Mathematical and computational models of immune-receptor signalling

Description of chemical systems by means of response functions

In this paper we introduce a formalism that allows to describe the response of a part of a biochemical system in terms of renewal equations. In particular, we examine under which conditions the interactions between the different parts of a chemical system, described by means of linear ODEs, can be represented in terms of renewal equations. We show also how to apply the formalism developed in this paper to some particular types of linear and non-linear ODEs, modelling some biochemical systems of interest in biology (for instance, some time-dependent versions of the classical Hopfield model of kinetic proofreading). We also analyse some of the properties of the renewal equations that we are interested in, as the long-time behaviour of their solution. Furthermore, we prove that the kernels characterising the renewal equations derived by biochemical system with reactions that satisfy the detail balance condition belong to the class of completely monotone functions.

Mathematical models of TCR initial triggering

T cell receptors (TCRs) play crucial roles in regulating T cell response by rapidly and accurately recognizing foreign and non-self antigens. The process involves multiple molecules and regulatory mechanisms, forming a complex network to achieve effective antigen recognition. Mathematical modeling techniques can help unravel the intricate network of TCR signaling and identify key regulators that govern it. In this review, we introduce and briefly discuss relevant mathematical models of TCR initial triggering, with a focus on kinetic proofreading (KPR) models with different modified structures. We compare the topology structures, biological hypotheses, parameter choices, and simulation performance of each model, and summarize the advantages and limitations of them. Further studies on TCR modeling design, aiming for an optimized balance of specificity and sensitivity, are expected to contribute to the development of new therapeutic strategies.

Quantitative Analysis of Therapeutic Antibody Interactions with Fcγ Receptors Using High-Speed Atomic Force Microscopy

This study employed high-speed atomic force microscopy to quantitatively analyze the interactions between therapeutic antibodies and Fcγ receptors (FcγRs). Antibodies are essential components of the immune system and are integral to biopharmaceuticals. The focus of this study was on immunoglobulin G molecules, which are crucial for antigen binding via the Fab segments and cytotoxic functions through their Fc portions. We conducted real-time, label-free observations of the interactions of rituximab and mogamulizumab with the recombinant FcγRIIIa and FcγRIIa. The dwell times of FcγR binding were measured at the single-molecule level, which revealed an extended interaction duration of mogamulizumab with FcγRIIIa compared with that of rituximab. This is linked to enhanced antibody-dependent cellular cytotoxicity that is attributed to the absence of the core fucosylation of Fc-linked N-glycan. This study also emphasizes the crucial role of the Fab segments in the interaction with FcγRIIa as well as that with FcγRIIIa. This approach provided quantitative insight into therapeutic antibody interactions and exemplified kinetic proofreading, where cellular discrimination relies on ligand residence times. Observing the dwell times of antibodies on the effector molecules has emerged as a robust indicator of therapeutic antibody efficacy. Ultimately, these findings pave the way for the development of refined therapeutic antibodies with tailored interactions with specific FcγRs. This research contributes to the advancement of biopharmaceutical antibody design and optimizing antibody-based treatments for enhanced efficacy and precision.

BioNetGMMFit: estimating parameters of a BioNetGen model from time-stamped snapshots of single cells

Mechanistic models are commonly employed to describe signaling and gene regulatory kinetics in single cells and cell populations. Recent advances in single-cell technologies have produced multidimensional datasets where snapshots of copy numbers (or abundances) of a large number of proteins and mRNA are measured across time in single cells. The availability of such datasets presents an attractive scenario where mechanistic models are validated against experiments, and estimated model parameters enable quantitative predictions of signaling or gene regulatory kinetics. To empower the systems biology community to easily estimate parameters accurately from multidimensional single-cell data, we have merged a widely used rule-based modeling software package BioNetGen, which provides a user-friendly way to code for mechanistic models describing biochemical reactions, and the recently introduced CyGMM, that uses cell-to-cell differences to improve parameter estimation for such networks, into a single software package: BioNetGMMFit. BioNetGMMFit provides parameter estimates of the model, supplied by the user in the BioNetGen markup language (BNGL), which yield the best fit for the observed single-cell, time-stamped data of cellular components. Furthermore, for more precise estimates, our software generates confidence intervals around each model parameter. BioNetGMMFit is capable of fitting datasets of increasing cell population sizes for any mechanistic model specified in the BioNetGen markup language. By streamlining the process of developing mechanistic models for large single-cell datasets, BioNetGMMFit provides an easily-accessible modeling framework designed for scale and the broader biochemical signaling community.

Modelling the Tumour Microenvironment, but What Exactly Do We Mean by "Model"?

The Oxford English Dictionary includes 17 definitions for the word "model" as a noun and another 11 as a verb. Therefore, context is necessary to understand the meaning of the word model. For instance, "model railways" refer to replicas of railways and trains at a smaller scale and a "model student" refers to an exemplary individual. In some cases, a specific context, like cancer research, may not be sufficient to provide one specific meaning for model. Even if the context is narrowed, specifically, to research related to the tumour microenvironment, "model" can be understood in a wide variety of ways, from an animal model to a mathematical expression. This paper presents a review of different "models" of the tumour microenvironment, as grouped by different definitions of the word into four categories: model organisms, in vitro models, mathematical models and computational models. Then, the frequencies of different meanings of the word "model" related to the tumour microenvironment are measured from numbers of entries in the MEDLINE database of the United States National Library of Medicine at the National Institutes of Health. The frequencies of the main components of the microenvironment and the organ-related cancers modelled are also assessed quantitatively with specific keywords. Whilst animal models, particularly xenografts and mouse models, are the most commonly used "models", the number of these entries has been slowly decreasing. Mathematical models, as well as prognostic and risk models, follow in frequency, and these have been growing in use.

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.

McSNAC: A software to approximate first-order signaling networks from mass cytometry data

Background

Mass cytometry (CyTOF) gives unprecedented opportunity to simultaneously measure up to 40 proteins in single cells, with a theoretical potential to reach 100 proteins. This high-dimensional single-cell information can be very useful in dissecting mechanisms of cellular activity. In particular, measuring abundances of signaling proteins like phospho-proteins can provide detailed information on the dynamics of single-cell signaling processes. However, computational analysis is required to reconstruct such networks with a mechanistic model.

Methods

We propose our Mass cytometry Signaling Network Analysis Code (McSNAC), a new software capable of reconstructing signaling networks and estimating their kinetic parameters from CyTOF data. McSNAC approximates signaling networks as a network of first-order reactions between proteins. This assumption often breaks down as signaling reactions can involve binding and unbinding, enzymatic reactions, and other nonlinear constructions. Furthermore, McSNAC may be limited to approximating indirect interactions between protein species, as cytometry experiments are only able to assay a small fraction of protein species involved in signaling.

Results

We carry out a series of in silico experiments here to show (1) McSNAC is capable of accurately estimating the ground-truth model in a scalable manner when given data originating from a first-order system; (2) McSNAC is capable of qualitatively predicting outcomes to perturbations of species abundances in simple second-order reaction models and in a complex in silico nonlinear signaling network in which some proteins are unmeasured.

Conclusions

These findings demonstrate that McSNAC can be a valuable screening tool for generating models of signaling networks from time-stamped CyTOF data.

Spatially resolved in silico modeling of NKG2D signaling kinetics suggests a key role of NKG2D and Vav1 Co-clustering in generating natural killer cell activation

Natural Killer (NK) cells provide key resistance against viral infections and tumors. A diverse set of activating and inhibitory NK cell receptors (NKRs) interact with cognate ligands presented by target host cells, where integration of dueling signals initiated by the ligand-NKR interactions determines NK cell activation or tolerance. Imaging experiments over decades have shown micron and sub-micron scale spatial clustering of activating and inhibitory NKRs. The mechanistic roles of these clusters in affecting downstream signaling and activation are often unclear. To this end, we developed a predictive in silico framework by combining spatially resolved mechanistic agent based modeling, published TIRF imaging data, and parameter estimation to determine mechanisms by which formation and spatial movements of activating NKG2D microclusters affect early time NKG2D signaling kinetics in a human cell line NKL. We show co-clustering of NKG2D and the guanosine nucleotide exchange factor Vav1 in NKG2D microclusters plays a dominant role over ligand (ULBP3) rebinding in increasing production of phospho-Vav1(pVav1), an activation marker of early NKG2D signaling. The in silico model successfully predicts several scenarios of inhibition of NKG2D signaling and time course of NKG2D spatial clustering over a short (~3 min) interval. Modeling shows the presence of a spatial positive feedback relating formation and centripetal movements of NKG2D microclusters, and pVav1 production offers flexibility towards suppression of activating signals by inhibitory KIR ligands organized in inhomogeneous spatial patterns (e.g., a ring). Our in silico framework marks a major improvement in developing spatiotemporal signaling models with quantitatively estimated model parameters using imaging data.

Natural Killer cells are lymphocytes of our innate immunity and provide important resistance against viral infections and tumors. NK cells scan the local environment with diverse activating and inhibitory NK cell receptors (NKRs) and remain tolerized or lyse target cells expressing cognate ligands to NKRs. NKRs have been found to form micron sized clusters (or microclusters) as they interact with cognate ligands, and mechanisms regarding how the formation and movements of these microclusters influence NK cell signaling and activation, specifically related to activating NKRs, are often unclear. To this end, we develop a predictive spatially resolved early-time NK cell signaling model to study the interplay between membrane-proximal biochemical signaling events and the kinetics of microclusters of activating NKG2D and inhibitory KIR2DL2 receptors. We used published TIRF imaging data to validate our in silico models and estimate model parameters. Predictions from multiple in silico models are tested against a variety of data obtained from published imaging experiments and immunoassays. Our analysis suggests co-clustering of NKG2D and the guanosine nucleotide exchange factor Vav1 in the microclusters plays a major role in enhancing downstream activating signals. The developed framework can be extended to describe spatiotemporal signaling for other activating NKRs including CD16.

Proofreading through spatial gradients

Key enzymatic processes use the nonequilibrium error correction mechanism called kinetic proofreading to enhance their specificity. The applicability of traditional proofreading schemes, however, is limited because they typically require dedicated structural features in the enzyme, such as a nucleotide hydrolysis site or multiple intermediate conformations. Here, we explore an alternative conceptual mechanism that achieves error correction by having substrate binding and subsequent product formation occur at distinct physical locations. The time taken by the enzyme–substrate complex to diffuse from one location to another is leveraged to discard wrong substrates. This mechanism does not have the typical structural requirements, making it easier to overlook in experiments. We discuss how the length scales of molecular gradients dictate proofreading performance, and quantify the limitations imposed by realistic diffusion and reaction rates. Our work broadens the applicability of kinetic proofreading and sets the stage for studying spatial gradients as a possible route to specificity.

Engineering AvidCARs for combinatorial antigen recognition and reversible control of CAR function

T cells engineered to express chimeric antigen receptors (CAR-T cells) have shown impressive clinical efficacy in the treatment of B cell malignancies. However, the development of CAR-T cell therapies for solid tumors is hampered by the lack of truly tumor-specific antigens and poor control over T cell activity. Here we present an avidity-controlled CAR (AvidCAR) platform with inducible and logic control functions. The key is the combination of (i) an improved CAR design which enables controlled CAR dimerization and (ii) a significant reduction of antigen-binding affinities to introduce dependence on bivalent interaction, i.e. avidity. The potential and versatility of the AvidCAR platform is exemplified by designing ON-switch CARs, which can be regulated with a clinically applied drug, and AND-gate CARs specifically recognizing combinations of two antigens. Thus, we expect that AvidCARs will be a highly valuable platform for the development of controllable CAR therapies with improved tumor specificity.

Cancer-induced immunosuppression can enable effectiveness of immunotherapy through bistability generation: A mathematical and computational examination

•

The presence of an immunological barrier in cancer- immune system interaction (CISI) is consistent with the bistability patterns in that system.

•

In CISI models, bistability patterns are consistent with immunosuppressive effects dominating immunoproliferative effects.

•

Bistability could be harnessed to devise effective combination immunotherapy approaches.

Cancer immunotherapies rely on how interactions between cancer and immune system cells are constituted. The more essential to the emergence of the dynamical behavior of cancer growth these interactions are, the more effectively they may be used as mechanisms for interventions. Mathematical modeling can help unearth such connections, and help explain how they shape the dynamics of cancer growth. Here, we explored whether there exist simple, consistent properties of cancer-immune system interaction (CISI) models that might be harnessed to devise effective immunotherapy approaches. We did this for a family of three related models of increasing complexity. To this end, we developed a base model of CISI, which captures some essential features of the more complex models built on it. We find that the base model and its derivates can plausibly reproduce biological behavior that is consistent with the notion of an immunological barrier. This behavior is also in accord with situations in which the suppressive effects exerted by cancer cells on immune cells dominate their proliferative effects. Under these circumstances, the model family may display a pattern of bistability, where two distinct, stable states (a cancer-free, and a full-grown cancer state) are possible. Increasing the effectiveness of immune-caused cancer cell killing may remove the basis for bistability, and abruptly tip the dynamics of the system into a cancer-free state. Additionally, in combination with the administration of immune effector cells, modifications in cancer cell killing may be harnessed for immunotherapy without the need for resolving the bistability. We use these ideas to test immunotherapeutic interventions in silico in a stochastic version of the base model. This bistability-reliant approach to cancer interventions might offer advantages over those that comprise gradual declines in cancer cell numbers.

Nonequilibrium Thermodynamics in Cell Biology: Extending Equilibrium Formalism to Cover Living Systems

We discuss new developments in the nonequilibrium dynamics and thermodynamics of living systems, giving a few examples to demonstrate the importance of nonequilibrium thermodynamics for understanding biological dynamics and functions. We study single-molecule enzyme dynamics, in which the nonequilibrium thermodynamic and dynamic driving forces of chemical potential and flux are crucial for the emergence of non-Michaelis-Menten kinetics. We explore single-gene expression dynamics, in which nonequilibrium dissipation can suppress fluctuations. We investigate the cell cycle and identify the nutrition supply as the energy input that sustains the stability, speed, and coherence of cell cycle oscillation, from which the different vital phases of the cell cycle emerge. We examine neural decision-making processes and find the trade-offs among speed, accuracy, and thermodynamic costs that are important for neural function. Lastly, we consider the thermodynamic cost for specificity in cellular signaling and adaptation.

Allostery and Kinetic Proofreading

Kinetic proofreading is an error correction mechanism present in the processes of the central dogma and beyond and typically requires the free energy of nucleotide hydrolysis for its operation. Though the molecular players of many biological proofreading schemes are known, our understanding of how energy consumption is managed to promote fidelity remains incomplete. In our work, we introduce an alternative conceptual scheme called "the piston model of proofreading" in which enzyme activation through hydrolysis is replaced with allosteric activation achieved through mechanical work performed by a piston on regulatory ligands. Inspired by Feynman's ratchet and pawl mechanism, we consider a mechanical engine designed to drive the piston actions powered by a lowering weight, whose function is analogous to that of ATP synthase in cells. Thanks to its mechanical design, the piston model allows us to tune the "knobs" of the driving engine and probe the graded changes and trade-offs between speed, fidelity, and energy dissipation. It provides an intuitive explanation of the conditions necessary for optimal proofreading and reveals the unexpected capability of allosteric molecules to beat the Hopfield limit of fidelity by leveraging the diversity of states available to them. The framework that we have built for the piston model can also serve as a basis for additional studies of driven biochemical systems.

Assessing the interactions between radiotherapy and antitumour immunity

Immunotherapy, specifically the introduction of immune checkpoint inhibitors, has transformed the treatment of cancer, enabling long-term tumour control even in individuals with advanced-stage disease. Unfortunately, only a small subset of patients show a response to currently available immunotherapies. Despite a growing consensus that combining immune checkpoint inhibitors with radiotherapy can increase response rates, this approach might be limited by the development of persistent radiation-induced immunosuppression. The ultimate goal of combining immunotherapy with radiotherapy is to induce a shift from an ineffective, pre-existing immune response to a long-lasting, therapy-induced immune response at all sites of disease. To achieve this goal and enable the adaptation and monitoring of individualized treatment approaches, assessment of the dynamic changes in the immune system at the patient level is essential. In this Review, we summarize the available clinical data, including forthcoming methods to assess the immune response to radiotherapy at the patient level, ranging from serum biomarkers to imaging techniques that enable investigation of immune cell dynamics in patients. Furthermore, we discuss modelling approaches that have been developed to predict the interaction of immunotherapy with radiotherapy, and highlight how they could be combined with biomarkers of antitumour immunity to optimize radiotherapy regimens and maximize their synergy with immunotherapy.

Learning-accelerated discovery of immune-tumour interactions

We present an integrated framework for enabling dynamic exploration of design spaces for cancer immunotherapies with detailed dynamical simulation models on high-performance computing resources. Our framework combines PhysiCell, an open source agent-based simulation platform for cancer and other multicellular systems, and EMEWS, an open source platform for extreme-scale model exploration. We build an agent-based model of immunosurveillance against heterogeneous tumours, which includes spatial dynamics of stochastic tumour-immune contact interactions. We implement active learning and genetic algorithms using high-performance computing workflows to adaptively sample the model parameter space and iteratively discover optimal cancer regression regions within biological and clinical constraints.

Data analysis to modeling to building theory in NK cell biology and beyond: How can computational modeling contribute?

The use of mathematical and computational tools in investigating Natural Killer (NK) cell biology and in general the immune system has increased steadily in the last few decades. However, unlike the physical sciences, there is a persistent ambivalence, which however is increasingly diminishing, in the biology community toward appreciating the utility of quantitative tools in addressing questions of biological importance. We survey some of the recent developments in the application of quantitative approaches for investigating different problems in NK cell biology and evaluate opportunities and challenges of using quantitative methods in providing biological insights in NK cell biology.

Quantitative Mechanistic Modeling in Support of Pharmacological Therapeutics Development in Immuno-Oncology

Following the approval, in recent years, of the first immune checkpoint inhibitor, there has been an explosion in the development of immuno-modulating pharmacological modalities for the treatment of various cancers. From the discovery phase to late-stage clinical testing and regulatory approval, challenges in the development of immuno-oncology (IO) drugs are multi-fold and complex. In the preclinical setting, the multiplicity of potential drug targets around immune checkpoints, the growing list of immuno-modulatory molecular and cellular forces in the tumor microenvironment-with additional opportunities for IO drug targets, the emergence of exploratory biomarkers, and the unleashed potential of modality combinations all have necessitated the development of quantitative, mechanistically-oriented systems models which incorporate key biology and patho-physiology aspects of immuno-oncology and the pharmacokinetics of IO-modulating agents. In the clinical setting, the qualification of surrogate biomarkers predictive of IO treatment efficacy or outcome, and the corresponding optimization of IO trial design have become major challenges. This mini-review focuses on the evolution and state-of-the-art of quantitative systems models describing the tumor vs. immune system interplay, and their merging with quantitative pharmacology models of IO-modulating agents, as companion tools to support the addressing of these challenges.

Hypersensitivity to monoclonal antibodies used for cancer and inflammatory or connective tissue diseases

OBJECTIVE

To review the medical literature on hypersensitivity reactions to therapeutic monoclonal antibodies for patients with malignant tumors and chronic inflammatory or connective tissues diseases.

DATA SOURCES

We searched the PubMed database using the terms monoclonal antibody, hypersensitivity, and allergy.

STUDY SELECTIONS

We selected case reports and cohort studies of patients with hypersensitivity reactions to monoclonal antibodies. We included selected review articles to glean expert opinion on issues for which high-quality data are available. We sought specific information on the incidence, clinical description, pathobiology, and treatment of reactions.

RESULTS

Hypersensitivity reactions to therapeutic monoclonal antibodies can be classic type I (mast cell mediated, perhaps IgE dependent) reactions, cytokine release reactions, or type IV cell-mediated reactions. There are limited data on the frequency of such reactions, and because new agents are added to the set at a relatively high rate, it is difficult to determine precisely the incidence of reactions to this class of drugs as a whole. The classification of a specific hypersensitivity reaction depends mainly on the medical history. Skin testing may be available but often is not validated and may be prohibitively expensive. Avoidance of the culpable agent is ideal, but if treatment with the responsible drug is necessary, rapid drug desensitization is an option for type I reactions. Desensitization is less likely to be effective for cytokine release reactions and is contraindicated for type IV reactions.

CONCLUSION

Hypersensitivity reactions to therapeutic monoclonal antibodies are heterogeneous. Management depends on accurate identification and thoughtful consideration of the pathobiologic features of the reaction.

Quantitative systems pharmacology of interferon alpha administration: A multi-scale approach

The therapeutic effect of a drug is governed by its pharmacokinetics which determine the downstream pharmacodynamic response within the cellular network. A complete understanding of the drug-effect relationship therefore requires multi-scale models which integrate the properties of the different physiological scales. Computational modelling of these individual scales has been successfully established in the past. However, coupling of the scales remains challenging, although it will provide a unique possibility of mechanistic and holistic analyses of therapeutic outcomes for varied treatment scenarios. We present a methodology to combine whole-body physiologically-based pharmacokinetic (PBPK) models with mechanistic intracellular models of signal transduction in the liver for therapeutic proteins. To this end, we developed a whole-body distribution model of IFN-α in human and a detailed intracellular model of the JAK/STAT signalling cascade in hepatocytes and coupled them at the liver of the whole-body human model. This integrated model infers the time-resolved concentration of IFN-α arriving at the liver after intravenous injection while simultaneously estimates the effect of this dose on the intracellular signalling behaviour in the liver. In our multi-scale physiologically-based pharmacokinetic/pharmacodynamic (PBPK/PD) model, receptor saturation is seen at low doses, thus giving mechanistic insights into the pharmacodynamic (PD) response. This model suggests a fourfold lower intracellular response after administration of a typical IFN-α dose to an individual as compared to the experimentally observed responses in in vitro setups. In conclusion, this work highlights clear differences between the observed in vitro and in vivo drug effects and provides important suggestions for future model-based study design.

Sensitivity analysis for reproducible candidate values of model parameters in signaling hub model

Mathematical models for signaling pathways are helpful for understanding molecular mechanism in the pathways and predicting dynamic behavior of the signal activity. To analyze the robustness of such models, local sensitivity analysis has been implemented. However, such analysis primarily focuses on only a certain parameter set, even though diverse parameter sets that can recapitulate experiments may exist. In this study, we performed sensitivity analysis that investigates the features in a system considering the reproducible and multiple candidate values of the model parameters to experiments. The results showed that although different reproducible model parameter values have absolute differences with respect to sensitivity strengths, specific trends of some relative sensitivity strengths exist between reactions regardless of parameter values. It is suggested that (i) network structure considerably influences the relative sensitivity strength and (ii) one might be able to predict relative sensitivity strengths specified in the parameter sets employing only one of the reproducible parameter sets.

A review of inflammatory mechanism in airway diseases

BACKGROUND

Inflammation in the lung is the body's natural response to injury. It acts to remove harmful stimuli such as pathogens, irritants, and damaged cells and initiate the healing process. Acute and chronic pulmonary inflammation are seen in different respiratory diseases such as; acute respiratory distress syndrome, chronic obstructive pulmonary disease (COPD), asthma, and cystic fibrosis (CF).

FINDINGS

In this review, we found that inflammatory response in COPD is determined by the activation of epithelial cells and macrophages in the respiratory tract. Epithelial cells and macrophages discharge transforming growth factor-β (TGF-β), which trigger fibroblast proliferation and tissue remodeling. Asthma leads to airway hyper-responsiveness, obstruction, mucus hyper-production, and airway-wall remodeling. Cytokines, allergens, chemokines, and infectious agents are the main stimuli that activate signaling pathways in epithelial cells in asthma. Mutation of the CF transmembrane conductance regulator (CFTR) gene results in CF. Mutations in CFTR influence the lung epithelial innate immune function that leads to exaggerated and ineffective airway inflammation that fails to abolish pulmonary pathogens. We present mechanistic computational models (based on ordinary differential equations, partial differential equations and agent-based models) that have been applied in studying the complex physiological and pathological mechanisms of chronic inflammation in different airway diseases.

CONCLUSION

The scope of the present review is to explore the inflammatory mechanism in airway diseases and highlight the influence of aging on airways' inflammation mechanism. The main goal of this review is to encourage research collaborations between experimentalist and modelers to promote our understanding of the physiological and pathological mechanisms that control inflammation in different airway diseases.

Evaluation of Parallel Tempering to Accelerate Bayesian Parameter Estimation in Systems Biology

Models of biological systems often have many unknown parameters that must be determined in order for model behavior to match experimental observations. Commonly-used methods for parameter estimation that return point estimates of the best-fit parameters are insufficient when models are high dimensional and under-constrained. As a result, Bayesian methods, which treat model parameters as random variables and attempt to estimate their probability distributions given data, have become popular in systems biology. Bayesian parameter estimation often relies on Markov Chain Monte Carlo (MCMC) methods to sample model parameter distributions, but the slow convergence of MCMC sampling can be a major bottleneck. One approach to improving performance is parallel tempering (PT), a physics-based method that uses swapping between multiple Markov chains run in parallel at different temperatures to accelerate sampling. The temperature of a Markov chain determines the probability of accepting an unfavorable move, so swapping with higher temperatures chains enables the sampling chain to escape from local minima. In this work we compared the MCMC performance of PT and the commonly-used Metropolis-Hastings (MH) algorithm on six biological models of varying complexity. We found that for simpler models PT accelerated convergence and sampling, and that for more complex models, PT often converged in cases MH became trapped in non-optimal local minima. We also developed a freely-available MATLAB package for Bayesian parameter estimation called PTEMPEST (http://github.com/RuleWorld/ptempest), which is closely integrated with the popular BioNetGen software for rule-based modeling of biological systems.

Dimensionality of Motion and Binding Valency Govern Receptor-Ligand Kinetics As Revealed by Agent-Based Modeling

Mathematical modeling and computer simulations have become an integral part of modern biological research. The strength of theoretical approaches is in the simplification of complex biological systems. We here consider the general problem of receptor–ligand binding in the context of antibody–antigen binding. On the one hand, we establish a quantitative mapping between macroscopic binding rates of a deterministic differential equation model and their microscopic equivalents as obtained from simulating the spatiotemporal binding kinetics by stochastic agent-based models. On the other hand, we investigate the impact of various properties of B cell-derived receptors—such as their dimensionality of motion, morphology, and binding valency—on the receptor–ligand binding kinetics. To this end, we implemented an algorithm that simulates antigen binding by B cell-derived receptors with a Y-shaped morphology that can move in different dimensionalities, i.e., either as membrane-anchored receptors or as soluble receptors. The mapping of the macroscopic and microscopic binding rates allowed us to quantitatively compare different agent-based model variants for the different types of B cell-derived receptors. Our results indicate that the dimensionality of motion governs the binding kinetics and that this predominant impact is quantitatively compensated by the bivalency of these receptors.

An experimental and computational framework for engineering multifunctional nanoparticles: designing selective anticancer therapies

A key challenge in the treatment of cancer with nanomedicine is to engineer and select nanoparticle formulations that lead to the desired selectivity between tumorigenic and non-tumorigenic cells. To this aim, novel designed nanomaterials, deep biochemical understanding of the mechanisms of interaction between nanomaterials and cells, and computational models are emerging as very useful tools to guide the design of efficient and selective nanotherapies. This works shows, using a combination of detailed experimental approaches and simulations, that the specific targeting of cancer cells in comparison to non-tumorigenic cells can be achieved through the custom design of multivalent nanoparticles. A theoretical model that provides simple yet quantitative predictions to tune the nanoparticles targeting and cytotoxic properties by their degree of functionalization is developed. As a case study, a system that included a targeting agent and a drug and is amenable to controlled experimental manipulation and theoretical analysis is used. This study shows how at defined functionalization levels multivalent nanoparticles can selectively kill tumor cells, while barely affecting non-tumorigenic cells. This work opens a way to the rational design of multifunctionalized nanoparticles with defined targeting and cytotoxic properties for practical applications.

Splitting nodes and linking channels: A method for assembling biocircuits from stochastic elementary units

Akin to electric circuits, we construct biocircuits that are manipulated by cutting and assembling channels through which stochastic information flows. This diagrammatic manipulation allows us to create a method which constructs networks by joining building blocks selected so that (a) they cover only basic processes; (b) it is scalable to large networks; (c) the mean and variance-covariance from the Pauli master equation form a closed system; and (d) given the initial probability distribution, no special boundary conditions are necessary to solve the master equation. The method aims to help with both designing new synthetic signaling pathways and quantifying naturally existing regulatory networks.

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.

The Dark Side of Cell Signaling: Positive Roles for Negative Regulators

Cell signaling is dominated by analyzing positive responses to stimuli. Signal activation is balanced by negative regulators that are generally considered to terminate signaling. Rather than exerting only negative effects, however, many such regulators play important roles in enhancing cell-signaling control. Considering responses downstream of selected cell-surface receptors, we discuss how receptor internalization affects signaling specificity and how rapid kinase/phosphatase and GTP/GDP cycles increase responsiveness and allow kinetic proofreading in receptor signaling. We highlight the blurring of distinctions between positive and negative signals, recasting signal termination as the response to a switch-like transition into a new cellular state.

Automated adaptive inference of phenomenological dynamical models

Dynamics of complex systems is often driven by large and intricate networks of microscopic interactions, whose sheer size obfuscates understanding. With limited experimental data, many parameters of such dynamics are unknown, and thus detailed, mechanistic models risk overfitting and making faulty predictions. At the other extreme, simple ad hoc models often miss defining features of the underlying systems. Here we develop an approach that instead constructs phenomenological, coarse-grained models of network dynamics that automatically adapt their complexity to the available data. Such adaptive models produce accurate predictions even when microscopic details are unknown. The approach is computationally tractable, even for a relatively large number of dynamical variables. Using simulated data, it correctly infers the phase space structure for planetary motion, avoids overfitting in a biological signalling system and produces accurate predictions for yeast glycolysis with tens of data points and over half of the interacting species unobserved.

Mechanistic modelling of dynamical phenomena with many degrees of freedom runs the risk of overfitting and making faulty predictions, whereas ad hoc models may miss defining features. Here the authors develop an approach to construct dynamical models that adapt their complexity to the amount of available data.

Modeling for (physical) biologists: an introduction to the rule-based approach

Models that capture the chemical kinetics of cellular regulatory networks can be specified in terms of rules for biomolecular interactions. A rule defines a generalized reaction, meaning a reaction that permits multiple reactants, each capable of participating in a characteristic transformation and each possessing certain, specified properties, which may be local, such as the state of a particular site or domain of a protein. In other words, a rule defines a transformation and the properties that reactants must possess to participate in the transformation. A rule also provides a rate law. A rule-based approach to modeling enables consideration of mechanistic details at the level of functional sites of biomolecules and provides a facile and visual means for constructing computational models, which can be analyzed to study how system-level behaviors emerge from component interactions.

Integrated signaling pathway and gene expression regulatory model to dissect dynamics of Escherichia coli challenged mammary epithelial cells

Cells transform external stimuli, through the activation of signaling pathways, which in turn activate gene regulatory networks, in gene expression. As more omics data are generated from experiments, eliciting the integrated relationship between the external stimuli, the signaling process in the cell and the subsequent gene expression is a major challenge in systems biology. The complex system of non-linear dynamic protein interactions in signaling pathways and gene networks regulates gene expression. The complexity and non-linear aspects have resulted in the study of the signaling pathway or the gene network regulation in isolation. However, this limits the analysis of the interaction between the two components and the identification of the source of the mechanism differentiating the gene expression profiles. Here, we present a study of a model of the combined signaling pathway and gene network to highlight the importance of integrated modeling. Based on the experimental findings we developed a compartmental model and conducted several simulation experiments. The model simulates the mRNA expression of three different cytokines (RANTES, IL8 and TNFα) regulated by the transcription factor NFκB in mammary epithelial cells challenged with E. coli. The analysis of the gene network regulation identifies a lack of robustness and therefore sensitivity for the transcription factor regulation. However, analysis of the integrated signaling and gene network regulation model reveals distinctly different underlying mechanisms in the signaling pathway responsible for the variation between the three cytokine's mRNA expression levels. Our key findings reveal the importance of integrating the signaling pathway and gene expression dynamics in modeling. Modeling infers valid research questions which need to be verified experimentally and can assist in the design of future biological experiments.

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.

Pleomorphic ensembles: formation of large clusters composed of weakly interacting multivalent molecules

Molecular interactions of importance to cell biology are subject to sol-gel transitions: large clusters of weakly interacting multivalent molecules (gel phase) are produced at a critical concentration of monomers. Examples include cell-cell and cell-matrix adhesions, nucleoprotein bodies, and cell signaling platforms. We use the term pleomorphic ensembles (PEs) to describe these clusters, because they have dynamic compositions and sizes and have rapid turnover of their molecular constituents; this plasticity can be highly responsive to cellular signals. The classical polymer physical chemistry theory developed by Flory and Stockmayer provides a brilliant framework for treating multivalent interactions for simple idealized systems. But the complexity and variability of PEs challenges existing modeling approaches. Here we describe and validate a computational algorithm that extends the Flory-Stockmayer formalism to overcome the limitations of analytic theories. We divide the problem by deterministically calculating the fraction of bound sites for each type of binding site, followed by the stochastic assignment of the bonds to a finite number of molecules. The method allows for high valency within many different kinds of interacting molecules and site types, permits simulation of steady-state distributions, as well as assembly kinetics, and can treat cooperative binding within one of the interacting molecules. We then apply our method to the analysis of interactions in the nephrin-Nck-N-Wasp signaling system, demonstrating how multivalent layered scaffolds produce PEs at low monomer concentrations despite weak binding interactions. We show how the experimental data for this system are most consistent with synergistic cooperative interactions between Nck and N-Wasp.

Changing face of vaccination in immunocompromised hosts

Infection prevention is a key component of care and an important determinant of clinical outcomes in a diverse population of immunocompromised hosts. Vaccination remains a fundamental preventative strategy, and clear guidelines exist for the vaccination of immunocompromised individuals and close contacts. Unfortunately, adherence to such guidelines is frequently suboptimal, with consequent missed opportunities to prevent infection. Additionally, vaccination of immunocompromised individuals is known to produce responses inferior to those observed in immunocompetent hosts. Multiple factors contribute to this finding, and developing improved vaccination strategies for those at high risk of infectious complications remains a priority of care providers. Herein, we review potential factors contributing to vaccine outcomes, focusing on host immune responses, and propose a means for applying modern, innovative systems biology technology to model critical determinants of vaccination success. With influenza vaccine in solid organ transplants used as a case in point, novel means for stratifying individuals using a host "immunophenotype" are explored, and strategies for individualizing vaccine approaches tailored to safely optimize vaccine responses in those most at risk are discussed.

A hierarchical Bayesian framework to infer the progression level to diabetes based on deficient clinical data

The increase in lifestyle-related diseases such as heart disease, diabetes, and high blood pressure is a challenging problem that should be resolved. The physiological mechanisms of the human body have long been studied using mathematical models. In particular, to study glucose metabolism, several models that infer insulin sensitivity and β-cell function have been developed. The use of mathematical models to assess progression to diabetes based on clinical data could be effective for preventing the onset of diabetes. However, to assess the progression level, we need clinical data including data from oral glucose tolerance tests, which are not typically performed on patients whose glucose tolerance may be impaired. To address this shortcoming, we developed a hierarchical Bayesian framework to infer the progression of glucose intolerance based on deficient data. We demonstrated how the framework infers the level of progression to diabetes and showed that glucose disposal capacity and insulin-secretory function depend on the fasting glucose and glycated hemoglobin (HbA1c) levels.

Receptor Pre-Clustering and T cell Responses: Insights into Molecular Mechanisms

T cell activation, initiated by T cell receptor (TCR) mediated recognition of pathogen-derived peptides presented by major histocompatibility complex class I or II molecules (pMHC), shows exquisite specificity and sensitivity, even though the TCR-pMHC binding interaction is of low affinity. Recent experimental work suggests that TCR pre-clustering may be a mechanism via which T cells can achieve such high sensitivity. The unresolved stoichiometry of the TCR makes TCR-pMHC binding and TCR triggering, an open question. We formulate a mathematical model to characterize the pre-clustering of T cell receptors (TCRs) on the surface of T cells, motivated by the experimentally observed distribution of TCR clusters on the surface of naive and memory T cells. We extend a recently introduced stochastic criterion to compute the timescales of T cell responses, assuming that ligand-induced cross-linked TCR is the minimum signaling unit. We derive an approximate formula for the mean time to signal initiation. Our results show that pre-clustering reduces the mean activation time. However, additional mechanisms favoring the existence of clusters are required to explain the difference between naive and memory T cell responses. We discuss the biological implications of our results, and both the compatibility and complementarity of our approach with other existing mathematical models.

Exact hybrid particle/population simulation of rule-based models of biochemical systems

Detailed modeling and simulation of biochemical systems is complicated by the problem of combinatorial complexity, an explosion in the number of species and reactions due to myriad protein-protein interactions and post-translational modifications. Rule-based modeling overcomes this problem by representing molecules as structured objects and encoding their interactions as pattern-based rules. This greatly simplifies the process of model specification, avoiding the tedious and error prone task of manually enumerating all species and reactions that can potentially exist in a system. From a simulation perspective, rule-based models can be expanded algorithmically into fully-enumerated reaction networks and simulated using a variety of network-based simulation methods, such as ordinary differential equations or Gillespie's algorithm, provided that the network is not exceedingly large. Alternatively, rule-based models can be simulated directly using particle-based kinetic Monte Carlo methods. This “network-free” approach produces exact stochastic trajectories with a computational cost that is independent of network size. However, memory and run time costs increase with the number of particles, limiting the size of system that can be feasibly simulated. Here, we present a hybrid particle/population simulation method that combines the best attributes of both the network-based and network-free approaches. The method takes as input a rule-based model and a user-specified subset of species to treat as population variables rather than as particles. The model is then transformed by a process of “partial network expansion” into a dynamically equivalent form that can be simulated using a population-adapted network-free simulator. The transformation method has been implemented within the open-source rule-based modeling platform BioNetGen, and resulting hybrid models can be simulated using the particle-based simulator NFsim. Performance tests show that significant memory savings can be achieved using the new approach and a monetary cost analysis provides a practical measure of its utility.

Rule-based modeling is a modeling paradigm that addresses the problem of combinatorial complexity in biochemical systems. The key idea is to specify only those components of a biological macromolecule that are directly involved in a biochemical transformation. Until recently, this “pattern-based” approach greatly simplified the process of model building but did nothing to improve the performance of model simulation. This changed with the introduction of “network-free” simulation methods, which operate directly on the compressed rule set of a rule-based model rather than on a fully-enumerated set of reactions and species. However, these methods represent every molecule in a system as a particle, limiting their use to systems containing less than a few million molecules. Here, we describe an extension to the network-free approach that treats rare, complex species as particles and plentiful, simple species as population variables, while retaining the exact dynamics of the model system. By making more efficient use of computational resources for species that do not require the level of detail of a particle representation, this hybrid particle/population approach can simulate systems much larger than is possible using network-free methods and is an important step towards realizing the practical simulation of detailed, mechanistic models of whole cells.

The past, present, and future of immune repertoire biology - the rise of next-generation repertoire analysis

T and B cell repertoires are collections of lymphocytes, each characterized by its antigen-specific receptor. We review here classical technologies and analysis strategies developed to assess immunoglobulin (IG) and T cell receptor (TR) repertoire diversity, and describe recent advances in the field. First, we describe the broad range of available methodological tools developed in the past decades, each of which answering different questions and showing complementarity for progressive identification of the level of repertoire alterations: global overview of the diversity by flow cytometry, IG repertoire descriptions at the protein level for the identification of IG reactivities, IG/TR CDR3 spectratyping strategies, and related molecular quantification or dynamics of T/B cell differentiation. Additionally, we introduce the recent technological advances in molecular biology tools allowing deeper analysis of IG/TR diversity by next-generation sequencing (NGS), offering systematic and comprehensive sequencing of IG/TR transcripts in a short amount of time. NGS provides several angles of analysis such as clonotype frequency, CDR3 diversity, CDR3 sequence analysis, V allele identification with a quantitative dimension, therefore requiring high-throughput analysis tools development. In this line, we discuss the recent efforts made for nomenclature standardization and ontology development. We then present the variety of available statistical analysis and modeling approaches developed with regards to the various levels of diversity analysis, and reveal the increasing sophistication of those modeling approaches. To conclude, we provide some examples of recent mathematical modeling strategies and perspectives that illustrate the active rise of a "next-generation" of repertoire analysis.

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.

Latent Tuberculosis: Models, Computational Efforts and the Pathogen's Regulatory Mechanisms during Dormancy

Latent tuberculosis is a clinical syndrome that occurs after an individual has been exposed to the Mycobacterium tuberculosis (Mtb) Bacillus, the infection has been established and an immune response has been generated to control the pathogen and force it into a quiescent state. Mtb can exit this quiescent state where it is unresponsive to treatment and elusive to the immune response, and enter a rapid replicating state, hence causing infection reactivation. It remains a gray area to understand how the pathogen causes a persistent infection and it is unclear whether the organism will be in a slow replicating state or a dormant non-replicating state. The ability of the pathogen to adapt to changing host immune response mechanisms, in which it is exposed to hypoxia, low pH, nitric oxide (NO), nutrient starvation, and several other anti-microbial effectors, is associated with a high metabolic plasticity that enables it to metabolize under these different conditions. Adaptive gene regulatory mechanisms are thought to coordinate how the pathogen changes their metabolic pathways through mechanisms that sense changes in oxygen tension and other stress factors, hence stimulating the pathogen to make necessary adjustments to ensure survival. Here, we review studies that give insights into latency/dormancy regulatory mechanisms that enable infection persistence and pathogen adaptation to different stress conditions. We highlight what mathematical and computational models can do and what they should do to enhance our current understanding of TB latency.

Spatiotemporal control and superselectivity in supramolecular polymers using multivalency

Multivalency has an important but poorly understood role in molecular self-organization. We present the noncovalent synthesis of a multicomponent supramolecular polymer in which chemically distinct monomers spontaneously coassemble into a dynamic, functional structure. We show that a multivalent recruiter is able to bind selectively to one subset of monomers (receptors) and trigger their clustering along the self-assembled polymer, behavior that mimics raft formation in cell membranes. This phenomenon is reversible and affords spatiotemporal control over the monomer distribution inside the supramolecular polymer by superselective binding of single-strand DNA to positively charged receptors. Our findings reveal the pivotal role of multivalency in enabling structural order and nonlinear recognition in water-soluble supramolecular polymers, and it offers a design principle for functional, structurally defined supramolecular architectures.

A large number of receptors may reduce cellular response time variation

Cells often have tens of thousands of receptors, even though only a few activated receptors can trigger full cellular responses. Reasons for the overabundance of receptors remain unclear. We suggest that, under certain conditions, the large number of receptors can result in a competition among receptors to be the first to activate the cell. The competition decreases the variability of the time to cellular activation, and hence results in a more synchronous activation of cells. We argue that, in simple models, this variability reduction does not necessarily interfere with the receptor specificity to ligands achieved by the kinetic proofreading mechanism. Thus cells can be activated accurately in time and specifically to certain signals. We predict the minimum number of receptors needed to reduce the coefficient of variation for the time to activation following binding of a specific ligand. Furthermore, we predict the maximum number of receptors so that the kinetic proofreading mechanism still can improve the specificity of the activation. These predictions fall in line with experimentally reported receptor numbers for multiple systems.

Population-expression models of immune response

The immune response to a pathogen has two basic features. The first is the expansion of a few pathogen-specific cells to form a population large enough to control the pathogen. The second is the process of differentiation of cells from an initial naive phenotype to an effector phenotype which controls the pathogen, and subsequently to a memory phenotype that is maintained and responsible for long-term protection. The expansion and the differentiation have been considered largely independently. Changes in cell populations are typically described using ecologically based ordinary differential equation models. In contrast, differentiation of single cells is studied within systems biology and is frequently modeled by considering changes in gene and protein expression in individual cells. Recent advances in experimental systems biology make available for the first time data to allow the coupling of population and high dimensional expression data of immune cells during infections. Here we describe and develop population-expression models which integrate these two processes into systems biology on the multicellular level. When translated into mathematical equations, these models result in non-conservative, non-local advection-diffusion equations. We describe situations where the population-expression approach can make correct inference from data while previous modeling approaches based on common simplifying assumptions would fail. We also explore how model reduction techniques can be used to build population-expression models, minimizing the complexity of the model while keeping the essential features of the system. While we consider problems in immunology in this paper, we expect population-expression models to be more broadly applicable.

Characterizing the relationship between steady state and response using analytical expressions for the steady states of mass action models

The steady states of cells affect their response to perturbation. Indeed, diagnostic markers for predicting the response to therapeutic perturbation are often based on steady state measurements. In spite of this, no method exists to systematically characterize the relationship between steady state and response. Mathematical models are established tools for studying cellular responses, but characterizing their relationship to the steady state requires that it have a parametric, or analytical, expression. For some models, this expression can be derived by the King-Altman method. However, King-Altman requires that no substrate act as an enzyme, and is therefore not applicable to most models of signal transduction. For this reason we developed py-substitution, a simple but general method for deriving analytical expressions for the steady states of mass action models. Where the King-Altman method is applicable, we show that py-substitution yields an equivalent expression, and at comparable efficiency. We use py-substitution to study the relationship between steady state and sensitivity to the anti-cancer drug candidate, dulanermin (recombinant human TRAIL). First, we use py-substitution to derive an analytical expression for the steady state of a published model of TRAIL-induced apoptosis. Next, we show that the amount of TRAIL required for cell death is sensitive to the steady state concentrations of procaspase 8 and its negative regulator, Bar, but not the other procaspase molecules. This suggests that activation of caspase 8 is a critical point in the death decision process. Finally, we show that changes in the threshold at which TRAIL results in cell death is not always equivalent to changes in the time of death, as is commonly assumed. Our work demonstrates that an analytical expression is a powerful tool for identifying steady state determinants of the cellular response to perturbation. All code is available at http://signalingsystems.ucsd.edu/models-and-code/ or as supplementary material accompanying this paper.