Rule-based modelling and simulation of biochemical systems with molecular finite automata.

The authors propose a theoretical formalism, molecular finite automata (MFA), to describe individual proteins as rule-based computing machines. The MFA formalism provides a framework for modelling individual protein behaviours and systems-level dynamics via construction of programmable and executable machines. Models specified within this formalism explicitly represent the context-sensitive dynamics of individual proteins driven by external inputs and represent protein-protein interactions as synchronised machine reconfigurations. Both deterministic and stochastic simulations can be applied to quantitatively compute the dynamics of MFA models. They apply the MFA formalism to model and simulate a simple example of a signal-transduction system that involves an MAP kinase cascade and a scaffold protein.

Determinants of bistability in induction of the Escherichia coli lac operon.

The authors have developed a mathematical model of regulation of expression of the Escherichia coli lac operon, and have investigated bistability in its steady-state induction behaviour in the absence of external glucose. Numerical analysis of equations describing regulation by artificial inducers revealed two natural bistability parameters that can be used to control the range of inducer concentrations over which the model exhibits bistability. By tuning these bistability parameters, the authors found a family of biophysically reasonable systems that are consistent with an experimentally determined bistable region for induction by thio-methylgalactoside (TMG) (in Ozbudak et al. Nature, 2004, 427; p. 737). To model regulation by lactose, the authors developed similar equations in which allolactose, a metabolic intermediate in lactose metabolism and a natural inducer of lac, is the inducer. For biophysically reasonable parameter values, these equations yield no bistability in response to induction by lactose - only systems with an unphysically small permease-dependent export effect can exhibit small amounts of bistability for limited ranges of parameter values. These results cast doubt on the relevance of bistability in the lac operon within the natural context of E. coli, and help shed light on the controversy among existing theoretical studies that address this issue. The results also motivate a deeper experimental characterisation of permease-independent transport of lac inducers, and suggest an experimental approach to address the relevance of bistability in the lac operon within the natural context of E. coli. The sensitivity of lac bistability to the type of inducer emphasises the importance of metabolism in determining the functions of genetic regulatory networks.

Dynamics of the nucleated polymerization model of prion replication.

The disease process for transmissible spongiform encephalopathies (TSEs), in one way or another, involves the conversion of a predominantly alpha-helical normal host-coded prion protein (PrP(C)) to an abnormally folded (predominantly beta sheet) protease resistant isoform (PrP(Sc)). Several alternative mechanisms have been proposed for this auto-catalytic process. Here the dynamical behavior of one of these models, the nucleated polymerization model, is studied by Monte Carlo discrete-event simulation of the explicit conversion reactions. These simulations demonstrate the characteristic dynamical behavior of this model for prion replication. Using estimates for the reaction rates and concentrations, time courses are estimated for concentration of PrP(Sc), PrP(Sc) aggregates, and PrP(C) as well as size distributions for the aggregates. The implications of these dynamics on protein misfolding cyclic amplification (PMCA) is discussed.

Combinatorial complexity and dynamical restriction of network flows in signal transduction.

The activities and interactions of proteins that govern the cellular response to a signal generate a multitude of protein phosphorylation states and heterogeneous protein complexes. Here, using a computational model that accounts for 307 molecular species implied by specified interactions of four proteins involved in signalling by the immunoreceptor FcepsilonRI, we determine the relative importance of molecular species that can be generated during signalling, chemical transitions among these species, and reaction paths that lead to activation of the protein tyrosine kinase (PTK) Syk. By all of these measures and over two- and ten-fold ranges of model parameters--rate constants and initial concentrations--only a small portion of the biochemical network is active. The spectrum of active complexes, however, can be shifted dramatically, even by a change in the concentration of a single protein, which suggests that the network can produce qualitatively different responses under different cellular conditions and in response to different inputs. Reduced models that reproduce predictions of the full model for a particular set of parameters lose their predictive capacity when parameters are varied over two-fold ranges.

Kinetic proofreading models for cell signaling predict ways to escape kinetic proofreading.

In the context of cell signaling, kinetic proofreading was introduced to explain how cells can discriminate among ligands based on a kinetic parameter, the ligand-receptor dissociation rate constant. In the kinetic proofreading model of cell signaling, responses occur only when a bound receptor undergoes a complete series of modifications. If the ligand dissociates prematurely, the receptor returns to its basal state and signaling is frustrated. We extend the model to deal with systems where aggregation of receptors is essential to signal transduction, and present a version of the model for systems where signaling depends on an extrinsic kinase. We also investigate the kinetics of signaling molecules, "messengers," that are generated by aggregated receptors but do not remain associated with the receptor complex. We show that the extended model predicts modes of signaling that exhibit kinetic discrimination for some range of parameters but for other parameter values show little or no discrimination and thus escape kinetic proofreading. We compare model predictions with experimental data.

Influence of follicular dendritic cells on decay of HIV during antiretroviral therapy.

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

Influence of follicular dendritic cells on HIV dynamics.

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

Dissociation of HIV-1 from follicular dendritic cells during HAART: mathematical analysis.

Follicular dendritic cells (FDC) provide a reservoir for HIV type 1 (HIV-1) that may reignite infection if highly active antiretroviral therapy (HAART) is withdrawn before virus on FDC is cleared. To estimate the treatment time required to eliminate HIV-1 on FDC, we develop deterministic and stochastic models for the reversible binding of HIV-1 to FDC via ligand-receptor interactions and examine the consequences of reducing the virus available for binding to FDC. Analysis of these models shows that the rate at which HIV-1 dissociates from FDC during HAART is biphasic, with an initial period of rapid decay followed by a period of slower exponential decay. The speed of the slower second stage of dissociation and the treatment time required to eradicate the FDC reservoir of HIV-1 are insensitive to the number of virions bound and their degree of attachment to FDC before treatment. In contrast, the expected time required for dissociation of an individual virion from FDC varies sensitively with the number of ligands attached to the virion that are available to interact with receptors on FDC. Although most virions may dissociate from FDC on the time scale of days to weeks, virions coupled to a higher-than-average number of ligands may persist on FDC for years. This result suggests that HAART may not be able to clear all HIV-1 trapped on FDC and that, even if clearance is possible, years of treatment will be required.

Steric effects on multivalent ligand-receptor binding: exclusion of ligand sites by bound cell surface receptors.

Steric effects can influence the binding of a cell surface receptor to a multivalent ligand. To account for steric effects arising from the size of a receptor and from the spacing of binding sites on a ligand, we extend a standard mathematical model for ligand-receptor interactions by introducing a steric hindrance factor. This factor gives the fraction of unbound ligand sites that are accessible to receptors, and thus available for binding, as a function of ligand site occupancy. We derive expressions for the steric hindrance factor for various cases in which the receptor covers a compact region on the ligand surface and the ligand expresses sites that are distributed regularly or randomly in one or two dimensions. These expressions are relevant for ligands such as linear polymers, proteins, and viruses. We also present numerical algorithms that can be used to calculate steric hindrance factors for other cases. These theoretical results allow us to quantify the effects of steric hindrance on ligand-receptor kinetics and equilibria.

Quantifying aggregation of IgE-FcepsilonRI by multivalent antigen.

Aggregation of cell surface receptors by multivalent ligand can trigger a variety of cellular responses. A well-studied receptor that responds to aggregation is the high affinity receptor for IgE (FcepsilonRI), which is responsible for initiating allergic reactions. To quantify antigen-induced aggregation of IgE-FcepsilonRI complexes, we have developed a method based on multiparameter flow cytometry to monitor both occupancy of surface IgE combining sites and association of antigen with the cell surface. The number of bound IgE combining sites in excess of the number of bound antigens, the number of bridges between receptors, provides a quantitative measure of IgE-FcepsilonRI aggregation. We demonstrate our method by using it to study the equilibrium binding of a haptenated fluorescent protein, 2,4-dinitrophenol-coupled B-phycoerythrin (DNP25-PE), to fluorescein isothiocyanate-labeled anti-DNP IgE on the surface of rat basophilic leukemia cells. The results, which we analyze with the aid of a mathematical model, indicate how IgE-FcepsilonRI aggregation depends on the total concentrations of DNP25-PE and surface IgE. As expected, we find that maximal aggregation occurs at an optimal antigen concentration. We also find that aggregation varies qualitatively with the total concentration of surface IgE as predicted by an earlier theoretical analysis.

Completely uncoupled and perfectly coupled gene expression in repressible systems.

Two forms of extreme coupling have been documented for the regulation of gene expression in repressible systems governed by a regulator protein. The first form, complete uncoupling, is distinguished by a constant level of regulator protein. The second form, perfect coupling, is distinguished by a level of regulator protein that varies coordinately with the level of the regulated enzyme. To determine how these two forms of coupling influence the performance of a system, so that we might predict the conditions under which each evolves through natural selection, we have used a mathematical approach to compare systems with complete uncoupling and perfect coupling. Our comparisons, which are controlled so that alternative systems are free from irrelevant differences, are based on a priori criteria that are related to various aspects of a system's performance, such as temporal responsiveness. By examining the influence of physical constraints that are related to the subunit structure of regulatory proteins and that limit the cooperativity of regulatory interactions, we have extended an early theory of gene circuitry for repressible systems. We obtain new results and testable predictions that can be summarized as follows. For typical systems with a low gain, performance is better with perfect coupling than with complete uncoupling if the mode of regulation is negative and better with complete uncoupling than with perfect coupling if the mode of regulation is positive. For systems with a high gain, these preferred forms of coupling are prevented by the physical constraints on cooperativity, and other forms of coupling can be expected. Tests of our predictions are illustrated by using data available in the literature.

Rules for coupled expression of regulator and effector genes in inducible circuits.

The induction of effector genes that encode enzymes is often controlled by the protein product of a regulator gene that is directly involved in the control of its own expression. This coupling of elementary gene circuits can lead to three patterns of regulator and effector gene expression. As effector gene expression increases, regulator gene expression can increase, remain the same, or decrease, and these are referred to as directly coupled, uncoupled, or inversely coupled patterns. To determine the relative merits of each pattern, we have constructed appropriate mathematical models for the alternative gene circuits and made well-controlled comparisons using a priori criteria to evaluate their functional effectiveness. We have considered both negatively and positively controlled systems that are induced by an intermediate of the regulated pathway. Different results are obtained in the two cases. Our results indicate that direct coupling is better than inverse coupling or uncoupling for negatively controlled systems, while inverse coupling is better than the other two patterns for positively controlled systems. These optimal forms of coupling promote a fast response to inducer. Our results also indicate that realization of the optimal forms of coupling is influenced by the subunit structure of regulator proteins and requires a low capacity for induction, i.e. the ratio of maximal to minimal level of effector gene expression is small. These results lead to testable predictions, which we have compared with experimental data from over 30 systems.

Subunit structure of regulator proteins influences the design of gene circuitry: analysis of perfectly coupled and completely uncoupled circuits.

Cells regulate expression their genome by means of a diverse repertoire of molecular mechanisms. However, little is known about their design principles or how these are influenced by underlying physical constraints. An early theory of gene regulation for inducible systems predicted that expression of the regulator and regulated proteins would be perfectly coupled (coordinate expression of regulator) when the regulator is a repressor and completely uncoupled (invariant expression of regulator) when the regulator is an activator. The experimental data then available tended to support these predictions, but there were notable exceptions. Here, we describe an extended theory, which takes into account the subunit structure of regulator proteins. The number of subunits determines the allowable range of values for the regulatory parameters, and, as a consequence, new rules for the prediction of gene circuitry emerge. The theory predicts perfectly coupled circuits with repressors, but only when the capacity for induction is "small"; it predicts completely uncoupled circuits with repressors when the capacity is "large". This theory also predicts completely uncoupled circuits with activators when the capacity for induction is small; it predicts perfectly coupled circuits with activators when the capacity is large. These new predictions are more fully in accord with available experimental evidence.