Mathematical modeling is a relatively new field. You may be more aware of a subfield of biology called bioinformatics or computational biology. These tend to deal with larger data sets, studying them using more algorithmic methods. Computational biology also refers to mathematical biology, or an even mixture of mathematics and computer science as applied to biology.
Mathematical biology involves more math and less programming than bioinformatics. It studies systems on a larger scale with less components involved. So when would you want to use this?
Many times experimentation reveals how one particular component interacts with other components in the same system or process. Conceptual models can help us process all the available information on the process, but mathematical models force us to specify the details involved, providing additional information.
- Causal inference
Experimentation gives information about causality, but often it is challenging to observe the effects of a change on multiple components within the system. Once a mathematical model has been established, the effects on all components can be predicted for a given cause. This also contrasts with statistical models, perhaps studying the effect of a gene knockout on growth – there may be feedback loops or more direct effects on growth than the specific component being experimentally manipulated. An established model of the system can help us to identify these direct causes.
- Decrease the number of experiments
Most of the time, the interesting parts of a system are exactly when it “breaks” – when you begin to observe unusual activity. This helps us identify the roles of specific components within a system. However, predicting the values at which this will occur is difficult when working with a set of experiments and a conceptual model of the system. A ‘breaking point’ will have some numerical value, and quantifying the conceptual model helps us predict this with much more accuracy. It has the additional benefit of helping us understand if/when a predicted ‘breaking point’ is not observed experimentally – an additional method to avoid the “maybe it just wasn’t enough” explanation when working with conceptual models.
- Avoid experimental designs that don’t fully characterize the system
Many times biologists perform a set of experiments that flesh out a portion of some logarithmic scale. Sometimes it is an order of magnitude above or below in vivo conditions. Reiterating the previous point, it is very difficult if at all possible to determine some numerical quantity based on a conceptual model of the system. In my experience, a mathematical model often demonstrates that the experimentalists’ choice is not close enough to those extreme values that produce interesting system dynamics.
- Understanding across system scales
Interpretation of conceptual models is often an exercise in intuition. The ability of intuition to correctly guess how things work might vary between people and knowledge levels, but certainly intuiting how multi-scale processes affect each other requires serious quantitation. The effect of a smaller-scale process on a larger-scale process can be understood as integrating, or summing, the smaller-scale process’ effect on the larger scale. Of course, this implies that the sum of effects on the small scale are correctly predicted, as well as their impact on at the large scale – inherently quantitative relationships.
- Incorporate all knowledge, not just biological
One of the strongest benefits of mathematical modeling is the ability to easily incorporate multi-disciplinary concepts, such as chemistry or biophysics. Biological systems are necessarily controlled by chemical and physical processes, and in certain cases these effects should not be ignored or hand-waved away. Mathematical modeling allows us to summarize all the available understandings of a process, which brings me to my next point:
- Effective communication
Conceptual models are normally developed by experts in their field. A conceptual model of the same process might look very different when it is drawn by a mathematician, biologist, physicist, chemist, or computer scientist, to say nothing of the variation between disciplinary subfields. Mathematics might be intimidating or unapproachable to some, but its logical, objective structure makes a mathematical model the perfect interdisciplinary communication tool. Mathematics allows you to communicate *how* things work, while avoiding potentially tedious or esoteric discussions of jargon.
- Shift from intuition of system to quantitation of knowledge
As mentioned previously, conceptual models encourage intuitive thinking, even when quantitative relationships are offered. Providing quantitative relationships *without* an objective, standardized way to integrate can produce misunderstandings of how components affect each other, and prevent scientists from identifying unexpected results in their own data.