Stochastic effects and bistability in T cell receptor signaling

The influence of circadian rhythms on CD8+ T cell activation upon vaccination: A mathematical modeling perspective

Circadian rhythms have been implicated in the modulation of many physiological processes, including those associated with the immune system. For example, these rhythms influence CD8+ T cell responses within the adaptive immune system. The mechanism underlying this immune-circadian interaction, however, remains unclear, particularly in the context of vaccination. Here, we devise a molecularly-explicit gene regulatory network model of early signaling in the naïve CD8+ T cell activation pathway, comprised of three axes (or subsystems) labeled ZAP70, LAT and CD28, to elucidate the molecular details of this immune-circadian mechanism and its relation to vaccination. This is done by coupling the model to a periodic forcing function to identify the molecular players targeted by circadian rhythms, and analyzing how these rhythms subsequently affect CD8+ T cell activation under differing levels of T cell receptor (TCR) phosphorylation, which we designate as vaccine load. By performing both bifurcation and parameter sensitivity analyses on the model at the single cell and ensemble levels, we find that applying periodic forcing on molecular targets within the ZAP70 axis is sufficient to create a day-night discrepancy in CD8+ T cell activation in a manner that is dependent on the bistable switch inherent in CD8+ T cell early signaling. We also demonstrate that the resulting CD8+ T cell activation is dependent on the strength of the periodic coupling as well as on the level of TCR phosphorylation. Our results show that this day-night discrepancy is not transmitted to certain downstream molecules within the LAT subsystem, such as mTORC1, suggesting a secondary, independent circadian regulation on that protein complex. We also corroborate experimental results by showing that the circadian regulation of CD8+ T cell primarily acts at a baseline, pre-vaccination state, playing a facilitating role in priming CD8+ T cells to vaccine inputs according to the time of day. By applying an ensemble level analysis using bifurcation theory and by including several hypothesized molecular targets of this circadian rhythm, we further demonstrate an increased variability between CD8+ T cells (due to heterogeneity) induced by its circadian regulation, which may allow an ensemble of CD8+ T cells to activate at a lower vaccine load, improving its sensitivity. This modeling study thus provides insights into the immune targets of the circadian clock, and proposes an interaction between vaccine load and the influence of circadian rhythms on CD8+ T cell activation.

Reliable ligand discrimination in stochastic multistep kinetic proofreading: First passage time vs. product counting strategies

Cellular signaling, crucial for biological processes like immune response and homeostasis, relies on specificity and fidelity in signal transduction to accurately respond to stimuli amidst biological noise. Kinetic proofreading (KPR) is a key mechanism enhancing signaling specificity through time-delayed steps, although its effectiveness is debated due to intrinsic noise potentially reducing signal fidelity. In this study, we reformulate the theory of kinetic proofreading (KPR) by convolving multiple intermediate states into a single state and then define an overall "processing" time required to traverse these states. This simplification allows us to succinctly describe kinetic proofreading in terms of a single waiting time parameter, facilitating a more direct evaluation and comparison of KPR performance across different biological contexts such as DNA replication and T cell receptor (TCR) signaling. We find that loss of fidelity for longer proofreading steps relies on the specific strategy of information extraction and show that in the first-passage time (FPT) discrimination strategy, longer proofreading steps can exponentially improve the accuracy of KPR at the cost of speed. Thus, KPR can still be an effective discrimination mechanism in the high noise regime. However, in a product concentration-based discrimination strategy, longer proofreading steps do not necessarily lead to an increase in performance. However, by introducing activation thresholds on product concentrations, can we decompose the product-based strategy into a series of FPT-based strategies to better resolve the subtleties of KPR-mediated product discrimination. Our findings underscore the importance of understanding KPR in the context of how information is extracted and processed in the cell.

Model reduction for the Chemical Master Equation: An information-theoretic approach

The complexity of mathematical models in biology has rendered model reduction an essential tool in the quantitative biologist's toolkit. For stochastic reaction networks described using the Chemical Master Equation, commonly used methods include time-scale separation, Linear Mapping Approximation, and state-space lumping. Despite the success of these techniques, they appear to be rather disparate, and at present, no general-purpose approach to model reduction for stochastic reaction networks is known. In this paper, we show that most common model reduction approaches for the Chemical Master Equation can be seen as minimizing a well-known information-theoretic quantity between the full model and its reduction, the Kullback-Leibler divergence defined on the space of trajectories. This allows us to recast the task of model reduction as a variational problem that can be tackled using standard numerical optimization approaches. In addition, we derive general expressions for propensities of a reduced system that generalize those found using classical methods. We show that the Kullback-Leibler divergence is a useful metric to assess model discrepancy and to compare different model reduction techniques using three examples from the literature: an autoregulatory feedback loop, the Michaelis-Menten enzyme system, and a genetic oscillator.

Theoretical quantification of the polyvalent binding of nanoparticles coated with peptide-major histocompatibility complex to T cell receptor-nanoclusters

Nanoparticles (NPs) coated with peptide-major histocompatibility complexes (pMHCs) can be used as a therapy to treat autoimmune diseases. They do so by inducing the differentiation and expansion of disease-suppressing T regulatory type 1 (Tr1) cells by binding to their T cell receptors (TCRs) expressed as TCR-nanoclusters (TCR_nc). Their efficacy can be controlled by adjusting NP size and number of pMHCs coated on them (referred to as valence). The binding of these NPs to TCR_nc on T cells is thus polyvalent and occurs at three levels: the TCR-pMHC, NP-TCR_nc and T cell levels. In this study, we explore how this polyvalent interaction is manifested and examine if it can facilitate T cell activation downstream. This is done by developing a multiscale biophysical model that takes into account the three levels of interactions and the geometrical complexity of the binding. Using the model, we quantify several key parameters associated with this interaction analytically and numerically, including the insertion probability that specifies the number of remaining pMHC binding sites in the contact area between T cells and NPs, the dwell time of interaction between NPs and TCR_nc, carrying capacity of TCR_nc, the distribution of covered and bound TCRs, and cooperativity in the binding of pMHCs within the contact area. The model was fit to previously published dose-response curves of interferon-γ obtained experimentally by stimulating a population of T cells with increasing concentrations of NPs at various valences and NP sizes. Exploring the parameter space of the model revealed that for an appropriate choice of the contact area angle, the model can produce moderate jumps between dose-response curves at low valences. This suggests that the geometry and kinetics of NP binding to TCR_nc can act in synergy to facilitate T cell activation.

Modelling the interplay between the CD4 + /CD8 + T-cell ratio and the expression of MHC-I in tumours

Describing the anti-tumour immune response as a series of cellular kinetic reactions from known immunological mechanisms, we create a mathematical model that shows the CD4+ /CD8+ T-cell ratio, T-cell infiltration and the expression of MHC-I to be interacting factors in tumour elimination. Methods from dynamical systems theory and non-equilibrium statistical mechanics are used to model the T-cell dependent anti-tumour immune response. Our model predicts a critical level of MHC-I expression which determines whether or not the tumour escapes the immune response. This critical level of MHC-I depends on the helper/cytotoxic T-cell ratio. However, our model also suggests that the immune system is robust against small changes in this ratio. We also find that T-cell infiltration and the specificity of the intra-tumour TCR repertoire will affect the critical MHC-I expression. Our work suggests that the functional form of the time evolution of MHC-I expression may explain the qualitative behaviour of tumour growth seen in patients.

Bistability in cell signalling and its significance in identifying potential drug targets

MOTIVATION

Bistability is one of the salient dynamical features in various all-or-none kinds of decision-making processes. The presence of bistability in a cell signalling network plays a key role in input-output (I/O) relation. Our study is aiming to capture and emphasise the role of motif structure influencing the I/O relation between two nodes in the context of bistability. Here, a model-based analysis is made to investigate the critical conditions responsible for the emergence of different bistable protein-protein interaction (PPI) motifs and their possible applications to find the potential drug targets.

RESULTS

The global sensitivity analysis is used to identify sensitive parameters and their role in maintaining the bistability. Additionally, the bistable switching through hysteresis is explored to develop an understanding of the underlying mechanisms involved in the cell signalling processes, when significant motifs exhibiting bistability have emerged. Further, we elaborate the application of the results by the implication of the emerged PPI motifs to identify potential drug-targets in three cancer networks, which is validated with existing databases. The influence of stochastic perturbations that could hinder desired functionality of any signalling networks is also described here.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Autophosphorylation and the Dynamics of the Activation of Lck

Lck (lymphocyte-specific protein tyrosine kinase) is an enzyme which plays a number of important roles in the function of immune cells. It belongs to the Src family of kinases which are known to undergo autophosphorylation. It turns out that this leads to a remarkable variety of dynamical behaviour which can occur during their activation. We prove that in the presence of autophosphorylation one phenomenon, bistability, already occurs in a mathematical model for a protein with a single phosphorylation site. We further show that a certain model of Lck exhibits oscillations. Finally, we discuss the relations of these results to models in the literature which involve Lck and describe specific biological processes, such as the early stages of T cell activation and the stimulation of T cell responses resulting from the suppression of PD-1 signalling which is important in immune checkpoint therapy for cancer.

Optimal transition paths of phenotypic switching in a non-Markovian self-regulation gene expression

Gene expression is a complex biochemical process involving multiple reaction steps, creating molecular memory because the probability of waiting time between consecutive reaction steps no longer follows exponential distributions. What effect the molecular memory has on metastable states in gene expression remains not fully understood. Here, we study transition paths of switching between bistable states for a non-Markovian model of gene expression equipped with a self-regulation. Employing the large deviation theory for this model, we analyze the optimal transition paths of switching between bistable states in gene expression, interestingly finding that dynamic behaviors in gene expression along the optimal transition paths significantly depend on the molecular memory. Moreover, we discover that the molecular memory can prolong the time of switching between bistable states in gene expression along the optimal transition paths. Our results imply that the molecular memory may be an unneglectable factor to affect switching between metastable states in gene expression.

How the T cell signaling network processes information to discriminate between self and agonist ligands

T cells exhibit remarkable sensitivity and selectivity in detecting and responding to agonist peptides (p) bound to MHC molecules in a sea of self pMHC molecules. Despite much work, understanding of the underlying mechanisms of distinguishing such ligands remains incomplete. Here, we quantify T cell discriminatory capacity using channel capacity, a direct measure of the signaling network's ability to discriminate between antigen-presenting cells (APCs) displaying either self ligands or a mixture of self and agonist ligands. This metric shows how differences in information content between these two types of peptidomes are decoded by the topology and rates of kinetic proofreading signaling steps inside T cells. Using channel capacity, we constructed numerically substantiated hypotheses to explain the discriminatory role of a recently identified slow LAT Y132 phosphorylation step. Our results revealed that in addition to the number and kinetics of sequential signaling steps, a key determinant of discriminatory capability is spatial localization of a minimum number of these steps to the engaged TCR. Biochemical and imaging experiments support these findings. Our results also reveal the discriminatory role of early negative feedback and necessary amplification conferred by late positive feedback.

Scaling methods for accelerating kinetic Monte Carlo simulations of chemical reaction networks

Various kinetic Monte Carlo algorithms become inefficient when some of the population sizes in a system are large, which gives rise to a large number of reaction events per unit time. Here, we present a new acceleration algorithm based on adaptive and heterogeneous scaling of reaction rates and stoichiometric coefficients. The algorithm is conceptually related to the commonly used idea of accelerating a stochastic simulation by considering a subvolume λΩ (0 < λ < 1) within a system of interest, which reduces the number of reaction events per unit time occurring in a simulation by a factor 1/λ at the cost of greater error in unbiased estimates of first moments and biased overestimates of second moments. Our new approach offers two unique benefits. First, scaling is adaptive and heterogeneous, which eliminates the pitfall of overaggressive scaling. Second, there is no need for an a priori classification of populations as discrete or continuous (as in a hybrid method), which is problematic when discreteness of a chemical species changes during a simulation. The method requires specification of only a single algorithmic parameter, N_c, a global critical population size above which populations are effectively scaled down to increase simulation efficiency. The method, which we term partial scaling, is implemented in the open-source BioNetGen software package. We demonstrate that partial scaling can significantly accelerate simulations without significant loss of accuracy for several published models of biological systems. These models characterize activation of the mitogen-activated protein kinase ERK, prion protein aggregation, and T-cell receptor signaling.

Towards multiscale modeling of the CD8+ T cell response to viral infections

The CD8⁺ T cell response is critical to the control of viral infections. Yet, defining the CD8⁺ T cell response to viral infections quantitatively has been a challenge. Following antigen recognition, which triggers an intracellular signaling cascade, CD8⁺ T cells can differentiate into effector cells, which proliferate rapidly and destroy infected cells. When the infection is cleared, they leave behind memory cells for quick recall following a second challenge. If the infection persists, the cells may become exhausted, retaining minimal control of the infection while preventing severe immunopathology. These activation, proliferation and differentiation processes as well as the mounting of the effector response are intrinsically multiscale and collective phenomena. Remarkable experimental advances in the recent years, especially at the single cell level, have enabled a quantitative characterization of several underlying processes. Simultaneously, sophisticated mathematical models have begun to be constructed that describe these multiscale phenomena, bringing us closer to a comprehensive description of the CD8⁺ T cell response to viral infections. Here, we review the advances made and summarize the challenges and opportunities ahead. This article is categorized under: Analytical and Computational Methods > Computational Methods Biological Mechanisms > Cell Fates Biological Mechanisms > Cell Signaling Models of Systems Properties and Processes > Mechanistic Models.

Immunobiochemical Reconstruction of Influenza Lung Infection-Melanoma Skin Cancer Interactions

It was recently reported that acute influenza infection of the lung promoted distal melanoma growth in the dermis of mice. Melanoma-specific CD8+ T cells were shunted to the lung in the presence of the infection, where they expressed high levels of inflammation-induced cell-activation blocker PD-1, and became incapable of migrating back to the tumor site. At the same time, co-infection virus-specific CD8+ T cells remained functional while the infection was cleared. It was also unexpectedly found that PD-1 blockade immunotherapy reversed this effect. Here, we proceed to ground the experimental observations in a mechanistic immunobiochemical model that incorporates T cell pathways that control PD-1 expression. A core component of our model is a kinetic motif, which we call a PD-1 Double Incoherent Feed-Forward Loop (DIFFL), and which reflects known interactions between IRF4, Blimp-1, and Bcl-6. The different activity levels of the PD-1 DIFFL components, as a function of the cognate antigen levels and the given inflammation context, manifest themselves in phenotypically distinct outcomes. Collectively, the model allowed us to put forward a few working hypotheses as follows: (i) the melanoma-specific CD8+ T cells re-circulating with the blood flow enter the lung where they express high levels of inflammation-induced cell-activation blocker PD-1 in the presence of infection; (ii) when PD-1 receptors interact with abundant PD-L1, constitutively expressed in the lung, T cells loose motility; (iii) at the same time, virus-specific cells adapt to strong stimulation by their cognate antigen by lowering the transiently-elevated expression of PD-1, remaining functional and mobile in the inflamed lung, while the infection is cleared. The role that T cell receptor (TCR) activation and feedback loops play in the underlying processes are also highlighted and discussed. We hope that the results reported in our study could potentially contribute to the advancement of immunological approaches to cancer treatment and, as well, to a better understanding of a broader complexity of fundamental interactions between pathogens and tumors.

A Step-by-Step Guide to Using BioNetFit

BioNetFit is a software tool designed for solving parameter identification problems that arise in the development of rule-based models. It solves these problems through curve fitting (i.e., nonlinear regression). BioNetFit is compatible with deterministic and stochastic simulators that accept BioNetGen language (BNGL)-formatted files as inputs, such as those available within the BioNetGen framework. BioNetFit can be used on a laptop or stand-alone multicore workstation as well as on many Linux clusters, such as those that use the Slurm Workload Manager to schedule jobs. BioNetFit implements a metaheuristic population-based global optimization procedure, an evolutionary algorithm (EA), to minimize a user-defined objective function, such as a residual sum of squares (RSS) function. BioNetFit also implements a bootstrapping procedure for determining confidence intervals for parameter estimates. Here, we provide step-by-step instructions for using BioNetFit to estimate the values of parameters of a BNGL-encoded model and to define bootstrap confidence intervals. The process entails the use of several plain-text files, which are processed by BioNetFit and BioNetGen. In general, these files include (1) one or more EXP files, which each contains (experimental) data to be used in parameter identification/bootstrapping; (2) a BNGL file containing a model section, which defines a (rule-based) model, and an actions section, which defines simulation protocols that generate GDAT and/or SCAN files with model predictions corresponding to the data in the EXP file(s); and (3) a CONF file that configures the fitting/bootstrapping job and that defines algorithmic parameter settings.

Mathematical modeling identifies Lck as a potential mediator for PD-1 induced inhibition of early TCR signaling

Programmed cell death-1 (PD-1) is an inhibitory immune checkpoint receptor that negatively regulates the functioning of T cell. Although the direct targets of PD-1 were not identified, its inhibitory action on the TCR signaling pathway was known much earlier. Recent experiments suggest that the PD-1 inhibits the TCR and CD28 signaling pathways at a very early stage ─ at the level of phosphorylation of the cytoplasmic domain of TCR and CD28 receptors. Here, we develop a mathematical model to investigate the influence of inhibitory effect of PD-1 on the activation of early TCR and CD28 signaling molecules. Proposed model recaptures several quantitative experimental observations of PD-1 mediated inhibition. Model simulations show that PD-1 imposes a net inhibitory effect on the Lck kinase. Further, the inhibitory effect of PD-1 on the activation of TCR signaling molecules such as Zap70 and SLP76 is significantly enhanced by the PD-1 mediated inhibition of Lck. These results suggest a critical role for Lck as a mediator for PD-1 induced inhibition of TCR signaling network. Multi parametric sensitivity analysis explores the effect of parameter uncertainty on model simulations.

Modeling Procedures

Being concerned by the understanding of the mechanism underlying chronic degenerative diseases , we presented in the previous chapter the medical systems biology conceptual framework that we present for that purpose in this volume. More specifically, we argued there the clear advantages offered by a state-space perspective when applied to the systems-level description of the biomolecular machinery that regulates complex degenerative diseases. We also discussed the importance of the dynamical interplay between the risk factors and the network of interdependencies that characterizes the biochemical, cellular, and tissue-level biomolecular reactions that underlie the physiological processes in health and disease. As we pointed out in the previous chapter, the understanding of this interplay (articulated around cellular phenotypic plasticity properties, regulated by specific kinds of gene regulatory networks) is necessary if prevention is chosen as the human-health improvement strategy (potentially involving the modulation of the patient's lifestyle). In this chapter we provide the medical systems biology mathematical and computational modeling tools required for this task.

Untangling the Hairball: Fitness-Based Asymptotic Reduction of Biological Networks

Complex mathematical models of interaction networks are routinely used for prediction in systems biology. However, it is difficult to reconcile network complexities with a formal understanding of their behavior. Here, we propose a simple procedure (called ϕ¯) to reduce biological models to functional submodules, using statistical mechanics of complex systems combined with a fitness-based approach inspired by in silico evolution. The ϕ¯ algorithm works by putting parameters or combination of parameters to some asymptotic limit, while keeping (or slightly improving) the model performance, and requires parameter symmetry breaking for more complex models. We illustrate ϕ¯ on biochemical adaptation and on different models of immune recognition by T cells. An intractable model of immune recognition with close to a hundred individual transition rates is reduced to a simple two-parameter model. The ϕ¯ algorithm extracts three different mechanisms for early immune recognition, and automatically discovers similar functional modules in different models of the same process, allowing for model classification and comparison. Our procedure can be applied to biological networks based on rate equations using a fitness function that quantifies phenotypic performance.

Contractile actomyosin arcs promote the activation of primary mouse T cells in a ligand-dependent manner

Mechano-transduction is an emerging but still poorly understood component of T cell activation. Here we investigated the ligand-dependent contribution made by contractile actomyosin arcs populating the peripheral supramolecular activation cluster (pSMAC) region of the immunological synapse (IS) to T cell receptor (TCR) microcluster transport and proximal signaling in primary mouse T cells. Using super resolution microscopy, OT1-CD8+ mouse T cells, and two ovalbumin (OVA) peptides with different affinities for the TCR, we show that the generation of organized actomyosin arcs depends on ligand potency and the ability of myosin 2 to contract actin filaments. While weak ligands induce disorganized actomyosin arcs, strong ligands result in organized actomyosin arcs that correlate well with tension-sensitive CasL phosphorylation and the accumulation of ligands at the IS center. Blocking myosin 2 contractility greatly reduces the difference in the extent of Src and LAT phosphorylation observed between the strong and the weak ligand, arguing that myosin 2-dependent force generation within actin arcs contributes to ligand discrimination. Together, our data are consistent with the idea that actomyosin arcs in the pSMAC region of the IS promote a mechano-chemical feedback mechanism that amplifies the accumulation of critical signaling molecules at the IS.

Phenotypic spandrel: absolute discrimination and ligand antagonism

We consider the general problem of sensitive and specific discrimination between biochemical species. An important instance is immune discrimination between self and not-self, where it is also observed experimentally that ligands just below the discrimination threshold negatively impact response, a phenomenon called antagonism. We characterize mathematically the generic properties of such discrimination, first relating it to biochemical adaptation. Then, based on basic biochemical rules, we establish that, surprisingly, antagonism is a generic consequence of any strictly specific discrimination made independently from ligand concentration. Thus antagonism constitutes a 'phenotypic spandrel': a phenotype existing as a necessary by-product of another phenotype. We exhibit a simple analytic model of discrimination displaying antagonism, where antagonism strength is linear in distance from the detection threshold. This contrasts with traditional proofreading based models where antagonism vanishes far from threshold and thus displays an inverted hierarchy of antagonism compared to simpler models. The phenotypic spandrel studied here is expected to structure many decision pathways such as immune detection mediated by TCRs and FCϵRIs, as well as endocrine signalling/disruption.

Mathematical Models for Immunology: Current State of the Art and Future Research Directions

The advances in genetics and biochemistry that have taken place over the last 10 years led to significant advances in experimental and clinical immunology. In turn, this has led to the development of new mathematical models to investigate qualitatively and quantitatively various open questions in immunology. In this study we present a review of some research areas in mathematical immunology that evolved over the last 10 years. To this end, we take a step-by-step approach in discussing a range of models derived to study the dynamics of both the innate and immune responses at the molecular, cellular and tissue scales. To emphasise the use of mathematics in modelling in this area, we also review some of the mathematical tools used to investigate these models. Finally, we discuss some future trends in both experimental immunology and mathematical immunology for the upcoming years.

Efficient Optimization of Stimuli for Model-Based Design of Experiments to Resolve Dynamical Uncertainty

This model-based design of experiments (MBDOE) method determines the input magnitudes of an experimental stimuli to apply and the associated measurements that should be taken to optimally constrain the uncertain dynamics of a biological system under study. The ideal global solution for this experiment design problem is generally computationally intractable because of parametric uncertainties in the mathematical model of the biological system. Others have addressed this issue by limiting the solution to a local estimate of the model parameters. Here we present an approach that is independent of the local parameter constraint. This approach is made computationally efficient and tractable by the use of: (1) sparse grid interpolation that approximates the biological system dynamics, (2) representative parameters that uniformly represent the data-consistent dynamical space, and (3) probability weights of the represented experimentally distinguishable dynamics. Our approach identifies data-consistent representative parameters using sparse grid interpolants, constructs the optimal input sequence from a greedy search, and defines the associated optimal measurements using a scenario tree. We explore the optimality of this MBDOE algorithm using a 3-dimensional Hes1 model and a 19-dimensional T-cell receptor model. The 19-dimensional T-cell model also demonstrates the MBDOE algorithm’s scalability to higher dimensions. In both cases, the dynamical uncertainty region that bounds the trajectories of the target system states were reduced by as much as 86% and 99% respectively after completing the designed experiments in silico. Our results suggest that for resolving dynamical uncertainty, the ability to design an input sequence paired with its associated measurements is particularly important when limited by the number of measurements.

Many mathematical models that have been developed for biological systems are limited because the complex systems are not well understood, the parameters are not known, and available data is limited and noisy. On the other hand, experiments to support model development are limited in terms of costs and time, feasible inputs and feasible measurements. MBDOE combines the mathematical models with experiment design to strategically design optimal experiments to obtain data that will contribute to the understanding of the systems. Our approach extends current capabilities of existing MBDOE techniques to make them more useful for scientists to resolve the trajectories of the system under study. It identifies the optimal conditions for stimuli and measurements that yield the most information about the system given the practical limitations. Exploration of the input space is not a trivial extension to MBDOE methods used for determining optimal measurements due to the nonlinear nature of many biological system models. The exploration of the system dynamics elicited by different inputs requires a computationally efficient and tractable approach. Our approach plans optimal experiments to reduce dynamical uncertainty in the output of selected target states of the biological system.

Modeling biomolecular site dynamics in immunoreceptor signaling systems

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.

Experimental and theoretical bases for mechanisms of antigen discrimination by T cells

Interaction only within specific molecules is a requisite for accurate operations of a biochemical reaction in a cell where bulk of background molecules exist. While structural specificity is a well-established mechanism for specific interaction, biophysical and biochemical experiments indicate that the mechanism is not sufficient for accounting for the antigen discrimination by T cells. In addition, the antigen discrimination by T cells also accompanies three intriguing properties other than the specificity: sensitivity, speed, and concentration compensation. In this work, we review experimental and theoretical works on the antigen discrimination by focusing on these four properties and show future directions towards understanding of the fundamental principle for molecular discrimination.

Modeling the T cell immune response: a fascinating challenge

The immune system is designed to protect the organism from infection and to repair damaged tissue. An effective response requires recognition of the threat, the appropriate effector mechanism to clear the pathogen and a return to homeostasis with minimal damage to self-tissues. T cells play a central role in orchestrating the immune response at all stages of the response and have been the subject of intense study by both experimental immunologists and modelers. This review examines some of the more critical questions in T cell biology and describes the latest attempts to address those questions using approaches that combine mathematical modeling and experiments.

Phosphorylation site dynamics of early T-cell receptor signaling

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.

An Interaction Library for the FcεRI Signaling Network

Antigen receptors play a central role in adaptive immune responses. Although the molecular networks associated with these receptors have been extensively studied, we currently lack a systems-level understanding of how combinations of non-covalent interactions and post-translational modifications are regulated during signaling to impact cellular decision-making. To fill this knowledge gap, it will be necessary to formalize and piece together information about individual molecular mechanisms to form large-scale computational models of signaling networks. To this end, we have developed an interaction library for signaling by the high-affinity IgE receptor, FcεRI. The library consists of executable rules for protein–protein and protein–lipid interactions. This library extends earlier models for FcεRI signaling and introduces new interactions that have not previously been considered in a model. Thus, this interaction library is a toolkit with which existing models can be expanded and from which new models can be built. As an example, we present models of branching pathways from the adaptor protein Lat, which influence production of the phospholipid PIP₃ at the plasma membrane and the soluble second messenger IP₃. We find that inclusion of a positive feedback loop gives rise to a bistable switch, which may ensure robust responses to stimulation above a threshold level. In addition, the library is visualized to facilitate understanding of network circuitry and identification of network motifs.

Multiple model-informed open-loop control of uncertain intracellular signaling dynamics

Computational approaches to tune the activation of intracellular signal transduction pathways both predictably and selectively will enable researchers to explore and interrogate cell biology with unprecedented precision. Techniques to control complex nonlinear systems typically involve the application of control theory to a descriptive mathematical model. For cellular processes, however, measurement assays tend to be too time consuming for real-time feedback control and models offer rough approximations of the biological reality, thus limiting their utility when considered in isolation. We overcome these problems by combining nonlinear model predictive control with a novel adaptive weighting algorithm that blends predictions from multiple models to derive a compromise open-loop control sequence. The proposed strategy uses weight maps to inform the controller of the tendency for models to differ in their ability to accurately reproduce the system dynamics under different experimental perturbations (i.e. control inputs). These maps, which characterize the changing model likelihoods over the admissible control input space, are constructed using preexisting experimental data and used to produce a model-based open-loop control framework. In effect, the proposed method designs a sequence of control inputs that force the signaling dynamics along a predefined temporal response without measurement feedback while mitigating the effects of model uncertainty. We demonstrate this technique on the well-known Erk/MAPK signaling pathway in T cells. In silico assessment demonstrates that this approach successfully reduces target tracking error by 52% or better when compared with single model-based controllers and non-adaptive multiple model-based controllers. In vitro implementation of the proposed approach in Jurkat cells confirms a 63% reduction in tracking error when compared with the best of the single-model controllers. This study provides an experimentally-corroborated control methodology that utilizes the knowledge encoded within multiple mathematical models of intracellular signaling to design control inputs that effectively direct cell behavior in open-loop.

Composing problem solvers for simulation experimentation: a case study on steady state estimation

Simulation experiments involve various sub-tasks, e.g., parameter optimization, simulation execution, or output data analysis. Many algorithms can be applied to such tasks, but their performance depends on the given problem. Steady state estimation in systems biology is a typical example for this: several estimators have been proposed, each with its own (dis-)advantages. Experimenters, therefore, must choose from the available options, even though they may not be aware of the consequences. To support those users, we propose a general scheme to aggregate such algorithms to so-called synthetic problem solvers, which exploit algorithm differences to improve overall performance. Our approach subsumes various aggregation mechanisms, supports automatic configuration from training data (e.g., via ensemble learning or portfolio selection), and extends the plugin system of the open source modeling and simulation framework James II. We show the benefits of our approach by applying it to steady state estimation for cell-biological models.

Experimental design for dynamics identification of cellular processes

We address the problem of using nonlinear models to design experiments to characterize the dynamics of cellular processes by using the approach of the Maximally Informative Next Experiment (MINE), which was introduced in W. Dong et al. (PLoS ONE 3(8):e3105, 2008) and independently in M.M. Donahue et al. (IET Syst. Biol. 4:249-262, 2010). In this approach, existing data is used to define a probability distribution on the parameters; the next measurement point is the one that yields the largest model output variance with this distribution. Building upon this approach, we introduce the Expected Dynamics Estimator (EDE), which is the expected value using this distribution of the output as a function of time. We prove the consistency of this estimator (uniform convergence to true dynamics) even when the chosen experiments cluster in a finite set of points. We extend this proof of consistency to various practical assumptions on noisy data and moderate levels of model mismatch. Through the derivation and proof, we develop a relaxed version of MINE that is more computationally tractable and robust than the original formulation. The results are illustrated with numerical examples on two nonlinear ordinary differential equation models of biomolecular and cellular processes.

Spontaneous NF-κB activation by autocrine TNFα signaling: a computational analysis

NF-κB is a key transcription factor that regulates innate immune response. Its activity is tightly controlled by numerous feedback loops, including two negative loops mediated by NF-κB inducible inhibitors, IκBα and A20, which assure oscillatory responses, and by positive feedback loops arising due to the paracrine and autocrine regulation via TNFα, IL-1 and other cytokines. We study the NF-κB system of interlinked negative and positive feedback loops, combining bifurcation analysis of the deterministic approximation with stochastic numerical modeling. Positive feedback assures the existence of limit cycle oscillations in unstimulated wild-type cells and introduces bistability in A20-deficient cells. We demonstrated that cells of significant autocrine potential, i.e., cells characterized by high secretion of TNFα and its receptor TNFR1, may exhibit sustained cytoplasmic-nuclear NF-κB oscillations which start spontaneously due to stochastic fluctuations. In A20-deficient cells even a small TNFα expression rate qualitatively influences system kinetics, leading to long-lasting NF-κB activation in response to a short-pulsed TNFα stimulation. As a consequence, cells with impaired A20 expression or increased TNFα secretion rate are expected to have elevated NF-κB activity even in the absence of stimulation. This may lead to chronic inflammation and promote cancer due to the persistent activation of antiapoptotic genes induced by NF-κB. There is growing evidence that A20 mutations correlate with several types of lymphomas and elevated TNFα secretion is characteristic of many cancers. Interestingly, A20 loss or dysfunction also leaves the organism vulnerable to septic shock and massive apoptosis triggered by the uncontrolled TNFα secretion, which at high levels overcomes the antiapoptotic action of NF-κB. It is thus tempting to speculate that some cancers of deregulated NF-κB signaling may be prone to the pathogen-induced apoptosis.

The duration of T cell stimulation is a critical determinant of cell fate and plasticity

Variations in T cell receptor (TCR) signal strength, as indicated by differential activation of downstream signaling pathways, determine the fate of naïve T cells after encounter with antigen. Low-strength signals favor differentiation into regulatory T (T(reg)) cells containing the transcription factor Foxp3, whereas high-strength signals favor generation of interleukin-2-producing T helper (T(H)) cells. We constructed a logic circuit model of TCR signaling pathways, a major feature of which is an incoherent feed-forward loop involving both TCR-dependent activation of Foxp3 and its inhibition by mammalian target of rapamycin (mTOR), leading to the transient appearance of Foxp3(+) cells under T(H) cell-generating conditions. Experiments confirmed this behavior and the prediction that the immunosuppressive cytokine TGF-β (transforming growth factor-β) could generate T(reg) cells even during continued Akt-mTOR signaling. We predicted that sustained mTOR activity could suppress FOXP3 expression upon TGF-β removal, suggesting a possible mechanism for the experimentally observed instability of Foxp3(+) cells. Our model predicted, and experiments confirmed, that transient stimulation of cells with high-dose antigen generated T(H), T(reg), and nonactivated cells in proportions depending on the duration of TCR stimulation. Experimental analysis of cells after antigen removal identified three populations that correlated with these T cell fates. Further analysis of simulations implicated a negative feedback loop involving Foxp3, the phosphatase PTEN, and Akt-mTOR in determining fate. These results suggest that there is a critical time after TCR stimulation during which heterogeneity in the differentiating population of cells leads to increased plasticity of cell fate.

In silico modeling: methods and applications to trauma and sepsis

OBJECTIVES

To familiarize clinicians with advances in computational disease modeling applied to trauma and sepsis.

DATA SOURCES

PubMed search and review of relevant medical literature.

SUMMARY

Definitions, key methods, and applications of computational modeling to trauma and sepsis are reviewed.

CONCLUSIONS

Computational modeling of inflammation and organ dysfunction at the cellular, organ, whole-organism, and population levels has suggested a positive feedback cycle of inflammation → damage → inflammation that manifests via organ-specific inflammatory switching networks. This structure may manifest as multicompartment "tipping points" that drive multiple organ dysfunction. This process may be amenable to rational inflammation reprogramming.

Stochastic responses may allow genetically diverse cell populations to optimize performance with simpler signaling networks

Two theories have emerged for the role that stochasticity plays in biological responses: first, that it degrades biological responses, so the performance of biological signaling machinery could be improved by increasing molecular copy numbers of key proteins; second, that it enhances biological performance, by enabling diversification of population-level responses. Using T cell biology as an example, we demonstrate that these roles for stochastic responses are not sufficient to understand experimental observations of stochastic response in complex biological systems that utilize environmental and genetic diversity to make cooperative responses. We propose a new role for stochastic responses in biology: they enable populations to make complex responses with simpler biochemical signaling machinery than would be required in the absence of stochasticity. Thus, the evolution of stochastic responses may be linked to the evolvability of different signaling machineries.

Toggle switch: noise determines the winning gene

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

Principles of adaptive sorting revealed by in silico evolution

Many biological networks have to filter out useful information from a vast excess of spurious interactions. In this Letter, we use computational evolution to predict design features of networks processing ligand categorization. The important problem of early immune response is considered as a case study. Rounds of evolution with different constraints uncover elaborations of the same network motif we name "adaptive sorting." Corresponding network substructures can be identified in current models of immune recognition. Our work draws a deep analogy between immune recognition and biochemical adaptation.

Systems model of T cell receptor proximal signaling reveals emergent ultrasensitivity

Receptor phosphorylation is thought to be tightly regulated because phosphorylated receptors initiate signaling cascades leading to cellular activation. The T cell antigen receptor (TCR) on the surface of T cells is phosphorylated by the kinase Lck and dephosphorylated by the phosphatase CD45 on multiple immunoreceptor tyrosine-based activation motifs (ITAMs). Intriguingly, Lck sequentially phosphorylates ITAMs and ZAP-70, a cytosolic kinase, binds to phosphorylated ITAMs with differential affinities. The purpose of multiple ITAMs, their sequential phosphorylation, and the differential ZAP-70 affinities are unknown. Here, we use a systems model to show that this signaling architecture produces emergent ultrasensitivity resulting in switch-like responses at the scale of individual TCRs. Importantly, this switch-like response is an emergent property, so that removal of multiple ITAMs, sequential phosphorylation, or differential affinities abolishes the switch. We propose that highly regulated TCR phosphorylation is achieved by an emergent switch-like response and use the systems model to design novel chimeric antigen receptors for therapy.

Recognition of antigen by the T cell antigen receptor (TCR) is a central event in the initiation of adaptive immune responses and for this reason the TCR has been extensively studied. Multiple studies performed over the past 20 years have revealed intriguing findings that include the observation that the TCR has multiple phosphorylation sites that are sequentially phosphorylated by the kinase Lck and that ZAP-70, a cytosolic kinase, binds to these sites with different affinities. The purpose of multiple sites, their sequential phosphorylation by Lck, and the differential binding affinities of ZAP-70 are unknown. Using a novel mechanistic model that incorporates a high level of molecular detail, we find, unexpectedly, that all factors are critical for producing ultrasensitivity (switch-like response) and therefore this signaling architecture exhibits systems-level emergent ultrasensitivity. We use the model to study existing therapeutic chimeric antigen receptors and in the design of novel ones. The work also has direct implications to the study of many other immune receptors.

Phenotypic model for early T-cell activation displaying sensitivity, specificity, and antagonism

Early T-cell activation is selected by evolution to discriminate a few foreign peptides rapidly from a vast excess of self-peptides, and it is unclear in quantitative terms how this is possible. We show that a generic proofreading cascade supplemented by a single negative feedback mediated by the Src homology 2 domain phosphatase-1 (SHP-1) accounts quantitatively for early T-cell activation, including the effects of antagonists. Modulation of the negative feedback with SHP-1 concentration explains counterintuitive experimental observations, such as the nonmonotonic behavior of receptor activity on agonist concentration, the digital vs. continuous behavior on certain parameters, and the loss of response for high SHP-1 concentration. New experiments validate predictions on the nontrivial joint dependence on binding time and concentration for the relative effect of two antagonists: We explain why strong antagonists behave as partial agonists at low concentration and predict that the relative effect of antagonists can invert as their concentrations are varied. By focusing on the phenotype, our model quantitatively fits a body of experimental data with minimal variables and parameters.

Leveraging modeling approaches: reaction networks and rules

We have witnessed an explosive growth in research involving mathematical models and computer simulations of intracellular molecular interactions, ranging from metabolic pathways to signaling and gene regulatory networks. Many software tools have been developed to aid in the study of such biological systems, some of which have a wealth of features for model building and visualization, and powerful capabilities for simulation and data analysis. Novel high-resolution and/or high-throughput experimental techniques have led to an abundance of qualitative and quantitative data related to the spatiotemporal distribution of molecules and complexes, their interactions kinetics, and functional modifications. Based on this information, computational biology researchers are attempting to build larger and more detailed models. However, this has proved to be a major challenge. Traditionally, modeling tools require the explicit specification of all molecular species and interactions in a model, which can quickly become a major limitation in the case of complex networks - the number of ways biomolecules can combine to form multimolecular complexes can be combinatorially large. Recently, a new breed of software tools has been created to address the problems faced when building models marked by combinatorial complexity. These have a different approach for model specification, using reaction rules and species patterns. Here we compare the traditional modeling approach with the new rule-based methods. We make a case for combining the capabilities of conventional simulation software with the unique features and flexibility of a rule-based approach in a single software platform for building models of molecular interaction networks.

Computational model for autophagic vesicle dynamics in single cells

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.

Type of noise defines global attractors in bistable molecular regulatory systems

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

A stochastic T cell response criterion

The adaptive immune system relies on different cell types to provide fast and coordinated responses, characterized by recognition of pathogenic challenge, extensive cellular proliferation and differentiation, as well as death. T cells are a subset of the adaptive immune cellular pool that recognize immunogenic peptides expressed on the surface of antigen-presenting cells by means of specialized receptors on their membrane. T cell receptor binding to ligand determines T cell responses at different times and locations during the life of a T cell. Current experimental evidence provides support to the following: (i) sufficiently long receptor–ligand engagements are required to initiate the T cell signalling cascade that results in productive signal transduction and (ii) counting devices are at work in T cells to allow signal accumulation, decoding and translation into biological responses. In the light of these results, we explore, with mathematical models, the timescales associated with T cell responses. We consider two different criteria: a stochastic one (the mean time it takes to have had N receptor–ligand complexes bound for at least a dwell time, τ, each) and one based on equilibrium (the time to reach a threshold number N of receptor–ligand complexes). We have applied mathematical models to previous experiments in the context of thymic negative selection and to recent two-dimensional experiments. Our results indicate that the stochastic criterion provides support to the thymic affinity threshold hypothesis, whereas the equilibrium one does not, and agrees with the ligand hierarchy experimentally established for thymic negative selection.

A minimal model for the mitochondrial rapid mode of Ca²+ uptake mechanism

Mitochondria possess a remarkable ability to rapidly accumulate and sequester Ca²⁺. One of the mechanisms responsible for this ability is believed to be the rapid mode (RaM) of Ca²⁺ uptake. Despite the existence of many models of mitochondrial Ca²⁺ dynamics, very few consider RaM as a potential mechanism that regulates mitochondrial Ca²⁺ dynamics. To fill this gap, a novel mathematical model of the RaM mechanism is developed herein. The model is able to simulate the available experimental data of rapid Ca²⁺ uptake in isolated mitochondria from both chicken heart and rat liver tissues with good fidelity. The mechanism is based on Ca²⁺ binding to an external trigger site(s) and initiating a brief transient of high Ca²⁺ conductivity. It then quickly switches to an inhibited, zero-conductive state until the external Ca²⁺ level is dropped below a critical value (∼100–150 nM). RaM's Ca²⁺- and time-dependent properties make it a unique Ca²⁺ transporter that may be an important means by which mitochondria take up Ca²⁺in situ and help enable mitochondria to decode cytosolic Ca²⁺ signals. Integrating the developed RaM model into existing models of mitochondrial Ca²⁺ dynamics will help elucidate the physiological role that this unique mechanism plays in mitochondrial Ca²⁺-homeostasis and bioenergetics.

A mathematical framework for analyzing T cell receptor scanning of peptides

T cells continuously search for antigenic peptides presented on major histocompatibility complexes expressed on nearly all nucleated cells. Because only a few antigenic peptides are presented in a sea of thousands of self-peptides, the T cells have a critical task in discriminating between self- and nonself-peptides. This search process for antigens must be performed with sufficient speed in order to induce a fast response against invading pathogens. This study presents a mathematical framework for analyzing the scanning process of peptides. The framework includes analytic expressions for calculating the sampling rate as well as continuous-systems- and stochastic-agent-based models. The results show that the scanning of self-peptides is a very fast process due to fast off-rates. The simulations also predict the existence of an optimal sampling rate for a certain range of on-rates based on the recently proposed confinement time model. Calculations reveal that most of the self-peptides located within a microdomain are scanned within just a few seconds, and that the T cell receptors have kinetics for self-peptides, facilitating fast scanning. The derived mathematical expressions within this study provide conceptual calculations for further investigations of how the T cell discriminates between self- and nonself-peptides.