Modeling breast cancer proliferation, drug synergies, and alternating therapies

Estrogen receptor positive (ER+) breast cancer is responsive to a number of targeted therapies used clinically. Unfortunately, the continuous application of targeted therapy often results in resistance, driving the consideration of combination and alternating therapies. Toward this end, we developed a mathematical model that can simulate various mono, combination, and alternating therapies for ER + breast cancer cells at different doses over long time scales. The model is used to look for optimal drug combinations and predicts a significant synergism between Cdk4/6 inhibitors in combination with the anti-estrogen fulvestrant, which may help explain the clinical success of adding Cdk4/6 inhibitors to anti-estrogen therapy. Furthermore, the model is used to optimize an alternating treatment protocol so it works as well as monotherapy while using less total drug dose.

Mathematically Modeling the Effect of Endocrine and Cdk4/6 Inhibitor Therapies on Breast Cancer Cells

Mathematical modeling of cancer systems is beginning to be used to design better treatment regimens, especially in chemotherapy and radiotherapy. The effectiveness of mathematical modeling to inform treatment decisions and identify therapy protocols, some of which are highly nonintuitive, is because it enables the exploration of a huge number of therapeutic possibilities. Considering the immense cost of laboratory research and clinical trials, these nonintuitive therapy protocols would likely never be found by experimental approaches. While much of the work to date in this area has involved high-level models, which look simply at overall tumor growth or the interaction of resistant and sensitive cell types, mechanistic models that integrate molecular biology and pharmacology can contribute greatly to the discovery of better cancer treatment regimens. These mechanistic models are better able to account for the effect of drug interactions and the dynamics of therapy. The aim of this chapter is to demonstrate the use of ordinary differential equation-based mechanistic models to describe the dynamic interactions between the molecular signaling of breast cancer cells and two key clinical drugs. In particular, we illustrate the procedure for building a model of the response of MCF-7 cells to standard therapies used in the clinic. Such mathematical models can be used to explore the vast number of potential protocols to suggest better treatment approaches.

Mathematical modelling of breast cancer cells in response to endocrine therapy and Cdk4/6 inhibition

Oestrogen receptor (ER)-positive breast cancer is responsive to a number of targeted therapies used clinically. Unfortunately, the continuous application of any targeted therapy often results in resistance to the therapy. Our ultimate goal is to use mathematical modelling to optimize alternating therapies that not only decrease proliferation but also stave off resistance. Toward this end, we measured levels of key proteins and proliferation over a 7-day time course in ER+ MCF-7 breast cancer cells. Treatments included endocrine therapy, either oestrogen deprivation, which mimics the effects of an aromatase inhibitor, or fulvestrant, an ER degrader. These data were used to calibrate a mathematical model based on key interactions between ER signalling and the cell cycle. We show that the calibrated model is capable of predicting the combination treatment of fulvestrant and oestrogen deprivation. Further, we show that we can add a new drug, palbociclib, to the model by measuring only two key proteins, cMyc and hyperphosphorylated RB1, and adjusting only parameters associated with the drug. The model is then able to predict the combination treatment of oestrogen deprivation and palbociclib. We illustrate the model's potential to explore protocols that limit proliferation and hold off resistance by not depending on any one therapy.

Abstract 674: Predicting cellular response to therapy in breast cancer using mathematical modeling

About 70% of all breast cancer tumors are estrogen receptor positive (ER+) and are treated with antiestrogen therapies. While the inevitability of developing resistance to these therapies remains uncontested, little is known about the mechanism and prevention of resistance. Our ultimate goal is to use mathematical modeling to optimize dynamic therapies that decrease proliferation and stave off resistance. In this initial study, we used MCF7 cells as a model of ER+ breast cancer and estrogen deprivation as a surrogate for aromatase inhibitors. We developed long-term estrogen deprived MCF7s (LTEDs) that proliferate similarly to untreated MCF7s but are resistant to antiestrogens. We collected time-course data for simultaneous gene expression and protein levels (NanoString Pan Cancer panel, a non-amplification based digital method) over 6 weeks to capture early molecular adaptations of deprived MCF7s and compare them to those present in LTEDs. PCA analysis of the mRNA and protein data shows a dramatic change in the first component due to estrogen deprivation, and a dramatic change in the second component associated with long-term resistance. Correlation analysis shows a large number of cell cycle genes that are similarly regulated, decreasing with estrogen deprivation and increasing with the onset of resistance. There is also a highly-correlated group of genes associated with resistance. This knowledge allowed us to hypothesize a molecular mechanism for resistance that can be tested experimentally. To begin building a dynamic model, we measured a 7-day time course of estrogen related proteins. The model is built around ER signaling and the cell cycle, and simulates protein and proliferation changes in response to deprivation and antiestrogen (ICI182,780; ICI) treatment. To determine which treatments to model in addition to estrogen deprivation and ICI, we looked for a promising sequential therapy to limit proliferation and, hopefully, stave off resistance compared to a single non-stop therapy. We found that alternating the targeting of CDK4/6 using Palbociclib (clinical anti-cancer therapy) with targeting of RUNX1 (gene that is increased at the onset of resistance in our model), could be a plausible strategy to inhibit endocrine resistance. Future work will involve extending the model to longer time scales and using it for treatment optimization.

Citation Format: Wei He, Diane M. Demas, Isabel Conde, Yassi Fallah, William T. Baumann, Ayesha N. Shajahan-Haq. Predicting cellular response to therapy in breast cancer using mathematical modeling [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2019; 2019 Mar 29-Apr 3; Atlanta, GA. Philadelphia (PA): AACR; Cancer Res 2019;79(13 Suppl):Abstract nr 674.

Systems biology: perspectives on multiscale modeling in research on endocrine-related cancers

Drawing on concepts from experimental biology, computer science, informatics, mathematics and statistics, systems biologists integrate data across diverse platforms and scales of time and space to create computational and mathematical models of the integrative, holistic functions of living systems. Endocrine-related cancers are well suited to study from a systems perspective because of the signaling complexities arising from the roles of growth factors, hormones and their receptors as critical regulators of cancer cell biology and from the interactions among cancer cells, normal cells and signaling molecules in the tumor microenvironment. Moreover, growth factors, hormones and their receptors are often effective targets for therapeutic intervention, such as estrogen biosynthesis, estrogen receptors or HER2 in breast cancer and androgen receptors in prostate cancer. Given the complexity underlying the molecular control networks in these cancers, a simple, intuitive understanding of how endocrine-related cancers respond to therapeutic protocols has proved incomplete and unsatisfactory. Systems biology offers an alternative paradigm for understanding these cancers and their treatment. To correctly interpret the results of systems-based studies requires some knowledge of how in silico models are built, and how they are used to describe a system and to predict the effects of perturbations on system function. In this review, we provide a general perspective on the field of cancer systems biology, and we explore some of the advantages, limitations and pitfalls associated with using predictive multiscale modeling to study endocrine-related cancers.

Predicting network modules of cell cycle regulators using relative protein abundance statistics

BACKGROUND

Parameter estimation in systems biology is typically done by enforcing experimental observations through an objective function as the parameter space of a model is explored by numerical simulations. Past studies have shown that one usually finds a set of "feasible" parameter vectors that fit the available experimental data equally well, and that these alternative vectors can make different predictions under novel experimental conditions. In this study, we characterize the feasible region of a complex model of the budding yeast cell cycle under a large set of discrete experimental constraints in order to test whether the statistical features of relative protein abundance predictions are influenced by the topology of the cell cycle regulatory network.

RESULTS

Using differential evolution, we generate an ensemble of feasible parameter vectors that reproduce the phenotypes (viable or inviable) of wild-type yeast cells and 110 mutant strains. We use this ensemble to predict the phenotypes of 129 mutant strains for which experimental data is not available. We identify 86 novel mutants that are predicted to be viable and then rank the cell cycle proteins in terms of their contributions to cumulative variability of relative protein abundance predictions. Proteins involved in "regulation of cell size" and "regulation of G1/S transition" contribute most to predictive variability, whereas proteins involved in "positive regulation of transcription involved in exit from mitosis," "mitotic spindle assembly checkpoint" and "negative regulation of cyclin-dependent protein kinase by cyclin degradation" contribute the least. These results suggest that the statistics of these predictions may be generating patterns specific to individual network modules (START, S/G2/M, and EXIT). To test this hypothesis, we develop random forest models for predicting the network modules of cell cycle regulators using relative abundance statistics as model inputs. Predictive performance is assessed by the areas under receiver operating characteristics curves (AUC). Our models generate an AUC range of 0.83-0.87 as opposed to randomized models with AUC values around 0.50.

CONCLUSIONS

By using differential evolution and random forest modeling, we show that the model prediction statistics generate distinct network module-specific patterns within the cell cycle network.

Abstract P4-08-03: Mathematical modeling of the unfolded protein response pathway in breast cancer cells

Over 70% of all breast cancers are estrogen receptor positive (ER+) and are treated with endocrine therapies such as aromatase inhibitors or ER-disruptors as part of the standard of care. However, endocrine resistance remains a major clinical challenge for the successful treatment of this disease. ER+ breast cancer cells can escape endocrine therapy-mediated cell death through adaptations in the unfolded protein response (UPR), an evolutionarily conserved cellular stress response pathway that is linked to many other pathways (autophagy, metabolism, anti-oxidant response). To better understand how the UPR pathway promotes endocrine resistance, we built a mathematical model from quantitative measurements of key UPR signaling molecules following treatment of ER+ MCF7 cells (endocrine sensitive, estrogen dependent) with dithiothreitol (DTT). DTT disrupts or prevents protein disulfide bonding and is a potent inducer of the UPR. The same measurements were also made in two MCF7-derived resistant cell lines: LCC1 (sensitive to ER-disruptors, estrogen independent) and LCC9 (resistant to ER-disruptors, estrogen independent). Determining the (minimal) changes to the MCF7 mathematical model needed to adequately capture the data from the resistant cell lines allows us to create specific hypotheses for how the UPR adapts to help produce resistance to endocrine therapies. These hypotheses, in turn, can lead to drug targets for reversing resistance.

Citation Format: Shajahan-Haq AN, Demas DM, Clarke R, Baumann WT. Mathematical modeling of the unfolded protein response pathway in breast cancer cells [abstract]. In: Proceedings of the 2016 San Antonio Breast Cancer Symposium; 2016 Dec 6-10; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2017;77(4 Suppl):Abstract nr P4-08-03.

A Model of Yeast Cell-Cycle Regulation Based on a Standard Component Modeling Strategy for Protein Regulatory Networks

To understand the molecular mechanisms that regulate cell cycle progression in eukaryotes, a variety of mathematical modeling approaches have been employed, ranging from Boolean networks and differential equations to stochastic simulations. Each approach has its own characteristic strengths and weaknesses. In this paper, we propose a “standard component” modeling strategy that combines advantageous features of Boolean networks, differential equations and stochastic simulations in a framework that acknowledges the typical sorts of reactions found in protein regulatory networks. Applying this strategy to a comprehensive mechanism of the budding yeast cell cycle, we illustrate the potential value of standard component modeling. The deterministic version of our model reproduces the phenotypic properties of wild-type cells and of 125 mutant strains. The stochastic version of our model reproduces the cell-to-cell variability of wild-type cells and the partial viability of the CLB2-dbΔ clb5Δ mutant strain. Our simulations show that mathematical modeling with “standard components” can capture in quantitative detail many essential properties of cell cycle control in budding yeast.

Dynamic Modeling of the Interaction Between Autophagy and Apoptosis in Mammalian Cells

Autophagy is a conserved biological stress response in mammalian cells that is responsible for clearing damaged proteins and organelles from the cytoplasm and recycling their contents via the lysosomal pathway. In cases of mild stress, autophagy acts as a survival mechanism, while in cases of severe stress cells may switch to programmed cell death. Understanding the decision process that moves a cell from autophagy to apoptosis is important since abnormal regulation of autophagy occurs in many diseases, including cancer. To integrate existing knowledge about this decision process into a rigorous, analytical framework, we built a mathematical model of cell fate decisions mediated by autophagy. Our dynamical model is consistent with existing quantitative measurements of autophagy and apoptosis in rat kidney proximal tubular cells responding to cisplatin-induced stress.

Linking autophagy with inflammation through IRF1 signaling in ER+ breast cancer

Resistance to antiestrogen therapy remains a critical determinant of mortality in patients affected by ER+ breast cancer. Our previous work identified autophagy and interferon regulatory factor 1 (IRF1) signaling as key regulators of this process. We have recently demonstrated a novel reciprocal interaction between IRF1 and ATG7, linking inflammation and autophagy.

Interferon regulatory factor-1 signaling regulates the switch between autophagy and apoptosis to determine breast cancer cell fate

Interferon regulatory factor-1 (IRF1) is a tumor suppressor that regulates cell fate in several cell types. Here, we report an inverse correlation in expression of nuclear IRF1 and the autophagy regulator ATG7 in human breast cancer cells that directly affects their cell fate. In mice harboring mutant Atg7, nuclear IRF1 was increased in mammary tumors, spleen, and kidney. Mechanistic investigations identified ATG7 and the cell death modulator beclin-1 (BECN1) as negative regulators of IRF1. Silencing ATG7 or BECN1 caused estrogen receptor-α to exit the nucleus at the time when IRF1 nuclear localization occurred. Conversely, silencing IRF1 promoted autophagy by increasing BECN1 and blunting IGF1 receptor and mTOR survival signaling. Loss of IRF1 promoted resistance to antiestrogens, whereas combined silencing of ATG7 and IRF1 restored sensitivity to these agents. Using a mathematical model to prompt signaling hypotheses, we developed evidence that ATG7 silencing could resensitize IRF1-attenuated cells to apoptosis through mechanisms that involve other estrogen-regulated genes. Overall, our work shows how inhibiting the autophagy proteins ATG7 and BECN1 can regulate IRF1-dependent and -independent signaling pathways in ways that engender a new therapeutic strategy to attack breast cancer.

Knockdown of estrogen receptor-α induces autophagy and inhibits antiestrogen-mediated unfolded protein response activation, promoting ROS-induced breast cancer cell death

Approximately 70% of all newly diagnosed breast cancers express estrogen receptor (ER)-α. Although inhibiting ER action using targeted therapies such as fulvestrant (ICI) is often effective, later emergence of antiestrogen resistance limits clinical use. We used antiestrogen-sensitive and -resistant cells to determine the effect of antiestrogens/ERα on regulating autophagy and unfolded protein response (UPR) signaling. Knockdown of ERα significantly increased the sensitivity of LCC1 cells (sensitive) and also resensitized LCC9 cells (resistant) to antiestrogen drugs. Interestingly, ERα knockdown, but not ICI, reduced nuclear factor (erythroid-derived 2)-like (NRF)-2 (UPR-induced antioxidant protein) and increased cytosolic kelch-like ECH-associated protein (KEAP)-1 (NRF2 inhibitor), consistent with the observed increase in ROS production. Furthermore, autophagy induction by antiestrogens was prosurvival but did not prevent ERα knockdown-mediated death. We built a novel mathematical model to elucidate the interactions among UPR, autophagy, ER signaling, and ROS regulation of breast cancer cell survival. The experimentally validated mathematical model explains the counterintuitive result that knocking down the main target of ICI (ERα) increased the effectiveness of ICI. Specifically, the model indicated that ERα is no longer present in excess and that the effect on proliferation from further reductions in its level by ICI cannot be compensated for by increased autophagy. The stimulation of signaling that can confer resistance suggests that combining autophagy or UPR inhibitors with antiestrogens would reduce the development of resistance in some breast cancers.

A stochastic model correctly predicts changes in budding yeast cell cycle dynamics upon periodic expression of CLN2

In this study, we focus on a recent stochastic budding yeast cell cycle model. First, we estimate the model parameters using extensive data sets: phenotypes of 110 genetic strains, single cell statistics of wild type and cln3 strains. Optimization of stochastic model parameters is achieved by an automated algorithm we recently used for a deterministic cell cycle model. Next, in order to test the predictive ability of the stochastic model, we focus on a recent experimental study in which forced periodic expression of CLN2 cyclin (driven by MET3 promoter in cln3 background) has been used to synchronize budding yeast cell colonies. We demonstrate that the model correctly predicts the experimentally observed synchronization levels and cell cycle statistics of mother and daughter cells under various experimental conditions (numerical data that is not enforced in parameter optimization), in addition to correctly predicting the qualitative changes in size control due to forced CLN2 expression. Our model also generates a novel prediction: under frequent CLN2 expression pulses, G1 phase duration is bimodal among small-born cells. These cells originate from daughters with extended budded periods due to size control during the budded period. This novel prediction and the experimental trends captured by the model illustrate the interplay between cell cycle dynamics, synchronization of cell colonies, and size control in budding yeast.

Mathematical models of the transitions between endocrine therapy responsive and resistant states in breast cancer

Endocrine therapy, targeting the oestrogen receptor pathway, is the most common treatment for oestrogen receptor-positive breast cancers. Unfortunately, these tumours frequently develop resistance to endocrine therapies. Among the strategies to treat resistant tumours are sequential treatment (in which second-line drugs are used to gain additional responses) and intermittent treatment (in which a 'drug holiday' is imposed between treatments). To gain a more rigorous understanding of the mechanisms underlying these strategies, we present a mathematical model that captures the transitions among three different, experimentally observed, oestrogen-sensitivity phenotypes in breast cancer (sensitive, hypersensitive and independent). To provide a global view of the transitions between these phenotypes, we compute the potential landscape associated with the model. We show how this oestrogen response landscape can be reshaped by population selection, which is a crucial force in promoting acquired resistance. Techniques from statistical physics are used to create a population-level state-transition model from the cellular-level model. We then illustrate how this population-level model can be used to analyse and optimize sequential and intermittent oestrogen-deprivation protocols for breast cancer. The approach used in this study is general and can also be applied to investigate treatment strategies for other types of cancer.

Modelling the effect of GRP78 on anti-oestrogen sensitivity and resistance in breast cancer

Understanding the origins of resistance to anti-oestrogen drugs is of critical importance to many breast cancer patients. Recent experiments show that knockdown of GRP78, a key gene in the unfolded protein response (UPR), can re-sensitize resistant cells to anti-oestrogens, and overexpression of GRP78 in sensitive cells can cause them to become resistant. These results appear to arise from the operation and interaction of three cellular systems: the UPR, autophagy and apoptosis. To determine whether our current mechanistic understanding of these systems is sufficient to explain the experimental results, we built a mathematical model of the three systems and their interactions. We show that the model is capable of reproducing previously published experimental results and some new data gathered specifically for this paper. The model provides us with a tool to better understand the interactions that bring about anti-oestrogen resistance and the effects of GRP78 on both sensitive and resistant breast cancer cells.

Optimization and model reduction in the high dimensional parameter space of a budding yeast cell cycle model

Background

Parameter estimation from experimental data is critical for mathematical modeling of protein regulatory networks. For realistic networks with dozens of species and reactions, parameter estimation is an especially challenging task. In this study, we present an approach for parameter estimation that is effective in fitting a model of the budding yeast cell cycle (comprising 26 nonlinear ordinary differential equations containing 126 rate constants) to the experimentally observed phenotypes (viable or inviable) of 119 genetic strains carrying mutations of cell cycle genes.

Results

Starting from an initial guess of the parameter values, which correctly captures the phenotypes of only 72 genetic strains, our parameter estimation algorithm quickly improves the success rate of the model to 105–111 of the 119 strains. This success rate is comparable to the best values achieved by a skilled modeler manually choosing parameters over many weeks. The algorithm combines two search and optimization strategies. First, we use Latin hypercube sampling to explore a region surrounding the initial guess. From these samples, we choose ∼20 different sets of parameter values that correctly capture wild type viability. These sets form the starting generation of differential evolution that selects new parameter values that perform better in terms of their success rate in capturing phenotypes. In addition to producing highly successful combinations of parameter values, we analyze the results to determine the parameters that are most critical for matching experimental outcomes and the most competitive strains whose correct outcome with a given parameter vector forces numerous other strains to have incorrect outcomes. These “most critical parameters” and “most competitive strains” provide biological insights into the model. Conversely, the “least critical parameters” and “least competitive strains” suggest ways to reduce the computational complexity of the optimization.

Conclusions

Our approach proves to be a useful tool to help systems biologists fit complex dynamical models to large experimental datasets. In the process of fitting the model to the data, the tool identifies suggestive correlations among aspects of the model and the data.

Endoplasmic reticulum stress, the unfolded protein response, autophagy, and the integrated regulation of breast cancer cell fate

How breast cancer cells respond to the stress of endocrine therapies determines whether they will acquire a resistant phenotype or execute a cell-death pathway. After a survival signal is successfully executed, a cell must decide whether it should replicate. How these cell-fate decisions are regulated is unclear, but evidence suggests that the signals that determine these outcomes are highly integrated. Central to the final cell-fate decision is signaling from the unfolded protein response, which can be activated following the sensing of stress within the endoplasmic reticulum. The duration of the response to stress is partly mediated by the duration of inositol-requiring enzyme-1 activation following its release from heat shock protein A5. The resulting signals appear to use several B-cell lymphoma-2 family members to both suppress apoptosis and activate autophagy. Changes in metabolism induced by cellular stress are key components of this regulatory system, and further adaptation of the metabolome is affected in response to stress. Here we describe the unfolded protein response, autophagy, and apoptosis, and how the regulation of these processes is integrated. Central topologic features of the signaling network that integrate cell-fate regulation and decision execution are discussed.

Oscillatory dynamics of cell cycle proteins in single yeast cells analyzed by imaging cytometry

Progression through the cell division cycle is orchestrated by a complex network of interacting genes and proteins. Some of these proteins are known to fluctuate periodically during the cell cycle, but a systematic study of the fluctuations of a broad sample of cell-cycle proteins has not been made until now. Using time-lapse fluorescence microscopy, we profiled 16 strains of budding yeast, each containing GFP fused to a single gene involved in cell cycle regulation. The dynamics of protein abundance and localization were characterized by extracting the amplitude, period, and other indicators from a series of images. Oscillations of protein abundance could clearly be identified for Cdc15, Clb2, Cln1, Cln2, Mcm1, Net1, Sic1, and Whi5. The period of oscillation of the fluorescently tagged proteins is generally in good agreement with the inter-bud time. The very strong oscillations of Net1 and Mcm1 expression are remarkable since little is known about the temporal expression of these genes. By collecting data from large samples of single cells, we quantified some aspects of cell-to-cell variability due presumably to intrinsic and extrinsic noise affecting the cell cycle.

Dynamic modelling of oestrogen signalling and cell fate in breast cancer cells

Cancers of the breast and other tissues arise from aberrant decision-making by cells regarding their survival or death, proliferation or quiescence, damage repair or bypass. These decisions are made by molecular signalling networks that process information from outside and from within the breast cancer cell and initiate responses that determine the cell's survival and reproduction. Because the molecular logic of these circuits is difficult to comprehend by intuitive reasoning alone, we present some preliminary mathematical models of the basic decision circuits in breast cancer cells that may aid our understanding of their susceptibility or resistance to endocrine therapy.

Stochastic exit from mitosis in budding yeast: model predictions and experimental observations

Unlike many mutants that are completely viable or inviable, the CLB2-dbΔ clb5Δ mutant of Saccharomyces cerevisiae is inviable in glucose but partially viable on slower growth media such as raffinose. On raffinose, the mutant cells can bud and divide but in each cycle there is a chance that a cell will fail to divide (telophase arrest), causing it to exit the cell cycle. This effect gives rise to a stochastic phenotype that cannot be explained by a deterministic model. We measure the inter-bud times of wild type and mutant cells growing on raffinose and compute statistics and distributions to characterize the mutant's behavior. We convert a detailed deterministic model of the budding yeast cell cycle to a stochastic model and determine the extent to which it captures the stochastic phenotype of the mutant strain. Predictions of the mathematical model are in reasonable agreement with our experimental data and suggest directions for improving the model. Ultimately, the ability to accurately model stochastic phenotypes may prove critical to understanding disease and therapeutic interventions in higher eukaryotes.

Exploring the roles of noise in the eukaryotic cell cycle

The DNA replication-division cycle of eukaryotic cells is controlled by a complex network of regulatory proteins, called cyclin-dependent kinases, and their activators and inhibitors. Although comprehensive and accurate deterministic models of the control system are available for yeast cells, reliable stochastic simulations have not been carried out because the full reaction network has yet to be expressed in terms of elementary reaction steps. As a first step in this direction, we present a simplified version of the control system that is suitable for exact stochastic simulation of intrinsic noise caused by molecular fluctuations and extrinsic noise because of unequal division. The model is consistent with many characteristic features of noisy cell cycle progression in yeast populations, including the observation that mRNAs are present in very low abundance (approximately 1 mRNA molecule per cell for each expressed gene). For the control system to operate reliably at such low mRNA levels, some specific mRNAs in our model must have very short half-lives (<1 min). If these mRNA molecules are longer-lived (perhaps 2 min), then the intrinsic noise in our simulations is too large, and there must be some additional noise suppression mechanisms at work in cells.