Models, Simplifying Assumptions, and Bounding#

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Models and simplifying assumptions

Creating models of real things

Life is complicated. To help us understand what we see around us—in both our everyday lives and in science or engineering—we often construct models. A common aphorism states: all models are wrong, but some are useful. No matter how sophisticated, all models are approximations of something real. While they are not the “real thing” (and are thus wrong), models are useful when they allow us to make predictions about real life.

Types of models

Models come in a variety of forms that include:

Physical models: These are 3-D objects that we can touch.
Drawings: These can be on paper or on the computer and either in 2-D or virtual 3-D. We mostly look at them.
Mathematical models: These describe something in real life in mathematical terms. We use these to calculate the behavior of the thing or process we want to understand and to make predictions.
Verbal or written models: We communicate these models in written or spoken language.
Mental models: We construct these models in our minds and we use these to create other types of models and to understand the world.

Simplifying assumptions

Usually, in science and everyday life, we prefer simple models over complex ones. Creating simple models of complex real things, however, requires us to make simplifying assumptions. By using simplifying assumptions in our models, we remove some complexities of the real thing and thus simplify analysis. For example, someone trying to model a basketball might make the simplifying assumption that the ball is a perfect sphere, ignoring the surface texture and seams between the leather panels. When a simplified model no longer predicts behavior of the real thing within acceptable bounds of an experiment, too many simplifying assumptions may have been made (or the model is wrong). When little predictive value is gained from adding more details to a model, it is likely overly complex. Let’s take a look at different types of models from different disciplines and point out their simplifying assumptions.

An example from physics: a block on a frictionless plane

Figure 1. Most students who have taken introductory physics (high school or college) will be familiar with this figure. The line drawing depicts a rectangular block (of any material) sitting on a generic incline plane (depicted as a triangle). In introductory physics these models almost always come with some simplifying assumptions. For instance, we usually start ignoring what materials make up the block and plane. Often, we might also, for convenience, assume that the plane is frictionless. The simplifying assumptions allow the student to practice thinking about how to balance the forces acting on the block when it is elevated in a gravity field and to see that the surface it is sitting on is not perpendicular to the gravity vector (mg). This simplifies the math and allows the student to focus on the geometry of the model and how to represent that mathematically. The model, and its simplifying assumptions, might do a reasonably good job of predicting the behavior of an ice cube sliding down a glass incline plane but would likely do a bad job of predicting the behavior of a wet sponge on an incline plane coated with sandpaper. The model would be oversimplified for the latter scenario.
Source: Created by Marc T. Facciotti (Own work)

An example from chemistry: a molecular line model of glucose

Figure 2. A line drawing of a glucose molecule. By convention, we understand the points where straight lines meet to represent carbon atoms, while we show other atoms explicitly (i.e. we write a letter). Given additional information about the atoms represented in the figure, this model can be useful for predicting some chemical properties of this molecule, including the potential reactions it might enter into with other molecules. The simplifying assumptions, however, hide the dynamics of the molecules.
Source: Created by Marc T. Facciotti (Own work)

An example from biology: a ribbon diagram of a protein — The transmembrane protein bacteriorhodopsin

Figure 3. A cartoon model of the transmembrane protein bacteriorhodopsin. The protein is represented as a light blue and purple ribbon (the different colors highlight alpha helix and beta sheet), a yellow sphere represents a chloride ion, red spheres represent water molecules, pink balls and sticks represent a retinal molecule, and orange balls-and-sticks represent other lipid molecules. The figure displays the model in two views. On the left the model is viewed “side on” while on the right it is viewed along its long axis from the extracellular side of the protein (rotated 90 deg out of the page from the view on the left). This model simplifies many of the atomic-level details of the protein. It also fails to represent the dynamics of the protein. The simplifying assumptions mean that while the model predicts the space occupied by the protein in the cellular membrane, and how far into the membrane the retinal sits, it does not help predict the time it takes for the protein to transport protons.
Source: Created by Marc T. Facciotti (own work), University of California, Davis
Derived from: PDBID:4FPD DOI: 10.1016/j.str.2012.12.018

An example from everyday life: a scale model of a Ferrari

Figure 4. A scale toy model of a Ferrari. In this model, there are many simplifications. Most of the simplifictions only make this model useful for predicting the general shape and relative proportions of the real car. This model gives us no predictive power about how quickly it stops from a speed of 300 km/hr, or how it feels to drive a precision automobile.
Source: Created by Marc T. Facciotti (Own work)

Possible NB Discussion Point

Describe a physical model that you use in everyday life. What does the model simplify from the real thing?

Possible NB Discussion Point

Describe a drawing that you use in science class to model something real. What does the model simplify from the real thing? What are the advantages and disadvantages of the simplifications?

The spherical cow

The spherical cow is a famous metaphor in physics that makes fun of physicists tendencies to create hugely simplified models for very complex things. Numerous jokes are associated with this metaphor and they go something like this:

"Milk production at a dairy farm was low, so the farmer wrote to the local university, asking for help from academia. A multidisciplinary team of professors was assembled, headed by a theoretical physicist, and two weeks of intensive on-site investigation took place. The scholars then returned to the university, notebooks crammed with data, where the task of writing the report was left to the team leader. Shortly thereafter the physicist returned to the farm, saying to the farmer, "I have the solution, but it only works in the case of spherical cows in a vacuum"." - Source: Wikipedia page on Spherical Cow - accessed November 23, 2015.

Figure 5. A cartoon representation of a spherical cow.
Source: https://upload.wikimedia.org/wikiped.../d2/Sphcow.jpg
By Ingrid Kallick (Own work) [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC BY 3.0 (http://creativecommons.org/licenses/by/3.0)], via Wikimedia Commons

The spherical cow is an amusing way to ridicule the process of creating simple models and it is quite likely that you will have your BIS2A instructor invoke the reference to the spherical cow when an overly simplified model of something in biology is being discussed. Be ready for it!

Bounding or asymptotic analysis

In BIS2A, we use models often. Sometimes we also like to imagine or test how well our models represent reality by comparing model predictions with the real life thing. If you need to know a lot of detail about a system, you create a detailed model. If you're willing to live with less detail, you will create a simpler model. Besides applying simplifying assumptions, it is often useful to assess your model using a technique we call bounding (aka. asymptotic analysis). The main idea of this technique is to use a model, complete with its simplifying assumptions, to evaluate how the real thing might behave at extreme conditions (e.g. test the model at the minimum and maximum values of key variables). Let’s examine a simple real life example of how this technique works.

Example: bounding

Problem setup
Imagine that you need to leave Davis, CA and get home to Selma, CA for the weekend. It's 5PM and you told your parents that you'd be home by 6:30. Selma is 200 miles (322 kM) from Davis. You're getting worried that you won't make it home on time. Can you get some estimate of whether it's even possible or if you'll be reheating your dinner?

Create simplified model and use of bounding
You can create a simplified model. In this case you can assume that the road between Davis and Selma is perfectly straight. You also assume that your car has only two speeds: 0 mph and 120 mph. These two speeds are the minimum and maximum speeds that you can travel—the bounding values. You can now estimate that even under assumptions of the theoretically "best case" scenario, where you would drive on a perfectly straight road with no obstacles or traffic at maximum speed, you will not make it home on time. At maximum speed you would only cover 180 of the required 200 miles in the 1.5 hours you have.

Interpretation
In this real life example a simplified model is created. In this case, one very important simplifying assumption is made: the road is assumed to be straight and free of obstacles or traffic. These assumptions allow you to reasonably assume that you could drive this road at full speed the whole distance. The simplifying assumptions simplified out a lot of what you know is actually there in the real world that would influence the speed you could travel and by extension the time it would take to make the trip. The use of bounding—or calculating the behavior of a system at minimum and maximum values of key variables—is a way of making quick predictions/estimates about what might happen in the real world.

We will use this tool periodically in BIS2A.

The importance of knowing key model assumptions

Knowing the simplifying assumptions inherent in a model is critical to judging how useful it is for predicting real life and for making a guess about where the model needs improving if it is not sufficiently predictive. In BIS2A, we will periodically ask you to create different models and to identify the simplifying assumptions and the impact of those assumptions on the utility and predictive ability of the model. We will also use models together with bounding exercises to try learning something about the potential behavior of a system.