Sanft R., Walter A. Exploring Mathematical Modeling in Biology Through Case Studies and Experimental Activities

New York: Academic Press, 2020. — 258 p.Exploring Mathematical Modeling in Biology Through Case Studies and Experimental Activities provides supporting materials for courses taken by students majoring in mathematics, computer science or in the life sciences. The book's cases and lab exercises focus on hypothesis testing and model development in the context of real data. The supporting mathematical, coding and biological background permit readers to explore a problem, understand assumptions, and the meaning of their results. The experiential components provide hands-on learning both in the lab and on the computer. As a beginning text in modeling, readers will learn to value the approach and apply competencies in other settings.
Included case studies focus on building a model to solve a particular biological problem from concept and translation into a mathematical form, to validating the parameters, testing the quality of the model and finally interpreting the outcome in biological terms. The book also shows how particular mathematical approaches are adapted to a variety of problems at multiple biological scales. Finally, the labs bring the biological problems and the practical issues of collecting data to actually test the model and/or adapting the mathematics to the data that can be collected.
Presents a single volume on mathematics and biological examples, with data and wet lab experiences suitable for non-expertsContains three real-world biological case studies and one wet lab for application of the mathematical modelsIncludes R code templates throughout the text, which are also available through an online repository, along with the necessary data files to complete all projects and labsTips for using this text - instructors
Tips for using this text - students
Online resourcesPreliminaries: models, R, and lab techniques
Bringing mathematics and biology together through modeling
R basics
RStudio layout
Simple calculations
Data structures
Vectors
Matrices
Data frames
Basic plotting
Reading data from files
Iteration
Fitting a linear regression model
Prelab lab: practicing the fundamentalsMaterials
Using the micropipettes
Serial dilution with dye solutions and water
Measuring absorbance
Graphing and data analysis
Introduction to modeling using difference equations
Discrete-time models
Solutions to first-order difference equations
Using linear regression to estimate parameters
Putting it all together: the whooping crane
Case study : Island biogeography
Background
Model formulation
Rakata story
Data
Parameter estimation
Model analysis
Modern approach: lineage data
Model
Parameter estimation
Model analysis
Back to MacArthur and Wilson: effects of distance and area
Case study : Pharmacokinetics model
Background
Pharmacokinetics: basic concepts and terminology
Caffeine
Formulating the model
Understanding the model
Parameter estimation
Model evaluation/analysis
Further exploration
Case study : Invasive plant species
Background
Model formulation
Parameter estimation
Model predictions
Management strategies
Wet lab: logistic growth model of bacterial population dynamicsModeling populations
The experiment
Preparation
Measuring growth rates by optical density
Measuring growth rate by serial dilution and plate count
Model calibration and analysis
Experiment part : effect of changing media
Differential equations: model formulation, nonlinear regression, and model selection
Biological background
Mathematical and R background
Differential equation-based model formulation
Constant flow rate
Relative rates
Mass action
Feedback
Solutions to ordinary differential equations
Investigating parameter space
Nonlinear fitting
Model selection
Case study : How leaf decomposition rates vary with anthropogenic nitrogen deposition
Background
The data
Model formulation
Parameter estimation
Model evaluation
Case study : Exploring models to describe tumor growth rates
Background
The data
Model formulation
Parameter estimation
Model evaluation: descriptive power
Model evaluation: predictive power
Case study : Predator responses to prey density vary with temperature
Background
Can predator-prey interactions predict invasive behavior?
Analysis of functional response data: determining the parameters
Exploring functional responses as a function of temperature
Wet lab: enzyme kinetics of catechol oxidase
Overview of activities
Introduction to enzyme catalyzed reaction kinetics
Deriving the model
Estimating KM and Vmax
Our enzyme: catechol oxidase
Experiment: collecting initial rates for the Michaelis-Menten model
Overview of the procedure
Materials
Enzyme preparation
Running the reactions
Analysis in R
Effects of inhibitors on enzyme kinetics
Experiment: measuring the effects of two catechol oxidase inhibitors, phenylthiourea and benzoic acid
Analysis in R
Differential equations: numerical solutions, model calibration, and sensitivity analysis
Biological background
Mathematical and R background
Numerical solutions to differential equations
Example: logistic growth (one variable)
Example: tumor growth model (a system of two differential equations)
Calibration: fitting models to data
Example: calibrating the logistic growth model
Sensitivity analysis
Global sensitivity
Local sensitivity analysis
Local sensitivity example: logistic model
Putting it all together: the dynamics of Ebola virus infecting cells
Numerical solution for the Ebola model
Fitting parameters to the Ebola model: calibrating the model
Local sensitivity analysis: assessing key parameters in the Ebola virus-cell system
Case study : Modeling the influenza pandemic
Background
The SIR model
Model simulations
Cumulative number of cases
Epidemic threshold
Public health interventions
HN influenza pandemic
Estimating parameters
Taking action
Case study : Optimizing immunotherapy in prostate cancer
Background
Model formulation
Model implementation
Parameter estimation
Vaccination protocols and model predictions
Sensitivity analysis
Simulating other treatment strategies
Case study : Quorum sensingModel formulation
Parameter estimation
Model simulations
Sensitivity analysis
The temporal response
Wet lab: hormones and homeostasis-keeping blood glucose concentrations stable
Overview of activities
Introduction to blood glucose regulation and its importance
Developing a model
Experiment: measuring blood glucose concentrations following glucose ingestion
Introduction to the procedure
Experimental procedure
Analysis
Thoughts to consider for potential follow-up experiments
Technical notes for laboratory activitiesPopulation growth
Bacterial growth media and tips
Optional plate counts and converting optical density to numbers of bacteria
Alternate organisms
Enzyme kinetics
Tips and solution preparation for catechol oxidase
Notes on data analysis
Notes on other enzymes or similar experiments
Predator-prey or the enzyme game
Other enzymes
Blood glucose monitoring
Tips for glucose monitoring
Example of a subject consent form
Other lab activities