Figure 1.
Dictyostelium cAMP Oscillations
(A) The model of [5] for the network underlying cAMP oscillations in Dictyostelium. The nominal parameter values for the model are taken from [6,9] and are given by: k1 = 2.0 min−1, k2 = 0.9 μM−1 min−1, k3 = 2.5 min−1, k4 = 1.5 min−1, k5 = 0.6 min−1, k6 = 0.8 μM−1 min−1, k7 = 1.0 μM min−1, k8 = 1.3 μM−1 min−1 , k9 = 0.3 min−1, k10 = 0.8 μM−1 min−1, k11 = 0.7 min−1, k12 = 4.9 min−1, k13 = 23.0 min−1, and k14 = 4.5 min−1. A perturbation of magnitude 2% in the model parameters which causes the oscillations to cease is given by [10]: k1 = 1.9600, k2 = 0.8820, k3 = 2.5500, k4 = 1.5300, k5 = 0.5880, k6 = 0.8160, k7 = 1.0200, k8 = 1.2740, k9 = 0.3060, k10 = 0.8160, k11 = 0.6860, k12 = 4.9980, k13 = 22.5400, and k14 = 4.5900.
(B) With the above perturbation in the parameter values, the deterministic model stops oscillating. The stochastic model, on the other hand, continues to exhibit stable and robust oscillations.
Figure 2.
Deterministic and Stochastic Simulations for the Same Worst Case Parameter Combinations Are Performed
The numbers of molecules are sampled with a 0.1 s interval for 10 h, and the distribution of each molecular species is compared. To avoid influences from the initial transient response, only the samples obtained after 5 h are considered when plotting the distributions. The inset of (E) is the distribution for the external cAMP. The noise effect is clearly significant in terms of generating oscillations.
Figure 3.
Deterministic Model: Robustness Analysis of the Period and Amplitude of the Internal cAMP Oscillations with Respect to Perturbations in the Model Parameters and Initial Conditions
The first row shows the distribution in the period of the deterministic model for one cell with 5%, 10%, and 20% perturbations, and the second row shows the amplitude distribution. The peak bar at 20 min for the period distributions represents the number of cells that are not oscillating. The proportion of cells that are not oscillating increases from 2% to 25% as the size of the perturbation increases. The distributions of the amplitudes also show a similar tendency, i.e., the mean value decreases and the standard deviation increases as the magnitude of the perturbation increases. Each plot is the result of 100 simulations for different random samples of the model parameters, the cell volume, and initial conditions using a uniform distribution about the nominal values.
Figure 4.
Stochastic Model: Robustness Analysis of the Period and Amplitude of the Internal cAMP Oscillations with Respect to Perturbations in the Model Parameters and Initial Conditions
The first row shows the period distribution of the stochastic model for one cell with 5%, 10%, and 20% perturbations, and the second row shows the amplitude distribution. The peak bar at 20 min for the period distributions represents the number of cells that are not oscillating. The proportion of cells that is not oscillating increases from 0% to 14% as the size of the perturbation increases, and is always significantly smaller than the proportion of non-oscillating cells found in the deterministic model. The standard deviations of the amplitudes are also much smaller, for the same magnitude of perturbation, than those seen in the deterministic model. Each plot is the result of 100 simulations for different random samples of the model parameters, the cell volume, and initial conditions using a uniform distribution about the nominal values.
Figure 5.
Synchronisation Is Realised through Diffusion of External cAMP
(A) Shows the synchronisation mechanism for the case of three interacting Dictyostelium cells. Each cell has a different set of kinetic constants. kij is the kj in the Laub-Loomis model for the i-th cell.
(B) For twenty individual cells with no interaction and a 10% level of variation in the initial conditions and the kinetic constants between the cells, the internal cAMP oscillations are completely out of phase with each other.
(C) For the extended model incorporating the diffusion mechanism, with the same level of variation between the cells, the oscillations are synchronised in less than 10 min.
(D) Even for a 20% level of variation between the cells, the extended model shows highly synchronised oscillations.
Figure 6.
Extended Stochastic Model (Five Cells): Robustness Analysis of the Period and Amplitude of the Internal cAMP Oscillations with Respect to Perturbations in the Model Parameters and Initial Conditions
The first row shows the period distribution of the stochastic model for three cells with 5%, 10%, and 20% perturbations, and the second row shows the amplitude distribution. The peak bar at 20 min for the period distributions represents the total number of cells that are not oscillating. The proportion of non-oscillating cells increases from 0% to 12% as the size of the perturbation increases, which is smaller than the proportion seen in either the deterministic or stochastic single cell models. The distributions of the amplitudes show a similar tendency, i.e., the mean decreases and the standard deviation increases as the magnitude of the perturbation increases. Each plot is the result of 100 simulations for different random samples of the model parameters, the cell volume, and initial conditions using a uniform distribution about the nominal values.
Figure 7.
Extended Stochastic Model (Ten Cells): Robustness Analysis of the Period and Amplitude of the Internal cAMP Oscillations with Respect to Perturbations in the Model Parameters and Initial Conditions
The first row shows the period distribution of the stochastic model for three cells with 5%, 10%, and 20% perturbations, and the second row shows the amplitude distribution. The peak bar at 20 min for the period distributions represents the total number of cells, which are not oscillating. The proportion of non-oscillating cells increases from 0% to 5% as the size of the perturbation increases, which is much smaller than the proportion seen in all other cases. The distributions of the amplitudes show a similar tendency, i.e., the mean decreases and the standard deviation increases as the magnitude of the perturbation increases. Each plot is the result of 100 simulations for different random samples of the model parameters, the cell volume, and initial conditions using a uniform distribution about the nominal values.