Math Insight

Synthesis project

Group members:
Total points: 1
Grading rubric

To earn credit, a project must meet the following criteria.

CriterionMetNot met
Demonstrate understanding of the strengths and weakness of each modeling approach.
Develop of convincing summary of how modeling can be useful in biology.
Project receives creditYESNO
Submitting project

Submit the following by the due date.

  1. This cover sheet
  2. Answers to the project questions (typed or handwritten)

Background

In this project you will synthesize your understanding from the projects you completed throughout the semester. It may be helpful to refer to the mapping between biology and math schematic throughout:

biology math schematic
  1. Module 1: Class-structured models (Sea turtle and gut microbiota projects)
    1. For each project, what specifically did you learn from the mathematical model about the biological system?
    2. What is one simplifying assumption of the model for each project?
    3. Brainstorm other specific biological contexts where using class-structured models might be appropriate. What do all of these have in common? Can you use this to state anything general about when a class-structured modeling approach can be used?
    4. Imagine you're writing a guide for someone who wants to use a class-structured model to study one of the biological systems you brainstormed above. Compare the steps you took in each project to get from the biological system/question to a biological answer. Write down a set of instructions for how to go through this process of mapping from biology to math and back for any biological system where you would want to use a class-structured model. Be as specific as you can, (e.g. instead of saying "analyze the model" explain how you might analyze the model).

  2. Module 2: Two-dimensional continuous dynamical systems (influenza and memory projects)
    1. For each project, what specifically did you learn from the mathematical model about the biological system?
    2. What is one simplifying assumption of the model for each project?
    3. Brainstorm other specific biological contexts where using continuous dynamical system models might be appropriate. What do all of these examples have in common? Can you use this to state anything general about when a continuous dynamical system modeling approach can be used?
    4. Imagine you're writing a guide for someone who wants to use a continuous dynamical system model to study one of the biological systems you brainstormed above. Compare the steps you took in each project to get from the biological system/question to a biological answer. Write down a set of instructions for how to go through this process of mapping from biology to math and back for any biological system where you would want to use a continuous dynamical system model.

  3. Module 3: Probabilistic modeling (neural decoding and tumor growth projects)
    1. For each project, what specifically did you learn from the mathematical model about the biological system?
    2. What is one simplifying assumption of the model for each project?
    3. Brainstorm other specific biological contexts where using probabilistic models might be appropriate. What do all of these examples have in common? Can you use this to state anything general about when using a probabilistic modeling approach can be used?
    4. Imagine you're writing a guide for someone who wants to use a probabilistic model to study one of the biological systems you brainstormed above. Compare the steps you took in each project to get from the biological system/question to a biological answer. Write down a set of instructions for how to go through this process of mapping from biology to math and back for any biological system where you would want to use a probabilistic model.

  4. Module 4: Spatial modeling (metapopulation and diffusion projects)
    1. For each project, what specifically did you learn from the mathematical model about the biological system?
    2. What is one simplifying assumption of the model for each project?
    3. Brainstorm other specific biological contexts where using either metapopulation, diffusion, or another spatial modeling approach might be appropriate. What do all of these examples have in common? Can you use this to state anything general about when using a spatial modeling approach can be used?
    4. Imagine you're writing a guide for someone who wants to use a spatial model to study one of the biological systems you brainstormed above. Compare the steps you took in each project to get from the biological system/question to a biological answer. Write down a set of instructions for how to go through this process of mapping from biology to math and back for any biological system where you would want to use a spatial model.

  5. Comparing modeling approaches.
    1. When would it be more appropriate to use a discrete time model (e.g. those in the class-structured models module) versus a continuous time model (e.g. those in the continuous dynamical systems module) and vice versa?
    2. When would it be more appropriate to use a probabilistic modeling approach rather than a deterministic (non-probabilistic) one? What are the pros and cons of using probabilistic or stochastic models instead of deterministic (non-probabilistic) ones?
    3. When would it be more appropriate to use a spatial modeling approach rather than a non-spatial one? What are the pros and cons of using spatial models instead of non-spatial ones?
    4. If you do build a spatial model, when would it be better to use a discrete space approach (e.g. metapopulation) and when would it be better to use a continuous approach (e.g. diffusion)?

  6. Summary.
    1. Must all models have simplifying assumptions that make them different than the biological system they describe? Why?
    2. Given your answer to (a), what criteria would you propose for determining how to evaluate whether a mathematical model is 'good'?
    3. A friend hears that you are taking this class and says “What does math have to do with biology?”. How would you respond? What is the role of mathematical modeling in biology?

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Math 2241, Spring 2023