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[Michaelis-Menten enzyme kinetics](https://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics) is a simple description of an enzyme ($E$) transforming a substrate ($S$) into a product ($P$). Under certain assumptions it can be simplified to a singe function (a Michaelis-Menten function) and used as a reaction rate. Here we instead present the full system model:
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[Michaelis-Menten enzyme kinetics](https://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics) is a simple description of an enzyme ($E$) transforming a substrate ($S$) into a product ($P$). Under certain assumptions, it can be simplified to a single function (a Michaelis-Menten function) and used as a reaction rate. Here we instead present the full system model:
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```@example crn_library_michaelis_menten
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using Catalyst
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mm_system = @reaction_network begin
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oplt = plot(osol; title = "Reaction rate equation (ODE)")
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splt = plot(ssol; title = "Chemical Langevin equation (SDE)")
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jplt = plot(jsol; title = "Stochastic chemical kinetics (Jump)")
plot(osol; title = "Reaction rate equation (ODE)")
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plot(osol; title = "Reaction rate equation (ODE)", size=(800,350))
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```
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Next we perform 3 different Jump simulations. Note that for the stochastic model, the occurrence of a outbreak is not certain. Rather, there is a possibility that it fizzles out without a noteworthy peak.
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Next, we perform 3 different Jump simulations. Note that for the stochastic model, the occurrence of an outbreak is not certain. Rather, there is a possibility that it fizzles out without a noteworthy peak.
The Wilhelm model was introduced in [*Wilhelm (2009)*](https://bmcsystbiol.biomedcentral.com/articles/10.1186/1752-0509-3-90) as the smallest CRN model (with constant rates) that exhibits bistability.
The simplest self-activation loop consist of a single species (here called $X$) which activates its own production. If its production rate is modelled with a hill function with $n>1$, the system may exhibit bistability.
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The simplest self-activation loop consists of a single species (here called $X$) which activates its own production. If its production rate is modelled with a hill function with $n>1$, the system may exhibit bistability.
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```@example crn_library_self_activation
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using Catalyst
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sa_loop = @reaction_network begin
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We simulate the self-activation loop from a single initial condition using both deterministic (ODE) and stochastic (jump) simulations. We note that while the deterministic simulation reaches a single steady state, the stochastic one switches between two different states.
The [Brusselator](https://en.wikipedia.org/wiki/Brusselator) is a wellknown (theoretical) CRN model able to produce oscillations (its name is a portmanteau of "Brussels" and "oscillator").
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The [Brusselator](https://en.wikipedia.org/wiki/Brusselator) is a well-known (theoretical) CRN model able to produce oscillations (its name is a portmanteau of "Brussels" and "oscillator").
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```@example crn_library_brusselator
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using Catalyst
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brusselator = @reaction_network begin
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1, X --> ∅
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end
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```
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It is generally known to (for reaction rate equation-based ODE simulations) produce oscillations when $B > 1 + A^2$. However, this results is based on models generated when *combinatorial adjustment of rates is not performed*. Since Catalyst automatically perform these adjustments, and one reaction contain a stoichiometric constant $>1$, the threshold will be different. Here, we trial two different values of $B$. In both cases, $B < 1 + A^2$, however, in he second case the system is able to generate oscillations.
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It is generally known to (for reaction rate equation-based ODE simulations) produce oscillations when $B > 1 + A^2$. However, this result is based on models generated when *combinatorial adjustment of rates is not performed*. Since Catalyst automatically perform these adjustments, and one reaction contains a stoichiometric constant $>1$, the threshold will be different. Here, we trial two different values of $B$. In both cases, $B < 1 + A^2$, however, in the second case the system can generate oscillations.
The repressilator was introduced in [*Elowitz & Leibler (2000)*](https://www.nature.com/articles/35002125) as a simple system that is able to generate oscillations (most notably, they demonstrated this both in a model and in a synthetic in vivo implementation in *Escherichia col*). It consists of three genes, repressing each other in a cycle. Here, we will implement it using three species ($X$, $Y$, and $Z$) which production rates are (repressing) [Hill functions](https://en.wikipedia.org/wiki/Hill_equation_(biochemistry)).
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The repressilator was introduced in [*Elowitz & Leibler (2000)*](https://www.nature.com/articles/35002125) as a simple system that can generate oscillations (most notably, they demonstrated this both in a model and in a synthetic in vivo implementation in *Escherichia col*). It consists of three genes, repressing each other in a cycle. Here, we will implement it using three species ($X$, $Y$, and $Z$) whose production rates are (repressing) [Hill functions](https://en.wikipedia.org/wiki/Hill_equation_(biochemistry)).
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```@example crn_library_brusselator
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using Catalyst
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repressilator = @reaction_network begin
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d, (X, Y, Z) --> ∅
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end
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```
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Whether it oscillates or not depends on its parameter values. Here, we will perform deterministic (ODE) simulations for two different values of $K$, showing that it oscillates for one value and not the other one. Next, we will perform stochastic (SDE) simulations for both $K$ values, showing that the stochastic model is able to sustain oscillations in both cases. This is an example of the phenomena of *noise-induced oscillation*.
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Whether it oscillates or not depends on its parameter values. Here, we will perform deterministic (ODE) simulations for two different values of $K$, showing that it oscillates for one value and not the other one. Next, we will perform stochastic (SDE) simulations for both $K$ values, showing that the stochastic model can sustain oscillations in both cases. This is an example of the phenomena of *noise-induced oscillation*.
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k7, Z --> ∅
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end
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```
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Here we first simulate the model for a single initial conditions, showing in both time-state space and phase space how how it reaches a [*strange attractor*](https://www.dynamicmath.xyz/strange-attractors/).
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Here we first simulate the model for a single initial condition, showing in both time-state space and phase space how it reaches a [*strange attractor*](https://www.dynamicmath.xyz/strange-attractors/).
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