The specification of SBML Level 1 is freely available from http://www.sbml.org/

Recent observations show that the single-cell response of p53 to ionizing radiation (IR) is ''digital'' in that it is the number of oscillations rather than the amplitude of p53 that shows dependence on the radiation dose. We present a model of this phenomenon. In our model, double-strand break (DSB) sites induced by IR interact with a limiting pool of DNA repair proteins, forming DSB-protein complexes at DNA damage foci. The persisting complexes are sensed by ataxia telangiectasia mutated (ATM), a protein kinase that activates p53 once it is phosphorylated by DNA damage. The ATM-sensing module switches on or off the downstream p53 oscillator, consisting of a feedback loop formed by p53 and its negative regulator, Mdm2. In agreement with experiments, our simulations show that by assuming stochasticity in the initial number of DSBs and the DNA repair process, p53 and Mdm2 exhibit a coordinated oscillatory dynamics upon IR stimulation in single cells, with a stochastic number of oscillations whose mean increases with IR dose. The damped oscillations previously observed in cell populations can be explained as the aggregate behavior of single cells.DNA damage response ͉ mathematical model of p53 ͉ p53 regulation ͉ p53 pathway C ells under stresses such as DNA damage, hypoxia, and aberrant oncogene signals trigger their internal self-defense machinery. One critical response is the activation of the tumor suppressor protein p53, which transcribes genes that induce cell cycle arrest, DNA repair, and apoptosis (1-4). A central node in the p53 network is the Mdm2 protein, the product of one of the p53 target genes and a negative regulator of p53. The negative feedback loop formed by p53 and Mdm2 can produce oscillatory dynamics. Indeed, damped oscillations of p53 and Mdm2 protein level have been observed upon ionizing radiation (IR)-induced DNA damage in cell populations (5). Intriguingly, recent in vivo fluorescence measurements in individual cells revealed that in response to IR, these two proteins exhibit a ''digital'' response that produces discrete pulses of p53 and Mdm2. The average height and duration of these pulses are fixed, whereas the mean number increases with the strength of DNA damage (6).Several models have been proposed (5,7,8) to explain the damped oscillations of p53 in cell populations. However, these modeling efforts did not explore sustained pulses as found in single-cell responses and did not attempt to characterize the signaling between DNA damage and the activation of the p53 oscillatory response.In this study, we present a model for the digital, undamped oscillatory p53 activity elicited by IR at the single-cell level consisting of three subsystems: a DNA damage repair module, an ataxia telangiectasia mutated (ATM) switch, and the p53-Mdm2 oscillator. We investigate the controlling role of ATM to set a threshold level of DNA damage during the radiation response, as suggested by growing biochemical evidence (9-12). Finally, by adding stochasticity to selected model parameters, we replicate the va...

Based on realistic mechanisms of Ca2+ buffering that include both stationary and mobile buffers, we derive and investigate models of Ca2+ diffusion in the presence of rapid buffers. We obtain a single transport equation for Ca2+ that contains the effects caused by both stationary and mobile buffers. For stationary buffers alone, we obtain an expression for the effective diffusion constant of Ca2+ that depends on local Ca2+ concentrations. Mobile buffers, such as fura-2, BAPTA, or small endogenous proteins, give rise to a transport equation that is no longer strictly diffusive. Calculations are presented to show that these effects can modify greatly the manner and rate at which Ca2+ diffuses in cells, and we compare these results with recent measurements by Allbritton et al. (1992). As a prelude to work on Ca2+ waves, we use a simplified version of our model of the activation and inhibition of the IP3 receptor Ca2+ channel in the ER membrane to illustrate the way in which Ca2+ buffering can affect both the amplitude and existence of Ca2+ oscillations.

Systems biology has experienced dramatic growth in the number, size, and complexity of computational models. To reproduce simulation results and reuse models, researchers must exchange unambiguous model descriptions. We review the latest edition of the Systems Biology Markup Language (SBML), a format designed for this purpose. A community of modelers and software authors developed SBML Level 3 over the past decade. Its modular form consists of a core suited to representing reaction‐based models and packages that extend the core with features suited to other model types including constraint‐based models, reaction‐diffusion models, logical network models, and rule‐based models. The format leverages two decades of SBML and a rich software ecosystem that transformed how systems biologists build and interact with models. More recently, the rise of multiscale models of whole cells and organs, and new data sources such as single‐cell measurements and live imaging, has precipitated new ways of integrating data with models. We provide our perspectives on the challenges presented by these developments and how SBML Level 3 provides the foundation needed to support this evolution.

No abstract