Transcriptional pulsing has been observed in both prokaryotes and eukaryotes and plays a crucial role in cell-to-cell variability of protein and mRNA numbers. An important issue is how the time constants associated with episodes of transcriptional bursting and mRNA and protein degradation rates lead to different cellular mRNA and protein distributions, starting from the transient regime leading to the steady state. We address this by deriving and then investigating the exact time-dependent solution of the master equation for a transcriptional pulsing model of mRNA distributions. We find a plethora of results. We show that, among others, bimodal and long-tailed ͑power-law͒ distributions occur in the steady state as the rate constants are varied over biologically significant time scales. Since steady state may not be reached experimentally we present results for the time evolution of the distributions. Because cellular behavior is determined by proteins, we also investigate the effect of the different mRNA distributions on the corresponding protein distributions using numerical simulations.

Immune responses to viral infection are stochastic processes, which initiate in a limited number of cells that then propagate the response. A key component of the response to viral infection entails the synthesis and secretion of type I interferons (IFNs), including the early induction of the gene encoding IFN-β (Ifnb1). With single-cell analysis and mathematical modeling, we investigated the mechanisms underlying how increases in the amount of Ifnb1 mRNA per cell and in the numbers of cells expressing Ifnb1 calibrate the response to viral infection. We used single-cell, single-molecule assays to quantify the early induction of Ifnb1 expression (the Ifnb1 response) in human monocyte-derived dendritic cells infected with Newcastle disease virus, thus retaining the physiological stoichiometry of transcriptional regulators to both alleles of the Ifnb1 gene. We applied computational methods to extract the stochastic features that underlie the cell-to-cell variations in gene expression over time. Integration of simulations and experiments identified the role of paracrine signaling in increasing the number of cells that express Ifnb1 over time and in calibrating the immune response to viral infection.

To gain a deeper understanding of the transmission of visual signals from retina through the lateral geniculate nucleus (LGN), we have used a simple leaky integrate and-fire model to simulate a relay cell in the LGN. The simplicity of the model was motivated by two questions: (1) Can an LGN model that is driven by a retinal spike train recorded as synaptic ('S') potentials, but does not include a diverse array of ion channels, nor feedback inputs from the cortex, brainstem, and thalamic reticular nucleus, accurately simulate the LGN discharge on a spike-for-spike basis? (2) Are any special synaptic mechanisms, beyond simple summation of currents, necessary to model experimental recordings? We recorded cat relay cell responses to spatially homogeneous small or large spots, with luminance that was rapidly modulated in a pseudo-random fashion. Model parameters for each cell were optimized with a Simplex algorithm using a short segment of the recording. The model was then tested on a much longer, distinct data set consisting of responses to numerous repetitions of the noisy stimulus. For LGN cells that spiked in response to a sufficiently large fraction of retinal inputs, we found that this simplified model accurately predicted the firing times of LGN discharges. This suggests that modulations of the efficacy of the retino-geniculate synapse by pre-synaptic facilitation or depression are not necessary in order to account for the LGN responses generated by our stimuli, and that post-synaptic summation is sufficient.

The induction of interferon beta (IFNB1) is a key event in the antiviral immune response. We studied the role of transcriptional noise in the regulation of the IFNB1 locus in primary cultures of human dendritic cells (DCs), which are important ‘first responders’ to viral infection. In single cell assays, IFNB1 mRNA expression in virus-infected DCs showed much greater cell-to-cell variation than that of a housekeeping gene, another induced transcript and viral RNA. We determined the contribution of intrinsic noise by measuring the allelic origin of transcripts in each cell and found that intrinsic noise is a very significant part of total noise. We developed a stochastic model to investigate the underlying mechanisms. We propose that the surprisingly high levels of IFNB1 transcript noise originate from the complexity of IFNB1 enhanceosome formation, which leads to a range up to many minutes in the differences within each cell in the time of activation of each allele.

No abstract

Computer simulations of large genetic networks are often extremely time consuming because, in addition to the biologically interesting translation and transcription reactions, many less interesting reactions like DNA binding and dimerizations have to be simulated. It is desirable to use the fact that the latter occur on much faster timescales than the former to eliminate the fast and uninteresting reactions and to obtain effective models of the slow reactions only. We use three examples of self-regulatory networks to show that the usual reduction methods where one obtains a system of equations of the Hill type fail to capture the fluctuations that these networks exhibit due to the small number of molecules; moreover, they may even miss describing the behavior of the average number of proteins. We identify the inclusion of fast-varying variables in the effective description as the cause for the failure of the traditional schemes. We suggest a different effective description, which entails the introduction of an additional species, not present in the original networks, that is slowly varying. We show that this description allows for a very efficient simulation of the reduced system while retaining the correct fluctuations and behavior of the full system. This approach ought to be applicable to a wide range of genetic networks.

In the first few hours following Newcastle disease viral infection of human monocyte-derived dendritic cells, the induction of IFNB1 is extremely low and the secreted type I interferon response is below the limits of ELISA assay. However, many interferon-induced genes are activated at this time, for example DDX58 (RIGI), which in response to viral RNA induces IFNB1. We investigated whether the early induction of IFNBI in only a small percentage of infected cells leads to low level IFN secretion that then induces IFN-responsive genes in all cells. We developed an agent-based mathematical model to explore the IFNBI and DDX58 temporal dynamics. Simulations showed that a small number of early responder cells provide a mechanism for efficient and controlled activation of the DDX58-IFNBI positive feedback loop. The model predicted distributions of single cell responses that were confirmed by single cell mRNA measurements. The results suggest that large cell-to-cell variation plays an important role in the early innate immune response, and that the variability is essential for the efficient activation of the IFNB1 based feedback loop.

We investigate numerically the scaling properties of spatiotemporal correlation functions in the onedimensional Burgers equation driven by noise with variance proportional to ͉k͉ ␤ . The long-distance behavior at ␤Ͻ0 is determined by shocks that lead to multifractality in the high-order structure functions and a dynamical exponent z close to unity. For ␤Ͼ0 earlier theoretical predictions for scaling exponents constrained by Galilean invariance obtain; these results are not expected to hold for ␤Ͻ0. Nevertheless, the continuation of the fixed point to ␤Ͻ0 correctly predicts some of the properties, an occurrence that we relate to the anomalous scaling of composite operators. ͓S1063-651X͑96͒05811-4͔PACS number͑s͒: 05.45.ϩb, 47.10.ϩg The Burgers equation ͓1͔ for a one-dimensional velocity field u(x,t) has served as a simple model for investigating a variety of interesting issues that arise in fluid turbulence. Recently, there has been renewed interest in the Burgers equation with stochastic noise ͓2-5͔