Generalizing Gillespie's Direct Method to Enable Network-Free Simulations

Parameter Estimation and Uncertainty Quantification for Systems Biology Models

Mathematical models can provide quantitative insights into immunoreceptor signaling, and other biological processes, but require parameterization and uncertainty quantification before reliable predictions become possible. We review currently available methods and software tools to address these problems. We consider gradient-based and gradient-free methods for point estimation of parameter values, and methods of profile likelihood, bootstrapping, and Bayesian inference for uncertainty quantification. We consider recent and potential future applications of these methods to systems-level modeling of immune-related phenomena.

PyBioNetFit and the Biological Property Specification Language

In systems biology modeling, important steps include model parameterization, uncertainty quantification, and evaluation of agreement with experimental observations. To help modelers perform these steps, we developed the software PyBioNetFit, which in addition supports checking models against known system properties and solving design problems. PyBioNetFit introduces Biological Property Specification Language (BPSL) for the formal declaration of system properties. BPSL allows qualitative data to be used alone or in combination with quantitative data. PyBioNetFit performs parameterization with parallelized metaheuristic optimization algorithms that work directly with existing model definition standards: BioNetGen Language (BNGL) and Systems Biology Markup Language (SBML). We demonstrate PyBioNetFit's capabilities by solving various example problems, including the challenging problem of parameterizing a 153-parameter model of cell cycle control in yeast based on both quantitative and qualitative data. We demonstrate the model checking and design applications of PyBioNetFit and BPSL by analyzing a model of targeted drug interventions in autophagy signaling.

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.

Using RuleBuilder to Graphically Define and Visualize BioNetGen-Language Patterns and Reaction Rules

RuleBuilder is a tool for drawing graphs that can be represented by the BioNetGen language (BNGL), which is used to formulate mathematical, rule-based models of biochemical systems. BNGL provides an intuitive plain text, or string, representation of such systems, which is based on a graphical formalism. Reactions are defined in terms of graph-rewriting rules that specify the necessary intrinsic properties of the reactants, a transformation, and a rate law. Rules also contain contextual constraints that restrict application of the rule. In some cases, the specification of contextual constraints can be verbose, making a rule difficult to read. RuleBuilder is designed to ease the task of reading and writing individual reaction rules or other BNGL patterns required for model formulation. The software assists in the reading of existing models by converting BNGL strings of interest into a graph-based representation composed of nodes and edges. RuleBuilder also enables the user to construct de novo a visual representation of BNGL strings using drawing tools available in its interface. As objects are added to the drawing canvas, the corresponding BNGL string is generated on the fly, and objects are similarly drawn on the fly as BNGL strings are entered into the application. RuleBuilder thus facilitates construction and interpretation of rule-based models.

A Probabilistic Model of the Germinal Center Reaction

Germinal centers (GCs) are specialized compartments within the secondary lymphoid organs, where B cells proliferate, differentiate, and mutate their antibody genes. Upon exit from the GC, B cells terminally differentiate into plasma cells or memory B cells. While we have a good comprehension of plasma cell differentiation, memory B cell differentiation is still incompletely understood. In this paper, we extend previous models of the molecular events underlying B cell differentiation with new findings regarding memory B cell formation, and present a quantitative stochastic model of the intracellular and extracellular dynamics governing B cell maturation and exit from the GC. To simulate this model, we develop a novel extension to the Gillespie algorithm that enables the efficient stochastic simulation of the system, while keeping track of individual cell properties. Our model is able to explain the dynamical shift from memory B cell to plasma cell production over the lifetime of a GC. Moreover, our results suggest that B cell fate selection can be explained as a process that depends fundamentally on antigen affinity.

Modeling cell line-specific recruitment of signaling proteins to the insulin-like growth factor 1 receptor

Receptor tyrosine kinases (RTKs) typically contain multiple autophosphorylation sites in their cytoplasmic domains. Once activated, these autophosphorylation sites can recruit downstream signaling proteins containing Src homology 2 (SH2) and phosphotyrosine-binding (PTB) domains, which recognize phosphotyrosine-containing short linear motifs (SLiMs). These domains and SLiMs have polyspecific or promiscuous binding activities. Thus, multiple signaling proteins may compete for binding to a common SLiM and vice versa. To investigate the effects of competition on RTK signaling, we used a rule-based modeling approach to develop and analyze models for ligand-induced recruitment of SH2/PTB domain-containing proteins to autophosphorylation sites in the insulin-like growth factor 1 (IGF1) receptor (IGF1R). Models were parameterized using published datasets reporting protein copy numbers and site-specific binding affinities. Simulations were facilitated by a novel application of model restructuration, to reduce redundancy in rule-derived equations. We compare predictions obtained via numerical simulation of the model to those obtained through simple prediction methods, such as through an analytical approximation, or ranking by copy number and/or K_D value, and find that the simple methods are unable to recapitulate the predictions of numerical simulations. We created 45 cell line-specific models that demonstrate how early events in IGF1R signaling depend on the protein abundance profile of a cell. Simulations, facilitated by model restructuration, identified pairs of IGF1R binding partners that are recruited in anti-correlated and correlated fashions, despite no inclusion of cooperativity in our models. This work shows that the outcome of competition depends on the physicochemical parameters that characterize pairwise interactions, as well as network properties, including network connectivity and the relative abundances of competitors.

Cells rely on networks of interacting biomolecules to sense and respond to environmental perturbations and signals. However, it is unclear how information is processed to generate appropriate and specific responses to signals, especially given that these networks tend to share many components. For example, receptors that detect distinct ligands and regulate distinct cellular activities commonly interact with overlapping sets of downstream signaling proteins. Here, to investigate the downstream signaling of a well-studied receptor tyrosine kinase (RTK), the insulin-like growth factor 1 (IGF1) receptor (IGF1R), we formulated and analyzed 45 cell line-specific mathematical models, which account for recruitment of 18 different binding partners to six sites of receptor autophosphorylation in IGF1R. The models were parameterized using available protein copy number and site-specific affinity measurements, and restructured to allow for network generation. We find that recruitment is influenced by the protein abundance profile of a cell, with different patterns of recruitment in different cell lines. Furthermore, in a given cell line, we find that pairs of IGF1R binding partners may be recruited in a correlated or anti-correlated fashion. We demonstrate that the simulations of the model have greater predictive power than protein copy number and/or binding affinity data, and that even a simple analytical model cannot reproduce the predicted recruitment ranking obtained via simulations. These findings represent testable predictions and indicate that the outputs of IGF1R signaling depend on cell line-specific properties in addition to the properties that are intrinsic to the biomolecules involved.

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.

MCell-R: A Particle-Resolution Network-Free Spatial Modeling Framework

Spatial heterogeneity can have dramatic effects on the biochemical networks that drive cell regulation and decision-making. For this reason, a number of methods have been developed to model spatial heterogeneity and incorporated into widely used modeling platforms. Unfortunately, the standard approaches for specifying and simulating chemical reaction networks become untenable when dealing with multistate, multicomponent systems that are characterized by combinatorial complexity. To address this issue, we developed MCell-R, a framework that extends the particle-based spatial Monte Carlo simulator, MCell, with the rule-based model specification and simulation capabilities provided by BioNetGen and NFsim. The BioNetGen syntax enables the specification of biomolecules as structured objects whose components can have different internal states that represent such features as covalent modification and conformation and which can bind components of other molecules to form molecular complexes. The network-free simulation algorithm used by NFsim enables efficient simulation of rule-based models even when the size of the network implied by the biochemical rules is too large to enumerate explicitly, which frequently occurs in detailed models of biochemical signaling. The result is a framework that can efficiently simulate systems characterized by combinatorial complexity at the level of spatially resolved individual molecules over biologically relevant time and length scales.

Intrinsic limits of information transmission in biochemical signalling motifs

All living things have evolved to sense changes in their environment in order to respond in adaptive ways. At the cellular level, these sensing systems generally involve receptor molecules at the cell surface, which detect changes outside the cell and relay those changes to the appropriate response elements downstream. With the advent of experimental technologies that can track signalling at the single-cell level, it has become clear that many signalling systems exhibit significant levels of 'noise,' manifesting as differential responses of otherwise identical cells to the same environment. This noise has a large impact on the capacity of cell signalling networks to transmit information from the environment. Application of information theory to experimental data has found that all systems studied to date encode less than 2.5 bits of information, with the majority transmitting significantly less than 1 bit. Given the growing interest in applying information theory to biological data, it is crucial to understand whether the low values observed to date represent some sort of intrinsic limit on information flow given the inherently stochastic nature of biochemical signalling events. In this work, we used a series of computational models to explore how much information a variety of common 'signalling motifs' can encode. We found that the majority of these motifs, which serve as the basic building blocks of cell signalling networks, can encode far more information (4-6 bits) than has ever been observed experimentally. In addition to providing a consistent framework for estimating information-theoretic quantities from experimental data, our findings suggest that the low levels of information flow observed so far in living system are not necessarily due to intrinsic limitations. Further experimental work will be needed to understand whether certain cell signalling systems actually can approach the intrinsic limits described here, and to understand the sources and purpose of the variation that reduces information flow in living cells.

Using Equation-Free Computation to Accelerate Network-Free Stochastic Simulation of Chemical Kinetics

The chemical kinetics of many complex systems can be concisely represented by reaction rules, which can be used to generate reaction events via a kinetic Monte Carlo method that has been termed network-free simulation. Here, we demonstrate accelerated network-free simulation through a novel approach to equation-free computation. In this process, variables are introduced that approximately capture system state. Derivatives of these variables are estimated using short bursts of exact stochastic simulation and finite differencing. The variables are then projected forward in time via a numerical integration scheme, after which a new exact stochastic simulation is initialized and the whole process repeats. The projection step increases efficiency by bypassing the firing of numerous individual reaction events. As we show, the projected variables may be defined as populations of building blocks of chemical species. The maximal number of connected molecules included in these building blocks determines the degree of approximation. Equation-free acceleration of network-free simulation is found to be both accurate and efficient.