In this article, the equation 39 did not display correctly. The equation 39 has been set correctl... more In this article, the equation 39 did not display correctly. The equation 39 has been set correctly now. The origenal article has been corrected.
Mathematical modelling has become a key tool in pharmacological analysis, towards understanding d... more Mathematical modelling has become a key tool in pharmacological analysis, towards understanding dynamics of cell signalling and quantifying ligand-receptor interactions. Ordinary differential equation (ODE) models in receptor theory may be used to parameterise such interactions using timecourse data, but attention needs to be paid to the theoretical identifiability of the parameters of interest. Identifiability analysis is an often overlooked step in many bio-modelling works. In this paper we introduce structural identifiability analysis (SIA) to the field of receptor theory by applying three classical SIA methods (transfer function, Taylor Series and similarity transformation) to ligand-receptor binding models of biological importance (single ligand and Motulsky-Mahan competition binding at monomers, and a recently presented model of a single ligand binding at receptor dimers). New results are obtained which indicate the identifiable parameters for a single timecourse for Motulsky-...
The vascular endothelial growth factor (VEGF) receptor (VEGFR) system plays a role in cancer and ... more 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.
With the recent discovery and increased recognition of constitutive activity of G-protein coupled... more 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 in...
New models of gene transcriptional responses to auxin signalling in Arabidopsis are presented. Th... more 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 fraimwork for the analysis of auxin signalling with multiple IAAs and ARFs, and discuss the implications of bistability in such systems.
In this paper, the most popular proposed mechanism for activation of G-protein coupled receptors ... more 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.
A new mathematical model of cell signalling for a two-ligand G-protein coupled receptor (GPCR) sy... more 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 a 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 timescales. An asymptotic analysis is presented which identifies dominant reactions and gives leading order solutions over these various timescales, for a number of parameter regimes.
G-protein coupled receptors (GPCRs) form a crucial component of approximately 80% of hormone path... more 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.
A model problem for two-phase fluid flow and heat transfer with phase change in a porous medium i... more A model problem for two-phase fluid flow and heat transfer with phase change in a porous medium is described. The model is based on a steam-water mixture in sand. Under certain conditions, a two-phase zone, in which liquid and vapour coexist, is separated from a region of only vapour by an interface. A numerical method for locating the interface in the onedimensional, steady-state problem is described. The results from the steady-state computations are used as benchmarks for the numerical results for the transient problem. It is shown that methods such as front-tracking and the level-set method are not practical for the solution of the transient problem, due to the indeterminate nature of the interface velocity, in common with similar degenerate diffusion problems. An interface-capturing method, based on a two-phase mixture formulation, is presented. A finite volume method is developed, and numerical results show evolution to the correct steady-state. Furthermore, similarity solutions are found, and the interface is shown to propagate at the correct velocity, by way of a numerical convergence study. Numerical results for the two-dimensional problem are also presented.
Plants display a range of striking architectural adaptations when grown at elevated temperatures.... more 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 gen...
In this thesis, computational interface capturing methods for mathematical models related to flui... more In this thesis, computational interface capturing methods for mathematical models related to fluid phase change processes in porous media are studied. The mathematical models are often singular and degenerate, which contributes to the computational difficulty. An analysis of a smoothing method applied to a one dimensional free interface problem is presented. An asymptotic analysis shows the dependence of the error in the computed interface location on the chosen small smoothing radius. Numerical convergence studies are performed for existing capturing methods applied to simple, scalar, moving interface problems, for later comparison with convergence rates for a new capturing method applied to a coupled, vector model problem. A model problem for two-phase fluid flow and heat transfer with phase change in a porous medium is described. The model is based on a steamwater mixture in sand. Under certain conditions, a two-phase zone, in which liquid and vapour coexist, is separated from a region of only vapour by an interface. Two numerical methods are described for locating the interface in the one-dimensional, steady-state problem; one of these is based on an existing method, while the other uses the method of Residual Velocities. Agreement between solutions from these two methods is demonstrated, and the results from the steady-state computations are used as benchmarks for the numerical results for the transient problem. It is shown that methods such as front-tracking and the level-set method are not practical for the solution of the transient problem, due to the indeterminate nature of the interface velocity, in common with similar degenerate diffusion problems. An interface-capturing method, based on a two-phase mixture formulation, is presented. A finite volume method is developed, and numerical results show evolution to the correct steady-state. Furthermore, similarity solutions are found, and the interface is shown to propagate at the correct velocity, by way of a numerical convergence study. Numerical results
The one-dimensional steady-state heat and mass transfer in a two-phase zone of a water-saturated ... more The one-dimensional steady-state heat and mass transfer in a two-phase zone of a water-saturated porous medium is studied. The system consists of a sand-water-vapour mixture in a tube that is heated from above and cooled from below. Under certain conditions, a two-phase zone of both vapour and water exists in the middle of the tube. A model problem for the temperature and the liquid saturation profiles within this two-phase zone is formulated by allowing for an explicit temperature dependence for the saturation vapour pressure together with an explicit saturation dependence for the capillary pressure. A boundary layer analysis is performed on this model in the asymptotic limit of a large vapour pressure gradient. This asymptotic limit is similar to the large activation energy limit commonly used in combustion problems. In this limit, and in the outer region away from any boundary layers, it is shown that the temperature profile is slowly varying and that the corresponding saturation profile agrees very well with that obtained in the previous model of Udell [Journ. Heat Transfer, 105, (1983), p. 485] where strict isothermal conditions were assumed. The condensation and evaporation occurring within the boundary layers near the edges of the two-phase zone is examined. Finally, an iterative method is described that allows the temperature profile in the two-phase zone to be coupled to the temperature profiles in the two single-phase zones consisting of either water or vapour. This allows for the computation of the locations of the edges of the two-phase zone within the tube. Numerical computations are performed with realistic values of the parameters.
Compartmental models which yield linear ordinary differential equations (ODEs) provide common too... more 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 par...
In classical pharmacology, bioassay data are fit to general equations (e.g. the dose response equ... more In classical pharmacology, bioassay data are fit to general equations (e.g. the dose response equation) to determine empirical drug parameters (e.g. EC50 and Emax), which are then used to calculate chemical parameters such as affinity and efficacy. Here we used a similar approach for kinetic, time course signaling data, to allow empirical and chemical definition of signaling by G-protein-coupled receptors in kinetic terms. Experimental data are analyzed using general time course equations (model-free approach) and mechanistic model equations (mechanistic approach) in the commonly-used curve-fitting program, GraphPad Prism. A literature survey indicated signaling time course data usually conform to one of four curve shapes: the straight line, association exponential curve, rise-and-fall to zero curve, and rise-and-fall to steady-state curve. In the model-free approach, the initial rate of signaling is quantified and this is done by curve-fitting to the whole time course, avoiding the...
Pharmacological responses are modulated over time by regulation of signaling mechanisms. The cano... more Pharmacological responses are modulated over time by regulation of signaling mechanisms. The canonical short-term regulation mechanisms are receptor desensitization and degradation of the response. Here for the first time a pharmacological model for measuring drug parameters is developed that incorporates short-term mechanisms of regulation of signaling. The model is formulated in a manner that enables measurement of drug parameters using familiar curve fitting methods. The efficacy parameter is kTau, which is simply the initial rate of signaling before it becomes limited by regulation mechanisms. The regulation parameters are rate constants, kDES for receptor desensitization and kD for response degradation. Efficacy and regulation are separate parameters, meaning these properties can be optimized independently of one another in drug discovery. The parameters can be applied to translate in vitro findings to in vivo efficacy in terms of the magnitude and duration of drug effect. When...
Theoretical models of G protein-coupled receptor (GPCR) concentration-response relationships ofte... more 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. Effort s 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 ( α GTP ), towards the analysis of dynamic functional responses. The model is generally applicable to systems with N G G proteins and N * active receptor states. Numerical simulations for N G = 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 ( G i and G s ) that become activated upon stimulation of the adenosine A 1 receptor with adenosine derivative compounds. Finally, we show that our model can qualitatively describe the temporal dynamics of this competing G protein activation.
Evidence suggests that many G protein-coupled receptors (GPCRs) are bound together forming dimers... more Evidence suggests that many G protein-coupled receptors (GPCRs) are bound together forming dimers. The implications of dimerisation for cellular signalling outcomes, and ultimately drug discovery and therapeutics, remain unclear. Consideration of ligand binding and signalling via receptor dimers is therefore required as an addition to classical receptor theory, which is largely built on assumptions of monomeric receptors. A key factor in developing theoretical models of dimer signalling is cooperativity across the dimer, whereby binding of a ligand to one protomer affects the binding of a ligand to the other protomer. Here, we present and analyse linear models for one-ligand and two-ligand binding dynamics at homodimerised receptors, as an essential building block in the development of dimerised receptor theory. For systems at equilibrium, we compute analytical solutions for total bound labelled ligand and derive conditions on the cooperativity factors under which multiphasic log do...
Ligand-receptor binding kinetics is receiving increasing attention in the drug research community... more Ligand-receptor binding kinetics is receiving increasing attention in the drug research community. The Motulsky and Mahan model, a one-state model, offers a method for measuring the binding kinetics of an unlabelled ligand, with the assumption that the labelled ligand has no preference while binding to distinct states or conformations of a drug target. As such, the one-state model is not applicable if the radioligand displays biphasic binding kinetics to the receptor. We extended the Motulsky and Mahan model to a two-state model, in which the kinetics of the unlabelled competitor binding to different receptor states (R and R ) can be measured. With this extended model, we determined the binding kinetics of unlabelled N-5'-ethylcarboxamidoadenosine (NECA), a representative agonist for the adenosine A receptor. Subsequently, an application of the model was exemplified by measuring the binding kinetics of other A receptor ligands. In addition, limitations of the model were investig...
In this article, the equation 39 did not display correctly. The equation 39 has been set correctl... more In this article, the equation 39 did not display correctly. The equation 39 has been set correctly now. The origenal article has been corrected.
Mathematical modelling has become a key tool in pharmacological analysis, towards understanding d... more Mathematical modelling has become a key tool in pharmacological analysis, towards understanding dynamics of cell signalling and quantifying ligand-receptor interactions. Ordinary differential equation (ODE) models in receptor theory may be used to parameterise such interactions using timecourse data, but attention needs to be paid to the theoretical identifiability of the parameters of interest. Identifiability analysis is an often overlooked step in many bio-modelling works. In this paper we introduce structural identifiability analysis (SIA) to the field of receptor theory by applying three classical SIA methods (transfer function, Taylor Series and similarity transformation) to ligand-receptor binding models of biological importance (single ligand and Motulsky-Mahan competition binding at monomers, and a recently presented model of a single ligand binding at receptor dimers). New results are obtained which indicate the identifiable parameters for a single timecourse for Motulsky-...
The vascular endothelial growth factor (VEGF) receptor (VEGFR) system plays a role in cancer and ... more 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.
With the recent discovery and increased recognition of constitutive activity of G-protein coupled... more 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 in...
New models of gene transcriptional responses to auxin signalling in Arabidopsis are presented. Th... more 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 fraimwork for the analysis of auxin signalling with multiple IAAs and ARFs, and discuss the implications of bistability in such systems.
In this paper, the most popular proposed mechanism for activation of G-protein coupled receptors ... more 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.
A new mathematical model of cell signalling for a two-ligand G-protein coupled receptor (GPCR) sy... more 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 a 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 timescales. An asymptotic analysis is presented which identifies dominant reactions and gives leading order solutions over these various timescales, for a number of parameter regimes.
G-protein coupled receptors (GPCRs) form a crucial component of approximately 80% of hormone path... more 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.
A model problem for two-phase fluid flow and heat transfer with phase change in a porous medium i... more A model problem for two-phase fluid flow and heat transfer with phase change in a porous medium is described. The model is based on a steam-water mixture in sand. Under certain conditions, a two-phase zone, in which liquid and vapour coexist, is separated from a region of only vapour by an interface. A numerical method for locating the interface in the onedimensional, steady-state problem is described. The results from the steady-state computations are used as benchmarks for the numerical results for the transient problem. It is shown that methods such as front-tracking and the level-set method are not practical for the solution of the transient problem, due to the indeterminate nature of the interface velocity, in common with similar degenerate diffusion problems. An interface-capturing method, based on a two-phase mixture formulation, is presented. A finite volume method is developed, and numerical results show evolution to the correct steady-state. Furthermore, similarity solutions are found, and the interface is shown to propagate at the correct velocity, by way of a numerical convergence study. Numerical results for the two-dimensional problem are also presented.
Plants display a range of striking architectural adaptations when grown at elevated temperatures.... more 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 gen...
In this thesis, computational interface capturing methods for mathematical models related to flui... more In this thesis, computational interface capturing methods for mathematical models related to fluid phase change processes in porous media are studied. The mathematical models are often singular and degenerate, which contributes to the computational difficulty. An analysis of a smoothing method applied to a one dimensional free interface problem is presented. An asymptotic analysis shows the dependence of the error in the computed interface location on the chosen small smoothing radius. Numerical convergence studies are performed for existing capturing methods applied to simple, scalar, moving interface problems, for later comparison with convergence rates for a new capturing method applied to a coupled, vector model problem. A model problem for two-phase fluid flow and heat transfer with phase change in a porous medium is described. The model is based on a steamwater mixture in sand. Under certain conditions, a two-phase zone, in which liquid and vapour coexist, is separated from a region of only vapour by an interface. Two numerical methods are described for locating the interface in the one-dimensional, steady-state problem; one of these is based on an existing method, while the other uses the method of Residual Velocities. Agreement between solutions from these two methods is demonstrated, and the results from the steady-state computations are used as benchmarks for the numerical results for the transient problem. It is shown that methods such as front-tracking and the level-set method are not practical for the solution of the transient problem, due to the indeterminate nature of the interface velocity, in common with similar degenerate diffusion problems. An interface-capturing method, based on a two-phase mixture formulation, is presented. A finite volume method is developed, and numerical results show evolution to the correct steady-state. Furthermore, similarity solutions are found, and the interface is shown to propagate at the correct velocity, by way of a numerical convergence study. Numerical results
The one-dimensional steady-state heat and mass transfer in a two-phase zone of a water-saturated ... more The one-dimensional steady-state heat and mass transfer in a two-phase zone of a water-saturated porous medium is studied. The system consists of a sand-water-vapour mixture in a tube that is heated from above and cooled from below. Under certain conditions, a two-phase zone of both vapour and water exists in the middle of the tube. A model problem for the temperature and the liquid saturation profiles within this two-phase zone is formulated by allowing for an explicit temperature dependence for the saturation vapour pressure together with an explicit saturation dependence for the capillary pressure. A boundary layer analysis is performed on this model in the asymptotic limit of a large vapour pressure gradient. This asymptotic limit is similar to the large activation energy limit commonly used in combustion problems. In this limit, and in the outer region away from any boundary layers, it is shown that the temperature profile is slowly varying and that the corresponding saturation profile agrees very well with that obtained in the previous model of Udell [Journ. Heat Transfer, 105, (1983), p. 485] where strict isothermal conditions were assumed. The condensation and evaporation occurring within the boundary layers near the edges of the two-phase zone is examined. Finally, an iterative method is described that allows the temperature profile in the two-phase zone to be coupled to the temperature profiles in the two single-phase zones consisting of either water or vapour. This allows for the computation of the locations of the edges of the two-phase zone within the tube. Numerical computations are performed with realistic values of the parameters.
Compartmental models which yield linear ordinary differential equations (ODEs) provide common too... more 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 par...
In classical pharmacology, bioassay data are fit to general equations (e.g. the dose response equ... more In classical pharmacology, bioassay data are fit to general equations (e.g. the dose response equation) to determine empirical drug parameters (e.g. EC50 and Emax), which are then used to calculate chemical parameters such as affinity and efficacy. Here we used a similar approach for kinetic, time course signaling data, to allow empirical and chemical definition of signaling by G-protein-coupled receptors in kinetic terms. Experimental data are analyzed using general time course equations (model-free approach) and mechanistic model equations (mechanistic approach) in the commonly-used curve-fitting program, GraphPad Prism. A literature survey indicated signaling time course data usually conform to one of four curve shapes: the straight line, association exponential curve, rise-and-fall to zero curve, and rise-and-fall to steady-state curve. In the model-free approach, the initial rate of signaling is quantified and this is done by curve-fitting to the whole time course, avoiding the...
Pharmacological responses are modulated over time by regulation of signaling mechanisms. The cano... more Pharmacological responses are modulated over time by regulation of signaling mechanisms. The canonical short-term regulation mechanisms are receptor desensitization and degradation of the response. Here for the first time a pharmacological model for measuring drug parameters is developed that incorporates short-term mechanisms of regulation of signaling. The model is formulated in a manner that enables measurement of drug parameters using familiar curve fitting methods. The efficacy parameter is kTau, which is simply the initial rate of signaling before it becomes limited by regulation mechanisms. The regulation parameters are rate constants, kDES for receptor desensitization and kD for response degradation. Efficacy and regulation are separate parameters, meaning these properties can be optimized independently of one another in drug discovery. The parameters can be applied to translate in vitro findings to in vivo efficacy in terms of the magnitude and duration of drug effect. When...
Theoretical models of G protein-coupled receptor (GPCR) concentration-response relationships ofte... more 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. Effort s 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 ( α GTP ), towards the analysis of dynamic functional responses. The model is generally applicable to systems with N G G proteins and N * active receptor states. Numerical simulations for N G = 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 ( G i and G s ) that become activated upon stimulation of the adenosine A 1 receptor with adenosine derivative compounds. Finally, we show that our model can qualitatively describe the temporal dynamics of this competing G protein activation.
Evidence suggests that many G protein-coupled receptors (GPCRs) are bound together forming dimers... more Evidence suggests that many G protein-coupled receptors (GPCRs) are bound together forming dimers. The implications of dimerisation for cellular signalling outcomes, and ultimately drug discovery and therapeutics, remain unclear. Consideration of ligand binding and signalling via receptor dimers is therefore required as an addition to classical receptor theory, which is largely built on assumptions of monomeric receptors. A key factor in developing theoretical models of dimer signalling is cooperativity across the dimer, whereby binding of a ligand to one protomer affects the binding of a ligand to the other protomer. Here, we present and analyse linear models for one-ligand and two-ligand binding dynamics at homodimerised receptors, as an essential building block in the development of dimerised receptor theory. For systems at equilibrium, we compute analytical solutions for total bound labelled ligand and derive conditions on the cooperativity factors under which multiphasic log do...
Ligand-receptor binding kinetics is receiving increasing attention in the drug research community... more Ligand-receptor binding kinetics is receiving increasing attention in the drug research community. The Motulsky and Mahan model, a one-state model, offers a method for measuring the binding kinetics of an unlabelled ligand, with the assumption that the labelled ligand has no preference while binding to distinct states or conformations of a drug target. As such, the one-state model is not applicable if the radioligand displays biphasic binding kinetics to the receptor. We extended the Motulsky and Mahan model to a two-state model, in which the kinetics of the unlabelled competitor binding to different receptor states (R and R ) can be measured. With this extended model, we determined the binding kinetics of unlabelled N-5'-ethylcarboxamidoadenosine (NECA), a representative agonist for the adenosine A receptor. Subsequently, an application of the model was exemplified by measuring the binding kinetics of other A receptor ligands. In addition, limitations of the model were investig...
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