Team:Calgary/Modelling/Method

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University of Calgary

MODELLING INDEX

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Membrane Computing Modelling

Differential Equation Modelling

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. More more information about sundials within Simbiology, please visit the Matlab website.

When sundials solver is selected, the program selects one of the 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
***Information provided by Matlab website

Rationale

The following initial conditions and k constants are estimated values. The modelling team was unable to find much literature regarding rate constants and initial conditions for our signalling system. With that being said, this model still serves a purpose in understanding the signalling pathway. We were able to demonstrate various trends when AI-2 and expression of specific proteins were adjusted. The output when inputs were changed showed significant trends that may be taken into consideration when the laboratory work is performed. This model provides a good starting place for understanding of the basic signalling system. Further research and work will be dedicated to provide a more accurate model. The main focus for this team is to find significant trends that will effect the AI-2 signalling cascade.

During the past few months, the modelling team developed the AI-2 signalling circuit within Simbiology. The following is a diagram that simplifies the AI-2 signalling cascade with interactions of different species:

AI-2 Signalling System Map Developed

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

The following parameters were assigned values based on the assumptions made from the knowledge of the system. The values are choosen on the basis of a comparison to each other. 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 equilibrium 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. 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	See Sigma54:LuxO:p:Pqrr4
Pqrr4	0.025	See Sigma54:LuxO:p:Pqrr4
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	1.0E-6	??
kPhosO	1.0E-6	??
kTranscription	0.1056	Transcription is fast and dependent on the size of the gene.
kTranslation	0.0017	The translation of mRNA to GFP is a very slow process due to the many processes needed and present to build GPF.
kProtDegrad	2.8756E-4	The value was calculated from the estimated half life of the GFP in the gene circuit.
kAI2bind	1.0	LuxP is a receptor for AI-2 therfore the binding is seen to be a fast process due to its specificity.
kAI2unbind	0.25	The unbinding of AI-2 from the receptor is seen as a slower process than the binding of AI-2 to the receptor.
kPQphosphatase	0.3	The dephosphorylation reaction is relatively fast due to the specificity of the protein LuxQ to LuxU:p.
kNSPU	0.09	This constant refers to the dephosphorylation of LuxU:p. This dephosphorylation is carried out by non-specific phosphatase.
kNSPO	0.09	The dephosphorylation of LuxO:p is carried out by non-specific phosphatase which is similar to the situation with kNSPU. For this reason , the two parameter values are equal.
kPqrr4Sig54unbind	0.002	This value is much smaller than the binding value because it is assumed that sigma54 stays bound to Pqrr4 most of the time.
kPqrr4Sig54bind	0.2	This is a fast reaction due to the complexes strong binding affinity.
kOPqrr4Unbind	1.0	The binding and unbinding is assumed to be a fast reaction , having equal probability of staying in both states.
kOPqrr4bind	1.0	See kOPqrr4Unbind.
kRNAdegrad	0.0048	???

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