Spontaneous Mirror Symmetry Breaking from Recycling in Enantioselective Polymerization

Abstract

A key challenge for origin of life research is understanding how the homochirality of extant biological systems may have emerged during the abiotic phase of chemical evolution. Living systems depend on bio-macromolecules made from chiral building blocks and a crucial question is the relationship of polymerization with the emergence of homochirality. We present a reaction scheme demonstrating how spontaneous mirror symmetry breaking (SMSB) can be achieved in enantioselective polymerization without chiral inhibition and without autocatalysis. The model is based on nucleated cooperative polymerization: nucleation, elongation, dissociation, fusion and fragmentation and monomer racemization. These are micro-reversible processes subject to constraints dictated by chemical thermodynamics. To maintain this closed system out of equilibrium, we model an external energy source which induces the irreversible breakage of the longest polymers in the system. Simulations reveal that SMSB can be achieved starting from the tiny intrinsic statistical fluctuations about the idealized mirror symmetric composition.

Appendix

The complete set of rate equations taking into account all the above processes can be expressed in terms of reaction rate fluxes and stoichiometric matrices. Introduce the reaction rate fluxes, where $\alpha = L,R$ is a chirality label:

$$\begin{aligned} \phi _j^{\alpha }= & {} \left\{ \begin{array}{c} k_1c_1^{\alpha }c_j^{\alpha }-k_{-1}c_{j+1}^{\alpha }, \qquad 1\le j\le n_{c}-1, \\ k_2c_1^{\alpha }c_j^{\alpha }-k_{-2}c_{j+1}^{\alpha }, \qquad n_c\le j \le N-1, \\ \end{array} \right. \end{aligned}$$

(26)

$$\begin{aligned} \phi _r^{L,R}= & {} k_r(c_1^R - c_1^L), \end{aligned}$$

(27)

$$\begin{aligned} \varPhi _k^{\alpha }= & {} \gamma c_k^{\alpha }, \end{aligned}$$

(28)

$$\begin{aligned} \varPsi _{m,n}^{\alpha }= & {} k_a c_m^{\alpha }c_n^{\alpha } - k_{-a}c_{m+n}^{\alpha }, \end{aligned}$$

(29)

for linear chain growth $(\phi )$, racemization $(\phi _r)$, mechanical breakage $(\varPhi )$ and fusion $(\varPsi )$, respectively. Although the individual fusion fluxes, Eq. (29), depend on two indices m, n, for purposes of counting they can be enumerated in a sequential fashion by imposing, and without loss of generality, that $m\le n$. The number of independent fusion fluxes is calculated to be

$$\begin{aligned} N_{\psi } = \sum _{k=i_{minf}}^{N-j_{minf}} \sum _{l=\max (k,j_{minf})}^{N-k}1, \end{aligned}$$

(30)

the lower limit of the second summand ensures that the flux $\varPsi _{m,n}=\varPsi _{n,m}$ is counted just once.

We can group all these fluxes together into a single flux vector:

$$\begin{aligned} \varvec{f}^{\alpha } = \big ( \{ \phi _j^{\alpha } \}_{j=1}^{N-1}, \phi ^{\alpha }_r, \{ \varPhi _k^{\alpha } \}_{k=1}^{N-n_{min}+1},\{ \varPsi _m^{\alpha } \}_{m=1}^{N_{\psi }} \big ). \end{aligned}$$

(31)

Then the differential rate equations for the i-th species can be written as follows:

$$\begin{aligned} \frac{d c^{\alpha }_i(t)}{dt} = \sum _{j=1}^{n_r} S _{i,j} f^{\alpha }_j, \qquad 1 \le i \le N, \qquad 1 \le j \le n_r, \end{aligned}$$

(32)

and $\varvec{S}$ is the stoichiometric matrix with elements $S _{i,j}$. The sum is over the number $n_r$ of reactions:

$$\begin{aligned} n_r = 2N-n_{min}+N_{\psi } + 1. \end{aligned}$$

(33)

Equations (32) represent the model to be solved numerically.