{"id":5625,"date":"2016-12-18T16:53:42","date_gmt":"2016-12-18T22:53:42","guid":{"rendered":"https:\/\/www.kabt.org\/?p=5625"},"modified":"2016-12-19T08:35:58","modified_gmt":"2016-12-19T14:35:58","slug":"curve-fitting-aka-model-fitting-the-end-goal","status":"publish","type":"post","link":"https:\/\/www.kabt.org\/?p=5625","title":{"rendered":"Curve Fitting AKA Model Fitting–the End Goal"},"content":{"rendered":"
Curve Fitting AKA Model Fitting:<\/div>\n
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When I started this series of posts my goal was to see if I could generate precise data with a proven classroom lab. \u00a0The data precision that is possible with the yeast catalase lab provides a unique opportunity where data analysis skills can be productively explored, practiced and understood. \u00a0My contention was that this is the ideal lab to focus not just on content, not just on experimental design, but also to introduce relatively sophisticated data analysis. \u00a0To be up front about it, I had only a hint of how rich this lab is for doing just that. \u00a0Partly , this is because in my years of teaching high school biology I covered most of the enzyme content in class activities and with 3D visualizations, focusing on the shape of enzymes but neglecting enzyme kinetics. \u00a0That would be different if I were teaching today—I’d focus more on the quantitative aspects. \u00a0Why? \u00a0Well, it isn’t just to introduce the skills but it has more to do with how quantitative methods help to build a deeper understanding of the phenomena you are trying to study. \u00a0My claim is that your students will develop a deeper understanding of enzymes and how enzymes work in the grand scheme of things if they follow learning paths that are guided and supported by quantitative data. \u00a0This post is an example.<\/div>\n
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The last post focused on plotting the data points as rates, along with some indication of the variability in each measurement in a plot like this.<\/div>\n
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As I said before, I would certainly be happy if most of my students got to this point as long as they understood how this graph helps them to describe enzyme reactions and interpret others work.<\/div>\n
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But a graph like this begs to have a line of best fit–a curve that perhaps plots the relationship implied by our data points.<\/div>\n
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Something like this.<\/div>\n
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One of the early lessons on model building in my current Research Methods course involves taking data we have generated with a manipulative model (radioactive decay) to generate a predictive model. \u00a0The students plot their data points and then try to find the mathematical expression that will describe the process best. \u00a0Almost always, my students ask EXCEL to generate a line of best fit based on the data. \u00a0Sometimes they pick linear plots, sometimes exponential, sometimes log plots and sometime power plots. \u00a0These are all options in EXCEL to try and fit the data to some mathematical expression. \u00a0It should be obvious that the process of exponential decay is not best predicted with multiple types of expressions. \u00a0There should be one type of expression that most closely fits the actual physical phenomenon–a way of capturing what is actually going on. \u00a0Just picking a “treandline” based on how well it visually fits the current data without considering the actual phenomenon is a very common error or misconception. \u00a0You see, to pick or develop the best expression requires a deep understanding of the process being described. \u00a0In my half-life exercise, I have the students go back and consider the fundamental things or core principles that are going on. \u00a0Much like the process described by Jungck, Gaff and Weisstein:<\/p>\n<\/div>\n

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“By linking mathematical manipulative models in a four-step process\u20141) use of physical manipulatives, 2) interactive exploration of computer simulations, 3) derivation of mathematical relationships from core principles, and 4) analysis of real data sets…”<\/div>\n
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Jungck, John R., Holly Gaff, and Anton E. Weisstein. “Mathematical manipulative models: In defense of \u201cBeanbag Biology\u201d.”\u00a0CBE-Life Sciences Education<\/i>\u00a09.3 (2010): 201-211.<\/div>\n
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The point is that we are really fitting curves or finding a curve of best fit–we are really trying to see how well our model will fit the real data. \u00a0And that is why fitting this model takes this lab to an entirely new level. \u00a0 But how are you going to build this mathematical model?<\/div>\n
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Remember that we started with models that were more conceptual or manipulative. \u00a0And we introduced a symbolic model as well that captured the core principles of enzyme action:<\/div>\n
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\nBy Thomas Shafee (Own work) [CC BY 4.0 (http:\/\/creativecommons.org\/licenses\/by\/4.0)], via Wikimedia Commons<\/span><\/div>\n
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Now how do we derive a mathematical expression from this? \u00a0I’m not suggesting that you should necessarily unless you feel comfortable doing so but I’ll bet there are kids in your class that can given a bit of guidance. \u00a0You may not feel comfortable providing the guidance. \u00a0But in this day of “just ask Google” you can provide that guidance in the form of a video discussion from the Khan Academy designed to help students prepare for the MCAT. \u00a0Don’t let that scare you off. \u00a0Here are two links that take the symbolic model and derive a mathematical expression–not just any expression—the Michaelis-Menten equation for enzyme kinetics. You or your students will no doubt need to view these more than once but the math is not that deep—not if your students are exploring calculus or advanced algebra. \u00a0It is really more about making assumptions and how those assumptions simplify things so that with regular algebra you can generate the Michaelis-Menten equation.<\/div>\n
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https:\/\/www.khanacademy.org\/test-prep\/mcat\/biomolecules\/enzyme-kinetics\/v\/an-introduction-to-enzyme-kinetics<\/a><\/div>\n
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https:\/\/www.khanacademy.org\/test-prep\/mcat\/biomolecules\/enzyme-kinetics\/v\/steady-states-and-the-michaelis-menten-equation<\/a><\/div>\n
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You can also find a worked out derivation here: \u00a0https:\/\/www.ncbi.nlm.nih.gov\/books\/NBK22430\/<\/a>\u00a0 in this text excerpt from Biochemistry, 5th ed.\u00a0Berg JM, Tymoczko JL, Stryer L.<\/div>\n
New York:\u00a0W H Freeman<\/a>; 2002.<\/div>\n
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Of course, you don’t even have to go through the derivation you could just provide the equation.<\/div>\n
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\nThe important thing is that students understand where this equation comes from—it doesn’t come out of thin air and it is based on the same core principles they uncovered or experienced if they did the toothpickase manipulation–it is just quantified now. \u00a0So how do I use this equation to actually see how well my data “fits”? \u00a0If it were a linear expression that would be easy in Excel or any spreadsheet package but what about non-linear trend lines? \u00a0I can tell you that this expression is not part of the trend line package you’ll find in spreadsheets.<\/div>\n
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I’ve got to admit, I spent too many years thinking that generating best-fit curves from non-linear expressions like the M-M equation was beyond the abilities of me or my students. \u00a0But again “Ask Google” comes to the rescue. \u00a0If you google “using solver for non-linear curve fitting regression” you’ll end up with lots of videos and even some specific to the Michaelis-Menten equation. \u00a0It turns out EXCEL (and I understand Google Sheets) has an add-on called Solver that helps you find the best fit line. \u00a0But what does that mean? \u00a0Well it means that you need to manipulate the parameters in the M-M equation to generate a line until it mostly fits your data–to see if the model is an accurate description of what you measured. \u00a0What parameters are these?<\/div>\n
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Look at the equation:<\/div>\n
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V0 equals the rate of the reaction at differing substrate concentrations–the vertical axis in the plots above.<\/div>\n
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Vmax equals the point at which all of the enzyme is complexed with the substrate–the maximum rate of the reaction with this particular enzyme at this particular enzyme concentration (that is enzyme concentration not substrate)<\/div>\n
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Km equals the concentration of the substrate where the rate of reaction is 1\/2 of\u00a0Vmax<\/p>\n<\/div>\n

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[S] \u00a0equals the substrate concentration, in this case the\u00a0H2O2<\/div>\n
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Two of these parameters are variables—one is our experimental or explanatory variable, the concentration of H2O2 and the other is our response variable, the rate of the reaction. Some folks prefer independent and dependent variable. This is what we graph on our axis.<\/div>\n
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The other two parameters are constants and the help to define the curve. More importantly, these are constants for this particular enzyme at this particular enzyme concentration for this particular reaction. These constants will be for different enzymes, different concentrations or reactions with inhibitors, competitors, etc. In other words it is these constants that help us to define our enzyme properties and provide a quantitative way to compare enzymes and enzyme reactions. You can google up tables of these values on the web. from: Biochemistry, 5th ed.\u00a0Berg JM, Tymoczko JL, Stryer L.<\/div>\n
New York:\u00a0W H Freeman<\/a>; 2002. https:\/\/www.ncbi.nlm.nih.gov\/books\/NBK22430\/table\/A1055\/?report=objectonly<\/a><\/div>\n
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So calculating these constants is a big deal and one that is not typically a goal in introductory biology but if you’ve come this far then why not?<\/div>\n
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This is where generating that line that best-fits the data based on the Michaelis-Menten equation comes in.<\/div>\n
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You can do this manually with some help from Solver in Excel. \u00a0(Google Sheets also is supposed to have a solver available but I haven’t tried it.<\/div>\n
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I have put together a short video on how to do this in Excel based on the data I generated for this lab.<\/div>\n
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