Stable IL- 1ß-Activation in an Inflammasome Signalling Model Depends on Positive and Negative Feedbacks and Tight Regulation of Protein Production.

INTRODUCTION: NLRP3-dependent inflammasome signalling is a key pathway during inflammatory processes and its deregulation is implicated in several diseases. NLRP3-inflammasome pathway activation leads to the rapid, phosphorylation-driven NF$\kappa$κB-pathway signalling, subsequently proceeds via slower transcription/translation process for producing pro-enzymes, and finally leads to the medium-speed enzymatic activation of the central inflammatory mediator IL-$1\beta$1ß[1] . We here were interested in how the timing of the rate-limiting step of transcription/translation and the presence of a positive and negative auto-regulation would pose conditions for meaningful and stable IL-$1\beta$1ß-activation. METHODS: We extracted the essential topology of the inflammasome pathway network using a linear chain of first-order reaction and a second-order reaction for inhibitory feedback. We then performed an analytical treatment of the resulting ODE set to obtain closed-form formulae. We therefore looked for the steady states and characterized their stability by using a Jacobian-based, local analysis. We employed the Small Gain Theorem from Control Theory as recently applied by us [2] and the Gershgorin Circle Theorem to obtain mathematically exact conditions for a positive ON state and stabilities for ON and OFF steady states. RESULTS: We identified an ON- and one OFF- steady state whose properties we characterized in terms of the kinetic parameters by closed-form formulae. We found that under the assumption of a first-order information flow through the network, the existence of a biologically reasonable ON steady state required the simultaneous presence of the positive and the negative feedback. Assuming non-competitivity between IL-$1\beta$1ß entities binding to different receptors, we found that a minimum kinetics for protein production is required to sustain a steady state with IL-$1\beta$1ß activation. Assuming competitivity between IL-$1\beta$1ß entities introduced additional restrictions on the maximum protein production speed to guarantee a biologically reasonable ON steady state. Finally, for both models, we ruled out bistability, suggesting that IL-$1\beta$1ß activation would undergo a smooth change upon alterations of its parameters. CONCLUSION: Exemplified by the core pathway of NLRP3-inflammasome signalling, we here demonstrate that a mostly linear activation cascade containing an intermediate rate limiting step poses kinetic restrictions on this step and requires positive and negative autoregulation for obtaining a meaningful ON steady state. Due to the generality of our framework, our results are important for a wide class of receptor mediated-pathways, where a fast initial phosphorylation cascade is followed by a (slower) transcriptional response and subsequent autoregulation. Our results may further provide important design principles for synthetic biological networks involving biochemical activation and transcription/translation, by relating timing considerations and autoregulation to stable pathway activation.

Biogas purification via optimal microalgae growth: A literature review.

In this review, we compare works from the current decade that address the CO2 -removal from biogas by means of microalgae. The microalgae culture process acts as a biochemical absorption process; it is potentially competitive with respect to classical and commercial absorption methods due to its additional benefits such as availing CO2 for the production of valuable microalgae biomass and being an environmentally friendly technique. Nevertheless, the low yield of biogas purification translates into the need to use optimal operation strategies that render the whole biogas production process economically feasible. A class of these strategies requires models capable of reproducing key traits of the dynamical behavior of microalgae growth. Thus, without overlooking the classical physico-chemical methods for biogas purification, our literature review addresses: (i) biogas purification via microalgae and different microalgae growth conditions, (ii) approaches that maximize microalgae growth, in order to increase CO2 -consumption, and (iii) different models that describe the representative characteristics of microalgae growth. This investigation traces a pathway to future considerations on optimal biogas purification alternatives by microalgae culture processes. © 2018 American Institute of Chemical Engineers Biotechnol. Prog., 34:1513-1532, 2018.

Order reduction of the chemical master equation via balanced realisation.

We consider a Markov process in continuous time with a finite number of discrete states. The time-dependent probabilities of being in any state of the Markov chain are governed by a set of ordinary differential equations, whose dimension might be large even for trivial systems. Here, we derive a reduced ODE set that accurately approximates the probabilities of subspaces of interest with a known error bound. Our methodology is based on model reduction by balanced truncation and can be considerably more computationally efficient than solving the chemical master equation directly. We show the applicability of our method by analysing stochastic chemical reactions. First, we obtain a reduced order model for the infinitesimal generator of a Markov chain that models a reversible, monomolecular reaction. Later, we obtain a reduced order model for a catalytic conversion of substrate to a product (a so-called Michaelis-Menten mechanism), and compare its dynamics with a rapid equilibrium approximation method. For this example, we highlight the savings on the computational load obtained by means of the reduced-order model. Furthermore, we revisit the substrate catalytic conversion by obtaining a lower-order model that approximates the probability of having predefined ranges of product molecules. In such an example, we obtain an approximation of the output of a model with 5151 states by a reduced model with 16 states. Finally, we obtain a reduced-order model of the Brusselator.

Exact probability distributions of selected species in stochastic chemical reaction networks.

Chemical reactions are discrete, stochastic events. As such, the species' molecular numbers can be described by an associated master equation. However, handling such an equation may become difficult due to the large size of reaction networks. A commonly used approach to forecast the behaviour of reaction networks is to perform computational simulations of such systems and analyse their outcome statistically. This approach, however, might require high computational costs to provide accurate results. In this paper we opt for an analytical approach to obtain the time-dependent solution of the Chemical Master Equation for selected species in a general reaction network. When the reaction networks are composed exclusively of zeroth and first-order reactions, this analytical approach significantly alleviates the computational burden required by simulation-based methods. By building upon these analytical solutions, we analyse a general monomolecular reaction network with an arbitrary number of species to obtain the exact marginal probability distribution for selected species. Additionally, we study two particular topologies of monomolecular reaction networks, namely (i) an unbranched chain of monomolecular reactions with and without synthesis and degradation reactions and (ii) a circular chain of monomolecular reactions. We illustrate our methodology and alternative ways to use it for non-linear systems by analysing a protein autoactivation mechanism. Later, we compare the computational load required for the implementation of our results and a pure computational approach to analyse an unbranched chain of monomolecular reactions. Finally, we study calcium ions gates in the sarco/endoplasmic reticulum mediated by ryanodine receptors.

The EGFR demonstrates linear signal transmission.

Cells sense information encoded in extracellular ligand concentrations and process it using intracellular signalling cascades. Using mathematical modelling and high-throughput imaging of individual cells, we studied how a transient extracellular growth factor signal is sensed by the epidermal growth factor receptor system, processed by downstream signalling, and transmitted to the nucleus. We found that transient epidermal growth factor signals are linearly translated into an activated epidermal growth factor receptor integrated over time. This allows us to generate a simplified model of receptor signaling where the receptor acts as a perfect sensor of extracellular information, while the nonlinear input-output relationship of EGF-EGFR triggered signalling is a consequence of the downstream MAPK cascade alone.

Equilibria and stability of a class of positive feedback loops.

Positive feedback loops are common regulatory elements in metabolic and protein signalling pathways. The length of such feedback loops determines stability and sensitivity to network perturbations. Here we provide a mathematical analysis of arbitrary length positive feedback loops with protein production and degradation. These loops serve as an abstraction of typical regulation patterns in protein signalling pathways. We first perform a steady state analysis and, independently of the chain length, identify exactly two steady states that represent either biological activity or inactivity. We thereby provide two formulas for the steady state protein concentrations as a function of feedback length, strength of feedback, as well as protein production and degradation rates. Using a control theory approach, analysing the frequency response of the linearisation of the system and exploiting the Small Gain Theorem, we provide conditions for local stability for both steady states. Our results demonstrate that, under some parameter relationships, once a biological meaningful on steady state arises, it is stable, while the off steady state, where all proteins are inactive, becomes unstable. We apply our results to a three-tier feedback of caspase activation in apoptosis and demonstrate how an intermediary protein in such a loop may be used as a signal amplifier within the cascade. Our results provide a rigorous mathematical analysis of positive feedback chains of arbitrary length, thereby relating pathway structure and stability.

Spatial Quantification of Cytosolic Ca²âº Accumulation in Nonexcitable Cells: An Analytical Study.

Calcium ions act as messengers in a broad range of processes such as learning, apoptosis, and muscular movement. The transient profile and the temporal accumulation of calcium signals have been suggested as the two main characteristics in which calcium cues encode messages to be forwarded to downstream pathways. We address the analytical quantification of calcium temporal-accumulation in a long, thin section of a nonexcitable cell by solving a boundary value problem. In these expressions we note that the cytosolic Ca(2+) accumulation is independent of every intracellular calcium flux and depends on the Ca(2+) exchange across the membrane, cytosolic calcium diffusion, geometry of the cell, extracellular calcium perturbation, and initial concentrations. In particular, we analyse the time-integrated response of cytosolic calcium due to i) a localised initial concentration of cytosolic calcium and ii) transient extracellular perturbation of calcium. In these scenarios, we conclude that i) the range of calcium progression is confined to the vicinity of the initial concentration, thereby creating calcium microdomains; and ii) we observe a low-pass filtering effect in the response driven by extracellular Ca(2+) perturbations. Additionally, we note that our methodology can be used to analyse a broader range of stimuli and scenarios.

Cumulative signal transmission in nonlinear reaction-diffusion networks.

Quantifying signal transmission in biochemical systems is key to uncover the mechanisms that cells use to control their responses to environmental stimuli. In this work we use the time-integral of chemical species as a measure of a network's ability to cumulatively transmit signals encoded in spatiotemporal concentrations. We identify a class of nonlinear reaction-diffusion networks in which the time-integrals of some species can be computed analytically. The derived time-integrals do not require knowledge of the solution of the reaction-diffusion equation, and we provide a simple graphical test to check if a given network belongs to the proposed class. The formulae for the time-integrals reveal how the kinetic parameters shape signal transmission in a network under spatiotemporal stimuli. We use these to show that a canonical complex-formation mechanism behaves as a spatial low-pass filter, the bandwidth of which is inversely proportional to the diffusion length of the ligand.

Positive feedback in the Akt/mTOR pathway and its implications for growth signal progression in skeletal muscle cells: an analytical study.

The IGF-1 mediated Akt/mTOR pathway has been recently proposed as mediator of skeletal muscle growth and a positive feedback between Akt and mTOR was suggested to induce homogeneous growth signals along the whole spatial extension of such long cells. Here we develop two biologically justified approximations which we study under the presence of four different initial conditions that describe different paradigms of IGF-1 receptor-induced Akt/mTOR activation. In first scenario the activation of the feedback cascade was assumed to be mild or protein turnover considered to be high. In turn, in the second scenario the transcriptional regulation was assumed to maintain defined levels of inactive pro-enzymes. For both scenarios, we were able to obtain closed-form formulas for growth signal progression in time and space and found that a localised initial signal maintains its Gaussian shape, but gets delocalised and exponentially degraded. Importantly, mathematical treatment of the reaction diffusion system revealed that diffusion filtered out high frequencies of spatially periodic initiator signals suggesting that the muscle cell is robust against fluctuations in spatial receptor expression or activation. However, neither scenario was consistent with the presence of stably travelling signal waves. Our study highlights the role of feedback loops in spatiotemporal signal progression and results can be applied to studies in cell proliferation, cell differentiation and cell death in other spatially extended cells.