A model of optimal timing for a predictive adaptive response.

Predictive adaptive responses (PARs) are a form of developmental plasticity in which the developmental response to an environmental cue experienced early in life is delayed and yet, at the same time, the induced phenotype anticipates (i.e., is completely developed before) exposure to the eventual environmental state predicted by the cue, in which the phenotype is adaptive. We model this sequence of events to discover, under various assumptions concerning the cost of development, what lengths of delay, developmental time, and anticipation are optimal. We find that in many scenarios modeled, development of the induced phenotype should be completed at the exact same time that the environmental exposure relevant to the induced phenotype begins: that is, in contrast to our observed cases of PARs, there should be no anticipation. Moreover, unless slow development is costly, development should commence immediately after the cue: there should be no delay. Thus, PARs, which normally have non-zero delays and/or anticipation, are highly unusual. Importantly, the exceptions to these predictions of zero delays and anticipation occurred when developmental time was fixed and delaying development was increasingly costly. We suggest, therefore, that PARs will only evolve under three kinds of circumstances: (i) there are strong timing constraints on the cue and the environmental status, (ii) delaying development is costly, and development time is either fixed or slow development is costly, or (iii) when the period between the cue and the eventual environmental change is variable and the cost of not completing development before the change is high. These predictions are empirically testable.

Modeling the effect of insulin-like growth factor-1 on human cell growth.

Insulin-like growth factor-1 (IGF-1) plays a key role in human growth and development. The interactions of IGF-1 with IGF-1 receptors and IGF-1 binding proteins (IGFBPs) regulate IGF-1 function. Recent research suggests that a metabolite of IGF-1, cyclo-glycyl-proline (cGP), has a role in regulating IGF-1 homeostasis. A component of this interaction is believed to be the competitive binding of IGF-1 and cGP to IGFBPs. In this paper we describe a mathematical model of the interaction between IGF-1 and cGP on human cell growth. The model can be used to understand the interaction between IGF-1, IGFBPs, cGP and IGF-1 receptors along with the kinetics of cell growth. An explicit model of the known interactions between IGF-1, cGP, IGFBPs, IGF-1 receptors explained a large portion of the variance in cell growth (R(2) = 0.83). An implicit model of the interactions between IGF-1, cGP, IGFBPs, IGF-1 receptors that included a hypothesized feedback of cGP on IGF-1 receptors explained nonlinear features of interaction between IGF-1 and cGP not described by the explicit model (R(2) = 0.84). The model also explained the effect of IGFBP antibody on the interaction between cGP and IGF-1 (R(2) = 0.78). This demonstrates that the competitive binding of IGF-1 and cGP to IGFBPs plays a large role in the interaction between IGF-1 and cGP, but that other factors potentially play a role in the interaction between cGP and IGF-1. These models can be used to predict the complex interaction between IGF-1 and cGP on human cell growth and form a basis for further research in this field.

Propagating longitudinal contractions in the ileum of the rabbit--efficiency of advective mixing.

Three longitudinal motions of the rabbit small intestine were modelled in the CFD software Polyflow using ex vivo experimental data previously reported in literature. Consideration was given to chyme rheology and mixing performance of the macro-scale lumenal motions, as triggered by the observed wall motions. Simulations were performed to qualitatively assess the flow behaviour. The advective properties of the flow were universally characterised by analysing the stretching ability of the flow. Two Newtonian fluids, with viscosities of µ = 1 Pa s and µ = 0.001 Pa s, and a non-Newtonian shear-thinning fluid (Bird-Carreau relationship with n = 0.41, λ = 0.1, η∞ = 5.01 × 10(-9) Pa s and η0 = 0.65 Pa s) were investigated. It was found that both the type of contraction and chyme rheology significantly affected the flow and subsequent efficiency of advective motions in the intestinal core. Results also showed that shear rates generated were too small to unveil the pseudo-plastic behaviour of the non-Newtonian fluid. Of the longitudinal motions analysed, the oral propagation was the one leading to the higher, but also the most localised levels of stretching in the rabbit small intestine. This oral propagation was largely characterised by an ordered axial flow and was able to facilitate mixing by stretching material elements in the vicinity of the intestinal wall, particularly in the case of a low viscous water like fluid.

Cyclic glycine-proline regulates IGF-1 homeostasis by altering the binding of IGFBP-3 to IGF-1.

The homeostasis of insulin-like growth factor-1 (IGF-1) is essential for metabolism, development and survival. Insufficient IGF-1 is associated with poor recovery from wounds whereas excessive IGF-1 contributes to growth of tumours. We have shown that cyclic glycine-proline (cGP), a metabolite of IGF-1, can normalise IGF-1 function by showing its efficacy in improving the recovery from ischemic brain injury in rats and inhibiting the growth of lymphomic tumours in mice. Further investigation in cell culture suggested that cGP promoted the activity of IGF-1 when it was insufficient, but inhibited the activity of IGF-1 when it was excessive. Mathematical modelling revealed that the efficacy of cGP was a modulated IGF-1 effect via changing the binding of IGF-1 to its binding proteins, which dynamically regulates the balance between bioavailable and non-bioavailable IGF-1. Our data reveal a novel mechanism of auto-regulation of IGF-1, which has physiological and pathophysiological consequences and potential pharmacological utility.

Modelling induced resistance to plant diseases.

Plant disease control has traditionally relied heavily on the use of agrochemicals despite their potentially negative impact on the environment. An alternative strategy is that of induced resistance (IR). However, while IR has proven effective in controlled environments, it has shown variable field efficacy, thus raising questions about its potential for disease management in a given crop. Mathematical modelling of IR assists researchers with understanding the dynamics of the phenomenon in a given plant cohort against a selected disease-causing pathogen. Here, a prototype mathematical model of IR promoted by a chemical elicitor is proposed and analysed. Standard epidemiological models describe that, under appropriate environmental conditions, Susceptible plants (S) may become Diseased (D) upon exposure to a compatible pathogen or are able to Resist the infection (R) via basal host defence mechanisms. The application of an elicitor enhances the basal defence response thereby affecting the relative proportion of plants in each of the S, R and D compartments. IR is a transient response and is modelled using reversible processes to describe the temporal evolution of the compartments. Over time, plants can move between these compartments. For example, a plant in the R-compartment can move into the S-compartment and can then become diseased. Once in the D-compartment, however, it is assumed that there is no recovery. The terms in the equations are identified using established principles governing disease transmission and this introduces parameters which are determined by matching data to the model using computer-based algorithms. These then give the best match of the model with experimental data. The model predicts the relative proportion of plants in each compartment and quantitatively estimates elicitor effectiveness. An illustrative case study will be given; however, the model is generic and will be applicable for a range of plant-pathogen-elicitor scenarios.

The statistical-mechanics of chromosome conformation capture.

Since Jacob and Monod's characterization of the role of DNA elements in gene control, it has been recognized that the linear organization of genome structure is important for the regulation of gene transcription and hence the manifestation of phenotypes. Similarly, it has long been hypothesized that the spatial organization (in three dimensions evolving through time), as part of the epigenome, makes a significant contribution to the genotype-phenotype transition. Proximity ligation assays commonly known as chromosome conformation capture (3C) and 3C based methodologies (e.g., GCC, HiC and ChIA-Pet) are increasingly being incorporated into empirical studies to investigate the role that three-dimensional genome structure plays in the regulation of phenotype. The apparent simplicity of these methodologies-crosslink chromatin, digest, dilute, ligate, detect interactions-belies the complexity of the data and the considerations that should be taken into account to ensure the generation and accurate interpretation of reliable data. Here we discuss the probabilistic nature of these methodologies and how this contributes to their endogenous limitations.

Optimal nutritional intake for fetal growth.

The regular nutritional intake of an expectant mother clearly affects the weight development of the fetus. Assuming the growth of the fetus follows a deterministic growth law, like a logistic equation, albeit dependent on the nutritional intake, the ideal solution is usually determined by the birth-weight being pre-assigned, for example, as a percentage of the mother's average weight. This problem can then be specified as an optimal control problem with the daily intake as the control, which appears in a Michaelis-Menten relationship, for which there are well-developed procedures to follow. The best solution is determined by requiring minimum total intake under which the preassigned birth weight is reached. The algorithm has been generalized to the case where the fetal weight depends in a detailed way on the cumulative intake, suitably discounted according to the history. The optimality system is derived and then solved numerically using an iterative method for the specific values of parameter. The procedure is generic and can be adapted to any growth law and any parameterisation obtained by the detailed physiology.

Decompression schedule optimization with an isoprobabilistic risk of decompression sickness.

INTRODUCTION: Divers use decompression schedules to reduce the probability of occurrence of decompression sickness when returning to the surface at the end of a dive. The probability of decompression sickness resulting from these schedules varies across different dives and the models used to generate them. Usually the diver is unaware of this variance in risk. This paper describes an investigation into the feasibility of producing optimized iso-probabilistic decompression schedules that minimize the time it takes for a diver to reach the surface. METHODS: The decompression schedules were optimized using the sequential quadratic programming method (SQP), which minimizes the ascent time for a given probability of decompression sickness. The U.S. linear-exponential multi-gas model was used to calculate an estimate of the probability of decompression sickness for a given dive. In particular 1.3-bar oxygen in helium rebreather bounce dives to between 18 m and 81 m were considered and compared against the UK Navy QinetiQ 90 tables for a similar estimate of probability of decompression sickness. RESULTS: The SQP method reliably produced schedules with fast and stable convergence to an optimized solution. Comparison of the optimized decompression schedules with the QinetiQ 90 schedules showed similar stop times for shallow dives to 18 m. For dives with a maximum depth of 39 m to 81 m, optimizing the decompression resulted in savings in decompression time of up to 30 min. CONCLUSIONS: This paper has shown that it is feasible to produce optimized iso-probabilistic decompression tables given a reliable risk model for decompression sickness and appropriate dive trials.

On a functional equation model of transient cell growth.

A cell-growth model with applications to modelling the size distribution of diatoms is examined. The analytic solution to the model without dispersion is found and is shown to display periodic exponential growth rather than asynchronous (or balanced) exponential growth. It is shown that a bounding envelope (hull) of the solution to the model without dispersion takes the same shape as the limiting steady-size distribution to the dispersive case as dispersion tends to zero. The effect of variable growth rate on the shape of the hull is also discussed.

Modelling the flow [corrected] cytometric data obtained from unperturbed human tumour cell lines: parameter fitting and comparison.

In this paper we firstly present three alternative formulations of a mathematical model for human tumour cell lines unperturbed by cancer therapy. The model counts the number density of cells in each phase of the cell cycle over time where cells are differentiated by their DNA content. Data are available from the Auckland Cancer Society Research Centre, Auckland, New Zealand, in the form of DNA histograms or profiles from 11 different human tumour cell lines (i.e. in vitro) unperturbed by cancer therapy. We then apply one (computationally fast) formulation of the model and discover that although in general different combinations of parameter values give rise to very different DNA profiles it is possible that different combinations of parameter values give rise to virtually identical profiles. Experimental estimates of the rate of transition from the G1-phase (growth) to the S-phase (DNA synthesis) enable us to uniquely determine other model parameters of interest that give the least square error between the model and data. We finally apply our model to each of the 11 different cell lines and compare cell cycle phase transit times. Although the DNA histograms of each of the cell lines have similar shapes these cell lines have different combinations of transit times to each other, which could explain why they often react very differently when exposed to anti-cancer therapies during laboratory experiments. An understanding of the in vitro situation may give an insight into why some human cancer patients do not respond to cancer therapy.

Targeted glycemic reduction in critical care using closed-loop control.

BACKGROUND: Critically ill patients are often hyperglycemic and extremely diverse in their dynamics. Consequently, fixed protocols and sliding scales can result in error and poor control. Tight glucose control has been shown to significantly reduce mortality in critical care. An improved physiological system model of the glucose-insulin dynamics of a critical care patient is used to develop an adaptive tight glucose control protocol that accounts for variable patient dynamics, and is verified in limited clinical trials. METHODS: A physiologically based two-compartment system model that accounts for time-varying insulin sensitivity, time-varying endogenous glucose removal, and two saturation kinetics mechanisms is developed. A bolus-based adaptive control protocol is developed that monitors the physiological status of a critically ill patient, enabling tight glycemic regulation to preset glycemic targets. The model and protocol are verified in three, 5-h preliminary proof-of-concept clinical trials. Ethics approval was granted by the Canterbury Ethics Committee (Christchurch, New Zealand). RESULTS: Preset glycemic targets are achieved with an average absolute error of 9%, with 75% of all targets achieved within the 7% measurement error. Absolute errors greater than 7% ranged from 17% to 21%. CONCLUSIONS: Tight stepwise control was exhibited in all cases, and the adaptive system was able to match the model and observed patient dynamics. Most errors are associated with external perturbations such as drug therapies, or mismodeled parameters that can be easily adjusted with longer trials and/or more data per hour. The overall result is targeted stepwise tight glycemic regulation using insulin boluses.

Adaptive bolus-based targeted glucose regulation of hyperglycaemia in critical care.

Tight regulation of blood glucose can significantly reduce mortality in critical illness. Critically ill patients are extremely diverse in the dynamics of their hyperglycaemia. Hence, responses can vary significantly, due to variations in insulin levels, effective insulin utilization, glucose absorption and other factors. Consequently, fixed protocols and sliding scales can result in error, given this large variation in patient dynamics. A two-compartment glucose-insulin system model that accounts for time-varying insulin sensitivity and endogenous glucose removal, along with two different saturation kinetics, is developed and tested in preliminary proof-of-concept clinical trials for adaptive control of blood glucose levels. The adaptive control algorithm developed in this research monitors the physiological status of a critically ill patient, allowing real-time, tight glycaemic regulation. The bolus-based insulin administration provides a safe approach to glucose level management. The ability to track changing physiological status and account for insulin transport and effect saturation enabled targeted stepwise reduction in glycaemic levels in three test cases.

Modeling and control of the agitation-sedation cycle for critical care patients.

Agitation-sedation cycling in critically ill patients, characterized by oscillations between states of agitation and over-sedation, is damaging to patient health, and increases length of stay and healthcare costs. The mathematical model presented captures the essential dynamics of the agitation-sedation system for the first time, and is statistically validated using recorded infusion data for 37 patients. Constant patient-specific patient parameters are used, illustrating the commonality of these fundamental dynamics over a broad range of patients. The validated model serves as a basis for comparison of sedation administration methods, devices, therapeutics and protocols. Heavy derivative feedback control is shown to be an effective means of managing agitation, given consistent agitation measurement. The improved agitation management reduces the modeled mean and peak agitation levels 68.4% and 52.9% on average, respectively. Some patients showed over 90% reduction in mean agitation level through increased control gains. This improved agitation management is achieved via heavy derivative feedback control of sedation administration, which provides an essentially bolus-driven management approach, aligned with recent sedation practices.

Modelling cell population growth with applications to cancer therapy in human tumour cell lines.

In this paper we present an overview of the work undertaken to model a population of cells and the effects of cancer therapy. We began with a theoretical one compartment size structured cell population model and investigated its asymptotic steady size distributions (SSDs) (On a cell growth model for plankton, MMB JIMA 21 (2004) 49). However these size distributions are not similar to the DNA (size) distributions obtained experimentally via the flow cytometric analysis of human tumour cell lines (data obtained from the Auckland Cancer Society Research Centre, New Zealand). In our one compartment model, size was a generic term, but in order to obtain realistic steady size distributions we chose size to be DNA content and devised a multi-compartment mathematical model for the cell division cycle where each compartment corresponds to a distinct phase of the cell cycle (J. Math. Biol. 47 (2003) 295). We then incorporated another compartment describing the possible induction of apoptosis (cell death) from mitosis phase (Modelling cell death in human tumour cell lines exposed to anticancer drug paclitaxel, J. Math. Biol. 2004, in press). This enabled us to compare our model to flow cytometric data of a melanoma cell line where the anticancer drug, paclitaxel, had been added. The model gives a dynamic picture of the effects of paclitaxel on the cell cycle. We hope to use the model to describe the effects of other cancer therapies on a number of different cell lines.