# Spontaneous Chiral Symmetry Breaking and Entropy Production in a Closed System

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{k}are the thermodynamic forces and J

_{k}are thermodynamic flows. A temperature gradient, for example, is the thermodynamic force, F

_{k}, that drives thermodynamic flow, J

_{k}, of heat current. The force that drives chemical reactions has been identified as affinity [2,5,6] and the corresponding flow is the rate of conversion form reactants to products. This flow is expressed as the time derivative dξ/dt (mol/s) of the extent of reaction ξ [5,6]. For an elementary chemical reaction step, the rate of entropy production can be written in terms of the forward reaction rate, R

_{f}, and the reverse reaction rate, R

_{r}[6]

_{r}, are non-zero. In considering kinetic equations of a chemical system, often the reverse rates have high reaction barriers and, correspondingly, very low reaction rates, and are assumed to be “zero” because they are negligible compared to the forward rates. Low reverse reaction rates keep the system from evolving to equilibrium state. In the model we will present below, the system is driven far from equilibrium by an inflow of radiation. In the absence of radiation, our system evolves to equilibrium—there are no very high reaction barriers to keep it from reaching its equilibrium state.

^{4}years [15]. These results show that a lot of interesting and important general conclusions can be arrived at through using theoretical models [16,17]. Along these lines, we investigate the thermodynamic aspects of systems that spontaneously break chiral symmetry, using a model presented below.

## 2. Materials and Methods

_{L}or X

_{D}(R2 and R3). The set of reactions R4a and R4b are elementary steps of an autocatalytic reaction for X

_{L}with the intermediate species S

_{L}; similar reaction steps result in the autocatalytic production of X

_{D}, as shown in R5a and R5b. Reaction steps R6, R7 and R8 show reactions through which the species decompose into starting material S and T. The scheme explicitly has all the steps needed for the system to reach chemical equilibrium in the absence of radiation.

(Ph)_{2} C = CH(R^{1}) + hv → [(Ph)_{2} C = CH(R^{1})]* | (R9) |

[(Ph)_{2} C = CH(R^{1})]* + (R^{2})OH → [(Ph)_{2} HC − CH(R^{1})(OR^{2})]_{L} | (R10) |

[(Ph)_{2} C = CH(R^{1})]* + (R^{2})OH → [(Ph)_{2} HC − CH(R^{1})(OR^{2})]_{D} | (R11) |

^{1}= CH

_{3}or C

_{2}H

_{5}and R

^{2}= CH

_{3}, C

_{2}H

_{5}or C

_{3}H

_{7}. In the reaction (R9), a photon is absorbed by the electrons in the C=C double bond and the molecule transitions to a reactive excited state [(Ph)

_{2}C = CH(R

^{1})]*. In the addition reaction shown in (R10) and (R11), the excited molecule reacts with an alcohol, (R

^{2}) OH, and produces a chiral compound (Ph)

_{2}HC −

**C**H(R

^{1})(OR

^{2}) in the L and D enantiomeric forms. In this compound, the carbon shown in boldface is a chiral carbon (its tetrahedral bonds to four different groups makes it so). Other examples of photoaddition reactions producing chiral products from achiral reactants can be found in [19]. We note that TE need not be an excited state; it could be a different, more reactive isomer of the T [19].

_{L}[t]

_{D}[t]

_{L}[t], R4ar = k4ar S

_{L}[t]

_{L}[t] TE[t], R4br = k4br (X

_{L}[t])

^{2}

_{D}[t], R5ar = k5ar S

_{D}[t]

_{D}[t] TE[t], R5br = k5br (X

_{L}[t])

^{2}

_{L}[t]X

_{D}[t], R6r = k6r P[t]W[t]

^{2}

^{2}

_{L}[t]/dt = R4af − R4ar − R4bf + R4br

_{D}[t]/dt = R5af − R5ar − R5bf + R5br

_{L}[t]/dt = R2f − R2r − R4af + R4ar + 2R4bf − 2R4br − R6f +

_{D}[t]/dt = R3f − R3r − R5af + R5ar + 2R5bf − 2R5br − R6f + R6r

_{i}},{t, t

_{min}, t

_{max}}], in which “Equations” are the differentials equations for the set of functions {y

_{i}} with t as the independent variable; numerical solutions are obtained in the range t

_{min}, to t

_{max}. More details can be found in the online documentation that comes with Mathematica. The rate constants used for the numerical solutions are summarized in Figure 1. In assigning values to rate constants, there are consistency conditions that must be met. For example, since reactions (R4a) and (R4b) together are equivalent to (R2), the products of the equilibrium constants of R4a and R4b must equal the equilibrium constant of R2. This gives us the following condition for the rate constants:

_{L}− X

_{D}), in the symmetric state α = 0 and in the asymmetric state α ≠ 0.

## 3. Results

^{4}s) to ensure the concentrations of all species have reached a steady state, which is the equilibrium state. At t = 10

^{4}s, the concentrations at equilibrium were: S = T = 8.478 × 10

^{−3}M, TE = 8.477 × 10

^{−6}M, S

_{L}= S

_{D}= 3.047 × 10

^{−6}M, X

_{L}= X

_{D}= 3.593 × 10

^{−5}M, and P = W = 7.188 × 10

^{−4}M. The conversion of the initial species S and T compared to other species was rather small for the numerical values of the rate constants shown in Figure 1. By choosing a different set of rate constants, the conversion could be increased. The numerical values confirm that complete symmetry of the system was maintained when no incident radiation is present. Figure 2 shows the time evolution of the chiral species S

_{L}, S

_{D}, X

_{L}, X

_{D}from t = 0 s, to t = 1000 s. As the system evolved to its equilibrium state, the entropy production σ was monitored; initially, it took a nonzero value but, as expected, its value decreased to zero at the equilibrium state.

^{−8}M) of X

_{L}was introduced into the system as a “random fluctuation”. If the system has the mechanism to break chiral symmetry, it will have a critical value II

_{C}. At values of II < II

_{C}, this excess 0.1% of X

_{L}will decrease and the system will again evolve into a steady state where X

_{L}= X

_{D}; at values of II > II

_{C,}the excess will increase and lead to a steady state in which X

_{L}> X

_{D}. In a real system, this slight perturbation may be due to a random fluctuation such as a local excess of one enantiomer that may then be amplified, resulting in a state of broken symmetry. The overall behavior of the system is summarized in Figure 3.

_{L}= X

_{D}and α = 0. Figure 4 shows the time evolution of the chiral species when II = 0.0030. A steady state is reached in approximately 600 s.

_{L}). As is well known in the study of stability of steady states [6,7,9], the initial exponential growth of the small enantiomeric excess depends on the eigenvalue of the unstable mode of the linearized equations derived from the set (14)−(22) around the initial state. These linearized equations are obtained by assuming a small perturbation of the concentrations, δC

_{k}, from the initial steady state. This leads to a set of linear equations of the type dδC

_{k}/dt = Σ

_{l}L

_{kl}δC

_{l}[6,7,9]. The eigenvalues of L

_{kl}with positive real parts are the exponents that determine the initial rate of growth of the enantiomeric excess. However, the later growth and leveling off at the steady state depend on the kinetics and rate constants. In general, near the critical point II

_{C}, the relaxation time is long, the so-called “critical slowing”, but as the value of II increases, the growth rate becomes faster and the relaxation time decreases.

_{C}(0.004 in this case), the relaxation to asymmetric steady state is slower and it becomes faster as the value of II increases. The exact quantitative relationship between relaxation time and II depends on the kinetics and rate constants and not of significance to the current study. In these runs, to obtain both positive and negative branches of α, the initial condition with a 0.1% excess of X

_{D}was also included. The dependence of α on II is shown in Figure 6, demonstrating the typical bifurcation of asymmetric states above the critical value II

_{C}= 0.004. As is expected, in a chiral symmetry breaking transition, the values of α above the critical point are parabolic.

_{C}. In a previous study [18] of an open system, σ behaved as entropy does in a second order phase transition: its slope is discontinuous at the transition point. In addition, we noted that its slope increased above the critical point. This behavior was consistent with the Maximum Entropy Production (MEP) hypothesis [35,36,37,38,39,40] that states that non-equilibrium steady states maximize the rate of entropy production. In other words, the breaking of symmetry was a reflection of the general tendency of non-equilibrium systems and, from this point of view, it was only to be expected. This would imply that biochemical asymmetry is a consequence of MEP. To date, there is no general proof for MEP; indeed, in our own investigation we found that MEP was valid for some systems but not all. MEP, in general, is a controversial hypothesis and its applications to various complex systems have been questioned [41]. Hence, we investigated MEP in the context of chiral symmetry breaking.

_{C}, if initially we set the system in a non-equilibrium symmetric state, which is an extrapolation of the symmetric state below II

_{C}, the system evolves to an asymmetric state in which σ is lower. The extrapolation of the symmetric state beyond the critical value II

_{C}is possible on a computer because, without a small perturbation in X

_{L}or X

_{D}, or other chiral species, the system stays in an unstable symmetric steady state. With a small perturbation, it evolves to the stable asymmetric state. The time evolution of the system from this initial state to its final asymmetric steady state results in the decrease in σ, thus indicating that the stable asymmetric state is associated with a lower value of σ compared with that of a symmetric state. Thus, we see that the entropy production in a closed system is not consistent with MEP, because MEP would predict higher values for σ in the asymmetric state.

## 4. Concluding Remarks

^{3}+ Bα + C, in which A, B and C are functions of the kinetic constants of the chiral symmetry breaking chemical reactions [11,12,13,15]. Finally, we show that the behavior of entropy production in this chiral-symmetry-breaking system is not consistent with the MEP hypothesis, because MEP implies that the state of broken symmetry will have a higher rate of entropy production compared to a symmetric state, but we find the opposite to be the case in this model. In a system with an inflow of reactants and outflow of products, however, the entropy production was higher in the asymmetric state. This indicates that MEP is valid for a certain class of systems, but what this class is has not yet been clearly identified.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The figure shows numerical values assigned to rate constants to obtain numerical solution for the rate equations of the system. Units of volume are assumed to be liters (L) and concentrations mol/L (M). Units of variable parameter II may be thought of as W/m

^{2}. If II is thought of as a radiation from the sun, its blackbody temperature is very high compared to the temperature of the system. All rate constants are assumed to have the appropriate units, though not written explicitly.

**Figure 2.**Time evolution of chiral species to their equilibrium state in the model reaction scheme when II = 0 and S = T = 0.01 M at t = 0. Concentrations of all other species was set to zero at t = 0. Due to the symmetry of the system, at equilibrium, S

_{L}= S

_{D}(dashed line) and X

_{L}= X

_{D}(solid line), so the curves for the two enantiomers overlap.

**Figure 3.**Schematic of reaction system. Radiation (shown as hν and denoted as II in the reaction scheme)) is incident on a closed chemical system. The radiation drives a generation–decomposition cycle of enantiomeric species X and other compounds, as shown in A. When II > II

_{C}, the system evolves to one of two asymmetric states, B or C. In state B, the amount of X

_{L}> X

_{D}, and in state C, X

_{D}> X

_{L}. The asymmetry is parametrized by α = (X

_{L}− X

_{D}).

**Figure 4.**Time evolution of chiral species using equilibrium values for initial conditions when II = 0.003. The solid line represents X

_{L}and X

_{D}and the dashed S

_{L}and S

_{D}. Since symmetry is not broken, the amounts of each chiral species are exactly equal, hence the overlapping curves (α = 0).

**Figure 5.**Time evolution of chiral species in an asymmetric state. Here, II = 0.008 and an asymmetric steady state is reached in about 7000 s. The blue solid and dashed lines represent X

_{D}and S

_{D}, respectively, and the red solid and dashed lines represent X

_{L}and S

_{L}, respectively. The black line represents α, which takes on a nonzero value once symmetry is broken.

**Figure 6.**Dependence of α on II. Units of α are M and II are Wm

^{−2}. Steady-state values of α are plotted as a function of II. When II > II

_{C}, α takes a positive (X

_{L}> X

_{D}, shown in green X) or negative (X

_{L}< X

_{D}, shown in red X) value, depending on the random perturbation that drives the system away from the unstable racemic state α = 0. The blue Xs show the symmetric branch which is unstable above the critical point. In the region II > II

_{C}, α increases in a characteristically parabolic manner.

**Figure 7.**Rate of entropy production in an asymmetric state when II = 0.008. Units of σ are JK

^{−1}L

^{−1}s

^{−1}. As in Figure 3, steady state is reached at about 7000 s. Although it may appear that σ is approaching 0, it is not so; σ maintains a nonzero value at steady state after symmetry is broken.

**Figure 8.**Rate of entropy production σ and α as a function of II. The critical value II

_{C}= 0.0040. Beyond this point, α takes on nonzero values and there is a decrease in the slope of σ. Units of II are Wm

^{−2}. The dashed arrow indicates a transition from an unstable state, where α = 0, to a stable asymmetric state where α is nonzero. In the region II > II

_{C}rate of entropy production of asymmetric state is lower than that of the symmetric state. The solid arrow shows that transition from an unstable symmetric state to a stable asymmetric state results in the lowering of σ, the rate of entropy production.

**Table 1.**A model photochemically driven reaction scheme in a closed system. The table lists all the elementary steps in the model. The excitation of T to the state TE drives the reaction that generate chiral species X

_{L}and X

_{D}. Reactions R4a, R4b, R5a, R5b are the autocatalytic steps for X

_{L}and X

_{D}. Autocatalysis and reaction R6 result in spontaneous breaking of chiral symmetry when the intensity of radiation, II, is above a critical intensity, II

_{C}.

Chemical Reactions | Number |
---|---|

S + II ⇄ TE | (R1) |

T ⇄ TE | (R1a) |

S + TE ⇄ X_{L} | (R2) |

S + TE ⇄ X_{D} | (R3) |

S + X_{L} ⇄ S_{L} | (R4a) |

S_{L} + TE ⇄ 2X_{L} | (R4b) |

S + X_{D} ⇄ S_{D} | (R5a) |

S_{D} + TE ⇄ 2X_{D} | (R5b) |

X_{L} + X_{D} ⇄ P + W | (R6) |

P ⇄ 2S | (R7) |

W ⇄ 2T | (R8) |

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Kondepudi, D.; Mundy, Z.
Spontaneous Chiral Symmetry Breaking and Entropy Production in a Closed System. *Symmetry* **2020**, *12*, 769.
https://doi.org/10.3390/sym12050769

**AMA Style**

Kondepudi D, Mundy Z.
Spontaneous Chiral Symmetry Breaking and Entropy Production in a Closed System. *Symmetry*. 2020; 12(5):769.
https://doi.org/10.3390/sym12050769

**Chicago/Turabian Style**

Kondepudi, Dilip, and Zachary Mundy.
2020. "Spontaneous Chiral Symmetry Breaking and Entropy Production in a Closed System" *Symmetry* 12, no. 5: 769.
https://doi.org/10.3390/sym12050769