Ingalls mathematics nov21

Local and global sensitivity analysis connect model parameters to system behaviour in a model of beta-cell metabolism Brian Ingalls, Rahul, Adam Stinchcombe Department of Applied Mathematics

Jamie Joseph School of Pharmacy University of Waterloo

Levels of glucose in the bloodstream must be carefully regulated.

A key regulator is the hormone insulin, which is secreted from beta cells in the pancreas.

http://courses.washington.edu/conj/protein/insulin.gif

 Beta-cells respond to increased glucose levels (e.g. after a meal) by increasing the rate of insulin secretion.  Insulin causes muscle, liver, and fat cells to absorb glucose from the bloodstream.  Malfunction of the beta-cell response causes type II (adult onset) diabetes.

http://1.bp.blogspot.com/-dIhWO-TnHYY/TkF6Y_yAosI/AAAAAAAAAKo/1zuPa1--b44/s1600/insulin+secretion.jpg

Our goal: to understand the metabolic coupling between glucose uptake and insulin secretion.

Our tool: a mechanistic mathematical model of the metabolic network.

Model Formulation Divide the cell into compartments, each of which is presumed well-stirred (by diffusion).

Describe the enzymatic rate of production and consumption of each metabolite in the network.

Formulate a mass-balance equation for each metabolite (ordinary differential equation).

Model Formulation

Reaction rates (from enzymological studies)

Balance equations

Model Components 24 metabolites 30 enzymatic reactions 123 kinetic parameters:

-- 89 from previous studies -- 34 fit to experiments

Model validation Established good fit to training data:

After calibration, the model was validated against a range of qualitative observations from the literature (test data)

Model Analysis Which system components have the greatest effect on the system output?

What perturbations will have the greatest effect on the system output? (potential drug targets) Model analysis techniques: local and global parametric sensitivity analysis

Parametric Sensitivity Analysis

Local Parametric Sensitivity Analysis slice

Scaled derivative:

Primary advantage: simple calculation and interpretation, one sensitivity coefficient per parameter Disadvantages: -- limited to describing the effect of small changes in parameter values -- captures only a single point in parameter space

Global Parametric Sensitivity Analysis -- Choose a region in parameter space -- Sample the model behaviour

L e.g. Latin hypercube sampling

L M H 3

M H 1 2

Summarize results in sensitivity measures: -- correlation coefficients (for monotonic relationship between parameer value and output) -- variance-based methods: relate variation in the output to variation in the parameter value(s): first order (individual) and total-effect (combined) sensitivities

Model Analysis: sensitivity rankings

Local sensitivity

Global Sensitivity, Variance method: first order effect

Global Sensitivity, correlation coefficients

Global Sensitivity, Variance method: Total effect

Conclusions --Identification of individual parameters that have the most influence on specific model outputs.

--Differences between local and global rankings reflect synergisms between parameters -- Differences between first order and total order rankings indicate which parameters combinations have significant effects -- consistent rankings between first order and total order effects indicate effects that are robust with respect to variability in other model parameters -- final conclusion: recommendation for optimal single and pairwise perturbations for effecting key insulin-secretion signals.