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

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
[math]\displaystyle{ A \xrightarrow{k_{1}} B \xrightarrow{k_{2}} C }[/math]	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: [math]\displaystyle{ \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} }[/math]	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
[math]\displaystyle{ E + S \begin{matrix} k_1 \\ \longrightarrow \\ \longleftarrow \\ k_{2} \end{matrix} ES \begin{matrix} k_3 \\ \longrightarrow\\ \end{matrix} E + P }[/math]	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: [math]\displaystyle{ \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} }[/math]

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:
  • [math]\displaystyle{ k_{1}=10^8 M^{-1} s^{-1} }[/math] ; [math]\displaystyle{ k_{2}= 100 s^{-1} }[/math] ; [math]\displaystyle{ k_{3}= 10^{-1} s^{-1} }[/math]
  • Initial Conditions: [math]\displaystyle{ [E]_{t=0}= 10^{-7} M }[/math] ; [math]\displaystyle{ [S]_{t=0}=10^{-5} M }[/math] ; [math]\displaystyle{ [P]_{t=0}=0 }[/math]
  • Open the Simulation Panel, set Time=150, NbPoints=1000.
• 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 [math]\displaystyle{ [S]_{t=0} }[/math].

• To investigate the influence of [math]\displaystyle{ [S]_{t=0} }[/math], we will make it vary between [math]\displaystyle{ 10^{-9} M }[/math] and 10^{-4} M</math> . The remaining parameters are kept as:
  • [math]\displaystyle{ k_{1}=10^8 M^{-1} s^{-1} }[/math] ; [math]\displaystyle{ k_{2}= 100 s^{-1} }[/math] ; [math]\displaystyle{ k_{3}= 10^{-1} s^{-1} }[/math]
  • Initial Conditions : [math]\displaystyle{ [E]_{t=0}= 10^{-7} M }[/math] ; [math]\displaystyle{ [P]_{t=0}=0 }[/math]
• 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 products 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 compounds 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 ( [math]\displaystyle{ \frac{d[ES]}{dt} =0 }[/math] ) 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 [math]\displaystyle{ \frac{[S]_{t=0}}{[S]_{t=0}+K_{m}} }[/math] ; what is the enzymatic constant [math]\displaystyle{ K_{m} }[/math] ?
  • 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 [math]\displaystyle{ \frac{d[P]}{dt}_{t=0} }[/math] is very useful to estimate the enzymatic constant [math]\displaystyle{ K_{m} }[/math] 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 [math]\displaystyle{ K_{m} }[/math] from these data?

Part IV: Additional Resources

• Law of Mass Action
  • Law of Mass Action (Wolfram's site)
  • Law of Mass Action in Chemistry (Wikipedia site)
  • Rate Law (from Wikipedia)

• Enzyme Kynetics
  • Michaelis-Menten_kinetics
  • Michaelis-Menten Formula Derivation
  • Steady State Approximation (from Wikipedia)