Insights into the dynamics of ligand-induced dimerisation via mathematical modelling and analysis.

The vascular endothelial growth factor (VEGF) receptor (VEGFR) system plays a role in cancer and many other diseases. It is widely accepted that VEGFR receptors dimerise in response to VEGF binding. However, analysis of these mechanisms and their implications for drug development still requires further exploration. In this paper, we present a mathematical model representing the binding of VEGF to VEGFR and the subsequent ligand-induced dimerisation. A key factor in this work is the qualitative and quantitative effect of binding cooperativity, which describes the effect that the binding of a ligand to a receptor has on the binding of that ligand to a second receptor, and the dimerisation of these receptors. We analyse the ordinary differential equation system at equilibrium, giving analytical solutions for the total amount of ligand bound. For time-course dynamics, we use numerical methods to explore possible behaviours under various parameter regimes, while perturbation analysis is used to understand the intricacies of these behaviours. Our simulation results show an excellent fit to experimental data, towards validating the model.

Exact solutions and equi-dosing regimen regions for multi-dose pharmacokinetics models with transit compartments.

Compartmental models which yield linear ordinary differential equations (ODEs) provide common tools for pharmacokinetics (PK) analysis, with exact solutions for drug levels or concentrations readily obtainable for low-dimensional compartment models. Exact solutions enable valuable insights and further analysis of these systems. Transit compartment models are a popular semi-mechanistic approach for generalising simple PK models to allow for delayed kinetics, but computing exact solutions for multi-dosing inputs to transit compartment systems leading to different final compartments is nontrivial. Here, we find exact solutions for drug levels as functions of time throughout a linear transit compartment cascade followed by an absorption compartment and a central blood compartment, for the general case of n transit compartments and M equi-bolus doses to the first compartment. We further show the utility of exact solutions to PK ODE models in finding constraints on equi-dosing regimen parameters imposed by a prescribed therapeutic range. This leads to the construction of equi-dosing regimen regions (EDRRs), providing new, novel visualisations which summarise the safe and effective dosing parameter space. EDRRs are computed for classical and transit compartment models with two- and three-dimensional parameter spaces, and are proposed as useful graphical tools for informing drug dosing regimen design.

Modelling and simulation of biased agonism dynamics at a G protein-coupled receptor.

Theoretical models of G protein-coupled receptor (GPCR) concentration-response relationships often assume an agonist producing a single functional response via a single active state of the receptor. These models have largely been analysed assuming steady-state conditions. There is now much experimental evidence to suggest that many GPCRs can exist in multiple receptor conformations and elicit numerous functional responses, with ligands having the potential to activate different signalling pathways to varying extents-a concept referred to as biased agonism, functional selectivity or pluri-dimensional efficacy. Moreover, recent experimental results indicate a clear possibility for time-dependent bias, whereby an agonist's bias with respect to different pathways may vary dynamically. Efforts towards understanding the implications of temporal bias by characterising and quantifying ligand effects on multiple pathways will clearly be aided by extending current equilibrium binding and biased activation models to include G protein activation dynamics. Here, we present a new model of time-dependent biased agonism, based on ordinary differential equations for multiple cubic ternary complex activation models with G protein cycle dynamics. This model allows simulation and analysis of multi-pathway activation bias dynamics at a single receptor for the first time, at the level of active G protein (αGTP), towards the analysis of dynamic functional responses. The model is generally applicable to systems with NG G proteins and N* active receptor states. Numerical simulations for NG=N*=2 reveal new insights into the effects of system parameters (including cooperativities, and ligand and receptor concentrations) on bias dynamics, highlighting new phenomena including the dynamic inter-conversion of bias direction. Further, we fit this model to 'wet' experimental data for two competing G proteins (Gi and Gs) that become activated upon stimulation of the adenosine A1 receptor with adenosine derivative compounds. Finally, we show that our model can qualitatively describe the temporal dynamics of this competing G protein activation.

Modelling of the activation of G-protein coupled receptors: drug free constitutive receptor activity.

G-protein coupled receptors (GPCRs) form a crucial component of approximately 80% of hormone pathways. In this paper, the most popular mechanism for activation of GPCRs-the shuttling mechanism-is modelled mathematically. An asymptotic analysis of this model clarifies the dynamics of the system in the absence of drug, in particular which reactions dominate during the different timescales. Equilibrium analysis of the model demonstrates the model's ability to predict constitutive receptor activity.

Impact of plant shoot architecture on leaf cooling: a coupled heat and mass transfer model.

Plants display a range of striking architectural adaptations when grown at elevated temperatures. In the model plant Arabidopsis thaliana, these include elongation of petioles, and increased petiole and leaf angles from the soil surface. The potential physiological significance of these architectural changes remains speculative. We address this issue computationally by formulating a mathematical model and performing numerical simulations, testing the hypothesis that elongated and elevated plant configurations may reflect a leaf-cooling strategy. This sets in place a new basic model of plant water use and interaction with the surrounding air, which couples heat and mass transfer within a plant to water vapour diffusion in the air, using a transpiration term that depends on saturation, temperature and vapour concentration. A two-dimensional, multi-petiole shoot geometry is considered, with added leaf-blade shape detail. Our simulations show that increased petiole length and angle generally result in enhanced transpiration rates and reduced leaf temperatures in well-watered conditions. Furthermore, our computations also reveal plant configurations for which elongation may result in decreased transpiration rate owing to decreased leaf liquid saturation. We offer further qualitative and quantitative insights into the role of architectural parameters as key determinants of leaf-cooling capacity.

Distinguishing possible mechanisms for auxin-mediated developmental control in Arabidopsis: models with two Aux/IAA and ARF proteins, and two target gene-sets.

New models of gene transcriptional responses to auxin signalling in Arabidopsis are presented. This work extends a previous model of auxin signalling to include networks of gene-sets which may control developmental responses along auxin gradients. Key elements of this new study include models of signalling pathways and networks involving two Aux-IAA proteins (IAAs), auxin response factors (ARFs) and gene targets. Hypotheses for the gene network topologies which may be involved in developmental responses have been tested against experimental observations for root hair growth in particular. In studying these models, we provide a framework for the analysis of auxin signalling with multiple IAAs and ARFs, and discuss the implications of bistability in such systems.

Modeling and simulation of inverse agonism dynamics.

With the recent discovery and increased recognition of constitutive activity of G-protein coupled receptors (GPCRs) and inverse agonists have come a number of important questions. The signaling mechanisms underlying inverse agonist effects on constitutively active systems need to be elucidated qualitatively. Furthermore, quantitative analysis is needed to support experimental observations, characterize the pharmacology of the ligands and systems of interest, and to provide numerical predictions of dynamic physiological responses to inverse agonists in an effort toward drug design. Here, we review the concept of inverse agonism and describe the application of mathematical and computational techniques to models of inverse agonists in GPCR systems. Numerical simulation results for active G-protein levels demonstrate a variety of dynamic features including inhibition of agonist-induced peak-plateau responses, undershoots, multiple time scales, and both surmountable and insurmountable inverse agonism.

Mathematical modelling of signalling in a two-ligand G-protein coupled receptor system: agonist-antagonist competition.

A new mathematical model of cell signalling for a two-ligand G-protein coupled receptor (GPCR) system is presented. This model extends the single-ligand cubic ternary complex to account for the possibility of an agonist and an antagonist competing for receptor sites. The G-protein cycle is included, and signalling as far as the dissociated G(alpha) subunit is considered. Numerical simulations are performed, and the effects on the system dynamics, such as peak and plateau behaviour, of antagonist "stickiness", and of the doses of agonist and antagonist, are discussed. Under certain parameter regimes, the plateau response is subject to surmountable antagonism, while the peak response is subject to insurmountable antagonism. The numerical results reveal responses evolving over a number of time-scales. An asymptotic analysis is presented which identifies dominant reactions and gives leading order solutions over these various time-scales, for a number of parameter regimes.

Modelling the activation of G-protein coupled receptors by a single drug.

In this paper, the most popular proposed mechanism for activation of G-protein coupled receptors (GPCRs)--the shuttling mechanism--is modelled mathematically. An asymptotic analysis of this model clarifies the dynamics of the system in the presence of a drug, in particular identifying which reactions dominate during the different timescales. The modelling also reveals challenging behaviour in the form of a peak response. This new mechanism gives simple explanations for complex, possibly misunderstood, behaviour.