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To increase situational awareness and support evidence-based policy-making, we formulated two types of mathematical models for COVID-19 transmission within a regional population. One is a fitting function that can be calibrated to reproduce an epidemic curve with two timescales (e.g., fast growth and slow decay). The other is a compartmental model that accounts for quarantine, self-isolation, social distancing, a non-exponentially distributed incubation period, asymptomatic individuals, and mild and severe forms of symptomatic disease. Using Bayesian inference, we have been calibrating our models daily for consistency with new reports of confirmed cases from the 15 most populous metropolitan statistical areas in the United States and quantifying uncertainty in parameter estimates and predictions of future case reports. This online learning approach allows for early identification of new trends despite considerable variability in case reporting. We infer new significant upward trends for five of the metropolitan areas starting between 19-April-2020 and 12-June-2020.

Although many persons in the United States have acquired immunity to COVID-19, either through vaccination or infection with SARS-CoV-2, COVID-19 will pose an ongoing threat to non-immune persons so long as disease transmission continues. We can estimate when sustained disease transmission will end in a population by calculating the population-specific basic reproduction number ℛ0, the expected number of secondary cases generated by an infected person in the absence of any interventions. The value of ℛ0 relates to a herd immunity threshold (HIT), which is given by 1 − 1/ℛ0. When the immune fraction of a population exceeds this threshold, sustained disease transmission becomes exponentially unlikely (barring mutations allowing SARS-CoV-2 to escape immunity). Here, we report state-level ℛ0 estimates obtained using Bayesian inference. Maximum a posteriori estimates range from 7.1 for New Jersey to 2.3 for Wyoming, indicating that disease transmission varies considerably across states and that reaching herd immunity will be more difficult in some states than others. ℛ0 estimates were obtained from compartmental models via the next-generation matrix approach after each model was parameterized using regional daily confirmed case reports of COVID-19 from 21-January-2020 to 21-June-2020. Our ℛ0 estimates characterize infectiousness of ancestral strains, but they can be used to determine HITs for a distinct, currently dominant circulating strain, such as SARS-CoV-2 variant Delta (lineage B.1.617.2), if the relative infectiousness of the strain can be ascertained. On the basis of Delta-adjusted HITs, vaccination data, and seroprevalence survey data, we find that no state has achieved herd immunity as of 20-September-2021.Significance StatementCOVID-19 will continue to threaten non-immune persons in the presence of ongoing disease transmission. We can estimate when sustained disease transmission will end by calculating the population-specific basic reproduction number ℛ0, which relates to a herd immunity threshold (HIT), given by 1 − 1/ℛ0. When the immune fraction of a population exceeds this threshold, sustained disease transmission becomes exponentially unlikely. Here, we report state-level ℛ0 estimates indicating that disease transmission varies considerably across states. Our ℛ0 estimates can also be used to determine HITs for the Delta variant of COVID-19. On the basis of Delta-adjusted HITs, vaccination data, and serological survey results, we find that no state has yet achieved herd immunity.

The effects of molecularly targeted drug perturbations on cellular activities and fates are difficult to predict using intuition alone because of the complex behaviors of cellular regulatory networks. An approach to overcoming this problem is to develop mathematical models for predicting drug effects. Such an approach beckons for co-development of computational methods for extracting insights useful for guiding therapy selection and optimizing drug scheduling. Here, we present and evaluate a generalizable strategy for identifying drug dosing schedules that minimize the amount of drug needed to achieve sustained suppression or elevation of an important cellular activity/process, the recycling of cytoplasmic contents through (macro)autophagy. Therapeutic targeting of autophagy is currently being evaluated in diverse clinical trials but without the benefit of a control engineering perspective. Using a nonlinear ordinary differential equation (ODE) model that accounts for activating and inhibiting influences among protein and lipid kinases that regulate autophagy (MTORC1, ULK1, AMPK and VPS34) and methods guaranteed to find locally optimal control strategies, we find optimal drug dosing schedules (open-loop controllers) for each of six classes of drugs and drug pairs. Our approach is generalizable to designing monotherapy and multi therapy drug schedules that affect different cell signaling networks of interest.

Systems biology has experienced dramatic growth in the number, size, and complexity of computational models. To reproduce simulation results and reuse models, researchers must exchange unambiguous model descriptions. We review the latest edition of the Systems Biology Markup Language (SBML), a format designed for this purpose. A community of modelers and software authors developed SBML Level 3 over the past decade. Its modular form consists of a core suited to representing reaction‐based models and packages that extend the core with features suited to other model types including constraint‐based models, reaction‐diffusion models, logical network models, and rule‐based models. The format leverages two decades of SBML and a rich software ecosystem that transformed how systems biologists build and interact with models. More recently, the rise of multiscale models of whole cells and organs, and new data sources such as single‐cell measurements and live imaging, has precipitated new ways of integrating data with models. We provide our perspectives on the challenges presented by these developments and how SBML Level 3 provides the foundation needed to support this evolution., Over the past two decades, scientists from different fields have been developing SBML, a standard format for encoding computational models in biology and medicine. This article summarizes recent progress and gives perspectives on emerging challenges.

In systems biology, qualitative data are often generated, but rarely used to parameterize models. We demonstrate an approach in which qualitative and quantitative data can be combined for parameter identification. In this approach, qualitative data are converted into inequality constraints imposed on the outputs of the model. These inequalities are used along with quantitative data points to construct a single scalar objective function that accounts for both datasets. To illustrate the approach, we estimate parameters for a simple model describing Raf activation. We then apply the technique to a more elaborate model characterizing cell cycle regulation in yeast. We incorporate both quantitative time courses (561 data points) and qualitative phenotypes of 119 mutant yeast strains (1647 inequalities) to perform automated identification of 153 model parameters. We quantify parameter uncertainty using a profile likelihood approach. Our results indicate the value of combining qualitative and quantitative data to parameterize systems biology models.

Epidermal growth factor receptor kinase mutations drive oncogenesis, but the molecular mechanism of pathological signal initiation is poorly understood. Using high-resolution microscopy methods, the authors reveal that these kinase mutations induce structural changes in the receptor ectodomain that lead to enhanced, ligand-independent dimerization., Mutations within the epidermal growth factor receptor (EGFR/erbB1/Her1) are often associated with tumorigenesis. In particular, a number of EGFR mutants that demonstrate ligand-independent signaling are common in non–small cell lung cancer (NSCLC), including kinase domain mutations L858R (also called L834R) and exon 19 deletions (e.g., ΔL747-P753insS), which collectively make up nearly 90% of mutations in NSCLC. The molecular mechanisms by which these mutations confer constitutive activity remain unresolved. Using multiple subdiffraction-limit imaging modalities, we reveal the altered receptor structure and interaction kinetics of NSCLC-associated EGFR mutants. We applied two-color single quantum dot tracking to quantify receptor dimerization kinetics on living cells and show that, in contrast to wild-type EGFR, mutants are capable of forming stable, ligand-independent dimers. Two-color superresolution localization microscopy confirmed ligand-independent aggregation of EGFR mutants. Live-cell Förster resonance energy transfer measurements revealed that the L858R kinase mutation alters ectodomain structure such that unliganded mutant EGFR adopts an extended, dimerization-competent conformation. Finally, mutation of the putative dimerization arm confirmed a critical role for ectodomain engagement in ligand-independent signaling. These data support a model in which dysregulated activity of NSCLC-associated kinase mutants is driven by coordinated interactions involving both the kinase and extracellular domains that lead to enhanced dimerization.

Proteins in cell signaling networks tend to interact promiscuously through low-affinity interactions. Consequently, evaluating the physiological importance of mapped interactions can be difficult. Attempts to do so have tended to focus on single, measurable physicochemical factors, such as affinity or abundance. For example, interaction importance has been assessed on the basis of the relative affinities of binding partners for a protein of interest, such as a receptor. However, multiple factors can be expected to simultaneously influence the recruitment of proteins to a receptor (and the potential of these proteins to contribute to receptor signaling), including affinity, abundance, and competition, which is a network property. Here, we demonstrate that measurements of protein copy numbers and binding affinities can be integrated within the framework of a mechanistic, computational model that accounts for mass action and competition. We use cell line-specific models to rank the relative importance of protein-protein interactions in the epidermal growth factor receptor (EGFR) signaling network for 11 different cell lines. Each model accounts for experimentally characterized interactions of six autophosphorylation sites in EGFR with proteins containing a Src homology 2 and/or phosphotyrosine-binding domain. We measure importance as the predicted maximal extent of recruitment of a protein to EGFR following ligand-stimulated activation of EGFR signaling. We find that interactions ranked highly by this metric include experimentally detected interactions. Proteins with high importance rank in multiple cell lines include proteins with recognized, well-characterized roles in EGFR signaling, such as GRB2 and SHC1, as well as a protein with a less well-defined role, YES1. Our results reveal potential cell line-specific differences in recruitment.

The high-affinity receptor for IgE expressed on the surface of mast cells and basophils interacts with antigens, via bound IgE antibody, and triggers secretion of inflammatory mediators that contribute to allergic reactions. To understand how past inputs (memory) influence future inflammatory responses in mast cells, a microfluidic device was used to precisely control exposure of cells to alternating stimulatory and non-stimulatory inputs. We determined that the response to subsequent stimulation depends on the interval of signaling quiescence. For shorter intervals of signaling quiescence, the second response is blunted relative to the first response, whereas longer intervals of quiescence induce an enhanced second response. Through an iterative process of computational modeling and experimental tests, we found that these memory-like phenomena arise from a confluence of rapid, short-lived positive signals driven by the protein tyrosine kinase Syk; slow, long-lived negative signals driven by the lipid phosphatase Ship1; and slower degradation of Ship1 co-factors. This work advances our understanding of mast cell signaling and represents a generalizable approach for investigating the dynamics of signaling systems.

Adaptor protein Grb2 binds phosphotyrosines in the epidermal growth factor (EGF) receptor (EGFR) and thereby links receptor activation to intracellular signaling cascades. Here, we investigated how recruitment of Grb2 to EGFR is affected by the spatial organization and quaternary state of activated EGFR. We used the techniques of image correlation spectroscopy (ICS) and lifetime-detected Förster resonance energy transfer (also known as FLIM-based FRET or FLIM–FRET) to measure ligand-induced receptor clustering and Grb2 binding to activated EGFR in BaF/3 cells. BaF/3 cells were stably transfected with fluorescently labeled forms of Grb2 (Grb2–mRFP) and EGFR (EGFR–eGFP). Following stimulation of the cells with EGF, we detected nanometer-scale association of Grb2–mRFP with EGFR–eGFP clusters, which contained, on average, 4 ± 1 copies of EGFR–eGFP per cluster. In contrast, the pool of EGFR–eGFP without Grb2–mRFP had an average cluster size of 1 ± 0.3 EGFR molecules per punctum. In the absence of EGF, there was no association between EGFR–eGFP and Grb2–mRFP. To interpret these data, we extended our recently developed model for EGFR activation, which considers EGFR oligomerization up to tetramers, to include recruitment of Grb2 to phosphorylated EGFR. The extended model, with adjustment of one new parameter (the ratio of the Grb2 and EGFR copy numbers), is consistent with a cluster size distribution where 2% of EGFR monomers, 5% of EGFR dimers

Ag-mediated crosslinking of IgE–FcεRI complexes activates mast cells and basophils, initiating the allergic response. Of 34 donors recruited having self-reported shrimp allergy, only 35% had significant levels of shrimp-specific IgE in serum and measurable basophil secretory responses to rPen a 1 (shrimp tropomyosin). We report that degranulation is linked to the number of FcεRI occupied with allergen-specific IgE, as well as the dose and valency of Pen a 1. Using clustered regularly interspaced palindromic repeat–based gene editing, human RBLrαKO cells were created that exclusively express the human FcεRIα subunit. Pen a 1–specific IgE was affinity purified from shrimp-positive plasma. Cells primed with a range of Pen a 1–specific IgE and challenged with Pen a 1 showed a bell-shaped dose response for secretion, with optimal Pen a 1 doses of 0.1–10 ng/ml. Mathematical modeling provided estimates of receptor aggregation kinetics based on FcεRI occupancy with IgE and allergen dose. Maximal degranulation was elicited when ∼2700 IgE–FcεRI complexes were occupied with specific IgE and challenged with Pen a 1 (IgE epitope valency of ≥8), although measurable responses were achieved when only a few hundred FcεRI were occupied. Prolonged periods of pepsin-mediated Pen a 1 proteolysis, which simulates gastric digestion, were required to diminish secretory responses. Recombinant fragments (60–79 aa), which together span the entire length of tropomyosin, were weak secretagogues. These fragments have reduced dimerization capacity, compete with intact Pen a 1 for binding to IgE–FcεRI complexes, and represent a starting point for the design of promising hypoallergens for immunotherapy.

BCR signaling regulates the activities and fates of B cells. BCR signaling encompasses two feedback loops emanating from Lyn and Fyn, which are Src family protein tyrosine kinases (SFKs). Positive feedback arises from SFK-mediated trans phosphorylation of BCR and receptor-bound Lyn and Fyn, which increases the kinase activities of Lyn and Fyn. Negative feedback arises from SFK-mediated cis phosphorylation of the transmembrane adapter protein PAG1, which recruits the cytosolic protein tyrosine kinase Csk to the plasma membrane, where it acts to decrease the kinase activities of Lyn and Fyn. To study the effects of the positive and negative feedback loops on the dynamical stability of BCR signaling and the relative contributions of Lyn and Fyn to BCR signaling, we consider in this study a rule-based model for early events in BCR signaling that encompasses membrane-proximal interactions of six proteins, as follows: BCR, Lyn, Fyn, Csk, PAG1, and Syk, a cytosolic protein tyrosine kinase that is activated as a result of SFK-mediated phosphorylation of BCR. The model is consistent with known effects of Lyn and Fyn deletions. We find that BCR signaling can generate a single pulse or oscillations of Syk activation depending on the strength of Ag signal and the relative levels of Lyn and Fyn. We also show that bistability can arise in Lyn- or Csk-deficient cells.

The pathogen Brucella melitensis secretes a Toll/interleukin-1 receptor (TIR) domain containing protein that abrogates host innate immune responses. In this study, we have characterized the biochemical interactions of Brucella TIR-like protein TcpB with host innate immune adaptor proteins. Using protein-fragment complementation assays based on Gaussia luciferase and green fluorescent protein, we find that TcpB interacts directly with MyD88 and that this interaction is significantly stronger than the interaction of TcpB with TIRAP, the only other adaptor protein that detectably interacts with TcpB. Surprisingly, the TcpB-MyD88 interaction depends on the death domain (DD) of MyD88, and TcpB does not interact with the isolated TIR domain of MyD88. TcpB disrupts MyD88(DD)-MyD88(DD), MyD88(DD)-MyD88(TIR) and MyD88(DD)-MyD88 interactions but not MyD88-MyD88 or MyD88(TIR)-MyD88(TIR) interactions. Structural models consistent with these results suggest how TcpB might inhibit TLR signaling by targeting MyD88 via a DD-TIR domain interface.

Receptor tyrosine kinases (RTKs) typically contain multiple autophosphorylation sites in their cytoplasmic domains. Once activated, these autophosphorylation sites can recruit downstream signaling proteins containing Src homology 2 (SH2) and phosphotyrosine-binding (PTB) domains, which recognize phosphotyrosine-containing short linear motifs (SLiMs). These domains and SLiMs have polyspecific or promiscuous binding activities. Thus, multiple signaling proteins may compete for binding to a common SLiM and vice versa. To investigate the effects of competition on RTK signaling, we used a rule-based modeling approach to develop and analyze models for ligand-induced recruitment of SH2/PTB domain-containing proteins to autophosphorylation sites in the insulin-like growth factor 1 (IGF1) receptor (IGF1R). Models were parameterized using published datasets reporting protein copy numbers and site-specific binding affinities. Simulations were facilitated by a novel application of model restructuration, to reduce redundancy in rule-derived equations. We compare predictions obtained via numerical simulation of the model to those obtained through simple prediction methods, such as through an analytical approximation, or ranking by copy number and/or KD value, and find that the simple methods are unable to recapitulate the predictions of numerical simulations. We created 45 cell line-specific models that demonstrate how early events in IGF1R signaling depend on the protein abundance profile of a cell. Simulations, facilitated by model restructuration, identified pairs of IGF1R binding partners that are recruited in anti-correlated and correlated fashions, despite no inclusion of cooperativity in our models. This work shows that the outcome of competition depends on the physicochemical parameters that characterize pairwise interactions, as well as network properties, including network connectivity and the relative abundances of competitors., Author summary Cells rely on networks of interacting biomolecules to sense and respond to environmental perturbations and signals. However, it is unclear how information is processed to generate appropriate and specific responses to signals, especially given that these networks tend to share many components. For example, receptors that detect distinct ligands and regulate distinct cellular activities commonly interact with overlapping sets of downstream signaling proteins. Here, to investigate the downstream signaling of a well-studied receptor tyrosine kinase (RTK), the insulin-like growth factor 1 (IGF1) receptor (IGF1R), we formulated and analyzed 45 cell line-specific mathematical models, which account for recruitment of 18 different binding partners to six sites of receptor autophosphorylation in IGF1R. The models were parameterized using available protein copy number and site-specific affinity measurements, and restructured to allow for network generation. We find that recruitment is influenced by the protein abundance profile of a cell, with different patterns of recruitment in different cell lines. Furthermore, in a given cell line, we find that pairs of IGF1R binding partners may be recruited in a correlated or anti-correlated fashion. We demonstrate that the simulations of the model have greater predictive power than protein copy number and/or binding affinity data, and that even a simple analytical model cannot reproduce the predicted recruitment ranking obtained via simulations. These findings represent testable predictions and indicate that the outputs of IGF1R signaling depend on cell line-specific properties in addition to the properties that are intrinsic to the biomolecules involved.

We investigate the ability of algorithms developed for reverse engineering of transcriptional regulatory networks to reconstruct metabolic networks from high-throughput metabolite profiling data. For this, we generate synthetic metabolic profiles for benchmarking purposes based on a well-established model for red blood cell metabolism. A variety of data sets is generated, accounting for different properties of real metabolic networks, such as experimental noise, metabolite correlations, and temporal dynamics. These data sets are made available online. We apply ARACNE, a mainstream transcriptional networks reverse engineering algorithm, to these data sets and observe performance comparable to that obtained in the transcriptional domain, for which the algorithm was originally designed., 14 pages, 3 figures. Presented at the DIMACS Workshop on Dialogue on Reverse Engineering Assessment and Methods (DREAM), Sep 2006

The immune system plays a central role in human health. The activities of immune cells, whether defending an organism from disease or triggering a pathological condition such as autoimmunity, are driven by the molecular machinery of cellular signaling systems. Decades of experimentation have elucidated many of the biomolecules and interactions involved in immune signaling and regulation, and recently developed technologies have led to new types of quantitative, systems-level data. To integrate such information and develop nontrivial insights into the immune system, computational modeling is needed, and it is essential for modeling methods to keep pace with experimental advances. In this chapter, we focus on the dynamic, site-specific, and context-dependent nature of interactions in immunoreceptor signaling (i.e., the biomolecular site dynamics of immunoreceptor signaling), the challenges associated with capturing these details in computational models, and how these challenges have been met through use of rule-based modeling approaches.

To investigate why responses of mast cells to antigen-induced IgE receptor (FceRI) aggregation depend nonlinearly on antigen dose, we characterized a new artificial ligand, DF3, through complementary modeling and experimentation. This ligand is a stable trimer of peptides derived from bacteriophage T4 fibritin, each conjugated to a hapten (DNP). We found low and high doses of DF3 at which degranulation of mast cells sensitized with DNP-specific IgE is minimal, but ligand-induced receptor aggregation is comparable to aggregation at an intermediate dose, optimal for degranulation. This finding makes DF3 an ideal reagent for studying the balance of negative and positive signaling in the FceRI pathway. We find that the lipid phosphatase SHIP and the protein tyrosine phosphatase SHP-1 negatively regulate mast cell degranulation over all doses considered. In contrast, SHP-2 promotes degranulation. With high DF3 doses, relatively rapid recruitment of SHIP to the plasma membrane may explain the reduced degranulation response. Our results demonstrate that optimal secretory responses of mast cells depend on the formation of receptor aggregates that promote sufficient positive signaling by Syk to override phosphatase-mediated negative regulatory signals.

In patients infected with human immunodeficiency virus type 1 (HIV-1), a large amount of virus is associated with follicular dendritic cells (FDCs) in lymphoid tissue. To assess the influence of FDCs on viral dynamics during antiretroviral therapy, we have developed a mathematical model for treatment of HIV-1 infection that includes FDCs. Here, we use this model to analyse measurements of HIV-1 dynamics in the blood and lymphoid tissue of a representative patient, who was treated with a combination of HIV-1 reverse transcriptase and protease inhibitors. We show that loss of virus from FDCs during therapy can make a much larger contribution to plasma virus than production of virus by infected cells. This result challenges the notion that long-lived infected cells are a significant source of HIV-1 during drug therapy. Due to release of FDC-associated virus, we find that it is necessary to revise upward previous estimates ofc, the rate at which free virus is cleared, andδ, the rate at which productively infected cells die. Furthermore, we find that potentially infectious virus, present before treatment, is released from FDCs during therapy and that the persistence of this virus can be affected by whether therapy includes reverse transcriptase inhibitors.

In adaptive immune responses, T-cell receptor (TCR) signaling impacts multiple cellular processes and results in T-cell differentiation, proliferation, and cytokine production. Although individual protein-protein interactions and phosphorylation events have been studied extensively, we lack a systems-level understanding of how these components cooperate to control signaling dynamics, especially during the crucial first seconds of stimulation. Here, we used quantitative proteomics to characterize reshaping of the T-cell phosphoproteome in response to TCR/CD28 co-stimulation, and found that diverse dynamic patterns emerge within seconds. We detected phosphorylation dynamics as early as 5 s and observed widespread regulation of key TCR signaling proteins by 30 s. Development of a computational model pointed to the presence of novel regulatory mechanisms controlling phosphorylation of sites with central roles in TCR signaling. The model was used to generate predictions suggesting unexpected roles for the phosphatase PTPN6 (SHP-1) and shortcut recruitment of the actin regulator WAS. Predictions were validated experimentally. This integration of proteomics and modeling illustrates a novel, generalizable framework for solidifying quantitative understanding of a signaling network and for elucidating missing links.

Mitochondria are dynamic organelles that undergo constant remodeling through the regulation of two opposing processes, mitochondrial fission and fusion. Although several key regulators and physiological stimuli have been identified to control mitochondrial fission and fusion, the role of mitochondrial morphology in the two processes remains to be determined. To address this knowledge gap, we investigated whether morphological features extracted from time-lapse live-cell images of mitochondria could be used to predict mitochondrial fate. That is, we asked if we could predict whether a mitochondrion is likely to participate in a fission or fusion event based on its current shape and local environment. Using live-cell microscopy, image analysis software, and supervised machine learning, we characterized mitochondrial dynamics with single-organelle resolution to identify features of mitochondria that are predictive of fission and fusion events. A random forest (RF) model was trained to correctly classify mitochondria poised for either fission or fusion based on a series of morphological and positional features for each organelle. Of the features we evaluated, mitochondrial perimeter positively correlated with mitochondria about to undergo a fission event. Similarly mitochondrial solidity (compact shape) positively correlated with mitochondria about to undergo a fusion event. Our results indicate that fission and fusion are positively correlated with mitochondrial morphological features; and therefore, mitochondrial fission and fusion may be influenced by the mechanical properties of mitochondrial membranes.

In colorectal cancer cells, APC, a tumor suppressor protein, is commonly expressed in truncated form. Truncation of APC is believed to disrupt degradation of β—catenin, which is regulated by a multiprotein complex called the destruction complex. The destruction complex comprises APC, Axin, β—catenin, serine/threonine kinases, and other proteins. The kinases and , which are recruited by Axin, mediate phosphorylation of β—catenin, which initiates its ubiquitination and proteosomal degradation. The mechanism of regulation of β—catenin degradation by the destruction complex and the role of truncation of APC in colorectal cancer are not entirely understood. Through formulation and analysis of a rule-based computational model, we investigated the regulation of β—catenin phosphorylation and degradation by APC and the effect of APC truncation on function of the destruction complex. The model integrates available mechanistic knowledge about site-specific interactions and phosphorylation of destruction complex components and is consistent with an array of published data. We find that the phosphorylated truncated form of APC can outcompete Axin for binding to β—catenin, provided that Axin is limiting, and thereby sequester β—catenin away from Axin and the Axin-recruited kinases and . Full-length APC also competes with Axin for binding to β—catenin; however, full-length APC is able, through its SAMP repeats, which bind Axin and which are missing in truncated oncogenic forms of APC, to bring β—catenin into indirect association with Axin and Axin-recruited kinases. Because our model indicates that the positive effects of truncated APC on β—catenin levels depend on phosphorylation of APC, at the first 20-amino acid repeat, and because phosphorylation of this site is mediated by , we suggest that is a potential target for therapeutic intervention in colorectal cancer. Specific inhibition of is predicted to limit binding of β—catenin to truncated APC and thereby to reverse the effect of APC truncation., Author Summary We asked the question, how can the effects of APC truncation, a very common mutation in colorectal cancer, be understood and reversed? We addressed this question by formulating a computational model for destruction complex function that incorporates site-specific details about protein-protein interactions and protein phosphorylation and examined the differences in predicted behaviors when APC is full length, as in normal cells, and truncated, as in colorectal cancer cells. Our model offers an explanation for how and why destruction complex function is altered by APC truncation. The model indicates that phosphorylation of the first 20-amino acid repeat in APC (which is usually the only 20-amino acid repeat that remains in truncated forms of APC) together with the absence of SAMP repeats (missing entirely because of truncation) allows truncated APC to act as a diversion sink. In other words, phosphorylated APC can outcompete Axin for binding to , provided Axin is limiting, and thereby prevent from associating with Axin and the Axin-associated kinases and , which initiate phosphorylation-dependent degradation of . Thus, the model identifies inhibition of APC phosphorylation, which is mediated by , as a potential means by which the oncogenic effect of APC truncation could be reversed.

Macroautophagy (autophagy) is a cellular recycling program essential for homeostasis and survival during cytotoxic stress. This process, which has an emerging role in disease etiology and treatment, is executed in four stages through the coordinated action of more than 30 proteins. An effective strategy for studying complicated cellular processes, such as autophagy, involves the construction and analysis of mathematical or computational models. When developed and refined from experimental knowledge, these models can be used to interrogate signaling pathways, formulate novel hypotheses about systems, and make predictions about cell signaling changes induced by specific interventions. Here, we present the development of a computational model describing autophagic vesicle dynamics in a mammalian system. We used time-resolved, live-cell microscopy to measure the synthesis and turnover of autophagic vesicles in single cells. The stochastically simulated model was consistent with data acquired during conditions of both basal and chemically-induced autophagy. The model was tested by genetic modulation of autophagic machinery and found to accurately predict vesicle dynamics observed experimentally. Furthermore, the model generated an unforeseen prediction about vesicle size that is consistent with both published findings and our experimental observations. Taken together, this model is accurate and useful and can serve as the foundation for future efforts aimed at quantitative characterization of autophagy.

Proteins with multiple binding sites play important roles in cell signaling systems by nucleating protein complexes in which, for example, enzymes and substrates are co-localized. Proteins that specialize in this function are called by a variety names, including adapter, linker and scaffold. Scaffold-mediated nucleation of protein complexes can be either constitutive or induced. Induced nucleation is commonly mediated by a docking site on a scaffold that is activated by phosphorylation. Here, by considering minimalist mathematical models, which recapitulate scaffold effects seen in more mechanistically detailed models, we obtain analytical and numerical results that provide insights into scaffold function. These results elucidate how recruitment of a pair of ligands to a scaffold depends on the concentrations of the ligands, on the binding constants for ligand-scaffold interactions, on binding cooperativity, and on the milieu of the scaffold, as ligand recruitment is affected by competitive ligands and decoy receptors. For the case of a bivalent scaffold, we obtain an expression for the unique scaffold concentration that maximally recruits a pair of monovalent ligands. Through simulations, we demonstrate that a bivalent scaffold can nucleate distinct sets of ligands to equivalent extents when the scaffold is present at different concentrations. Thus, the function of a scaffold can potentially change qualitatively with a change in copy number. We also demonstrate how a scaffold can change the catalytic efficiency of an enzyme and the sensitivity of the rate of reaction to substrate concentration. The results presented here should be useful for understanding scaffold function and for engineering scaffolds to have desired properties., Comment: 12 pages, 8 figures

Rule-based modeling involves the representation of molecules as structured objects and molecular interactions as rules for transforming the attributes of these objects. The approach is notable in that it allows one to systematically incorporate site-specific details about protein-protein interactions into a model for the dynamics of a signal-transduction system, but the method has other applications as well, such as following the fates of individual carbon atoms in metabolic reactions. The consequences of protein-protein interactions are difficult to specify and track with a conventional modeling approach because of the large number of protein phosphoforms and protein complexes that these interactions potentially generate. Here, we focus on how a rule-based model is specified in the BioNetGen language (BNGL) and how a model specification is analyzed using the BioNetGen software tool. We also discuss new developments in rule-based modeling that should enable the construction and analyses of comprehensive models for signal transduction pathways and similarly large-scale models for other biochemical systems.

Kinetic proofreading is an intrinsic property of the cell signaling process. It arises as a consequence of the multiple interactions that occur after a ligand triggers a receptor to initiate a ignaling cascade and it ensures that false signals do not propagate to completion. In order for an active signaling complex to form after a ligand binds to a cell surface receptor, a sequence of binding and phosphorylation events must occur that are rapidly reversed if the ligand dissociates from the receptor. This gives rise to a mechanism by which cells can discriminate among ligands that bind to the same receptor but form ligand-receptor complexes with different lifetimes. We review experiments designed to test for kinetic proofreading and models that exhibit kinetic proofreading.

The stochastic dynamics of T cell receptor (TCR) signaling are studied using a mathematical model intended to capture kinetic proofreading (sensitivity to ligand–receptor binding kinetics) and negative and positive feedback regulation mediated, respectively, by the phosphatase SHP1 and the MAP kinase ERK. The model incorporates protein–protein interactions involved in initiating TCR-mediated cellular responses and reproduces several experimental observations about the behavior of TCR signaling, including robust responses to as few as a handful of ligands (agonist peptide–MHC complexes on an antigen-presenting cell), distinct responses to ligands that bind TCR with different lifetimes, and antagonism. Analysis of the model indicates that TCR signaling dynamics are marked by significant stochastic fluctuations and bistability, which is caused by the competition between the positive and negative feedbacks. Stochastic fluctuations are such that single-cell trajectories differ qualitatively from the trajectory predicted in the deterministic approximation of the dynamics. Because of bistability, the average of single-cell trajectories differs markedly from the deterministic trajectory. Bistability combined with stochastic fluctuations allows for switch-like responses to signals, which may aid T cells in making committed cell-fate decisions.

We constructed a mechanistic, computational model for regulation of (macro)autophagy and protein synthesis (at the level of translation). The model was formulated to study the system-level consequences of interactions among the following proteins: two key components of MTOR complex 1 (MTORC1), namely the protein kinase MTOR (mechanistic target of rapamycin) and the scaffold protein RPTOR; the autophagy-initiating protein kinase ULK1; and the multimeric energy-sensing AMP-activated protein kinase (AMPK). Inputs of the model include intrinsic AMPK kinase activity, which is taken as an adjustable surrogate parameter for cellular energy level or AMP:ATP ratio, and rapamycin dose, which controls MTORC1 activity. Outputs of the model include the phosphorylation level of the translational repressor EIF4EBP1, a substrate of MTORC1, and the phosphorylation level of AMBRA1 (activating molecule in BECN1-regulated autophagy), a substrate of ULK1 critical for autophagosome formation. The model incorporates reciprocal regulation of mTORC1 and ULK1 by AMPK, mutual inhibition of MTORC1 and ULK1, and ULK1-mediated negative feedback regulation of AMPK. Through analysis of the model, we find that these processes may be responsible, depending on conditions, for graded responses to stress inputs, for bistable switching between autophagy and protein synthesis, or relaxation oscillations, comprising alternating periods of autophagy and protein synthesis. A sensitivity analysis indicates that the prediction of oscillatory behavior is robust to changes of the parameter values of the model. The model provides testable predictions about the behavior of the AMPK-MTORC1-ULK1 network, which plays a central role in maintaining cellular energy and nutrient homeostasis.

In HIV-infected patients, large quantities of HIV are associated with follicular dendritic cells (FDCs) in lymphoid tissue. During antiretroviral therapy, most of this virus disappears after six months of treatment, suggesting that FDC-associated virus has little influence on the eventual outcome of long-term therapy. However, a recent theoretical study using a stochastic model for the interaction of HIV with FDCs indicated that some virus may be retained on FDCs for years, where it can potentially reignite infection if treatment is interrupted. In that study, an approximate expression was used to estimate the time an individual virion remains on FDCs during therapy. Here, we determine the conditions under which this approximation is valid, and we develop expressions for the time a virion spends in any bound state and for the effect of rebinding on retention. We find that rebinding, which is influenced by diffusion, may play a major role in retention of HIV on FDCs. We also consider the possibility that HIV is retained on B cells during therapy, which like FDCs also interact with HIV. We find that virus associated with B cells is unlikely to persist during therapy.

Drug treatment of HIV type 1 (HIV-1) infection leads to a rapid initial decay of plasma virus followed by a slower second phase of decay. To investigate the role of HIV-1 retained on follicular dendritic cells (FDCs) in this process, we have developed and analyzed a mathematical model for HIV-1 dynamics in lymphoid tissue (LT) that includes FDCs. Analysis of clinical data using this model indicates that decay of HIV-1 during therapy may be influenced by release of FDC-associated virus. The biphasic character of viral decay can be explained by reversible multivalent binding of HIV-1 to receptors on FDCs, indicating that the second phase of decay is not necessarily caused by long-lived or latently infected cells. Furthermore, viral clearance and death of short-lived productively infected cells may be faster than previously estimated. The model, with reasonable parameter values, is consistent with kinetic measurements of viral RNA in plasma, viral RNA on FDCs, productively infected cells in LT, and CD4+T cells in LT during therapy.

Heterogeneity in responses of cells to a stimulus, such as a pathogen or allergen, can potentially play an important role in deciding the fate of the responding cell population and the overall systemic response. Measuring heterogeneous responses requires tools capable of interrogating individual cells. Cell signaling studies commonly do not have single-cell resolution because of the limitations of techniques used such as Westerns, ELISAs, mass spectrometry, and DNA microarrays. Microfluidics devices are increasingly being used to overcome these limitations. Here, we report on a microfluidic platform for cell signaling analysis that combines two orthogonal single-cell measurement technologies: on-chip flow cytometry and optical imaging. The device seamlessly integrates cell culture, stimulation, and preparation with downstream measurements permitting hands-free, automated analysis to minimize experimental variability. The platform was used to interrogate IgE receptor (FcεRI) signaling, which is responsible for triggering allergic reactions, in RBL-2H3 cells. Following on-chip crosslinking of IgE-FcεRI complexes by multivalent antigen, we monitored signaling events including protein phosphorylation, calcium mobilization and the release of inflammatory mediators. The results demonstrate the ability of our platform to produce quantitative measurements on a cell-by-cell basis from just a few hundred cells. Model-based analysis of the Syk phosphorylation data suggests that heterogeneity in Syk phosphorylation can be attributed to protein copy number variations, with the level of Syk phosphorylation being particularly sensitive to the copy number of Lyn.

RAS is the most frequently mutated gene across human cancers, but developing inhibitors of mutant RAS has proven to be challenging. Given the difficulties of targeting RAS directly, drugs that impact the other components of pathways where mutant RAS operates may potentially be effective. However, the system-level features, including different localizations of RAS isoforms, competition between downstream effectors, and interlocking feedback and feed-forward loops, must be understood to fully grasp the opportunities and limitations of inhibiting specific targets. Mathematical modeling can help us discern the system-level impacts of these features in normal and cancer cells. New technologies enable the acquisition of experimental data that will facilitate development of realistic models of oncogenic RAS behavior. In light of the wealth of empirical data accumulated over decades of study and the advancement of experimental methods for gathering new data, modelers now have the opportunity to advance progress toward realization of targeted treatment for mutant RAS-driven cancers.

The principle of microscopic reversibility implies detailed balance—the statement that, at thermodynamic equilibrium, each individual reaction is balanced. That is, at equilibrium each individual reaction occurs with equal forward and backward fluxes. A reaction system that satisfies detailed balance does not consume or dissipate free energy at thermodynamic equilibrium (1). Although reaction systems with no closed loops (or acyclic systems) always satisfy detailed balance, most complex reaction schemes involve reaction cycles, and satisfying detailed balance requires that the product of equilibrium constants around a reaction cycle equals one. Colquhoun et al. (2) recently presented methods to impose detailed balance on complex reaction mechanisms, such as ion channel kinetic schemes modeled with finite Markov chains. The methods rely on finding a fundamental cycle basis with respect to a spanning tree of a graph representing the reaction topology. In this comment, we discuss an alternative method that balances the forward and backward fluxes of each reversible reaction and does not require direct consideration of reaction cycles. The method will be discussed for single molecule dynamics and mass-action kinetics. We will also discuss techniques for identifying a minimal reaction network for imposing detailed balance.

Aggregation of FcεRI on mast cells and basophils leads to autophosphorylation and increased activity of the cytosolic protein tyrosine kinase Syk. We investigated the roles of the Src kinase Lyn, the immunoreceptor tyrosine-based activation motifs (ITAMs) on the β and γ subunits of FcεRI, and Syk itself in the activation of Syk. Our approach was to build a detailed mathematical model of reactions involving FcεRI, Lyn, Syk, and a bivalent ligand that aggregates FcεRI. We applied the model to experiments in which covalently cross-linked IgE dimers stimulate rat basophilic leukemia cells. The model makes it possible to test the consistency of mechanistic assumptions with data that alone provide limited mechanistic insight. For example, the model helps sort out mechanisms that jointly control dephosphorylation of receptor subunits. In addition, interpreted in the context of the model, experimentally observed differences between the β- and γ-chains with respect to levels of phosphorylation and rates of dephosphorylation indicate that most cellular Syk, but only a small fraction of Lyn, is available to interact with receptors. We also show that although the β ITAM acts to amplify signaling in experimental systems where its role has been investigated, there are conditions under which the β ITAM will act as an inhibitor.

Clinically used RAF inhibitors are ineffective in RAS-mutant tumors because they enhance homo- and heterodimerization of RAF kinases, leading to paradoxical activation of ERK signaling. Overcoming enhanced RAF dimerization and the resulting resistance is a challenge for drug design. Combining multiple inhibitors could be more effective, but it is unclear how the best combinations can be chosen. We built a next-generation mechanistic dynamic model to analyze combinations of structurally different RAF inhibitors, which can efficiently suppress MEK/ERK signaling. This rule-based model of the RAS/ERK pathway integrates thermodynamics and kinetics of drug-protein interactions, structural elements, post-translational modifications and cell mutational status as model rules to predict RAF inhibitor combinations for inhibiting ERK activity in oncogenic RAS and/or BRAFV600E backgrounds. Predicted synergistic inhibition of ERK signaling was corroborated by experiments in mutant NRAS, HRAS and BRAFV600E cells, and inhibition of oncogenic RAS signaling was associated with reduced cell proliferation and colony formation. European Commission Horizon 2020 Science Foundation Ireland