# 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
$\displaystyle{ A \xrightarrow{k_{1}} B \xrightarrow{k_{2}} C }$
• 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'

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

\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} }
• 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.
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
$\displaystyle{ E + S \begin{matrix} k_1 \\ \longrightarrow \\ \longleftarrow \\ k_{2} \end{matrix} ES \begin{matrix} k_3 \\ \longrightarrow\\ \end{matrix} E + P }$
• 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:

\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} }

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:
• $\displaystyle{ k_{1}=10^8 M^{-1} s^{-1} }$ ; $\displaystyle{ k_{2}= 100 s^{-1} }$ ; $\displaystyle{ k_{3}= 10^{-1} s^{-1} }$
• Initial Conditions: $\displaystyle{ [E]_{t=0}= 10^{-7} M }$ ; $\displaystyle{ [S]_{t=0}=10^{-5} M }$ ; $\displaystyle{ [P]_{t=0}=0 }$
• 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 $\displaystyle{ [S]_{t=0} }$.

• To investigate the influence of $\displaystyle{ [S]_{t=0} }$, we will make it vary between $\displaystyle{ 10^{-9} M }$ and 10^{-4} M[/itex] . The remaining parameters are kept as:
• $\displaystyle{ k_{1}=10^8 M^{-1} s^{-1} }$ ; $\displaystyle{ k_{2}= 100 s^{-1} }$ ; $\displaystyle{ k_{3}= 10^{-1} s^{-1} }$
• Initial Conditions : $\displaystyle{ [E]_{t=0}= 10^{-7} M }$ ; $\displaystyle{ [P]_{t=0}=0 }$
• 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

• 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 ( $\displaystyle{ \frac{d[ES]}{dt} =0 }$ ) 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 $\displaystyle{ \frac{[S]_{t=0}}{[S]_{t=0}+K_{m}} }$
• What is the enzymatic constant $\displaystyle{ K_{m} }$ ?
• How does the rate of production of P depend on the initial concentration of enzyme $\displaystyle{ [E]_{t=0} }$ ? 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 $\displaystyle{ \frac{d[P]}{dt}_{t=0} }$ is very useful to estimate the enzymatic constant $\displaystyle{ K_{m} }$ 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 $\displaystyle{ K_{m} }$ from these data?