Institute of Fundamental Technological Research

An overwhelming majority of mathematical models of regulatory pathways, including the intensively studied NF-κB pathway, remains non-identifiable, meaning that their parameters may not be determined by existing data. The existing NF-κB models that are capable of reproducing experimental data contain non-identifiable parameters, whereas simplified models with a smaller number of parameters exhibit dynamics that differs from that observed in experiments. Here, we reduced an existing model of the canonical NF-κB pathway by decreasing the number of equations from 15 to 6. The reduced model retains two negative feedback loops mediated by IκBα and A20, and in response to both tonic and pulsatile TNF stimulation exhibits dynamics that closely follow that of the original model. We carried out the sensitivity-based linear analysis and Monte Carlo-based analysis to demonstrate that the resulting model is both structurally and practically identifiable given measurements of 5 model variables from a simple TNF stimulation protocol. The reduced model is capable of reproducing different types of responses that are characteristic to regulatory motifs controlled by negative feedback loops: nearly-perfect adaptation as well as damped and sustained oscillations. It can serve as a building block of more comprehensive models of the immune response and cancer, where NF-κB plays a decisive role. Our approach, although may not be automatically generalized, suggests that models of other regulatory pathways can be transformed to identifiable, while retaining their dynamical features.

Type I interferons (IFNs) are key coordinators of the innate immune response to viral infection, which, through activation of the transcriptional regulators STAT1 and STAT2 (STAT1/2) in bystander cells, induce the expression of IFN-stimulated genes (ISGs). Here, we showed that in cells transfected with poly(I:C), an analog of viral RNA, the transcriptional activity of STAT1/2 was terminated because of depletion of the interferon-β (IFN-β) receptor, IFNAR. Activation of RNase L and PKR, products of two ISGs, not only hindered the replenishment of IFNAR but also suppressed negative regulators of IRF3 and NF-κB, consequently promoting IFNB transcription. We incorporated these findings into a mathematical model of innate immunity. By coupling signaling through the IRF3–NF-κB and STAT1/2 pathways with the activities of RNase L and PKR, the model explains how poly(I:C) switches the transcriptional program from being STAT1/2 induced to being IRF3 and NF-κB induced, which converts IFN-β–responding cells to IFN-β–secreting cells.

Alternations in the p53 regulatory network may render cancer cells resistant to the radiation-induced apoptosis. In this theoretical study we search for the best protocols combining targeted therapy with radiation to treat cancers with wild-type p53, but having downregulated expression of PTEN or overexpression of Wip1 resulting in resistance to radiation monotherapy. Instead of using the maximum tolerated dose paradigm, we exploit stochastic computational model of the p53 regulatory network to calculate apoptotic fractions for both normal and cancer cells. We consider combination protocols, with irradiations repeated every 12, 18, 24, or 36 h to find that timing between Mdm2 inhibitor delivery and irradiation significantly influences the apoptotic cell fractions. We assume that uptake of the inhibitor is higher by cancer than by normal cells and that cancer cells receive higher irradiation doses from intersecting beams. These two assumptions were found necessary for the existence of protocols inducing massive apoptosis in cancer cells without killing large fraction of normal cells neighboring tumor. The best found protocols have irradiations repeated every 24 or 36 h with two inhibitor doses per irradiation cycle, and allow to induce apoptosis in more than 95% of cancer cells, killing less than 10% of normal cells.

The innate immune system processes pathogen-induced signals into cell fate decisions. How information is turned to decision remains unknown. By combining stochastic mathematical modelling and experimentation, we demonstrate that feedback interactions between the IRF3, NF-κB and STAT pathways lead to switch-like responses to a viral analogue, poly(I:C), in contrast to pulse-like responses to bacterial LPS. Poly(I:C) activates both IRF3 and NF-κB, a requirement for induction of IFNβ expression. Autocrine IFNβ initiates a JAK/STAT-mediated positive-feedback stabilising nuclear IRF3 and NF-κB in first responder cells. Paracrine IFNβ, in turn, sensitises second responder cells through a JAK/STAT-mediated positive feedforward pathway that upregulates the positive-feedback components: RIG-I, PKR and OAS1A. In these sensitised cells, the 'live-or-die' decision phase following poly(I:C) exposure is shorter—they rapidly produce antiviral responses and commit to apoptosis. The interlinked positive feedback and feedforward signalling is key for coordinating cell fate decisions in cellular populations restricting pathogen spread.

cellular signalling networks, innate immunity, regulatory networks, stochastic modelling

We formulated a computational model for a MAPK signaling cascade downstream of the EGF receptor to investigate how interlinked positive and negative feedback loops process EGF signals into ERK pulses of constant amplitude but dose-dependent duration and frequency. A positive feedback loop involving RAS and SOS, which leads to bistability and allows for switch-like responses to inputs, is nested within a negative feedback loop that encompasses RAS and RAF, MEK, and ERK that inhibits SOS via phosphorylation. This negative feedback, operating on a longer time scale, changes switch-like behavior into oscillations having a period of 1 hour or longer. Two auxiliary negative feedback loops, from ERK to MEK and RAF, placed downstream of the positive feedback, shape the temporal ERK activity profile but are dispensable for oscillations. Thus, the positive feedback introduces a hierarchy among negative feedback loops, such that the effect of a negative feedback depends on its position with respect to the positive feedback loop. Furthermore, a combination of the fast positive feedback involving slow-diffusing membrane components with slower negative feedbacks involving faster diffusing cytoplasmic components leads to local excitation/global inhibition dynamics, which allows the MAPK cascade to transmit paracrine EGF signals into spatially non-uniform ERK activity pulses.

MAPK signaling, oscillations, mathematical modelling

Background: In favorable conditions bacterial doubling time is less than 20 min, shorter than DNA replication time. In E. coli a single round of genome replication lasts about 40 min and it must be accomplished about 20 min before cell division. To achieve such fast growth rates bacteria perform multiple replication rounds simultaneously. As a result, when the division time is as short as 20 min E. coli has about 8 copies of origin of replication (ori) and the average copy number of the genes situated close to ori can be 4 times larger than those near the terminus of replication (ter). It implies that shortening of cell cycle may influence dynamics of regulatory pathways involving genes placed at distant loci. Results: We analyze this effect in a model of a genetic toggle switch, i.e. a system of two mutually repressing genes, one localized in the vicinity of ori and the other localized in the vicinity of ter. Using a stochastic model that accounts for cell growth and divisions we demonstrate that shortening of the cell cycle can induce switching of the toggle to the state in which expression of the gene placed near ter is suppressed. The toggle bistability causes that the ratio of expression of the competing genes changes more than two orders of magnitude for a two-fold change of the doubling time. The increasing stability of the two toggle states enhances system sensitivity but also its reaction time. Conclusions: By fusing the competing genes with fluorescent tags this mechanism could be tested and employed to create an indicator of the doubling time. By manipulating copy numbers of the competing genes and locus of the gene situated near ter, one can obtain equal average expression of both genes for any doubling time T between 20 and 120 min. Such a toggle would accurately report departures of the doubling time from T.

Mathematical modeling, Stochastic simulations, Regulatory pathways, Bistability, DNA replication, Gene copy number

Bistable regulatory elements are important for nongenetic inheritance, increase of cell-to-cell heterogeneity allowing adaptation, and robust responses at the population level. Here, we study computationally the bistable genetic toggle switch—a small regulatory network consisting of a pair of mutual repressors—in growing and dividing bacteria. We show that as cells with an inhibited growth exhibit high stability of toggle states, cell growth and divisions lead to a dramatic increase of toggling rates. The toggling rates were found to increase with rate of cell growth, and can be up to six orders of magnitude larger for fast growing cells than for cells with the inhibited growth. The effect is caused mainly by the increase of protein and mRNA burst sizes associated with the faster growth. The observation that fast growth dramatically destabilizes toggle states implies that rapidly growing cells may vigorously explore the epigenetic landscape enabling nongenetic evolution, while cells with inhibited growth adhere to the local optima. This can be a clever population strategy that allows the slow growing (but stress resistant) cells to survive long periods of unfavorable conditions. Simultaneously, at favorable conditions, this stress resistant (but slowly growing—or not growing) subpopulation may be replenished due to a high switching rate from the fast growing population.

Gene expression, Bistability, Stochastic processes, Genetic toggle switch, Cell growth and division

Transitions between steady states of a multi-stable stochastic system in the perfectly mixed chemical reactor are possible only because of stochastic switching. In realistic cellular conditions, where diffusion is limited, transitions between steady states can also follow from the propagation of travelling waves. Here, we study the interplay between the two modes of transition for a prototype bistable system of kinase–phosphatase interactions on the plasma membrane. Within microscopic kinetic Monte Carlo simulations on the hexagonal lattice, we observed that for finite diffusion the behaviour of the spatially extended system differs qualitatively from the behaviour of the same system in the well-mixed regime. Even when a small isolated subcompartment remains mostly inactive, the chemical travelling wave may propagate, leading to the activation of a larger compartment. The activating wave can be induced after a small subdomain is activated as a result of a stochastic fluctuation. Such a spontaneous onset of activity is radically more probable in subdomains characterized by slower diffusion. Our results show that a local immobilization of substrates can lead to the global activation of membrane proteins by the mechanism that involves stochastic fluctuations followed by the propagation of a semi-deterministic travelling wave.

multi-stability, Markov process, spatially extended system, kinetic Monte Carlo on the lattice, kinase autophosphorylation, cell signalling

The aim of this study is to demonstrate that in molecular dynamical systems with the underlying bi- or multistability, the type of noise determines the most strongly attracting steady state or stochastic attractor. As an example we consider a simple stochastic model of autoregulatory gene with a nonlinear positive feedback, which in the deterministic approximation has two stable steady state solutions. Three types of noise are considered: transcriptional and translational – due to the small number of gene product molecules and the gene switching noise – due to gene activation and inactivation transitions. We demonstrate that the type of noise in addition to the noise magnitude dictates the allocation of probability mass between the two stable steady states. In particular, we found that when the gene switching noise dominates over the transcriptional and translational noise (which is characteristic of eukaryotes), the gene preferentially activates, while in the opposite case, when the transcriptional noise dominates (which is characteristic of prokaryotes) the gene preferentially remains inactive. Moreover, even in the zero-noise limit, when the probability mass generically concentrates in the vicinity of one of two steady states, the choice of the most strongly attracting steady state is noise type-dependent. Although the epigenetic attractors are defined with the aid of the deterministic approximation of the stochastic regulatory process, their relative attractivity is controlled by the type of noise, in addition to noise magnitude. Since noise characteristics vary during the cell cycle and development, such mode of regulation can be potentially employed by cells to switch between alternative epigenetic attractors.

Gene expression, Bistability, Stochastic processes, Epigenetic attractors

Bistable regulatory elements enhance heterogeneity in cell populations and, in multicellular organisms, allow cells to specialize and specify their fate. Our study demonstrates that in a system of bistable genetic switch, the noise characteristics control in which of the two epigenetic attractors the cell population will settle. We focus on two types of noise: the gene switching noise and protein dimerization noise. We found that the change of magnitudes of these noise components for one of the two competing genes introduces a large asymmetry of the protein stationary probability distribution and changes the relative probability of individual gene activation. Interestingly, an increase of noise associated with a given gene can either promote or suppress the activation of the gene, depending on the type of noise. Namely, each gene is repressed by an increase of its gene switching noise and activated by an increase of its protein-product dimerization noise. The observed effect was found robust to the large, up to fivefold deviations of the model parameters. In summary, we demonstrated that noise itself may determine the relative strength of the epigenetic attractors, which may provide a unique mode of control of cell fate decisions.

Gene expression, Bistability, Stochastic processes, Genetic toggle switch

Living cells may be considered as biochemical reactors of multiple steady states. Transitions between these states are enabled by noise, or, in spatially extended systems, may occur due to the traveling wave propagation. We analyze a one-dimensional bistable stochastic birth–death process by means of potential and temperature fields. The potential is defined by the deterministic limit of the process, while the temperature field is governed by noise. The stable steady state in which the potential has its global minimum defines the global deterministic attractor. For the stochastic system, in the low noise limit, the stationary probability distribution becomes unimodal, concentrated in one of two stable steady states, defined in this study as the global stochastic attractor. Interestingly, these two attractors may be located in different steady states. This observation suggests that the asymptotic behavior of spatially extended stochastic systems depends on the substrate diffusivity and size of the reactor. We confirmed this hypothesis within kinetic Monte Carlo simulations of a bistable reaction–diffusion model on the hexagonal lattice. In particular, we found that although the kinase–phosphatase system remains inactive in a small domain, the activatory traveling wave may propagate when a larger domain is considered.

multi-stability, Markov process, spatially extended system, kinetic Monte Carlo on the lattice, cell signalling

The aim of this study is to demonstrate that in dynamical systems with underlying bistability the type of noise qualitatively influences the stationary probability distribution (SPD). Specifically, we consider a simplified model of gene expression with the nonlinear positive feedback, which in the deterministic approximation has two stable steady state solutions. Two types of noise are considered; transcriptional - due to the limited number of protein molecules, and gene switching noise - due to gene activation and inactivation. In the limit of zero noise, the SPD generically concentrates in the decreasing vicinity of one of the two stable steady states. We demonstrated that for a range of parameters the SPD corresponding to the system with transcriptional noise only concentrates around a different steady state than SPD corresponding to the system with gene switching noise only.

Gene expression, Bistability, Stochastic processes, Epigenetic attractors