The modelling framework I used for the phototransduction modelling is a toolbox calledIQMtools in MATLAB (IQMtoolbox Website). MATLAB stands for Matrix Laboratory and is a programme that can be used for various kinds of numerical calculations and data analysis. For the analyses in this thesis, I used MATLAB version R2017b unless otherwise specified. Simulations were run on a desktop computer or on the CARL high performance computing cluster1 of the university of Oldenburg.
IQMtools is a toolbox specifically developed for modelling of biochemical and biomedical systems, which offers a vast treasure of methods ready for use. I used IQMtools version 1.2.2.2 for the analyses presented in the following.
The original Dell’Orco 2009 and Invergo 2014 models were implemented in the predecessor of IQMtools called SBTOOLBOX2 (Schmidt and Jirstrand, 2005). I converted these models as well as the simulation scripts to reproduce the figures from (Dell’Orco et al., 2009) and (Invergo et al., 2014) to the newer IQMtools syntax.
I then used the IQMmodel structure to develop the different models presented in this thesis. In the IQMmodel structure, a model is defined using either
differen-1See https://uol.de/en/school5/sc/high-perfomance-computing/hpc-facilities/
carl for the cluster’s specifications.
tial equations or reaction equations. The initial conditions, parameter values and additional variables are also stored in the model. Models of this structure can be combined with IQMexperiments, where parameter or state values can be altered at defined points in time.
This is an excerpt from the Dell’Orco 2009 model in the IQMtools syntax:
********** MODEL STATE INFORMATION R(0) = 3.6e9
PDE(0) = 1.335e7 Gt(0) = 3.6e8 ...
********** MODEL PARAMETERS Rtot = 3.6e9
PDEtot = 1.335e7 Gtot = 3.6e8 ...
********** MODEL VARIABLES
E = PDE a Ga GTP + 2 * Ga GTP a PDE a Ga GTP
J = 2/(2 + fCa)*power((cGMP/cGMPdark),ncg)*Jdark + fCa/(fCa + 2)*...
(Ca2 free-Ca2 0)/(Ca2dark-Ca2 0)*Jdark deltaJ = Jdark - J
...
********** MODEL REACTIONS R => R0 : v r1
vf = stimulus * R/Rtot R0 + RK <=> R0 RKpre : v r2 0
vf = kRK1 0 * RK * R0 vr = kRK2 * R0 RKpre R1 + RK <=> R1 RKpre : v r2 1
vf = kRK1 1 * RK * R1 vr = kRK2 * R1 RKpre
First, the initial conditions for all states (i.e. explicitly simulated molecular species) are defined underModel State Information. The initial conditions only need to be explicitly written down when they are not zero - otherwise, zero is the default. Next, parameters are declared in Model Parameters. Those can be, for example, total molecule numbers for given species or the reaction rate constants.
In Model Variables, the variables of the model are given as formulas - here for example the effector and the circulating current, or the reaction rate constants that
depend on the phosphorylation state of rhodopsin. Finally, in Model Reactions, the reactions are defined in terms of their reactants and products as well as the forward and backward rates. All the parameters and variables that play into the rates need to be defined in the appropriate sections above.
After defining the model, simulations can be performed. The necessary functions for this are also supplied by the IQMtools package.
For the deterministic simulations, I used IQMsimulateand IQMPsimulate, which carry out a simulation of the given model for a specified time vector. They are based on MATLAB’s own differential equation solver ode23s, which uses an im-plicit Runge-Kutta approach with adaptive step sizes. For an explanation of this method, please refer to section 2.2.3.
The difference between the two functions lies in the speed: while IQMsimulate carries out the simulation within MATLAB, IQMPsimulate first automatically converts the simulation to C, which executes more rapidly, and then translates the result back into the MATLAB environment.
For stochastic simulations, I used IQMstochsim, which uses a similar approach to the Gillespie algorithm as explained in section 2.2.4 by simulating the chemical Master equation.
The models can be combined with experiments to simulate specific conditions, for example stimulus paradigms or genetic modification. Below is one example for an experiment:
********** EXPERIMENT INITIAL PARAMETER AND STATE SETTINGS flashBG = 100
flashMag = 900 flashDur = 0.01 flashDel = 10
This experiment defines a stimulus. It consists of a background of 100 (flashBG), which translates to 233 photons/µm2s using the collecting area of 0.43µm2, as well as a flash. The flash starts at t= 10 s (flashDel), lasts for 0.01 s, and has a mag-nitude of 2093 photons/µm2 (again dividing by the collecting area).
This is another possible experiment:
********** EXPERIMENT INITIAL PARAMETER AND STATE SETTINGS
RK(0) = 2e5 RecR Ca RK(0) = 0 RecR Ca(0) = 0 RecT(0) = 0
********** EXPERIMENT PARAMETER CHANGES
********** EXPERIMENT STATE CHANGES
This experiment implements a knockout of recoverin. In a genetic knockout, the corresponding species is not produced by the animal and therefore also needs to be removed completely in the model to imitate the condition. Thus, all possible species containing recoverin are set to zero as an initial condition: the two differ-ent recoverin states with and without calcium RecT and RecR·Ca2+ as well as the calcium-bound form that also binds the rhodopsin kinase RecR·Ca2+·RK. Since this removes some of the rhodopsin kinase from the model, the initial condition for the rhodopsin kinase has to be adapted accordingly.
Please note that it is possible to define different initial settings for parameters, variables and states using the experiment, but also parameter or state changes that occur during the simulation time, as can be seen in the lower example.
After defining the experiment, model and experiment can be combined for simu-lation using the command
model exp = IQMmergemodexp(model, experiment);
An alternative way to define changed initial conditions is by using the func-tion IQMinitialconditions or indeed by directly changing the initial conditions within the model, with the following alternative syntaxes:
model = IQMinitialconditions(model, ’R0’, 1);
model.states(stateindexIQM(model, ’R0’)).initialCondition = 1;
In both cases, model is modified so that the initial condition for R0 - unphospho-rylated activated rhodopsin - is one molecule. This is one option to set the initial conditions for the single photon response.
Using the models as specified earlier and the tools described here, we have all the necessary tools to perform deterministic and stochastic simulations of the photo-transduction cascade in different conditions.
In this chapter, I am explaining the new developments of the deterministic models that I carried out during my PhD. I show results from the new modelling, which are compared to current and novel experimental data.
The main deterministic model builds up on the mammalian phototransduction model from Invergo et al. (Invergo et al., 2014), which in turn builds on the amphibian phototransduction model developed by Dell’Orco et al. (Dell’Orco et al., 2009).