#### By Katherine Todd-Brown, USA @KatheMathBio

Computational and mathematical modeling are increasingly in demand as society tries to apply current scientific understanding to future soil management practices. The flux of carbon from the soil to atmosphere in particular is a major uncertainty of future climate projections. Soil carbon is also a critical component of soil health. In addition, many funding agencies are interested in coupling experimental and modeling approaches to better target scientific inquiry and extrapolate into murky futures.

Computational and mathematical models are generally thought of as tools for projecting into the future, but they can also help design and run experiments that might otherwise be costly or even impossible to carry out in the lab/field. You can turn off gravity in a model. While I say this with a bit of tongue in check, imagine directly exploring the effects of capillary action vs gravitational forces in soil wetting. Something that is completely impractical in the real world can become almost trivial in the computer.

Models are frequently seen as this magic box that, if you can just figure out how to plug your data in, will tell the rest of the world how critically important your research is. As a modeler, I’ve found that my colleagues who work primarily in the field and/or on the bench are frequently at a bit of a loss how to wave this magic wand. Fully characterizing a site to run an off-the-shelf model like DayCent can be overwhelming or just plain intractable depending on the location. Plus experimental treatments or particular measurements will often not have direct representation in these off-the-shelf models, precisely because their effects are novel or uncertain and thus of interest.

**So where does that leave us?**

Fortunately, if you design experiments and interoperate their results, you are most of the way there; even if you are a field or laboratory scientist who hasn’t written mathematical equation since your college math class. **Numerical models are formalizations of conceptual frameworks that every scientists use on a daily basis.** We tell stories of how we think the world works in our grant proposals and manuscripts. Those stories are models in their most basic forms; English, Mandarin, mathematics, and computer code are just a different languages that can be used to convey some scientific understanding to the broader world. By telling stories of how we think the world works, scientists are creating models.

The main difference between conceptual models and numerical models is precision. Human language is generally very fuzzy leaving quite a bit of room for interpretation to the listener. Mathematics, on the other hand, is extremely precise. It’s one thing to say that the change in soil carbon is the difference between the inputs and outputs, and another to state: . This mathematical formulation has several embedded assumptions in it like: soil carbon does not change with space, and inputs change over time but that the decay rate doesn’t. Mathematics does not allow you to sweep anything under the rug, instead you must explicitly state what your assumptions are and under what conditions those hold true. While you can be as precise with an English description of your conceptual framework, it can be challenging to attain the level of detail you get from a mathematical formalization. But underpinning all this is still your original conceptual understanding of the research system.

**On a practical level how do you move from this conceptual framework to a precise mathematical formulation?**

One way to do this is to collaborate with a mathematician or computer scientists. Be aware however that this will frequently require several conversations since you are likely talking with someone who has no soils training. The nutrient, parent material, and biological differences between a tropical oxisol and temperate molisol might be obvious to you but not to someone trained to locate bifurcation points in dynamical systems. Often times the different soil conditions will tie directly to different mathematical assumptions in ways that are not obvious at first glance. Lobbing your data over the discipline dividing wall won’t cut it. Instead, these collaborations require a lively and ongoing exchange.

Alternatively you can develop your own model. Frequently if you have a strong hypothesis driven experimental design, you can frame this hypothesis as a numerical model and probably already do on some level. Many statistical tests like linear regressions are, themselves, numerical models. A common problem with these statistical tests is that they assume normally distributed data and linear relationships, frequently not the case in soil systems. Diagramming the mass or energy path you are interested in will often lead to a set of differential equations. Once you have this mathematical formalization you can apply model-data integration techniques to fit descriptive parameters to your data.

By using numerical models informed by your scientific understanding you can dramatically increase the power of your statistical tests, compete different hypothesis, explore the relative importance of entwined mechanisms, and extrapolate findings into management scenarios. Models allow you to convey your scientific understanding, supported by your experimental data, with a precision that is difficult to achieve in a standard scientific narrative. While the end goal is scientific understanding not mathematical poetry, models are playing an increasingly important role in biogeochemistry.

Mathematics is the language of size, shape and order […] ~ Lancelot Hogben (1936)

*Kathe Todd-Brown is a computational biogeochemist at the Pacific Northwest National Laboratory in Richland, Washington. She initially trained as a mathematician but it was a bit too clean. She transitioned to soil carbon cycling and couldn’t be happier to work in one of the most interesting systems on the planet.*