Team:Calgary/Modelling/Method

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University of Calgary

UNIVERSITY OF CALGARY



MODELLING INDEX
Overview

Membrane Computing Modelling
Differential Equation Modelling

A TOUR OF THE UNIVERSITY OF CALGARY iGEM TEAM


We've reached modelling, the fifth stop on our tour! We've looked in to two different methods of modelling our system: Differential Equation Based Modelling and Membrane Computing. Here, you can explore the similarities and differences, as well as the functions of each method. As well, you can find the results of our characterization of the signalling pathway. Once you're done, we'll move on to the Second Life component of the project HERE.


DIFFERENTIAL EQUATIONS MODELLING METHODS
The simbiology interface from Matlab was used to simulate the differential equations model. Chemical Kinetic equations were used to build the model for simulation.


Fig : The Reaction of Species A with B to produce C and D



Fig : The Chemical Kinetic Rate Equation


k is the kinetic rate constant. The size of k will determine the speed of the reaction. A smaller value of k will produce a slow reaction rate while a larger value of k will produce a fast reaction rate.

[A] is the amount of reactant A present.

The simulations were run for 50000 seconds . It was considered to be enough time for the system to reach equilibrium after disturbance.

Sundial Solver The sundial solver (SUNDIALS) was developed so that robust time integrators and non-linear solvers can be easily combined with already existing simulation codes. Minimal information from user is required and this solver allow users to easily supply their own data structures. The Sundials solvers are part of a third-party package developed at Lawrence Livermore National Laboratory. Built-in ordinary differential equation (ODE) solvers (ode45 and ode15s) are also part of the interface.

When sundials solver is selected, the program selects one of teh two sundials solvers that suits your model: CVODE or IDA. CVODE is used for systems of ODEs (stiff or nonstiff) and this type of solver is usually used for a model that has no algebraic rules. IDA is a differential-algebraic equation (DAE) solver and it is usually used when there is one more algebraic rules. Since our model incorporates an event (the addition of autoinducer-II (AI-2)), this type of solver was used in our model. More information can be found here: https://computation.llnl.gov/casc/sundials/description/description.html

The Reactions
The system was represented by the following reactions. The reactions with double headed arrows have two rate constants(forward/ reverse rate constant). All reactions were assumed to be elementary reactions.

Parameter Rationale
Table: Initial Values of the Species in the System

Species Initial Value Rationale
AI-2 0 Initially the amount of AI-2 is constant at 0. After an equilibruim is established variable amounts of AI-2 are added at different simulations.

LuxPQ 10 The amount of LuxPQ varies depending on the simulation run.
AI-2:LuxPQ 0 This value is kept at time = 0 because the initial concentration of AI-2 is 0.

LuxU:p 2 ----
LuxU 1000 There is a lot of this species present in the cell in nature. To signify plenty a value of 1000 is assigned.
LuxO:p 2 Equal amounts of LuxO:p and LuxU:p was considered in the model because LuxU:p phosphorylates LuxO . The phosphorylation reaction is considered to be a fast reaction therefore there are equal amounts of the two proteins present.
LuxO 1000 There are equal amounts of LuxO and LuxU present since they are encoded within the same operon in the cell.
p 10.0658 We assume that there is enough p is the environment that it doesn’t become a limiting factor. For that reason we assign p as a constant value in simbiology. (It doesn’t really matter that the initial amount is presented as a comparatively small number in this case. )

sigma54 0.14183 This amount is kept at a constant value to ensure that this value does not become a limiting factor.
sigma54:LuxO:p:Pqrr4 0.63 There is only 1 copy of Pqrr4 present in each cell . since in the reaction equations Pqrr4 is shared between 2 other equations we decided to break the concentration of Pqrr4 between 3 species: sigma54:LuxO:p:Pqrr4 , Pqrr4 , sigma54:Pqrr4 . The initial values of the three species add up to one. The fractions of the Pqrr4 combination species are weighted differently . Since the Pqrr4 promotor stays on most of the time we decided the sigma54:LuxO:p:Pqrr4 complex should recieve the most weight. Pqrr4 is assumed to stay unbound from any complex for the least amount of time therefore Pqrr4 initial amount is the smallest.

Sigma54:Pqrr4 0.345
Pqrr4 0.025
GFP 0 The model assumes that initially we have no GFP present . The simulation is allowed to run till the protein reaches equilibrium . The AI-2 is added after equilibrium conditions and a drop in GFP levels is observed.
mRNA 0 The justification for the initial value of mRNA is the same as GFP

Table: The Kinetic Rate Constant Values

Rate Constants Constant Value Rationale
kPhosU    
kPhosO    
kTranscription    
kTranslation    
kProtDegrad    
kAI2bind    
kAI2unbind    
kPQphosphatase    
kNSPU    
kNSPO    
kPqrr4Sig54unbind    
kPqrr4Sig54bind    
kOPqrr4Unbind    
kOPqrr4bind    
kRNAdegrad    

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