Math Modeling, Young Modelers

Dr Ingo Dreyer: Lessons on Modeling Biological Systems

On Oct 2, Dr Ingo Dreyer talked to the Modeling and Beyond community about his work modeling the dynamics of nutrient transport across membrane sandwiches. Throughout his talk, he mentioned many things of interest to modelers, which I will be summarizing here. If you want to know more about the modeling or biological specifics, you can check out his two related papers here and here. Alternatively, I have some notes on that, we could discuss too!

A little background

The motivation of this modeling project came from experimental information about the potassium battery in plant vascular tissues. Ingo’s background in physics informed his approach of focusing on the energetics of this battery. This battery involves transport of sugars, potassium establishing electrical gradients across the membrane. This establishes the battery as a system of coupled dynamics that must be considered together.

Dr Dreyer began his exploration of modeling the system by asking this question:

Can we learn something from computational cell biology?

The following is Ingo’s method of answering this question, and communicating that well to interdisciplinary audiences.

1. Categorize the system

Before beginning a modeling project, we must start by categorizing the biological system. This comprises 80-90% of effort in a modeling project. Ingo cautions us that we may be tempted to sketch out a model, run something quick – but this will probably end up in wasted time. Modeling is driven by the biological questions, and a poor understanding of the biological phenomena we are seeking to represent, infer, and predict will produce poor modeling.

2. Describe all processes mathematically

Only after we have a handle on the system, components, and parameters of interest, we can begin to think about how we will model the system. Sometimes we may need new math or computational tools to complete our model. Again, we should make sure to exhaust existing resources first – Ingo took advantage of existing platforms like VCell to do some simulations and study of the model (more on this later).

a. Take advantage of robustness and remove redundancies.

Redundancies and robustness exist, not only the model, but also in the biological system. This makes it important to consider areas where we can simplify or reduce the number of components to a set of functionally unique components or mechanisms. A model should be as simple as possible but complex as necessary. When constructing our model representations of biological systems, we seek to make a complex system manageable without losing important features.

b. Utilize black boxes to reduce free parameters.

One common simplification method is the inclusion of “black boxes” in a model. Black boxes can represent an entire suite of interactions, that produce some function that feeds into our system of interest. This can be used to simplify the complexity of the model, or to represent unknown processes. The model can be used to obtain more inference on these black boxes, too!

3. Program the model into an in silico system

Before creating our own math or computational tools, we can think about using existing infrastructure such as VCell. VCell is a free software that lets us create simple graphical representations of systems, creates a skeleton-model, and lets us do some model-based experimentation too.

At this point, we should know that “the model could not be so wrong” – time to make some inference and predictions! (Not sure? Read more about what makes a model good here.)

4. Model-based experimentation

Some methods of model-based experimentation include checking for their impact on system dynamics while performing the following:

  1. Changing parameter values
  2. Produce a response surface by changing 2 parameter values, holding others fixed
  3. Parameter sensitivity
    • This refers to the impact any one parameter has on the system dynamics. The more sensitive a parameter is, the more ‘important’ it is (and the easier it could be to get an estimate of its true value).
    • Its a good idea to check initial conditions too.
  4. Parameter estimation and optimization; and model selection
    • If data exists describing all or portions of your system, you can estimate parameter values by fitting or other simulation methods
    • Model selection between a set of differently-complex models can be done via certain optimization criteria, usually determined by how well the models represent data. This is challenging, as model complexity doesn’t necessarily mean easier to fit, for nonlinear ODEs and other model types
  5. Other mathematical properties
    • If your model is ‘tractable’ enough (could be solved by hand), then you can check for properties like bi-stability, equilibria, or other solutions like R-naught of an epidemic system.

Producing model prediction and inference

We produce model predictions and inference when we simulate different conditions and hypotheticals using the above suite of model ‘experiments’. Model-based inference is a type of prediction, specifically about how different model components or systems interact together, and those properties. Models can also predict different experimental designs, including knock-outs, over expression, experimental conditions, measurement length or frequency, or stressor magnitude.

5. Share your results

Sharing your modeling results is probably the hardest part. Many people are intimidated by the math and coding required to model a system, only to find out that trying to communicate these ideas successfully to biologists, mathematicians, and other modelers is even harder. It can be challenging to get proper feedback in this area too – one reason we created our modeling community. Fortunately for us, Ingo illustrated some great methods of interdisciplinary communication throughout his talk.

Throughout the presentation, Ingo took us through each simulation, carefully specifying the research question that led to that simulation, and how inference is gathered from the abstract graphs produced via simulation. He emphasized how model predictions were validated by experiments. The main set of model predictions immediately tested involved knockout of membrane transporters by his collaborators. In one case, however, a separate research group performed an experiment that validated some of the model predictions. This emphasizes the value of coming back to models and updating them; and the value of models in themselves, even if the experiments to fully test or validate them are a long time coming.

a. Emphasize model questions

Even minor results are inferences the model provides by answering some question. Consider these carefully, and clearly state the question: Why is this happening? Why do we require these constraints? How does this trade get implemented? How is this system being maintained?

b. Inference is part of model predictions

Sometimes model inference is so intuitive that we forget it is actually a critical result. When interpreting model results or simulations, consider what literature and experimental observations it agrees with. These should be explicitly stated and emphasized for an audience of biologists or experimentalists.

c. Use figures to explain the model

Illustrate steps, relationships, mechanisms within the system with figures, instead of discussing equations, or mathematical or biological relationships. Line and stick diagrams are a common method used to insinuate relationships. The nature of those relationships can be explained middle or late talk.

d. Explain simulations carefully

When showing figures of model simulations or model experiments, Ingo slowly and carefully took us through the interpretation using animations, arrows, and accessory figures. As modelers, it might be intuitive to interpret such graphs, but its important to carefully take the audience through the interpretation.

For example, Ingo took us through the parameter flux plots, showing how he chose the baseline parameter set, and what perturbations above and below meant biologically. In another set of slides, he showed a 3D parameter response plot to illustrate a multi-parameter effect space. He used animations to take us through what the curvature of the surface meant biologically and in terms of model simulations. This enabled the audience to better interpret the figure.

e. Use analogies

Ingo summarized his conclusions and put them into context using some popular concepts from game theory and microeconomics. As an analogy, he explained the prisoner’s dilemma – when two actors are competing, acting in each’s best interest may result in overall poor or suboptimal trade. In the case of plant-fungal symbiosis, this would likely result in a failed symbiotic relationship. He analyzed the trade between the plant and symbiotic fungal membranes involved in the sandwich by calculating the marginal cost and revenue to the plant, connecting these results back to the equilibria and parameter analysis shown previously.

A self organizing system

Tying together the experimental results, model experiments and parameter spaces, and pulling from game theory and microeconomics, Ingo showed us that this system is self-organizing and self-maintaining. He used a variety of methods to successfully communicate all of the information synthesized to produce that inference: biological background, experiments and results; model construction, analysis, and experimentation; diagrams and animations to explain the system and simulation figures; analogies and theories to maximize inference.

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