This is part III of an ongoing DoE series that I would have loved to read when I was first running these types of experiments in the lab.
You can start at the beginning, with part I: a general overview of what DoE is, here.
You’re running a fermentation experiment. You want to figure out how pH, temperature, and glucose residual affect biomass yield. You sit down to design a DoE. You plug in low and high levels for each factor—maybe 6.75 and 7.15 for pH, 30°C and 37°C for temperature, 5 g/L and 15 g/L for glucose. You run the experiment. You plot your data. The results are… fine. But something feels off. The middle point (the one you ran just for fun) gave the highest yield. That wasn’t in your model. What gives?
That’s curvature.
What is curvature?
Curvature means that the relationship between a factor and a response isn’t a straight line. If you plotted the response vs. one of your inputs and the line bows, arches, or dips, that’s curvature. It means the system behaves differently in the middle of the range than you’d expect from just connecting the low and high dots.
In fermentation, this happens all the time. Maybe yield increases with glucose… until it doesn’t. Maybe biomass grows faster at 33°C than at 30°C or 37°C. Biological systems are full of nonlinear behavior. That’s curvature.
Why does it matter?
If your design only looks at the ends—low and high levels—you’re assuming the system is linear. If there’s curvature and you don’t test for it, your model will miss the best part of the space. That can lead you to:
- Misinterpret your data
- Choose suboptimal conditions
- Build poor predictive models
The center of the range might be where the best action is happening, but your linear model won’t see it.
How do you test for it?
You include center points in your design.
Let’s say your factor is temperature, ranging from 30°C to 37°C. You include a few replicates at 33.5°C. If the average yield at that center point is higher (or lower) than the average of the low and high points, that’s a sign of curvature.
Most software will test this for you. JMP, for example, includes a formal “lack of fit” test when you include center points. If it says there’s significant curvature, it’s a clue you should move to a design that models curvature explicitly—like a response surface model (RSM).
When should you worry about curvature?
- If your system involves biology, chemistry, or ….
- Im being cheeky here, but realistically look for anything that saturates, peaks, or inhibits.
- If you’re working across a wide range of any factor
- If you’re optimizing, not just screening
What to do about it
If curvature is detected:
- Add squared terms in your model (e.g., pH² or temperature²)
- Use contour plots or profilers to explore the response surface
- Move to a design like Central Composite Design (CCD) or Box-Behnken.
These approaches can model the arch, the dip, the peak. They can help you find the sweet spot.
To sum it up
Curvature isn’t a bug in your system, its a feature. It’s a clue that your biology is doing something interesting, and can be insanely valuable when you follow its lead.
Leave a Reply