Model of influenza A virus infection: dynamics of viral antagonism and innate immune response.

Viral antagonism of host responses is an essential component of virus pathogenicity. The study of the interplay between immune response and viral antagonism is challenging due to the involvement of many processes acting at multiple time scales. Here we develop an ordinary differential equation model to investigate the early, experimentally measured, responses of human monocyte-derived dendritic cells to infection by two H1N1 influenza A viruses of different clinical outcomes: pandemic A/California/4/2009 and seasonal A/New Caledonia/20/1999. Our results reveal how the strength of virus antagonism, and the time scale over which it acts to thwart the innate immune response, differs significantly between the two viruses, as is made clear by their impact on the temporal behavior of a number of measured genes. The model thus sheds light on the mechanisms that underlie the variability of innate immune responses to different H1N1 viruses.

Power-laws in interferon-B mRNA distribution in virus-infected dendritic cells.

Interferon-beta (IFNB1) mRNA shows very large cell-to-cell variability in primary human dendritic cells infected by Newcastle disease virus, with copy numbers varying from a few to several thousands. Analysis of data from the direct measurement of the expression of this gene in its natural chromatin environment in primary human cells shows that the distribution of mRNA across cells follows a power law with an exponent close to -1, and thus encompasses a range of variation much more extensive than a Gaussian. We also investigate the single cell levels of IFNB1 mRNA induced by infection with Texas influenza A mutant viruses, which vary in their capacity to inhibit the signaling pathways responsible for activation of this gene. Here as well we observe power-law behavior for the distribution of IFNB1 mRNA, albeit over a truncated range of values, with exponents similar to the one for cells infected by Newcastle disease virus. We propose a model of stochastic enhanceosome and preinitiation complex formation that incorporates transcriptional pulsing. Analytical and numerical results show good agreement with the observed power laws, and thus support the existence of transcriptional pulsing of an unmodified, intact gene regulated by a natural stimulus.

Stochasticity of gene products from transcriptional pulsing.

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-law) 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.

Shared kinase fluctuations between two enzymatic reactions.

Kinases serve crucial roles in many cellular signaling pathways that process and transfer information. When signaling kinases phosphorylate two targets, these can serve as branch points that distribute information among two pathways. Responses to stimuli transmitted by activated kinases show high levels of cell-to-cell variation that influence cellular function. We ask how fluctuations around a steady state, due to kinase fluctuations and intrinsic noise, are distributed between two reactions with substrates phosphorylated by a shared kinase. We develop the formalism to answer this question and, for a realistic set of biological constants, we illustrate various features of fluctuations and relaxation times to a steady state. We find that the steady-state response determines the size and range in enzyme concentration of phosphorylated substrate fluctuations, and that the choice of an operating point can have a large impact on how shared kinase noise is distributed among two available pathways.

NF-kappaB oscillations and cell-to-cell variability.

Oscillations in the transcriptional activator NF-kappaB localized in the nucleus have been observed when a cell is stimulated by an external agent. A negative feedback based on the protein IkappaB whose expression is controlled by NF-kappaB is known to be responsible for these oscillations. We study NF-kappaB oscillations, which have been observed both for cell populations by Hoffmann et al. [2002. The IkappaB-NF-kappaB signaling module: temporal control and selective gene activation. Science 298, 1241-1245] and for single cells by Nelson et al. [2004. Oscillations in NF-kappaB signaling control the dynamics of gene expression. Science 306, 704-708]. In order to study cell-to-cell variability we use Gillespie's algorithm, applied to a simplified version of the model proposed by Hoffmann et al. (2002). We consider the amounts of cellular NF-kappaB and activated IKK as external parameters. When these are fixed, we show that intrinsic fluctuations are small in a model with strong transcription, as is the case of the Hoffmann et al. (2002) model, whether transcription is quadratic or linear in the number of NF-kappaB molecules. Intrinsic fluctuations can however be large when transcription is weak, as we illustrate in a model variant. The effect of extrinsic fluctuations can be significant: cell-to-cell fluctuations of the initial amount of cellular NF-kappaB affect mainly the amplitude of nuclear NF-kappaB oscillations, at least when transcription is linear in the number of NF-kappaB molecules, while fluctuations in the amount of activated IKK affect both their amplitude and period, whatever the mode of transcription. In this case model results are in qualitative agreement with the considerable cell-to-cell variability of NF-kappaB oscillations observed by Nelson et al. (2004).

A feedforward loop motif in transcriptional regulation: induction and repression.

We study the dynamical behavior of a unit of three positive transcriptional regulators which occurs frequently in biological networks of yeast and bacteria as a feedforward loop. We investigate numerically a set of reactions incorporating the basic features of transcription and translation. We determine (i) how the feedforward loop motif functions as a computational element such as an AND gate in the presence of stochastic fluctuations, and (ii) the robustness of the motif when transcription at the primary level is suddenly repressed. We highlight the effective time-scales which underlie both of these aspects of the feedforward loop motif. We show how threshold behavior of the motif output arises as a function of the number of external inducers as well as the time over which the inducer acts. We discuss how individual cell behavior can deviate significantly from average behavior, due to intrinsic fluctuations in the small number of molecules present in a cell.

The linear noise approximation for molecular fluctuations within cells.

We study the applicability of Van Kampen's linear noise approximation to the calculation of fluctuations in cells due to small number of molecules for simple genetic systems not previously considered. These systems include dimer formation and feedback. We explain why the linear noise approximation can be surprisingly effective, but also illustrate how it fails in a simple example when a protein probability distribution is not purely Gaussian.

Fluctuations and slow variables in genetic networks.

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.

The role of dimerization in noise reduction of simple genetic networks.

Fluctuations are an intrinsic property of genetic networks due to the small number of interacting molecules. We study the role of dimerization reactions in controlling these fluctuations in a simple genetic circuit with negative feedback. We compare two different pathways. In the dimeric pathway the proteins to be regulated form dimers in solution that afterward bind to an operator site and inhibit transcription. In the monomeric pathway monomers bind to the operator site and then recruit another monomer to form a dimer directly on the DNA. We find that while both pathways implement the same negative feedback mechanism, the protein number fluctuations in the dimeric pathway are drastically reduced compared to the monomeric pathway. This difference in the ability to reduce fluctuations may be of importance in the design of genetic networks.

Modeling corticofugal feedback and the sensitivity of lateral geniculate neurons to orientation discontinuity.

We model feedback from primary visual cortex to the dorsal lateral geniculate nucleus (dLGN). This feedback makes dLGN neurons sensitive to orientation discontinuity (Sillito et al., 1993; Cudeiro & Sillito, 1996). In the model, each dLGN neuron receives retinotopic input driven by layer 6 cortical neurons in a full set of orientation columns. Excitation is monosynaptic, while inhibition is through perigeniculate neurons and dLGN interneurons. The stimulus consists of drifting gratings,k one within and the other outside a circular region centered over the receptive field of the model dLGN relay neuron we study. They appear as a single grating when they are aligned with equal contrast. The model reproduces experimental results showing an increasing inhibitory effect of feedback on the firing rate of dLGN neurons as the two gratings move towards the aligned position. Moreover, enhancement of dLGN cell center-surround antagonism by feedback is revealed by measuring the responses to drifting gratings inside a circular window, as a function of window radius. This effect is related to the observed length tuning of dLGN cells. Sensitivity to orientation discontinuity could be mediated in the model by feedback from either simple or complex cells. The model puts constraints on the feedback synaptic footprint and shows that its elongated shape does not play a crucial role in sensitivity to orientation discontinuity. The inhibitory component of feedback must predominate overall, but the feedback signal from a cortical neuron to a dLGN neuron with the same or nearby receptive-field center can be dominated by excitation. Predictions of the model include (1) robust stimuli for layer 6 cortical neurons give pronounced nonlinearities in the responses of dLGN neurons; (2) the sensitivity to orientation discontinuity at low contrast is twice that at high contrast.