A Provably Convergent Control Closure Scheme for the Method of Moments of the Chemical Master Equation.

In this article, we introduce a novel moment closure scheme based on concepts from model predictive control (MPC) to accurately describe the time evolution of the statistical moments of the solution of the chemical master equation (CME). The method of moments, a set of ordinary differential equations frequently used to calculate the first nm moments, is generally not closed since lower-order moments depend on higher-order moments. To overcome this limitation, we interpret the moment equations as a nonlinear dynamical system, where the first nm moments serve as states, and the closing moments serve as the control input. We demonstrate the efficacy of our approach using three example systems and show that it outperforms existing closure schemes. For polynomial systems, which encompass all mass-action systems, we provide probability bounds for the error between true and estimated moment trajectories. We achieve this by combining the convergence properties of a priori moment estimates from stochastic simulations with guarantees for nonlinear reference tracking MPC. Our proposed method offers an effective solution to accurately predict the time evolution of moments of the CME, which has wide-ranging implications for many fields, including biology, chemistry, and engineering.

Model-Based and Data-Driven Control of Event- and Self-Triggered Discrete-Time Linear Systems.

The present paper considers the model-based and data-driven control of unknown discrete-time linear systems under event-triggering and self-triggering transmission schemes. To this end, we begin by presenting a dynamic event-triggering scheme (ETS) based on periodic sampling, and a discrete-time looped-functional approach, through which a model-based stability condition is derived. Combining the model-based condition with a recent data-based system representation, a data-driven stability criterion in the form of linear matrix inequalities (LMIs) is established, which also offers a way of co-designing the ETS matrix and the controller. To further alleviate the sampling burden of ETS due to its continuous/periodic detection, a self-triggering scheme (STS) is developed. Leveraging precollected input-state data, an algorithm for predicting the next transmission instant is given, while achieving system stability. Finally, numerical simulations showcase the efficacy of ETS and STS in reducing data transmissions as well as practicality of the proposed co-design methods.

Cell cycle progression and transmitotic apoptosis resistance promote escape from extrinsic apoptosis.

Extrinsic apoptosis relies on TNF-family receptor activation by immune cells or receptor-activating drugs. Here, we monitored cell cycle progression at a resolution of minutes to relate apoptosis kinetics and cell-to-cell heterogeneities in death decisions to cell cycle phases. Interestingly, we found that cells in S phase delay TRAIL receptor-induced death in favour of mitosis, thereby passing on an apoptosis-primed state to their offspring. This translates into two distinct fates, apoptosis execution post mitosis or cell survival from inefficient apoptosis. Transmitotic resistance is linked to Mcl-1 upregulation and its increased accumulation at mitochondria from mid-S phase onwards, which allows cells to pass through mitosis with activated caspase-8, and with cells escaping apoptosis after mitosis sustaining sublethal DNA damage. Antagonizing Mcl-1 suppresses cell cycle-dependent delays in apoptosis, prevents apoptosis-resistant progression through mitosis and averts unwanted survival after apoptosis induction. Cell cycle progression therefore modulates signal transduction during extrinsic apoptosis, with Mcl-1 governing decision making between death, proliferation and survival. Cell cycle progression thus is a crucial process from which cell-to-cell heterogeneities in fates and treatment outcomes emerge in isogenic cell populations during extrinsic apoptosis. This article has an associated First Person interview with the first author of the paper.

Robust and optimal predictive control of the COVID-19 outbreak.

We investigate adaptive strategies to robustly and optimally control the COVID-19 pandemic via social distancing measures based on the example of Germany. Our goal is to minimize the number of fatalities over the course of two years without inducing excessive social costs. We consider a tailored model of the German COVID-19 outbreak with different parameter sets to design and validate our approach. Our analysis reveals that an open-loop optimal control policy can significantly decrease the number of fatalities when compared to simpler policies under the assumption of exact model knowledge. In a more realistic scenario with uncertain data and model mismatch, a feedback strategy that updates the policy weekly using model predictive control (MPC) leads to a reliable performance, even when applied to a validation model with deviant parameters. On top of that, we propose a robust MPC-based feedback policy using interval arithmetic that adapts the social distancing measures cautiously and safely, thus leading to a minimum number of fatalities even if measurements are inaccurate and the infection rates cannot be precisely specified by social distancing. Our theoretical findings support various recent studies by showing that (1) adaptive feedback strategies are required to reliably contain the COVID-19 outbreak, (2) well-designed policies can significantly reduce the number of fatalities compared to simpler ones while keeping the amount of social distancing measures on the same level, and (3) imposing stronger social distancing measures early on is more effective and cheaper in the long run than opening up too soon and restoring stricter measures at a later time.

Sample-based modeling reveals bidirectional interplay between cell cycle progression and extrinsic apoptosis.

Apoptotic cell death can be initiated through the extrinsic and intrinsic signaling pathways. While cell cycle progression promotes the responsiveness to intrinsic apoptosis induced by genotoxic stress or spindle poisons, this has not yet been studied conclusively for extrinsic apoptosis. Here, we combined fluorescence-based time-lapse monitoring of cell cycle progression and cell death execution by long-term time-lapse microscopy with sampling-based mathematical modeling to study cell cycle dependency of TRAIL-induced extrinsic apoptosis in NCI-H460/geminin cells. In particular, we investigated the interaction of cell death timing and progression of cell cycle states. We not only found that TRAIL prolongs cycle progression, but in reverse also that cell cycle progression affects the kinetics of TRAIL-induced apoptosis: Cells exposed to TRAIL in G1 died significantly faster than cells stimulated in S/G2/M. The connection between cell cycle state and apoptosis progression was captured by developing a mathematical model, for which parameter estimation revealed that apoptosis progression decelerates in the second half of the cell cycle. Similar results were also obtained when studying HCT-116 cells. Our results therefore reject the null hypothesis of independence between cell cycle progression and extrinsic apoptosis and, supported by simulations and experiments of synchronized cell populations, suggest that unwanted escape from TRAIL-induced apoptosis can be reduced by enriching the fraction of cells in G1 phase. Besides novel insight into the interrelation of cell cycle progression and extrinsic apoptosis signaling kinetics, our findings are therefore also relevant for optimizing future TRAIL-based treatment strategies.

Reconstructing temporal and spatial dynamics from single-cell pseudotime using prior knowledge of real scale cell densities.

Modern cytometry methods allow collecting complex, multi-dimensional data sets from heterogeneous cell populations at single-cell resolution. While methods exist to describe the progression and order of cellular processes from snapshots of such populations, these descriptions are limited to arbitrary pseudotime scales. Here we describe MAPiT, an universal transformation method that recovers real-time dynamics of cellular processes from pseudotime scales by utilising knowledge of the distributions on the real scales. As use cases, we applied MAPiT to two prominent problems in the flow-cytometric analysis of heterogeneous cell populations: (1) recovering the kinetics of cell cycle progression in unsynchronised and thus unperturbed cell populations, and (2) recovering the spatial arrangement of cells within multi-cellular spheroids prior to spheroid dissociation for cytometric analysis. Since MAPiT provides a theoretic basis for the relation of pseudotime values to real temporal and spatial scales, it can be used broadly in the analysis of cellular processes with snapshot data from heterogeneous cell populations.

Moment Dynamics of Zirconia Particle Formation for Optimizing Particle Size Distribution.

We study the particle formation process of Zirconia ( ZrO 2 )-based material. With a model-based description of the particle formation process we aim for identifying the main growth mechanisms for different process parameters. After the introduction of a population balance based mathematical model, we derive the moment dynamics of the particle size distribution and compare the model to experimental data. From the fitted model we conclude that growth by molecular addition of Zr-tetramers or Zr-oligomers to growing particles as well as size-independent particle agglomeration takes place. For the purpose of depositing zirconia-based material (ZrbM) on a substrate, we determine the optimal process parameters such that the mineralization solution contains preferably a large number of nanoscaled particles leading to a fast and effective deposition on the substrate. Besides the deposition of homogeneous films, this also enables mineralization of nanostructured templates in a bioinspired mineralization process. The developed model is also transferable to other mineralization systems where particle growth occurs through addition of small molecular species or particle agglomeration. This offers the possibility for a fast determination of process parameters leading to an efficient film formation without carrying out extensive experimental investigations.

Analysis of primitive genetic interactions for the design of a genetic signal differentiator.

We study the dynamic and static input-output behavior of several primitive genetic interactions and their effect on the performance of a genetic signal differentiator. In a simplified design, several requirements for the linearity and time-scales of processes like transcription, translation and competitive promoter binding were introduced. By experimentally probing simple genetic constructs in a cell-free experimental environment and fitting semi-mechanistic models to these data, we show that some of these requirements can be verified, while others are only met with reservations in certain operational regimes. Analyzing the linearized model of the resulting genetic network, we conclude that it approximates a differentiator with relative degree one. Taking also the discovered nonlinearities into account and using a describing function approach, we further determine the particular frequency and amplitude ranges where the genetic differentiator can be expected to behave as such.

Sensitization of glioblastoma cells to TRAIL-induced apoptosis by IAP- and Bcl-2 antagonism.

Due to the lack of effective treatments for glioblastoma (GBM), we here studied the responsiveness of GBM cell lines to the combination of death ligand, TRAIL and the IAP antagonist, TL32711 (Birinapant). Responses were highly heterogeneous, with synergistic apoptosis as well as treatment resistance observed. Caspase-8 and Bid, together with caspase-3, form a nonlinear signalling hub that efficiently induced apoptosis in responder cell lines. Cells resistant to TRAIL/TL32711 expressed low amounts of procaspase-8 and Bid and poorly activated caspase-3. We therefore hypothesised that improving caspase-8 activation or sensitising mitochondria to truncated Bid (tBid) could convert non-responder GBM cell lines to responders. Mathematical simulations of both strategies predicted mitochondrial sensitization to tBid would outperform enhancing caspase-8 activation. Indeed, antagonising Bcl-2 by ABT-199 allowed TRAIL/TL32711 response synergies to manifest in otherwise TRAIL resistant cell lines. These findings were further corroborated in experiments with a translationally relevant hexavalent TRAIL variant. Our study therefore demonstrates that a high caspase-8/Bid signature is associated with synergistic TRAIL/TL32711-induced apoptosis in GBM cells and outlines Bcl-2 antagonism as a highly potent intervention to sensitize highly TRAIL-resistant GBM cells to TRAIL/TL32711 combination treatment.

Bcl-2-mediated control of TRAIL-induced apoptotic response in the non-small lung cancer cell line NCI-H460 is effective at late caspase processing steps.

Dysregulation of the mitochondrial signaling pathway of apoptosis induction represents a major hurdle in tumor therapy. The objective of the presented work was to investigate the role of the intrinsic (mitochondrial) apoptotic pathway in the non-small lung cancer cell line NCI-H460 upon induction of apoptosis using the highly bioactive TRAIL derivative Db-scTRAIL. NCI-H460 cells were TRAIL sensitive but an only about 3 fold overexpression of Bcl-2 was sufficient to induce a highly TRAIL resistant phenotype, confirming that the mitochondrial pathway is crucial for TRAIL-induced apoptosis induction. TRAIL resistance was paralleled by a strong inhibition of caspase-8, -9 and -3 activities and blocked their full processing. Notably, especially the final cleavage steps of the initiator caspase-8 and the executioner caspase-3 were effectively blocked by Bcl-2 overexpression. Caspase-9 knockdown failed to protect NCI-H460 cells from TRAIL-induced cell death, suggesting a minor role of this initiator caspase in this apoptotic pathway. Rather, knockdown of the XIAP antagonist Smac resulted in enhanced caspase-3 degradation after stimulation of cells with TRAIL. Of note, downregulation of XIAP had only limited effects on TRAIL sensitivity of wild-type NCI-H460 cells, but resensitized Bcl-2 overexpressing cells for TRAIL-induced apoptosis. In particular, XIAP knockdown in combination with TRAIL allowed the final cleavage step of caspase-3 to generate the catalytically active p17 fragment, whose production was otherwise blocked in Bcl-2 overexpressing cells. Together, our data strongly suggest that XIAP-mediated inhibition of final caspase-3 processing is the last and major hurdle in TRAIL-induced apoptosis in NCI-H460 cells, which can be overcome by Smac in a Bcl-2 level dependent manner. Quantitative investigation of the XIAP/Smac interplay using a mathematical model approach corroborates our experimental data strengthening the suggested roles of XIAP and Smac as critical determinants for TRAIL sensitivity.

A resource dependent protein synthesis model for evaluating synthetic circuits.

Reliable in silico design of synthetic gene networks necessitates novel approaches to model the process of protein synthesis under the influence of limited resources. We present such a novel protein synthesis model which originates from the Ribosome Flow Model and among other things describes the movement of RNA-polymerase and ribosomes on mRNA and DNA templates, respectively. By analyzing the convergence properties of this model based upon geometric considerations, we present additional insights into the dynamic mechanisms of the process of protein synthesis. Further, we demonstrate how this model can be used to evaluate the performance of synthetic gene circuits under different loading scenarios.

The circuit-breaking algorithm for monotone systems.

In earlier work, we have introduced the circuit-breaking algorithm (CBA) for the analysis of intracellular regulation networks. This algorithm uses the network topology to construct a one-dimensional circuit-characteristic whose zeros correspond to the fixed points of the system. In this study, we apply the CBA to monotone systems whose flow preserves a partial order with respect to some orthant cone. We consider relations between stability of fixed points and the derivative of the corresponding zeros of the circuit-characteristic. In particular, we derive sufficient conditions for instability in case of global asymptotic stability of the open-loop system. Furthermore, we fully characterize stability of the fixed points if in addition the system is monotone. Combined with the theory of monotone systems, our results are used to characterize the long-term behavior of two models for different intracellular regulation processes.

Pain modulators regulate the dynamics of PKA-RII phosphorylation in subgroups of sensory neurons.

Knowledge about the molecular structure of protein kinase A (PKA) isoforms is substantial. In contrast, the dynamics of PKA isoform activity in living primary cells has not been investigated in detail. Using a high content screening microscopy approach, we identified the RIIß subunit of PKA-II to be predominantly expressed in a subgroup of sensory neurons. The RIIß-positive subgroup included most neurons expressing nociceptive markers (TRPV1, NaV1.8, CGRP, IB4) and responded to pain-eliciting capsaicin with calcium influx. Isoform-specific PKA reporters showed in sensory-neuron-derived F11 cells that the inflammatory mediator PGE2 specifically activated PKA-II but not PKA-I. Accordingly, pain-sensitizing inflammatory mediators and activators of PKA increased the phosphorylation of RII subunits (pRII) in subgroups of primary sensory neurons. Detailed analyses revealed basal pRII to be regulated by the phosphatase PP2A. Increase of pRII was followed by phosphorylation of CREB in a PKA-dependent manner. Thus, we propose RII phosphorylation to represent an isoform-specific readout for endogenous PKA-II activity in vivo, suggest RIIß as a novel nociceptive subgroup marker, and extend the current model of PKA-II activation by introducing a PP2A-dependent basal state.

A new model to simulate and analyze proliferating cell populations in BrdU labeling experiments.

BACKGROUND: This paper presents a novel model for proliferating cell populations in labeling experiments. It is especially tailored to the technique of Bromodeoxyuridine (BrdU), which is taken up by dividing cells and thus accumulates with increasing division number during uplabeling. The study of the evolving label intensities of BrdU labeled cell populations is aimed at quantifying proliferation properties such as division and death rates. RESULTS: In contrast to existing models, our model considers a labeling efficacy that follows a distribution, rather than a uniform value. It thereby allows to account for noise as well as possibly space-dependent heterogeneity in the effective label uptake of the individual cells in a population. Furthermore, it enables more informative comparison with experimental data: The population-level label distribution is provided as a model output, thereby increasing the information content compared to existing models that give the fraction of labeled cells or the mean label intensity. CONCLUSION: The presented model is to our knowledge the first one that predicts the full label distribution for BrdU labeling experiments. Thus, it can exploit more information, namely the full intensity distribution, from labeling measurements, and thereby opens up new quantitative insights into cell proliferation.

Wound-healing growth factor, basic FGF, induces Erk1/2-dependent mechanical hyperalgesia.

UNLABELLED: Growth factors such as nerve growth factor and glial cell line-derived neurotrophic factor are known to induce pain sensitization. However, a plethora of other growth factors is released during inflammation and tissue regeneration, and many of them are essential for wound healing. Which wound-healing factors also alter the sensitivity of nociceptive neurons is not well known. We studied the wound-healing factor, basic fibroblast growth factor (bFGF), for its role in pain sensitization. Reverse transcription polymerase chain reaction showed that the receptor of bFGF, FGFR1, is expressed in lumbar rat dorsal root ganglia (DRG). We demonstrated presence of FGFR1 protein in DRG neurons by a recently introduced quantitative automated immunofluorescent microscopic technique. FGFR1 was expressed in all lumbar DRG neurons as quantified by mixture modeling. Corroborating the mRNA and protein expression data, bFGF induced Erk1/2 phosphorylation in nociceptive neurons, which could be blocked by inhibition of FGF receptors. Furthermore, bFGF activated Erk1/2 in a dose- and time-dependent manner. Using single-cell electrophysiological recordings, we found that bFGF treatment of DRG neurons increased the current-density of NaV1.8 channels. Erk1/2 inhibitors abrogated this increase. Importantly, intradermal injection of bFGF in rats induced Erk1/2-dependent mechanical hyperalgesia. PERSPECTIVE: Analyzing intracellular signaling dynamics in nociceptive neurons has proven to be a powerful approach to identify novel modulators of pain. In addition to describing a new sensitizing factor, our findings indicate the potential to investigate wound-healing factors for their role in nociception.

A visual analytics approach for models of heterogeneous cell populations.

In recent years, cell population models have become increasingly common. In contrast to classic single cell models, population models allow for the study of cell-to-cell variability, a crucial phenomenon in most populations of primary cells, cancer cells, and stem cells. Unfortunately, tools for in-depth analysis of population models are still missing. This problem originates from the complexity of population models. Particularly important are methods to determine the source of heterogeneity (e.g., genetics or epigenetic differences) and to select potential (bio-)markers. We propose an analysis based on visual analytics to tackle this problem. Our approach combines parallel-coordinates plots, used for a visual assessment of the high-dimensional dependencies, and nonlinear support vector machines, for the quantification of effects. The method can be employed to study qualitative and quantitative differences among cells. To illustrate the different components, we perform a case study using the proapoptotic signal transduction pathway involved in cellular apoptosis.

Threshold-free population analysis identifies larger DRG neurons to respond stronger to NGF stimulation.

Sensory neurons in dorsal root ganglia (DRG) are highly heterogeneous in terms of cell size, protein expression, and signaling activity. To analyze their heterogeneity, threshold-based methods are commonly used, which often yield highly variable results due to the subjectivity of the individual investigator. In this work, we introduce a threshold-free analysis approach for sparse and highly heterogeneous datasets obtained from cultures of sensory neurons. This approach is based on population estimates and completely free of investigator-set parameters. With a quantitative automated microscope we measured the signaling state of single DRG neurons by immunofluorescently labeling phosphorylated, i.e., activated Erk1/2. The population density of sensory neurons with and without pain-sensitizing nerve growth factor (NGF) treatment was estimated using a kernel density estimator (KDE). By subtraction of both densities and integration of the positive part, a robust estimate for the size of the responsive subpopulations was obtained. To assure sufficiently large datasets, we determined the number of cells required for reliable estimates using a bootstrapping approach. The proposed methods were employed to analyze response kinetics and response amplitude of DRG neurons after NGF stimulation. We thereby determined the portion of NGF responsive cells on a true population basis. The analysis of the dose dependent NGF response unraveled a biphasic behavior, while the study of its time dependence showed a rapid response, which approached a steady state after less than five minutes. Analyzing two parameter correlations, we found that not only the number of responsive small-sized neurons exceeds the number of responsive large-sized neurons--which is commonly reported and could be explained by the excess of small-sized cells--but also the probability that small-sized cells respond to NGF is higher. In contrast, medium-sized and large-sized neurons showed a larger response amplitude in their mean Erk1/2 activity.

Heterogeneity reduces sensitivity of cell death for TNF-stimuli.

BACKGROUND: Apoptosis is a form of programmed cell death essential for the maintenance of homeostasis and the removal of potentially damaged cells in multicellular organisms. By binding its cognate membrane receptor, TNF receptor type 1 (TNF-R1), the proinflammatory cytokine Tumor Necrosis Factor (TNF) activates pro-apoptotic signaling via caspase activation, but at the same time also stimulates nuclear factor κB (NF-κB)-mediated survival pathways. Differential dose-response relationships of these two major TNF signaling pathways have been described experimentally and using mathematical modeling. However, the quantitative analysis of the complex interplay between pro- and anti-apoptotic signaling pathways is an open question as it is challenging for several reasons: the overall signaling network is complex, various time scales are present, and cells respond quantitatively and qualitatively in a heterogeneous manner. RESULTS: This study analyzes the complex interplay of the crosstalk of TNF-R1 induced pro- and anti-apoptotic signaling pathways based on an experimentally validated mathematical model. The mathematical model describes the temporal responses on both the single cell level as well as the level of a heterogeneous cell population, as observed in the respective quantitative experiments using TNF-R1 stimuli of different strengths and durations. Global sensitivity of the heterogeneous population was quantified by measuring the average gradient of time of death versus each population parameter. This global sensitivity analysis uncovers the concentrations of Caspase-8 and Caspase-3, and their respective inhibitors BAR and XIAP, as key elements for deciding the cell's fate. A simulated knockout of the NF-κB-mediated anti-apoptotic signaling reveals the importance of this pathway for delaying the time of death, reducing the death rate in the case of pulse stimulation and significantly increasing cell-to-cell variability. CONCLUSIONS: Cell ensemble modeling of a heterogeneous cell population including a global sensitivity analysis presented here allowed us to illuminate the role of the different elements and parameters on apoptotic signaling. The receptors serve to transmit the external stimulus; procaspases and their inhibitors control the switching from life to death, while NF-κB enhances the heterogeneity of the cell population. The global sensitivity analysis of the cell population model further revealed an unexpected impact of heterogeneity, i.e. the reduction of parametric sensitivity.

Identification of models of heterogeneous cell populations from population snapshot data.

BACKGROUND: Most of the modeling performed in the area of systems biology aims at achieving a quantitative description of the intracellular pathways within a "typical cell". However, in many biologically important situations even clonal cell populations can show a heterogeneous response. These situations require study of cell-to-cell variability and the development of models for heterogeneous cell populations. RESULTS: In this paper we consider cell populations in which the dynamics of every single cell is captured by a parameter dependent differential equation. Differences among cells are modeled by differences in parameters which are subject to a probability density. A novel Bayesian approach is presented to infer this probability density from population snapshot data, such as flow cytometric analysis, which do not provide single cell time series data. The presented approach can deal with sparse and noisy measurement data. Furthermore, it is appealing from an application point of view as in contrast to other methods the uncertainty of the resulting parameter distribution can directly be assessed. CONCLUSIONS: The proposed method is evaluated using artificial experimental data from a model of the tumor necrosis factor signaling network. We demonstrate that the methods are computationally efficient and yield good estimation result even for sparse data sets.

Bridging time scales in cellular decision making with a stochastic bistable switch.

BACKGROUND: Cellular transformations which involve a significant phenotypical change of the cell's state use bistable biochemical switches as underlying decision systems. Some of these transformations act over a very long time scale on the cell population level, up to the entire lifespan of the organism. RESULTS: In this work, we aim at linking cellular decisions taking place on a time scale of years to decades with the biochemical dynamics in signal transduction and gene regulation, occurring on a time scale of minutes to hours. We show that a stochastic bistable switch forms a viable biochemical mechanism to implement decision processes on long time scales. As a case study, the mechanism is applied to model the initiation of follicle growth in mammalian ovaries, where the physiological time scale of follicle pool depletion is on the order of the organism's lifespan. We construct a simple mathematical model for this process based on experimental evidence for the involved genetic mechanisms. CONCLUSIONS: Despite the underlying stochasticity, the proposed mechanism turns out to yield reliable behavior in large populations of cells subject to the considered decision process. Our model explains how the physiological time constant may emerge from the intrinsic stochasticity of the underlying gene regulatory network. Apart from ovarian follicles, the proposed mechanism may also be of relevance for other physiological systems where cells take binary decisions over a long time scale.