title: PDE/ODE modeling and simulation to determine the role of diffusion in long-term and -range cellular signaling.
abstract: BACKGROUND
  We study the relevance of diffusion for the dynamics of signaling pathways. Mathematical modeling of cellular diffusion leads to a coupled system of differential equations with Robin boundary conditions which requires a substantial knowledge in mathematical theory. Using our new developed analytical and numerical techniques together with modern experiments, we analyze and quantify various types of diffusive effects in intra- and inter-cellular signaling. The complexity of these models necessitates suitable numerical methods to perform the simulations precisely and within an acceptable period of time.
  
  
  METHODS
  The numerical methods comprise a Galerkin finite element space discretization, an adaptive time stepping scheme and either an iterative operator splitting method or fully coupled multilevel algorithm as solver.
  
  
  RESULTS
  The simulation outcome allows us to analyze different biological aspects. On the scale of a single cell, we showed the high cytoplasmic concentration gradients in irregular geometries. We found an 11 % maximum relative total STAT5-concentration variation in a fibroblast and a 70 % maximum relative pSTAT5-concentration variation in a fibroblast with an irregular cell shape. For pSMAD2 the maximum relative variation was 18 % in a hepatocyte with a box shape and 70 % in an irregular geometry. This result can be also obtained in a cell with a box shape if the molecules diffuse slowly (with D=1 μm(2)/s instead of D=15 μm(2)/s). On a scale of cell system in the lymph node, our simulations showed an inhomogeneous IL-2 pattern with an amount over three orders of magnitude (10(-3)-1 pM) and high gradients in face of its fast diffusivity. We observed that 20 out of 125 cells were activated after 9 h and 33 in the steady state. Our in-silico experiments showed that the insertion of 31 regulatory T cells in our cell system can completely downregulate the signal.
  
  
  CONCLUSIONS
  We quantify the concentration gradients evolving from the diffusion of the molecules in several signaling pathways. For intracellular signaling pathways with nuclear accumulation the size of cytoplasmic gradients does not indicate the change in gene expression which has to be analyzed separately in future. For intercellular signaling the high cytokine concentration gradients play an essential role in the regulation of the molecular mechanism of the immune response. Furthermore, our simulation results can give the information on which signaling pathway diffusion may play a role. We conclude that a PDE model has to be considered for cells with an irregular shape or for slow diffusing molecules. Also the high gradients inside a cell or in a cell system can play an essential role in the regulation of the molecular mechanisms.
authors.affiliation: Department of Applied Mathematics, Im Neuenheimer Feld 294, Heidelberg, Germany.
authors.forename: Elfriede
authors.initials: E
authors.lastname: Friedmann
keywords: Cellular signaling; Numerical simulation; Reaction-diffusion systems; Systems of coupled differential equations