# Chiral Oscillations and Spontaneous Mirror Symmetry Breaking in a Simple Polymerization Model

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## Abstract

**:**

## 1. Introduction

## 2. The Expanded APED Model

## 3. Bifurcation Analysis

#### 3.1. Trimer Model

#### 3.2. Tetramer and Pentamer Models

## 4. Minimum Initial Enantiomeric Excess

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Stable oscillations of the concentrations of LLL and DDD as well as the enantiomeric excess (ee) for the parameter values ${\alpha}_{1}={\alpha}_{2}=20,a={h}_{1}={e}_{1}={p}_{1}={p}_{2}={h}_{2}={e}_{2}=1,b={\beta}_{1}={\gamma}_{1}={\beta}_{2}={\gamma}_{2}=0,$c = 0.5 and $e{e}_{init}=0.01$. For simplicity we only display [LLL] and [DDD], although all concentrations oscillate. The units for time are dimensionless, because we do not use literature values for the reaction rates.

**Figure 2.**The bifurcation diagrams for ${\alpha}_{1}$ (

**a**) and ${\alpha}_{2}$ (

**b**) and the joint parameter where ${\alpha}_{1}$ = ${\alpha}_{2}$ (

**c**) illustrate that heterodimers and heterotrimers must be formed preferentially for oscillations to occur. Other parameters are $a={p}_{1}={h}_{1}={e}_{1}={p}_{2}={h}_{2}={e}_{2}=1,\phantom{\rule{4pt}{0ex}}b={\beta}_{1}={\gamma}_{1}={\beta}_{2}={\gamma}_{2}=0,\phantom{\rule{4pt}{0ex}}{\alpha}_{2}=20$ (

**a**), ${\alpha}_{1}=20$ (

**b**), c = 0.5, $e{e}_{init}=0.01$.

**Figure 3.**The bifurcation diagrams for ${\gamma}_{1}$ (

**a**), ${\gamma}_{2}$ (

**b**), and the joint parameter where ${\gamma}_{1}$=${\gamma}_{2}$. Panel (

**c**) illustrates the flexibility of the range of values for the stereoselectivity of epimerization that give rise to oscillations. Under these parameter values the range of allowed values for stereoselectivity of epimerization extend past unity. Other parameters are $a={p}_{1}={h}_{1}={p}_{2}={h}_{2}={e}_{2}=1$, $b={\beta}_{1}={\beta}_{2}={\gamma}_{1}={\gamma}_{2}=0$, ${\alpha}_{1}={\alpha}_{2}=50,{e}_{1}=0.5$ (

**a**) and (

**c**); ${e}_{1}=1$ (

**b**), c = 0.5, $e{e}_{init}=0.01$.

**Figure 4.**The bifurcation diagrams of ${\beta}_{1}$ (

**a**), ${\beta}_{2}$ (

**b**), and the joint parameter where ${\beta}_{1}$=${\beta}_{2}$ (

**c**) depicts the range of parameter values for the stereoselectivity of hydrolysis that result in oscillations. Under this set of parameter values, the range of allowed values is not as flexible for hydrolysis as for epimerization. Other parameters are $a={p}_{1}={h}_{1}={e}_{1}={p}_{2}={e}_{2}=1,\phantom{\rule{4pt}{0ex}}b={\gamma}_{1}={\gamma}_{2}={\beta}_{2}=0$, ${\alpha}_{1}={\alpha}_{2}=50,\phantom{\rule{4pt}{0ex}}{h}_{2}=1$ (

**a**), ${h}_{2}=0.5$ (

**b**), and (

**c**), $c=0.5$, $e{e}_{init}=0.01$. Panel (

**d**) is the joint bifurcation diagram of ${\beta}_{1}$ and ${\beta}_{2}$ with the parameters ${h}_{1}={h}_{2}=0.2,{e}_{1}={e}_{2}=0.1$. The three values are ${\alpha}_{1}={\alpha}_{2}=50$ (green), ${\alpha}_{1}={\alpha}_{2}=100$ (red), ${\alpha}_{1}={\alpha}_{2}=150$ (blue).

**Figure 5.**The joint bifurcation diagrams for ${\gamma}_{1}$ and ${\beta}_{1}$ (

**a**) and ${\gamma}_{2}$ and ${\beta}_{2}$ (

**b**) illustrate that sustained oscillations occur even when dimers or trimers favor homochiral epimerization and hydrolysis (i.e., ${\gamma}_{1}={\beta}_{1}>1,\phantom{\rule{4pt}{0ex}}{\gamma}_{2}={\beta}_{2}>1$). Other parameters are $a={p}_{1}={p}_{2}=c=1,\phantom{\rule{4pt}{0ex}}{h}_{1}={h}_{2}=0.2,{e}_{1}={e}_{2}=0.1,\phantom{\rule{4pt}{0ex}}{\gamma}_{2}={\beta}_{2}=0$ (

**a**), ${\gamma}_{1}={\beta}_{1}=0$ (

**b**), and $e{e}_{init}=0.01$. Panel (

**c**) is the joint bifurcation diagram for ${\gamma}_{2}$ and ${\beta}_{2}$ with a different set of parameters, including ${\gamma}_{1}={\beta}_{1}=1,\phantom{\rule{4pt}{0ex}}{h}_{1}={h}_{2}=0.1\phantom{\rule{4pt}{0ex}}{e}_{1}={e}_{2}=0.02$. This diagram shows that oscillations occur even in the absence of stereoselectivity for all hydrolysis and epimerization terms.

**Figure 6.**The joint bifurcation diagrams for the tetramer (left) and pentamer (right) activation—polymerization—epimerization—depolymerization (APED) models for polymerization (panels (

**a**,

**b**)), epimerization (panels (

**c**,

**d**)), and hydrolysis (panels (

**e**,

**f**)). Parameters in (

**a**,

**b**) are $a={p}_{1,2,3,4}={h}_{1,2,3,4}={e}_{1,2,3,4}=1$, $b={\beta}_{1,2,3,4}={\gamma}_{1,2,3,4}=0$, $e{e}_{init}=0.01$. Parameters in (

**c**,

**d**) are same as (

**a**,

**b**) except for ${\alpha}_{1,2,3,4}=50$ and ${e}_{1,2,3,4}=0.5$, and parameters in (

**e**,

**f**) are the same as (

**c**,

**d**) except for ${e}_{1,2,3,4}=0.1$ and ${h}_{1,2,3,4}=0.2$.

**Figure 7.**Plots of the minimum initial enantiomeric excesses (ee) required for oscillations and amplification in the minimal and expanded APED models for varying ${\alpha}_{1}={\alpha}_{2}={\alpha}_{3}={\alpha}_{4}$ (

**a**) and varying ${\gamma}_{1}={\gamma}_{2}={\gamma}_{3}={\gamma}_{4}$ (

**b**). Other parameters are $a={p}_{1,2,3,4}={h}_{1,2,3,4}=c=1,\phantom{\rule{4pt}{0ex}}{e}_{1,2,3,4}=0.5,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}b={\beta}_{1,2,3,4}={\gamma}_{1,2,3,4}=0,$ and all other stereoselectivities (represented by greek letters) are zero. Parameters for (

**b**) are same as (

**a**) with ${\alpha}_{1,2,3,4}=50$.

**Figure 8.**Oscillations in overall enantiomeric excess for the minimal and expanded APED models. Each simulation began with an initial ee of ${10}^{-6}$, and the amplitudes of the oscillations for each model are as follows; dimer: −0.62, 0.62; trimer: −0.91, 0.91; tetramer: −0.96, 0.96; pentamer: −0.98, 0.98. Other parameters are $a={p}_{1,2,3,4}={h}_{1,2,3,4}=1,\phantom{\rule{4pt}{0ex}}{e}_{1,2,3,4}=0.5,\phantom{\rule{4pt}{0ex}}b={\beta}_{1,2,3,4}={\gamma}_{1,2,3,4}=0,\phantom{\rule{4pt}{0ex}}{\alpha}_{1,2,3,4}=50,\phantom{\rule{4pt}{0ex}}c=0.35$. Similarly to Figure 1, the units of time are dimensionless.

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**MDPI and ACS Style**

Bock, W.; Peacock-López, E.
Chiral Oscillations and Spontaneous Mirror Symmetry Breaking in a Simple Polymerization Model. *Symmetry* **2020**, *12*, 1388.
https://doi.org/10.3390/sym12091388

**AMA Style**

Bock W, Peacock-López E.
Chiral Oscillations and Spontaneous Mirror Symmetry Breaking in a Simple Polymerization Model. *Symmetry*. 2020; 12(9):1388.
https://doi.org/10.3390/sym12091388

**Chicago/Turabian Style**

Bock, William, and Enrique Peacock-López.
2020. "Chiral Oscillations and Spontaneous Mirror Symmetry Breaking in a Simple Polymerization Model" *Symmetry* 12, no. 9: 1388.
https://doi.org/10.3390/sym12091388