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JK and DGB conceived the initial idea for the study. JK designed and performed the numerical experiments. IP checked the mathematical derivations. DGB and PHH provided biological interpretations of the results. All authors contributed to writing the paper.

The authors have declared that no competing interests exist.

Stable and robust oscillations in the concentration of adenosine 3′, 5′-cyclic monophosphate (cAMP) are observed during the aggregation phase of starvation-induced development in ^{2+} oscillations, etc.) can only be done reliably by using stochastic simulations, even in the case where molecular concentrations are very high.

^{5} cells ([

In [

The model, which is written as a set of nonlinear ordinary differential equations, exhibits spontaneous cAMP oscillations of the correct period and amplitude, and also reproduces the experimentally observed interactions of the MAP kinase ERK2 and protein kinase PKA with the cAMP oscillations [

In this paper, we attempt to resolve this apparent paradox by showing how a stochastic representation of the deterministic model proposed in [

The original model for cAMP oscillations given in [

(A) The model of [_{1} = 2.0 min^{−1}, _{2} = 0.9 ^{−1} min^{−1}, _{3} = 2.5 min^{−1}, _{4} = 1.5 min^{−1}, _{5} = 0.6 min^{−1}, _{6} = 0.8 ^{−1} min^{−1}, _{7} = 1.0 ^{−1}, _{8} = 1.3 ^{−1} min^{−1} , _{9} = 0.3 min^{−1}, _{10} = 0.8 ^{−1} min^{−1}, _{11} = 0.7 min^{−1}, _{12} = 4.9 min^{−1}, _{13} = 23.0 min^{−1}, and _{14} = 4.5 min^{−1}. A perturbation of magnitude 2% in the model parameters which causes the oscillations to cease is given by [_{1} = 1.9600, _{2} = 0.8820, _{3} = 2.5500, _{4} = 1.5300, _{5} = 0.5880, _{6} = 0.8160, _{7} = 1.0200, _{8} = 1.2740, _{9} = 0.3060, _{10} = 0.8160, _{11} = 0.6860, _{12} = 4.9980, _{13} = 22.5400, and _{14} = 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.

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.

To systematically compare the robustness properties of the two models, we generated 100 random samples of kinetic constants, the cell volume, and initial conditions from uniform distributions around the nominal values for several different uncertainty ranges. The period distributions of the deterministic model for three uncertainty ranges, i.e., 5%, 10%, and 20%, are shown in

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.

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.

One important mechanism, which is missing in the model of [

To investigate the effect of synchronisation on the robustness of cAMP oscillations in

(A) Shows the synchronisation mechanism for the case of three interacting ^{i}_{j}_{j}

(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.

Robustness analysis results for the extended model in the case of five and ten interacting cells are shown in

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.

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.

Note that for computational reasons the number of interacting cells considered in the above analysis was limited to ten. In nature, some 10^{5}

As well as resolving an apparent paradox concerning the robustness of a proposed model for cAMP oscillations in

The deterministic model for cAMP oscillations used in this study is taken from [

To transform the above ordinary differential equations into the corresponding stochastic model, the following fourteen chemical reactions are deduced [_{A} is Avogadro's number, 6.023 × 10^{23}, 10^{−6} is a multiplication factor due to the unit ^{−14} _{1} × CAR1 and _{2}/_{A}/^{−6} × ACA × PKA, respectively. The probabilities for all the other reactions are defined similarly. Based on these, the chemical master equation is obtained and solved using standard numerical routines [

To consider synchronisation between multiple cells, Equation 2 is extended under the assumption that the distance between cells is small enough that diffusion is fast and uniform. In this case, the above reactions for each individual cell just need to be augmented with one reaction that includes the effect of external cAMP emitted by all the other cells. Since the external cAMP diffuses fast and uniformly, the reaction involving _{13} is modified as follows:
_{c} − 1, _{c}, where cAMPe is the total number of external cAMP molecules emitted by all the interacting cells, _{c} is the total number of cells, ^{i}_{13} is the _{i}

Note that the diffusion constant, ^{−4} cm^{2}/s [^{5} cells/cm^{2} [^{2}/(6

To ensure a consistent procedure for checking the robustness of both the deterministic and stochastic models, the Monte-Carlo simulation technique is used. The kinetic constants are sampled uniformly from the following:
_{c} − 1, _{c} and _{j}_{δ}^{i}_{j}_{c} − 1, _{c}, where ^{i}^{i}_{cAMPi} is a uniformly distributed random number between −1 and +1. The sampling for the other molecules is defined similarly. The nominal initial value for each molecule is given by [_{c} − 1, _{c}, where ^{−14}

Although some of the nominal parameter values in the model were derived from (inherently noisy) biological data, others were tuned to values which generated the required oscillatory behaviour.

Thus, we have very little a priori information on the likely distributions of the parameters as a result of environmental variations and modelling uncertainty. In such cases, the uniform distribution is the standard choice for the type of statistical robustness analysis performed in this paper. Indeed, this is the approach adopted in several previous studies of robustness in biomolecular networks, [

The simulations for the deterministic model and the stochastic model are performed using the Runge-Kutta 5^{th}-order adaptive algorithm and the τ-leap complex algorithm [^{−4} and 5 × 10^{−5} respectively, which are implemented in the software Dizzy, version 1.11.4 [

From the simulations, the time series of the internal cAMP concentration is obtained with a sampling interval of 0.01 min from 0 to 200 min. Taking the Fourier transform using the fast Fourier transform command in MATLAB [

adenosine 3′, 5′-cyclic monophosphate