Imperial College/Courses/2010/Synthetic Biology/Computer Modelling Practicals/Practical 1

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Practical 1


Foreword: "All models are wrong, but some of them are useful, George Box"

  • Possible Way to understand this: Modelling = Catching the Trend and Explaining it
    • Analysis of a problem identifies the most important process shaping the problem
    • The effect of each process is described with some equations (or any tools borrowed from mathematics) - their combination is then simulated.
    • Successful Modelling:
      • Predictive Power: the outcome to simulation is very close to the outcome in real life
      • Reusability: the model can be reused in another, similar case
  • Another Possible Explanation:
    • when data deviate from predictions, something interesting may be happening
    • New effect at work?
  • Models are therefore wrong but sometimes useful!


Part I: Getting to know CellDesigner


  • Thanks to Dr V rouilly for the Cell Designer Tutorial!!!
  • Read through the tutorial example, and get familiar with CellDesigner features. Official CellDesigner Tutorial
  • Please Note: the link redirects you to the 2008 tutorials. Make sure that when you are done you come back to this page!!!
  • Open a sample file: File -> Open -> Samples/...
  • Select items, move them around, delete, undo...


Part II: Your First Model: A --> B --> C



  • Now is the time to build your first model from scratch with CellDesigner, and to run a simulation.
  • The model explored describe a system where a compound 'A' is transformed into a compound 'B', which is consequently transformed into a compound 'C'.
  • To start, launch the CellDesigner Application: Double Click on the Icon found on your Desktop.
  • Then follow the instructions below to build the model.


Model CellDesigner Instructions

A \xrightarrow{k_{1}} B \xrightarrow{k_{2}} C
  • Define the topology of the reaction network:
    • Open a NEW document: File -> New.
    • Create 3 compounds A, B, and C (help).
    • Create Reaction_1 linking 'A' to 'B' (help).
    • Create Reaction_2 linking 'B' to 'C'
  • Save your model

Following the Law of Mass action, the dynamic of the system is described as:


\begin{alignat}{2}
\frac{d[A]}{dt} & = - k_{1}*[A] \\
\frac{d[B]}{dt} & =  k_{1}*[A] -k_{2}*[B] \\
\frac{d[C]}{dt} & =  k_{2}*[B] 
\end{alignat}
  • Edit Reaction_1, Create a NEW local parameter called k1, value equals 1.0 (help).
  • Create a kinetic law for Reaction_1, according to the dynamical system (help).
  • Edit Reaction_2, Create a NEW local parameter called k2, value equals 10.0
  • Create a kinetic law for Reaction_2, according to the dynamical system.
  • Save your model.
Simulate the dynamical behaviour
  • Open Simulation Panel (help)
  • In the top left panel set the End Time as 10 seconds
  • Set the number of points as 1000 (gives a nice smooth curve)
  • The panel below will be on the species tab, set Initial quantity of A as 10
  • Press Execute, and check results.


You are now ready to analyse the behaviour of the biochemical network A --> B --> C.


Questions

The following qustions are to be addressed in Section A of your coursework.

  • Question 1: Description of the Dynamics
    • Q1.1: Plot and Describe the evolution with time of the concentrations of A, B and C, using these default parameters?
    • Q1.2: Now swap the values of k1 and k2 (k1=10 and k2=1)under the parameters tab
      • How does this alter the formation of C?
      • How does B change?
      • Explain these results
  • Question 2: Now, let us place ourselves in the position of an experimentalist.
    • Q2.1: If you had real life data showing the accumulation of C for an A-B-C reaction you could fit the data using this model and two rate constants would be returned. Could you assign these rate constants to k1 or k2 (yes or no)?
    • Q2.2: What additional data would you need to assign k1 and k2? (Explain how you would extract k1 and k2)


Part III: Simulating an Enzymatic Reaction


  • An enzyme converts a substrate into a product, this is usually an irreversible reaction.
  • An enzyme reaction constitutes a dynamic process and can be studied as such.
  • One may look at the time courses of the reactants, or look at the steady-states and their stability properties.
  • A convenient approaximation, called the Michaelis-Menten approaximation is often used -sometimes wrongly!


Model CellDesigner Instructions

 E + S
  \begin{matrix}
    k_1 \\
    \longrightarrow \\
    \longleftarrow  \\
    k_{2}
  \end{matrix}
 ES
  \begin{matrix}
    k_3 \\
    \longrightarrow\\
     
  \end{matrix}
 E + P
  • Download this File on your desktop.
  • Open the file with CellDesigner.
  • 1 reaction network topology is described in this file, no kinetics information is yet defined.
Following law of mass action, we can write:


\begin{alignat}{2}
\frac{d[E]}{dt} & = k_{2}[ES] - k_{1}[E][S] + k_{3}[ES] \\
\frac{d[S]}{dt} & = k_{2}[ES] - k_{1}[E][S] \\
\frac{d[ES]}{dt} & = k_{1}[E][S] - k_{2}[ES] - k_{3}[ES] \\
\frac{d[P]}{dt} & = k_{3}[ES] 
\end{alignat}

Simple Enzymatic Reaction

Recommended Simulations


Now that you have a network representation and a system of ODEs, you can run simulations and try to understand the dynamics of enzymatic reactions.

  • Typical Simulation: let us consider the following realistic values:
    • k1 = 108M − 1s − 1 ; k2 = 100s − 1 ; k3 = 10 − 1s − 1
    • Initial Conditions: [E]t = 0 = 10 − 7M ; [S]t = 0 = 10 − 5M ; [P]t = 0 = 0
    • Open the Simulation Panel, set Time=2000, NbPoints=10000. NB: values have been changed - they should be correct now
  • Run the simulation - Get the feel for the behaviour of the system
    • Pay special attention to the formation and decay of the [ES] complex.
    • Note that this is a full simulation of the reaction scheme and does not rely on any assumption.

A critical input of the system is the initial concentration of substrate [S]t = 0.

  • To investigate the influence of [S]t = 0, we will make it vary between 10 − 9M and 10^{-4} M</math> . The remaining parameters are kept as:
    • k1 = 108M − 1s − 1 ; k2 = 100s − 1 ; k3 = 10 − 1s − 1
    • Initial Conditions : [E]t = 0 = 10 − 7M ; [P]t = 0 = 0
  • Run the simulation
    • Again pay special attention to the formation and decay of the [ES] complex.
    • This time pay also attention to the profile of [P], especially at the start of the simulation (time close to 0)


Questions

The following questions must be addressed in your coursework (Section B).

  • Question 1: Apply the law of mass action and derive the system of ODE
  • Question 2: Describe and explain the evolution with time of the species involved in the enzymatic reaction
    • Use the first (typical) simulation - You can comment directly on the graph(s) if you want
    • Do not forget that the evolution of the species is linked
  • Question 3: Describe and explain (in qulalitative terms) the impact of the initial concentration of substrate on the dynamics of the enzymatic reaction
  • Question 4: The Michaelis-Menten MM) Approximation. MM assumes that the enzyme complex ES is in a dynamic steady state (  \frac{d[ES]}{dt} =0 ) over the course of the reaction (that is while there is some substrate to turn into product).
    • Q4.1: Show that if this is verified then the product is created at a constant rate and that substrate disappears at the same rate.
    • Q4.2: Show that this rate is proportional to a fraction  \frac{[S]_{t=0}}{[S]_{t=0}+K_{m}}
      • What is the enzymatic constant Km ?
      • How does the rate of production of P depend on the initial concentration of enzyme [E]t = 0 ? What does this mean in practice?
    • Q4.3: Sketch the evolution with time of the compounds
    • Q4.4: Given the simulations you have run, when do you think MM is justified? (Bonus points if you can show this analytically)
  • Question 5: The gradient at the origin  \frac{d[P]}{dt}_{t=0} is very useful to estimate the enzymatic constant Km from data
    • Q5.1: How do you measure d[P]/dt from a simulation graph or experimental data?
    • Q5.2: Now imagine that you have conducted a set of enzymatic experiments where the initial concentration of enzyme was kept constant and the amount of substrate was made to vary over a large range of concentrations. Often you can not directly record the evolution with time of the product P: P(t), but let us say you have been able to convert the experimental data into a reliable estimate of P(t). How would you estimate Km from these data?



Part IV: Additional Resources


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