Session D39: Focus Session: Noise and Fluctuations in Biochemical Networks

Stochasticity in cell biology: Modeling across levels

Effective modeling of biological processes requires focusing on a particular level of description, and this requires summarizing de details of lower levels into effective variables and properly accounting for the constrains that other levels impose. In the context of stochasticity in gene expression, I will show how the details of the stochastic process can be characterized by a few effective parameters, which facilitates modeling but complicates interpretation of current experiments. I will show how the resulting noise can provide advantageous or deleterious phenotypic fluctuation and how noise control in the copy number control system of plasmids can change the selective pressures. This system illustrates the direct connection between molecular dynamics and evolutionary dynamics.   

Mean-field vs. Stochastic Models for Transcriptional Regulation

We introduce a minimal model description for the dynamics of transcriptional regulatory networks. It is studied within a mean-field approximation, i.e., by deterministic ode's representing the reaction kinetics, and by stochastic simulations employing the Gillespie algorithm. We elucidate the different results both approaches can deliver, depending on the network under study, and in particular depending on the level of detail retained in the respective description. Two examples are addressed in detail: the repressilator, a transcriptional clock based on a three-gene network realized experimentally in E. coli, and a bistable two-gene circuit under external driving, a transcriptional network motif recently proposed to play a role in cellular development.   

The stochastic spectral analysis of transcriptional regulatory cascades

Modeling the dynamics of biological networks while respecting the intrinsic stochasticity requires accounting for intrinsic fluctuations arising from the low copy count of the constituent particles. Traditional simulation-based approaches to computing the probability distribution, rather than by direct solution of its master equation, are fundamentally limited by long runtimes and the need to estimate a the distribution from samples. We obviate both limitations by directly solving for the distribution using a fast and accurate method that exploits the natural basis of the uncoupled problem from the same class. We illustrate our method on a ubiquitous biological example: linear signaling cascades. The huge efficiency gains permit optimization of information transmission over input and regulatory parameters, revealing design properties of the most informative cascades. We find, for threshold regulation, that a cascade of strong regulations converts a unimodal input to a bimodal output, that multimodal inputs are no more informative than bimodal inputs, and that a chain of “DC” up-regulations outperforms a chain of “AC” down-regulations.   

Tuning stochastic transition rates in a bistable genetic network.

We investigate the stochastic dynamics of a simple genetic network, a toggle switch, in which the system makes transitions between the two alternative states. Our interest is in exploring whether such stochastic transitions, which occur due to the intrinsic noise such as transcriptional and degradation events, can be slowed down/speeded up, without changing the mean expression levels of the two genes, which comprise the toggle network. Such tuning is achieved by linking a signaling network to the toggle switch. The signaling network comprises of a protein, which can exist either in an active (phosphorylated) or inactive (dephosphorylated) form, and where its state is determined by one of the genetic network components. The active form of the protein in turn feeds back on the dynamics of the genetic network. We find that the rate of stochastic transitions from one state to the other, is determined essentially by the speed of phosphorylation, and hence the rate can be modulated by varying the phosphatase levels. We hypothesize that such a network architecture can be implemented as a general mechanism for controlling transition rates and discuss applications in population studies of two differentiated cell lineages, ex: the myeloid/erythroid lineage in hematopoiesis.   

Optimizing information flow in small genetic networks

Many of the biological networks inside cells can be thought of as transmitting information from the inputs (e.g., the concentrations of transcription factors or other signaling molecules) to their outputs (e.g., the expression levels of various genes). On the molecular level, the relatively small concentrations of the relevant molecules and the intrinsic randomness of chemical reactions provide sources of noise that set physical limits on this information transmission. Given these limits, not all networks perform equally well, and maximizing information transmission provides a candidate design principle from which we might hope to derive the properties of real regulatory networks. As a starting point, I will consider the simple case of one input transcription factor that controls many genes. I will discuss the properties of these specific small networks that can transmit the maximum information. Concretely, I will show how the form of molecular noise drives predictions not just of the qualitative network topology but also the quantitative paramaters for the input/output relations at the nodes of the network. In an attempt to link these general theoretical considerations to real biological systems, I will illustrate the predictions on the example of transmission of positional information in the early development of the fly embryo.   

Characterizing noise in genetic oscillatory systems

Quantitative understanding of fluctuations in genetic circuits is crucial for understanding living systems. Despite the recent advances in the subject, however, fluctuations in non-stationary activities such as molecular oscillations have not been much investigated yet. Here we quantify the fluctuations in periods and amplitudes of oscillation and the noise propagation in the genetic oscillatory system, the repressilator, using exact stochastic simulation. At the single protein level, we found that the fluctuation in oscillation amplitudes is larger than that in oscillation periods. Noise propagation is studied in terms of the correlations in the successive periods and amplitudes, respectively, which decay exponentially down the regulatory cascades. We then study the extended repressilator system to investigate the effect of extra component and identify the combinatoric regulation pattern that reduces the fluctuations in oscillatory activities significantly.   

Purely stochastic binary decisions in cell signaling models without underlying deterministic bistabilities

Detection of different extra-cellular stimuli leading to functionally distinct outcomes is common in cell biology, and is often mediated by differential regulation of positive and negative feedback loops that are a part of the signaling network. For cellular responses stimulated by small numbers of molecules, the stochastic effects are important. Therefore, we studied the influence of stochastic fluctuations on a simple signaling model with dueling positive and negative feedback loops. The class of models we have studied is characterized by single deterministic steady states for all parameter values, but the stochastic response is bimodal; a behavior that is distinctly different from models studied in the context of gene regulation. For small numbers of signaling molecules, stochastic effects result in a bimodal distribution for this quantity, with neither mode corresponding to the deterministic solution; i.e., cells are in ``on'' or ``off'' states, not in some intermediate state. For a large number of molecules, the stochastic solution converges to the mean-field result. When fluctuations are important, we find that signal output scales with control parameters ``anomalously'' compared to mean-field predictions.   

Fitness effects of fluctuations in biochemical networks

The concentration of many cellular components fluctuates not only as a response to external and internal inputs but also due to random birth and death events of individual molecules. This biochemical noise affects the capacity of every individual cell in a population to respond and adapt to the environment. While the sources and effects of biochemical fluctuations on individual cells have been intensively studied, the effects of noise on the growth rate of a population of cells are much less understood. We present a model of the cell cycle in which the growth and division of individual cells are coupled with the noisy dynamics of their internal components. The model allows us to compute the contribution of the biochemical noise to the average growth rate of a population of cells as a function of the noise strength and the correlation time of the fluctuations. We show that, due to fluctuations, the growth rate of a population of cells is always larger than the average growth rate of a individual cell and can be larger even than a corresponding deterministic model. In most relevant cases it is assumed that the average concentration of a cellular component is close to a value that maximizes the population growth as given by the external, environmental, conditions and the internal cellular regulation. In such cases we show that contribution of fluctuations to the growth rate is negative and increases with the sensitivity of the biochemical networks to the noise sources and the noise correlation time. We also discuss how the selection pressure due to fluctuations affects the structure and parameters of genetic regulatory networks.   

Effects of delay and noise in a negative feedback regulatory motif

The small copy number of the molecules involved in gene regulation can induce nontrivial stochastic phenomena such as noise-induced oscillations. An often neglected aspect of regulation dynamics are the delays involved in transcription and translation. Delays introduce analytical and computational complications because the dynamics is non-Markovian. We study the interplay of noise and delays in a negative feedback model of the p53 core regulatory network. Recent experiments have found pronounced oscillations in the concentrations of proteins p53 and Mdm2 in individual cells subjected to DNA damage. Similar oscillations occur in the Hes-1 and NK-kB systems, and in circadian rhythms. Several mechanisms have been proposed to explain this oscillatory behaviour, such as deterministic limit cycles, with and without delay, or noise-induced excursions in excitable models. We consider a generic delayed Master Equation incorporating the activation of Mdm2 by p53 and the Mdm2-promoted degradation of p53. In the deterministic limit and for large delays, the model shows a Hopf bifurcation. Via exact stochastic simulations, we find strong noise-induced oscillations well outside the limit-cycle region. We propose that this may be a generic mechanism for oscillations in gene regulatory systems.   

Individuals in the crowd: studying bacterial quorum-sensing at the single-cell level

Like many bacterial species, the marine bacterium \textit{Vibrio fischeri} can detect its own population density through a quorum sensing (QS) mechanism. The bacterium releases a small molecule signal -- the autoinducer (AI) -- into its environment: high AI concentration indicates high population density and triggers a genetic switch that, in \textit{V.fischeri}, leads to bioluminescence. Although the QS behavior of bulk cultures of \textit{V.fischeri }has been extensively studied, little is known about either the response of individual cells to AI signal levels or the role of noise and local diffusion in QS signaling. We have used a photon-counting camera to record the luminescence of individual \textit{V.fischeri} cells immobilized in a flow cell and subject to varying concentrations of AI. We observe that light output by individual cells varies not only with bulk AI concentration, but also over time, between cells, with local (micron-scale) population density, and even with the flow rate of the medium. Most of these variations would not be evident in a bulk culture. We will present an analysis of this heterogeneity at the cell level and its implications for the role of noise in QS signaling.