slightest amount of nonautocatalytic production of $\mathrm{D}$ and $\mathrm{L}$ molecules makes the racemic state (green dot) the global attractor of the dynamics.

Figure 3

Comparison between the stationary distribution, Eq. (29), (dashed lines) and Gillespie simulations of reactions reaction (9) (markers), for different values of $\alpha$. Simulation parameters: $N={10}^3$, $k_a=k_n=k_d=1$.

Figure 4

Parameter ${\alpha }_c^{\text{patch}}$ in the two-patch system as a function of the diffusion rate $\delta$. Gillespie simulations (markers) are compared against Eq. (56) (solid blue line) and Eq. (58) (dashed red line). Simulation parameters as in Fig. 3.

Figure 5

Gillespie simulation of scheme reaction (42) for a one-dimensional system of $M=100$ patches, starting from racemic state and ending with all the patches in the same homochiral state $\omega =-1$. Simulation parameters: $N=1000$, $k_a=k_d=1$, $\delta ={10}^{-3}$, and $k_n=0$.

Figure 6

Mean switching time, $〈t〉$, from one homochiral state to the other as a function of $\alpha$. Analytical result (solid red line) from Eq. (B19) compared with the Gillespie simulation results (blue points). The analytical expression is valid for small $\alpha$ where the system is homochiral. For $\alpha$ close to 1 or greater (where the system is expected to stay racemic), to accurately predict the mean switching time, one needs to keep track of higher order term (in $1/N$) in the corresponding Fokker-Planck equation. Simulation parameters: $k_d=1$, $k_a=1$, $k_n=1$, $N=1000$, and $V=\alpha$.