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# Petri net simulation lab #1 - Dimerization kinetics

edited June 2020

From:

• Darren J. Wilkinson, Stochastic Modelling for Systems Biology, Taylor & Francis, New York, 2006. Good introduction to stochastic Petri nets, with applications to gene expression and the Lotka-Volterra equations for predator-prey interactions.

Chapter on case studies has the following applications, with simulation lab exercises:

• Dimerization kinetics
• Michaelis-Menten enzyme kinetics
• An auto-regulatory genetic network
• The lac operon

In this thread we'll look into the one on dimerization kinetics.

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1.
edited June 2020

A dimer is an oligomer consisting of two monomers joined by bonds that can be either strong or weak, covalent or intermolecular. The term homodimer is used when the two molecules are identical and heterodimer when they are not. The reverse of dimerization is often called dissociation.

We'll look at the homodimer case, which is just the reaction $$X + X = 2X \rightarrow X_2$$.

For instance, the potassium dimer $$K_2$$ is a molecule consisting of two potassium atoms, and the dimerization of potassium would be the reaction where two $$K$$ atoms combine into one $$K_2$$ molecule: $$2K \rightarrow K_2$$.

The reverse reaction, where the dimer splits into two monomers, is called dissociation: $$X_2 \rightarrow 2X$$.

Comment Source:* [Dimer](https://en.wikipedia.org/wiki/Dimer_(chemistry)), Wikipedia. > A dimer is an oligomer consisting of two monomers joined by bonds that can be either strong or weak, covalent or intermolecular. The term homodimer is used when the two molecules are identical and heterodimer when they are not. The reverse of dimerization is often called dissociation. We'll look at the homodimer case, which is just the reaction \$$X + X = 2X \rightarrow X_2\$$. For instance, the potassium dimer \$$K_2\$$ is a molecule consisting of two potassium atoms, and the dimerization of potassium would be the reaction where two \$$K\$$ atoms combine into one \$$K_2\$$ molecule: \$$2K \rightarrow K_2\$$. The reverse reaction, where the dimer splits into two monomers, is called dissociation: \$$X_2 \rightarrow 2X\$$. 
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2.
edited June 2020

For this lab, the reaction system consists of two reactions, dimerization and dissociation:

• $$2X \rightarrow X_2$$, with rate coefficient $$k_1$$ = 50,000
• $$X_2 \rightarrow 2X$$, with rate coefficient $$k_2$$ = 0.2

Let:

• $$X[t]$$ be the count of the number of monomers $$X$$ at time $$t$$
• $$X_2[t]$$ be the count of the number of dimers $$X_2$$ at time $$t$$
Comment Source:For this lab, the reaction system consists of two reactions, dimerization and dissociation: * \$$2X \rightarrow X_2\$$, with rate coefficient \$$k_1\$$ = 50,000 * \$$X_2 \rightarrow 2X\$$, with rate coefficient \$$k_2\$$ = 0.2 Let: * \$$X[t]\$$ be the count of the number of monomers \$$X\$$ at time \$$t\$$ * \$$X_2[t]\$$ be the count of the number of dimers \$$X_2\$$ at time \$$t\$$ 
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3.

So we have a "tug of war" between the two reactions, with dimerization working to build up the count of dimers $$X_2[t]$$ and reduce the count of monomers $$X[t]$$ - and dissociation doing just the opposite.

Comment Source:So we have a "tug of war" between the two reactions, with dimerization working to build up the count of dimers \$$X_2[t]\$$ and reduce the count of monomers \$$X[t]\$$ - and dissociation doing just the opposite.
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4.
edited June 2020

The dynamics of the system is expressed by the two time series $$X[t]$$ and $$X_2[t]$$.

Intuitively, we would expect that given a starting state $$S[0] = (X[0], X_2[0])$$, it will eventually approach some limiting state $$S[{\infty}] = (X', X_2')$$ as $$t$$ advances - that is the equilibrium state towards which it is bound.

Comment Source:The _dynamics_ of the system is expressed by the two time series \$$X[t]\$$ and \$$X_2[t]\$$. Intuitively, we would expect that given a starting state \$$S[0] = (X[0], X_2[0])\$$, it will eventually approach some limiting state \$$S[{\infty}] = (X', X_2')\$$ as \$$t\$$ advances - that is the _equilibrium_ state towards which it is bound. 
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5.
edited June 2020

The proportion between the two species in the equilibrium state will depend on the relative strengths of the two reactions - see the reaction coefficients. If dissociation had a coefficient of zero, it would be effectively gone, and the equilibrium state would consist 100% of dimers 0% of monomers. And similarly if it was all dissociation equilibrium would be at 100% momers.

The rate coefficients given above give a might higher strength to dimerization, and so we would expect equilibrium to consist mostly of dimers.

Comment Source:The proportion between the two species in the equilibrium state will depend on the relative strengths of the two reactions - see the reaction coefficients. If dissociation had a coefficient of zero, it would be effectively gone, and the equilibrium state would consist 100% of dimers 0% of monomers. And similarly if it was all dissociation equilibrium would be at 100% momers. The rate coefficients given above give a might higher strength to dimerization, and so we would expect equilibrium to consist _mostly_ of dimers. 
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6.

Our goal is now to understand the dynamics of the system. This means understanding the evolution of the state function S[t].

Comment Source:Our goal is now to understand the dynamics of the system. This means understanding the evolution of the state function S[t]. 
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7.
edited June 2020

We have a choice on the menu: deterministic or stochastic modeling of S[t].

In a deterministic model, S[t] is a definite function of the continuous parameter $$t$$. Once the initial condition is specified, by giving a value for S[0], then the future values of S[t] are uniquely determined.

This is a "flow." (Semi-flow actually, as S[t] is defined for non-negative t.)

Comment Source:We have a choice on the menu: deterministic or stochastic modeling of S[t]. In a deterministic model, S[t] is a definite function of the continuous parameter \$$t\$$. Once the initial condition is specified, by giving a value for S[0], then the future values of S[t] are uniquely determined. This is a "flow." (Semi-flow actually, as S[t] is defined for non-negative t.) 
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8.

In the stochastic model, $$t$$ is still a continuous parameter, but the state S[t] makes little "jumps" at random times, when one of the reactions randomly "fires." Every time the dimerization reaction fires, the count of monomers goes down by two, and the count of dimers goes up by one. When the opposite reaction viz. dissociation fires, the dimer count goes down by one and the monomer count goes up by two.

Comment Source: In the stochastic model, \$$t\$$ is still a continuous parameter, but the state S[t] makes little "jumps" at random times, when one of the reactions randomly "fires." Every time the dimerization reaction fires, the count of monomers goes down by two, and the count of dimers goes up by one. When the opposite reaction viz. dissociation fires, the dimer count goes down by one and the monomer count goes up by two.
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9.
edited June 2020

Where do the reaction coefficients show up in these models?

In the deterministic case, the semi-flow will be specified by differential equations, and the reaction coefficients will turn up as coefficients in the equations.

In the stochastic model, a reaction coefficient will turn up as a weight affecting the probability that the reaction fires within a small interval of time.

Comment Source:Where do the reaction coefficients show up in these models? In the deterministic case, the semi-flow will be specified by differential equations, and the reaction coefficients will turn up as coefficients in the equations. In the stochastic model, a reaction coefficient will turn up as a weight affecting the probability that the reaction fires within a small interval of time. 
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10.

Either way, we have a second choice on the menu: an analytic approach to the dynamics, or an experimental approach via simulation.

Comment Source:Either way, we have a second choice on the menu: an analytic approach to the dynamics, or an experimental approach via simulation.
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11.
edited June 2020

Let's start with the deterministic case. This applies well to chemical reactions involving large numbers of molecules, where the different species involved in the reaction are "well-mixed." (E.g. no clumps of dimers surrounded by zones of monomers.)

Comment Source:Let's start with the deterministic case. This applies well to chemical reactions involving large numbers of molecules, where the different species involved in the reaction are "well-mixed." (E.g. no clumps of dimers surrounded by zones of monomers.)