Computation of steady-state probability distributions in stochastic models of cellular networks

Cellular processes are "noisy". In each cell, concentrations of molecules are subject to random fluctuations due to the small numbers of these molecules and to environmental perturbations. While noise varies with time, it is often measured at steady state, for example by flow cytometry. When interrogating aspects of a cellular network by such steady-state measurements of network components, a key need is to develop efficient methods to simulate and compute these distributions. We describe innovations in stochastic modeling coupled with approaches to this computational challenge: first, an approach to modeling intrinsic noise via solution of the chemical master equation, and second, a convolution technique to account for contributions of extrinsic noise. We show how these techniques can be combined in a streamlined procedure for evaluation of different sources of variability in a biochemical network. Evaluation and illustrations are given in analysis of two well-characterized synthetic gene circuits, as well as a signaling network underlying the mammalian cell cycle entry.     

Multivariate analysis of noise in genetic regulatory networks

Stochasticity is an intrinsic property of genetic regulatory networks due to the low copy numbers of the major molecular species, such as, DNA, mRNA, and regulatory proteins. Therefore, investigation of the mechanisms that reduce the stochastic noise is essential in understanding the reproducible behaviors of real organisms and is also a key to design synthetic genetic regulatory networks that can reliably work. We use an analytical and systematic method, the linear noise approximation of the chemical master equation along with the decoupling of a stoichiometric matrix. In the analysis of fluctuations of multiple molecular species, the covariance is an important measure of noise. However, usually the representation of a covariance matrix in the natural coordinate system, i.e. the copy numbers of the molecular species, is intractably complicated because reactions change copy numbers of more than one molecular species simultaneously. Decoupling of a stoichiometric matrix, which is a transformation of variables, significantly simplifies the representation of a covariance matrix and elucidates the mechanisms behind the observed fluctuations in the copy numbers. We apply our method to three types of fundamental genetic regulatory networks, that is, a single-gene autoregulatory network, a two-gene autoregulatory network, and a mutually repressive network. We have found that there are multiple noise components differently originating. Each noise component produces fluctuation in the characteristic direction. The resulting fluctuations in the copy numbers of the molecular species are the sum of these fluctuations. In the examples, the limitation of the negative feedback in noise reduction and the trade-off of fluctuations in multiple molecular species are clearly explained. The analytical representations show the full parameter dependence. Additionally, the validity of our method is tested by stochastic simulations.     

Formulas for intrinsic noise evaluation in oscillatory genetic networks

The linear noise approximation is a useful method for stochastic noise evaluations in genetic regulatory networks, where the covariance equation described as a Lyapunov equation plays a central role. We discuss the linear noise approximation method for evaluations of an intrinsic noise in autonomously oscillatory genetic networks; in such oscillatory networks, the covariance equation becomes a periodic differential equation that provides generally an unbounded covariance matrix, so that the standard method of noise evaluation based on the covariance matrix cannot be adopted directly. In this paper, we develop a new method of noise evaluation in oscillatory genetic networks; first, we investigate structural properties, e.g., orbital stability and periodicity, of the solutions to the covariance equation given as a periodic Lyapunov differential equation by using the Floquet-Lyapunov theory, and propose a global measure for evaluating stochastic amplitude fluctuations on the periodic trajectory; we also derive an evaluation formula for the period fluctuation. Finally, we apply our method to a model of circadian oscillations based on negative auto-regulation of gene expression, and show validity of our method by comparing the evaluation results with stochastic simulations.     

Exponential Signaling Gain at the Receptor Level Enhances Signal-to-Noise Ratio in Bacterial Chemotaxis

Cellular signaling systems show astonishing precision in their response to external stimuli despite strong fluctuations in the molecular components that determine pathway activity. To control the effects of noise on signaling most efficiently, living cells employ compensatory mechanisms that reach from simple negative feedback loops to robustly designed signaling architectures. Here, we report on a novel control mechanism that allows living cells to keep precision in their signaling characteristics – stationary pathway output, response amplitude, and relaxation time – in the presence of strong intracellular perturbations. The concept relies on the surprising fact that for systems showing perfect adaptation an exponential signal amplification at the receptor level suffices to eliminate slowly varying multiplicative noise. To show this mechanism at work in living systems, we quantified the response dynamics of the E. coli chemotaxis network after genetically perturbing the information flux between upstream and downstream signaling components. We give strong evidence that this signaling system results in dynamic invariance of the activated response regulator against multiplicative intracellular noise. We further demonstrate that for environmental conditions, for which precision in chemosensing is crucial, the invariant response behavior results in highest chemotactic efficiency. Our results resolve several puzzling features of the chemotaxis pathway that are widely conserved across prokaryotes but so far could not be attributed any functional role.     

Stochastic environmental fluctuations drive epidemiology in experimental host–parasite metapopulations

Environmental fluctuations are important for parasite spread and persistence. However, the effects of the spatial and temporal structure of environmental fluctuations on host–parasite dynamics are not well understood. Temporal fluctuations can be random but positively autocorrelated, such that the environment is similar to the recent past (red noise), or random and uncorrelated with the past (white noise). We imposed red or white temporal temperature fluctuations on experimental metapopulations of Paramecium caudatum, experiencing an epidemic of the bacterial parasite Holospora undulata. Metapopulations (two subpopulations linked by migration) experienced fluctuations between stressful (5°C) and permissive (23°C) conditions following red or white temporal sequences. Spatial variation in temperature fluctuations was implemented by exposing subpopulations to the same (synchronous temperatures) or different (asynchronous temperatures) temporal sequences. Red noise, compared with white noise, enhanced parasite persistence. Despite this, red noise coupled with asynchronous temperatures allowed infected host populations to maintain sizes equivalent to uninfected populations. It is likely that this occurs because subpopulations in permissive conditions rescue declining subpopulations in stressful conditions. We show how patterns of temporal and spatial environmental fluctuations can impact parasite spread and host population abundance. We conclude that accurate prediction of parasite epidemics may require realistic models of environmental noise.     

A new method for choosing the computational cell in stochastic reaction–diffusion systems

How to choose the computational compartment or cell size for the stochastic simulation of a reaction–diffusion system is still an open problem, and a number of criteria have been suggested. A generalized measure of the noise for finite-dimensional systems based on the largest eigenvalue of the covariance matrix of the number of molecules of all species has been suggested as a measure of the overall fluctuations in a multivariate system, and we apply it here to a discretized reaction–diffusion system. We show that for a broad class of first-order reaction networks this measure converges to the square root of the reciprocal of the smallest mean species number in a compartment at the steady state. We show that a suitably re-normalized measure stabilizes as the volume of a cell approaches zero, which leads to a criterion for the maximum volume of the compartments in a computational grid. We then derive a new criterion based on the sensitivity of the entire network, not just of the fastest step, that predicts a grid size that assures that the concentrations of all species converge to a spatially-uniform solution. This criterion applies for all orders of reactions and for reaction rate functions derived from singular perturbation or other reduction methods, and encompasses both diffusing and non-diffusing species. We show that this predicts the maximal allowable volume found in a linear problem, and we illustrate our results with an example motivated by anterior-posterior pattern formation in Drosophila, and with several other examples.     

Protein signaling networks from single cell fluctuations and information theory profiling

Protein signaling networks among cells play critical roles in a host of pathophysiological processes, from inflammation to tumorigenesis. We report on an approach that integrates microfluidic cell handling, in situ protein secretion profiling, and information theory to determine an extracellular protein-signaling network and the role of perturbations. We assayed 12 proteins secreted from human macrophages that were subjected to lipopolysaccharide challenge, which emulates the macrophage-based innate immune responses against Gram-negative bacteria. We characterize the fluctuations in protein secretion of single cells, and of small cell colonies (n = 2, 3,···), as a function of colony size. Measuring the fluctuations permits a validation of the conditions required for the application of a quantitative version of the Le Chatelier's principle, as derived using information theory. This principle provides a quantitative prediction of the role of perturbations and allows a characterization of a protein-protein interaction network.     

Bayesian Markov Random Field Analysis for Protein Function Prediction Based on Network Data

Inference of protein functions is one of the most important aims of modern biology. To fully exploit the large volumes of genomic data typically produced in modern-day genomic experiments, automated computational methods for protein function prediction are urgently needed. Established methods use sequence or structure similarity to infer functions but those types of data do not suffice to determine the biological context in which proteins act. Current high-throughput biological experiments produce large amounts of data on the interactions between proteins. Such data can be used to infer interaction networks and to predict the biological process that the protein is involved in. Here, we develop a probabilistic approach for protein function prediction using network data, such as protein-protein interaction measurements. We take a Bayesian approach to an existing Markov Random Field method by performing simultaneous estimation of the model parameters and prediction of protein functions. We use an adaptive Markov Chain Monte Carlo algorithm that leads to more accurate parameter estimates and consequently to improved prediction performance compared to the standard Markov Random Fields method. We tested our method using a high quality S.cereviciae validation network with 1622 proteins against 90 Gene Ontology terms of different levels of abstraction. Compared to three other protein function prediction methods, our approach shows very good prediction performance. Our method can be directly applied to protein-protein interaction or coexpression networks, but also can be extended to use multiple data sources. We apply our method to physical protein interaction data from S. cerevisiae and provide novel predictions, using 340 Gene Ontology terms, for 1170 unannotated proteins and we evaluate the predictions using the available literature.     

Systematic multiscale parameterization of heterogeneous elastic network models of proteins

We present a method to parameterize heterogeneous elastic network models (heteroENMs) of proteins to reproduce the fluctuations observed in atomistic simulations. Because it is based on atomistic simulation, our method allows the development of elastic coarse-grained models of proteins under different conditions or in different environments. The method is simple and applicable to models at any level of coarse-graining. We validated the method in three systems. First, we computed the persistence length of ADP-bound F-actin, using a heteroENM model. The value of 6.1 ± 1.6 μm is consistent with the experimentally measured value of 9.0 ± 0.5 μm. We then compared our method to a uniform elastic network model and a realistic extension algorithm via covariance Hessian (REACH) model of carboxy myoglobin, and found that the heteroENM method more accurately predicted mean-square fluctuations of α-carbon atoms. Finally, we showed that the method captures critical differences in effective harmonic interactions for coarse-grained models of the N-terminal Bin/amphiphysin/Rvs (N-BAR) domain of amphiphysin, by building models of N-BAR both bound to a membrane and free in solution.     

Predicting the asymmetric response of a genetic switch to noise

We present a simple analytical tool which gives an approximate insight into the stationary behavior of nonlinear systems undergoing the influence of a weak and rapid noise from one dominating source, e.g. the kinetic equations describing a genetic switch with the concentration of one substrate fluctuating around a constant mean. The proposed method allows for predicting the asymmetric response of the genetic switch to noise, arising from the noise-induced shift of stationary states. The method has been tested on an example model of the lac operon regulatory network: a reduced Yildirim-Mackey model with fluctuating extracellular lactose concentration. We calculate analytically the shift of the system's stationary states in the presence of noise. The results of the analytical calculation are in excellent agreement with the results of numerical simulation of the noisy system. The simulation results suggest that the structure of the kinetics of the underlying biochemical reactions protects the bistability of the lactose utilization mechanism from environmental fluctuations. We show that, in the consequence of the noise-induced shift of stationary states, the presence of fluctuations stabilizes the behavior of the system in a selective way: Although the extrinsic noise facilitates, to some extent, switching off the lactose metabolism, the same noise prevents it from switching on.     

Noise Propagation in Gene Regulation Networks Involving Interlinked Positive and Negative Feedback Loops

It is well known that noise is inevitable in gene regulatory networks due to the low-copy numbers of molecules and local environmental fluctuations. The prediction of noise effects is a key issue in ensuring reliable transmission of information. Interlinked positive and negative feedback loops are essential signal transduction motifs in biological networks. Positive feedback loops are generally believed to induce a switch-like behavior, whereas negative feedback loops are thought to suppress noise effects. Here, by using the signal sensitivity (susceptibility) and noise amplification to quantify noise propagation, we analyze an abstract model of the Myc/E2F/MiR-17-92 network that is composed of a coupling between the E2F/Myc positive feedback loop and the E2F/Myc/miR-17-92 negative feedback loop. The role of the feedback loop on noise effects is found to depend on the dynamic properties of the system. When the system is in monostability or bistability with high protein concentrations, noise is consistently suppressed. However, the negative feedback loop reduces this suppression ability (or improves the noise propagation) and enhances signal sensitivity. In the case of excitability, bistability, or monostability, noise is enhanced at low protein concentrations. The negative feedback loop reduces this noise enhancement as well as the signal sensitivity. In all cases, the positive feedback loop acts contrary to the negative feedback loop. We also found that increasing the time scale of the protein module or decreasing the noise autocorrelation time can enhance noise suppression; however, the systems sensitivity remains unchanged. Taken together, our results suggest that the negative/positive feedback mechanisms in coupled feedback loop dynamically buffer noise effects rather than only suppressing or amplifying the noise.     

Stochastic Actin Polymerization and Steady Retrograde Flow Determine Growth Cone Advancement

Neuronal growth is an extremely complex yet reliable process that is directed by a dynamic lamellipodial structure at the tip of every growing neurite, called the growth cone. Lamellipodial edge fluctuations are controlled by the interplay between actin polymerization pushing the edge forward and molecular motor driven retrograde actin flow retracting the actin network. The leading edge switches randomly between extension and retraction processes. We identify switching of “on/off” states in actin polymerization as the main determinant of lamellipodial advancement. Our analysis of motility statistics allows for a prediction of growth direction. This was used in simulations explaining the amazing signal detection capabilities of neuronal growth by the experimentally found biased stochastic processes. Our measurements show that the intensity of stochastic fluctuations depend on changes in the underlying active intracellular processes and we find a power law η = a^∗x^α with exponent α = 2.63 ± 0.12 between noise intensity η and growth cone activity x, defined as the sum of protrusion and retraction velocity. Differences in the lamellipodial dynamics between primary neurons and a neuronal cell line further suggests that active processes tune the observed stochastic fluctuations. This hints at a possible role of noise intensity in determining signal detection sensitivity.     

A relatively low level of ribosome depurination by mutant forms of ricin toxin A chain can trigger protein synthesis inhibition,cell signaling and apoptosis in mammalian cells

The A chain of the plant toxin ricin (RTA) is an N-glycosidase that inhibits protein synthesis by removing a specific adenine from the 28S rRNA. RTA also induces ribotoxic stress, which activates stress-induced cell signaling cascades and apoptosis. However, the mechanistic relationship between depurination, protein synthesis inhibition and apoptosis remains an open question. We previously identified two RTA mutants that suggested partial independence of these processes in a yeast model. The goals of this study were to establish an endogenous RTA expression system in mammalian cells and utilize RTA mutants to examine the relationship between depurination, protein synthesis inhibition, cell signaling and apoptosis in mammalian cells. The non-transformed epithelial cell line MAC-T was transiently transfected with plasmid vectors encoding precursor (pre) or mature forms of wild-type (WT) RTA or mutants. PreRTA was glycosylated indicating that the native signal peptide targeted RTA to the ER in mammalian cells. Mature RTA was not glycosylated and thus served as a control to detect changes in catalytic activity. Both pre- and mature WT RTA induced ribosome depurination, protein synthesis inhibition, activation of cell signaling and apoptosis. Analysis of RTA mutants showed for the first time that depurination can be reduced by 40% in mammalian cells with minimal effects on inhibition of protein synthesis, activation of cell signaling and apoptosis. We further show that protein synthesis inhibition by RTA correlates more linearly with apoptosis than ribosome depurination.     

Extrinsic noise passing through a Michaelis-Menten reaction: A universal response of a genetic switch

The study of biochemical pathways usually focuses on a small section of a protein interactions network. Two distinct sources contribute to the noise in such a system: intrinsic noise, inherent in the studied reactions, and extrinsic noise generated in other parts of the network or in the environment. We study the effect of extrinsic noise entering the system through a nonlinear uptake reaction which acts as a nonlinear filter. Varying input noise intensity varies the mean of the noise after the passage through the filter, which changes the stability properties of the system. The steady-state displacement due to small noise is independent on the kinetics of the system but it only depends on the nonlinearity of the input function.For monotonically increasing and concave input functions such as the Michaelis-Menten uptake rate, we give a simple argument based on the small-noise expansion, which enables qualitative predictions of the steady-state displacement only by inspection of experimental data: when weak and rapid noise enters the system through a Michaelis-Menten reaction, then the graph of the system's steady states vs. the mean of the input signal always shifts to the right as noise intensity increases.We test the predictions on two models of lac operon, where TMG/lactose uptake is driven by a Michaelis-Menten enzymatic process. We show that as a consequence of the steady state displacement due to fluctuations in extracellular TMG/lactose concentration the lac switch responds in an asymmetric manner: as noise intensity increases, switching off lactose metabolism becomes easier and switching it on becomes more difficult.     

Populational Heterogeneity vs. Temporal Fluctuation in Escherichia coli Flagellar Motor Switching

The dynamic switching of the bacterial flagellar motor regulates cell motility in bacterial chemotaxis. It has been reported under physiological conditions that the switching bias of the flagellar motor undergoes large temporal fluctuations, which reflects noise propagating in the chemotactic signaling network. On the other hand, nongenetic heterogeneity is also observed in flagellar motor switching, as a large group of switching motors show different switching bias and frequency under the same physiological condition. In this work, we present simultaneous measurement of groups of Escherichia coli flagellar motor switching and compare them to long time recording of single switching motors. Consistent with previous studies, we observed temporal fluctuations in switching bias in long time recording experiments. However, the variability in switching bias at the populational level showed much higher volatility than its temporal fluctuation. These results suggested stable individuality in E. coli motor switching. We speculate that uneven expression of key regulatory proteins with amplification by the ultrasensitive response of the motor can account for the observed populational heterogeneity and temporal fluctuations.