IFNB1-DDX58 Feedback Loop Cell-to-Cell Variability Figure 8. Processes, Descriptions and Rates. The processes, descriptions and rates for each of the six reactions in the stochastic simulations. doi:10.1371/journal.pone.0016614.g008 Figure 8 (w indicates a source or a sink, used to denote creation or degradation of molecules, respectively): The values used for the rate constants are based on experimental results, and were changed to fit the experimental data. Specifically, in the simulation results shown in the paper the 1 1 1 1 valuesb~75, c~7000, dDDX ~104sec, dIFN ~104sec, and 1 dRIG~104sec were used. In order to account for some of the cell-to-cell variation, the transcription rate constants for each cell in the simulation were chosen from a Gaussian distribution around 1 1 the values KDDX ~20sec, KRIG~20sec, with standard deviations of 0.01, and values below 0.01 and above 0.05 were ignored and re-sampled. Inter-cellular Diffusion Simulation The inter-cellular medium is simulated by a Monte-Carlo simulation on a 2d square lattice. The simulations were performed in 2D to reduce computational overhead and because, as discussed below, the conclusions are applicable to the 3D case. The inter- cellular simulation follows the diffusion of interferon molecules in the medium, which is described by the diffusion equation. Thus the average displacement of each molecule from its origin increases as the square root of time. This result holds for 3d systems, as well as for 2d ones. The difference between 2D and 3D can be accounted for by a change in the value of the diffusion coefficient. In order to preserve the average time for the diffusion to a neighboring cell, we keep the average distance between neighboring cells equal to the one found in 3d. Besides the distance between nearest neighbors, the interaction between secreted interferon molecules and a neighboring cell also depends on the probability of binding to cell-surface receptors, and the dimensionality of the system. By changing the cell surface binding probability, we can replace the 3d system by an equivalent 2d system, in which the interaction of neighboring cells is unchanged. We recognize that there are differences between the 2D and 3D systems. For example, in random walks in 2d as compared to 3d, the probability of a molecule to return to its origin is higher. Therefore, in our simulations, the amount of paracrine signaling is underestimated compared to autocrine signaling. However, this difference is immaterial for the questions addressed and for the conclusions of the present study. The 2d lattice is simulated using periodic boundary conditions, and its grid size is set to be the diameter of a single cell, denoted L, which is approximately 30 mm. Cells are distributed inside the matrix randomly with a density that ensures the correct average distance to nearest neighbors (as explained above). Each cell has a probability of 50% of being infected or uninfected, in accordance with the experimental MOI of 0.5. Uninfected cells do not transcribe interferon. At each grid point the simulation follows the amount of interferon molecules, denoted I, and uses the number of free and bound interferon receptors obtained from each cell, denoted F, and B, respectively. Grid squares that do not contain a cell have zero F and B at all times. Receptor binding and unbinding by IFNb are adapted from work described in reference [30]. At each time step interferon molecules in lattice squares containing cells may bind to free surface receptors, while bound receptors may unbind. The binding rate is given by KON :F :I , 1:25 where KON ~ is the binding rate constant. The unbinding 106sec 1 rate constant was found to be KOFF ~ . The rates per 103sec receptor are slow enough so that for the chosen time-step we have KON :I :Dtvv1 and KOFF :Dtvv1. Therefore we approximate the binding probability of each free receptor in a time step by KON :I :Dt, and the unbinding probability for every bound receptor by KOFF :Dt. The distribution of molecules in a diffusion-governed system with initial condition of all the molecules concentrated at the origin is expressed by a Gaussian function f ðx,y,tÞ~ 1 e{ðx2zy2Þ=4Dt, allowing calculation of the time by which 4ptD half the molecules pass a distance of a cell radius, giving approximately 9 seconds (solving the equation Ð{L=L2=2 f (j,g,t)djdg~0:5, with L = 30 mm). We use this time interval as the basic time-step in the simulation, during which each interferon molecule has a probability of 50% to pass to a neighboring lattice square. As a result, in each simulation step each interferon molecule in each cell has a probability of 50% of moving to one of the four neighboring cells. This ensures a random walk behavior for a small number of molecules, and a diffusion behavior for large numbers of molecules. PLoS ONE | www.plosone.org 10 February 2011 | Volume 6 | Issue 2 | e16614