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This reaction is one in which a molecule of species A interacts with a molecule of species B. The A molecule is converted into a B molecule. The final product consists of the original B molecule plus the B molecule created in the reaction.

This reaction is one in which a molecule of species A interacts with a molecule of species B. The A molecule is converted into a B molecule. The final product consists of the original B molecule plus the B molecule created in the reaction.

 

A similar example exists for living chemical systems. The sun provides energy to green plants. The green plants are food for other living chemical systems. The energy absorbed by plants and converted into chemical energy generates a system on earth that is orderly and far from chemical equilibrium. Here, the difference from chemical equilibrium is determined by an excess of reactants over the equilibrium amount. Once again, order on earth is generated at the expense of entropy increase of the sun. The total entropy of the earth and the rest of the universe increases, consistent with the Second Law.

A similar example exists for living chemical systems. The sun provides energy to green plants. The green plants are food for other living chemical systems. The energy absorbed by plants and converted into chemical energy generates a system on earth that is orderly and far from chemical equilibrium. Here, the difference from chemical equilibrium is determined by an excess of reactants over the equilibrium amount. Once again, order on earth is generated at the expense of entropy increase of the sun. The total entropy of the earth and the rest of the universe increases, consistent with the Second Law.

The key feature of these rate equations is that they are [[nonlinear]]; the second term on the right varies as the square of the concentration of B. This feature can lead to multiple fixed points of the system, much like a [[quadratic equation]] can have multiple roots. Multiple fixed points allow for multiple states of the system. A system existing in multiple [[macroscopic]] states is more orderly (has lower entropy) than a system in a single state.

The key feature of these rate equations is that they are [[nonlinear]]; the second term on the right varies as the square of the concentration of B. This feature can lead to multiple fixed points of the system, much like a [[quadratic equation]] can have multiple roots. Multiple fixed points allow for multiple states of the system. A system existing in multiple [[macroscopic]] states is more orderly (has lower entropy) than a system in a single state.

 

The graph for these equations is a [[sigmoid function|sigmoid curve]] (specifically a [[logistic function]]), which is typical for autocatalytic reactions: these chemical reactions proceed slowly at the start (the [[induction period]]) because there is little catalyst present, the rate of reaction increases progressively as the reaction proceeds as the amount of catalyst increases and then it again slows down as the reactant concentration decreases. If the concentration of a reactant or product in an experiment follows a sigmoid curve, the reaction may be autocatalytic.

The graph for these equations is a [[sigmoid function|sigmoid curve]] (specifically a [[logistic function]]), which is typical for autocatalytic reactions: these chemical reactions proceed slowly at the start (the [[induction period]]) because there is little catalyst present, the rate of reaction increases progressively as the reaction proceeds as the amount of catalyst increases and then it again slows down as the reactant concentration decreases. If the concentration of a reactant or product in an experiment follows a sigmoid curve, the reaction may be autocatalytic.

 

A chemical reaction cannot oscillate about a position of final equilibrium because the second law of thermodynamics requires that a thermodynamic system approach equilibrium and not recede from it. For a closed system at constant temperature and pressure, the Gibbs free energy must decrease continuously and not oscillate. However it is possible that the concentrations of some reaction intermediates oscillate, and also that the rate of formation of products oscillates.

A chemical reaction cannot oscillate about a position of final equilibrium because the second law of thermodynamics requires that a thermodynamic system approach equilibrium and not recede from it. For a closed system at constant temperature and pressure, the Gibbs free energy must decrease continuously and not oscillate. However it is possible that the concentrations of some reaction intermediates oscillate, and also that the rate of formation of products oscillates.

These kinetic equations apply for example to the acid-catalyzed hydrolysis of some [[ester]]s to [[carboxylic acid]]s and [[alcohol]]s.<ref name=Moore/> There must be at least some acid present initially to start the catalyzed mechanism; if not the reaction must start by an alternate uncatalyzed path which is usually slower. The above equations for the catalyzed mechanism would imply that the concentration of acid product remains zero forever.<ref name=Moore/>

These kinetic equations apply for example to the acid-catalyzed hydrolysis of some [[ester]]s to [[carboxylic acid]]s and [[alcohol]]s.<ref name=Moore/> There must be at least some acid present initially to start the catalyzed mechanism; if not the reaction must start by an alternate uncatalyzed path which is usually slower. The above equations for the catalyzed mechanism would imply that the concentration of acid product remains zero forever.<ref name=Moore/>

 

This system of rate equations is known as the [[Lotka–Volterra equation]] and is most closely associated with [[population dynamics]] in predator–prey relationships. This system of equations can yield oscillating concentrations of the reaction intermediates X and Y. The amplitude of the oscillations depends on the concentration of A (which decreases without oscillation). Such oscillations are a form of emergent temporal order that is not present in equilibrium.

This system of rate equations is known as the [[Lotka–Volterra equation]] and is most closely associated with [[population dynamics]] in predator–prey relationships. This system of equations can yield oscillating concentrations of the reaction intermediates X and Y. The amplitude of the oscillations depends on the concentration of A (which decreases without oscillation). Such oscillations are a form of emergent temporal order that is not present in equilibrium.

 

The Brusselator in the unstable regime. A=1. B=2.5. X(0)=1. Y(0)=0. The system approaches a [[limit cycle. For B<1+A the system is stable and approaches a fixed point.]]

The Brusselator in the unstable regime. A=1. B=2.5. X(0)=1. Y(0)=0. The system approaches a [[limit cycle. For B<1+A the system is stable and approaches a fixed point.]]

where, for convenience, the rate constants have been set to 1.

where, for convenience, the rate constants have been set to 1.

 

<math>{d \over dt}[ X_2 ] = [A ] + [ X _2]^2 [Y_2 ]  - [B ] [X_2 ] - [X_2 ]  + D_x\left( X_1 - X_2 \right)\,</math>

<math>{d \over dt}[ X_2 ] = [A ] + [ X _2]^2 [Y_2 ]  - [B ] [X_2 ] - [X_2 ]  + D_x\left( X_1 - X_2 \right)\,</math>

The Brusselator has a fixed point at

The Brusselator has a fixed point at

 

Noting that photo-polymerization rate is proportional to intensity and that refractive index is proportional to molecular weight, the positive feedback between intensity and photo-polymerization establishes the auto-catalytic behavior. Optical auto-catalysis has been shown to result on spontaneous pattern formation in photopolymers. Hosein and co-workers discovered that optical autocatalysis can also occur in photoreactive polymer blends, and that the process can induce binary phase morphologies with the same pattern as the light profile. Glycolysis consists of the degradation of one molecule of glucose and the overall production of two molecules of ATP. The process is therefore of great importance to the energetics of living cells. The global glycolysis reaction involves glucose, ADP, NAD, pyruvate, ATP, and NADH.

Noting that photo-polymerization rate is proportional to intensity and that refractive index is proportional to molecular weight, the positive feedback between intensity and photo-polymerization establishes the auto-catalytic behavior. Optical auto-catalysis has been shown to result on spontaneous pattern formation in photopolymers. Hosein and co-workers discovered that optical autocatalysis can also occur in photoreactive polymer blends, and that the process can induce binary phase morphologies with the same pattern as the light profile. Glycolysis consists of the degradation of one molecule of glucose and the overall production of two molecules of ATP. The process is therefore of great importance to the energetics of living cells. The global glycolysis reaction involves glucose, ADP, NAD, pyruvate, ATP, and NADH.

注意到光聚合速率与强度成正比，折射率与分子量成正比，强度与光聚合之间的正反馈建立了自催化行为。光学自催化已被证明是光聚合物自发形成图案的原因。Hosein 和他的合作者发现，光学自催化也可以发生在光反应聚合物共混物中，这一过程可以诱导形成与光型相同的二元相形态。糖酵解是由一个葡萄糖分子的降解和两个 ATP 分子的全部产生组成。因此，这个过程对活细胞的能量学来说是非常重要的。糖酵解反应包括葡萄糖、 ADP、 NAD、丙酮酸、 ATP 和 NADH。

注意到光聚合速率与强度成正比，折射率与分子量成正比，强度与光聚合之间的正反馈建立了自催化行为。光学自催化已被证明是光聚合物自发形成图案的原因。Hosein和他的合作者发现，光学自催化也可以发生在光反应聚合物共混物中，这一过程可以诱导形成与光型相同的二元相形态。糖酵解是由一个葡萄糖分子的降解和两个 ATP 分子的全部产生组成。因此，这个过程对活细胞的能量学来说是非常重要的。糖酵解反应包括葡萄糖、 ADP、 NAD、丙酮酸、 ATP 和 NADH。

}} Dynamics of the Brusselator

}} Dynamics of the Brusselator

The initial amounts of reactants determine the distance from a chemical equilibrium of the system. The greater the initial concentrations the further the system is from equilibrium. As the initial concentration increases, an abrupt change in order occurs. This abrupt change is known as phase transition. At the phase transition, fluctuations in macroscopic quantities, such as chemical concentrations, increase as the system oscillates between the more ordered state (lower entropy, such as ice) and the more disordered state (higher entropy, such as liquid water). Also, at the phase transition, macroscopic equations, such as the rate equations, fail. Rate equations can be derived from microscopic considerations. The derivations typically rely on a mean field theory approximation to microscopic dynamical equations. Mean field theory breaks down in the presence of large fluctuations (see Mean field theory article for a discussion).  Therefore, since large fluctuations occur in the neighborhood of a phase transition, macroscopic equations, such as rate equations, fail. As the initial concentration increases further, the system settles into an ordered state in which fluctuations are again small. (see Prigogine reference)

The initial amounts of reactants determine the distance from a chemical equilibrium of the system. The greater the initial concentrations the further the system is from equilibrium. As the initial concentration increases, an abrupt change in order occurs. This abrupt change is known as phase transition. At the phase transition, fluctuations in macroscopic quantities, such as chemical concentrations, increase as the system oscillates between the more ordered state (lower entropy, such as ice) and the more disordered state (higher entropy, such as liquid water). Also, at the phase transition, macroscopic equations, such as the rate equations, fail. Rate equations can be derived from microscopic considerations. The derivations typically rely on a mean field theory approximation to microscopic dynamical equations. Mean field theory breaks down in the presence of large fluctuations (see Mean field theory article for a discussion).  Therefore, since large fluctuations occur in the neighborhood of a phase transition, macroscopic equations, such as rate equations, fail. As the initial concentration increases further, the system settles into an ordered state in which fluctuations are again small. (see Prigogine reference)

反应物的初始量决定了与体系化学平衡的距离。初始浓度越大，系统离平衡越远。随着初始浓度的增加，顺序发生突变。这种突变被称为相变。在相变阶段，宏观量的波动，如化学浓度，随着系统在更有序的状态（低熵，如冰）和更无序的状态（更高的熵，如液态水）之间振荡而增加。同样，在相变过程中，宏观方程，如速率方程，会失效。速率方程可以从微观角度推导出来。推导通常依赖于对微观动力学方程的平均场理论近似。平均场理论在大波动的存在下会崩溃（见平均场理论文章的讨论）。因此，由于大的波动发生在相变附近，宏观方程，如速率方程，失败了。随着初始浓度的进一步增加，系统进入有序状态，在这种状态下波动又很小(见 Prigogine 参考文献)

反应物的初始量决定了与体系化学平衡的距离。初始浓度越大，系统离平衡越远。随着初始浓度的增加，顺序发生突变。这种突变被称为相变。在相变阶段，宏观量的波动，如化学浓度，随着系统在更有序的状态（低熵，如冰）和更无序的状态（更高的熵，如液态水）之间振荡而增加。同样，在相变过程中，宏观方程，如速率方程，会失效。速率方程可以从微观角度推导出来。推导通常依赖于对微观动力学方程的平均场理论近似。平均场理论在大波动的存在下会崩溃（见平均场理论文章的讨论）。因此，由于大的波动发生在相变附近，例如速率方程等的宏观方程便失灵了。随着初始浓度的进一步增加，系统进入有序状态，在这种状态下波动又很小(见 Prigogine 参考文献)

:<math>{d \over dt}[ Y_2 ] =  [B ] [X_2 ] - [ X_2 ]^2 [Y_2 ]  + D_y\left( Y_1 - Y_2\right)  \,</math>

:<math>{d \over dt}[ Y_2 ] =  [B ] [X_2 ] - [ X_2 ]^2 [Y_2 ]  + D_y\left( Y_1 - Y_2\right)  \,</math>

Here, the numerical subscripts indicate which box the material is in. There are additional terms proportional to the diffusion coefficient D that account for the exchange of material between boxes.

Here, the numerical subscripts indicate which box the material is in. There are additional terms proportional to the diffusion coefficient D that account for the exchange of material between boxes.

 

Asymmetric autocatalysis occurs when the reaction product is chiral and thus acts as a chiral catalyst for its own production. Reactions of this type, such as the Soai reaction, have the property that they can amplify a very small enantiomeric excess into a large one. This has been proposed as an important step in the origin of biological homochirality.

Asymmetric autocatalysis occurs when the reaction product is chiral and thus acts as a chiral catalyst for its own production. Reactions of this type, such as the Soai reaction, have the property that they can amplify a very small enantiomeric excess into a large one. This has been proposed as an important step in the origin of biological homochirality.

If the system is initiated with the same conditions in each box, then a small fluctuation will lead to separation of materials between the two boxes. One box will have a predominance of X, and the other will have a predominance of Y.

If the system is initiated with the same conditions in each box, then a small fluctuation will lead to separation of materials between the two boxes. One box will have a predominance of X, and the other will have a predominance of Y.

 

Real examples of [[clock reaction]]s are the [[Belousov–Zhabotinsky reaction]] (BZ reaction), the [[Briggs–Rauscher reaction]], the [[Bray–Liebhafsky reaction]] and the [[iodine clock reaction]]. These are oscillatory reactions, and the concentration of products and reactants can be approximated in terms of [[damping|damped]] [[oscillation]]s.

Real examples of [[clock reaction]]s are the [[Belousov–Zhabotinsky reaction]] (BZ reaction), the [[Briggs–Rauscher reaction]], the [[Bray–Liebhafsky reaction]] and the [[iodine clock reaction]]. These are oscillatory reactions, and the concentration of products and reactants can be approximated in terms of [[damping|damped]] [[oscillation]]s.

 

In 1995 Stuart Kauffman proposed that life initially arose as autocatalytic chemical networks.

In 1995 Stuart Kauffman proposed that life initially arose as autocatalytic chemical networks.