Figure 1: Modeling in physics

But like any large-scale construction project (like building a skyscraper or launching a rocket to the Moon), science is constructed from the “ground up”. Each piece of the structure rests on that which came before it. In this tutorial, we stress that each individual tutorials is the foundation for that which follows. And that’s the way science succeeds . . . one step at a time.

 Cycle of modeling

Biological modeling is no different that what we do with physics; the only difference is that we are used to describing physical phenomenon in mathematical terms. But modern biology – and especially the increased interdisciplinary nature of science – requires more and more mathematics. So biological modeling is not really all that different from that done with physics for the past 400 years.

Figure 2: Cycle of modeling (click the numbers to read more)

Take, for example, a simple real-world problem. The first step is to rough out an approximate biological model of the problem. What is happening here? How long does it take? What steps appear to take place as the behavior progresses? After the basic schematic of the problem is laid out, we now approach it from a mathematical or computation point-of-view. We attempt to mathematically define each of the previously-stated questions from a quantitative standpoint. Then we develop a computer simulation – using each of the mathematical rules for the model – and let it run.

We then evaluate the result and compare it to the biological model. If the model reasonably matches or predicts that model, we analyze our model with respect to the original real-world problem. And, again, if the results are reasonably accurate, we know that we are on the right track. Of course, no model is complete . . . life has too many variables (and surprises) for us to incorporate fully. Instead, we modify and merge and tweak and twist our computational model into shape, always comparing and contrasting the model to that of the real world. And if we are able to develop a predictive model that works, science progresses. If not, we take our licks, pick ourselves up again, and prepare to do battle with nature once more.

So how does the cycle of modeling – and biological modeling, in particular – work?

	Going from a Real World problem to a Biological Model The first step is to rough out an approximate biological model of the problem. What is happening here? How long does it take? What steps appear to take place as the behavior progresses? What assumptions and/or simplifications can we make in order to focus on the behavior under study?
	Going from a Biological Model to a Mathematical and Computational Model After the basic schematic of the problem is laid out, we now approach it from a mathematical or computation point-of-view. We attempt to mathematically define each of the previously-stated questions from a quantitative standpoint. Then we develop a computer simulation – using each of the mathematical rules for the model – and let it run. Analysis Now we can employ a variety of mathematical and computation tools; most notably, statistics and probability. Here’s where the use of random numbers, Monte Carlo simulations, and Markov chains play a key role in examining the model. Conclusions and Predictions What are our results? Are they valid (in a mathematical sense; that is, are they statistically significant? Can they be reproduced with little or no variation?)?
	Interpreting Mathematical Conclusions and Predictions in the Biological Context After evaluating the success (or failure) of our mathematical model, it is time to compare it to the biological model we first developed.
	Interpreting Biological Conclusions and Predictions in terms of a Real life Problem So how did we do? How does the biological model compare (and contrast) to that of the real life problem we started with? How relevant and appropriate were the original assumptions we made in order to model our real life problem? If the results are reasonably accurate, we know that we are on the right track. If not, we start the process all over again.
	Once we have analyzed the model and compared it to the real life problem, it’s time to modify and adjust and improve our model. We make new assumptions, or change one or more parameters, or possibly throw out the original model as we seek to improve our understanding of the problem.

Of course, no model is complete . . . life has too many variables (and surprises) for us to incorporate fully. Instead, we modify and merge and tweak and twist our computational model into shape, always comparing and contrasting the model to that of the real world. And if we are able to develop a predictive model that works, science progresses. If not, we take our licks, pick ourselves up again, and prepare to do battle with nature once more.

Mathematical Foundations and Biological Applications

The understanding of biological processes begins with observation. We start be collecting and recording observations. Once we have gathered enough data we try to come up with the rules governing that biological process: we hypothesize. Here is where modeling comes into play.

For instance, we may be interested in understanding how a disease develops in humans. (Of course, we cannot do experiments with humans.) We could start by studying how this disease develops in animals or in individual cells or in test tubes. These approaches are called animal modeling, in vivo modeling and in vitro modeling respectively.

Which approach is better than which? They are all good, but they are applied depending on the scenario. In general, animal modeling is used for system-wide studies, in vivo modeling for cellular level studies and in vitro for molecular scale studies. Animal modeling is ideal to study, for instance, immunological response (tissue wide) to an infection or how hormones regulate mood, body weight and sexual behavior. Modeling in vivo is ideal for studying, for example, cell movement and division, cellular localization of biomolecules and organelle rearrangements. Modeling in vitro is best to study, for example, protein folding, interactions within biomolecular complexes, and molecular mechanisms. Usually these approaches are used in combination.

All these three types of modeling share something in common: they are experimental approaches that we use to test our understanding (our hypothesis). For instance, we can propose that the presence of a certain protein is essential for progression of a particular disease. That would be our hypothesis or rule; now we need to test it. What we can do in the laboratory is deplete the protein from cells and see if this actually stops the disease. If it does, the hypothesis is ratified, if it does not, the hypothesis is rectified.

What if the experiment is too difficult or impossible to do? For instance, imagine a simple experiment that must be done in the absence of gravity; that would require going to space. At a molecular scale there are also examples: making a protein to be completely rigid or move only when we want it is impossible. What about studies of natural disasters and spread of contagious diseases? There must better alternatives than destroying levees and infecting populations for the sake of doing experiments. Mathematical modeling can help in all the examples above.

Furthermore, a very valuable feature of mathematical modeling is that it is inherently quantitative. It can potentially tell us not only what happens but how much and when. In the case of a contagious disease outbreak, we may need to predict not only whether a  whole city will be hit but also how many hospital beds will be needed and how long it will take to reach the crisis peak. Because we have the equations and we can keep track of the numbers, we essentially know it all in a mathematical simulation. In experimental approaches, we can only record the things we purposely measure and not everything is measurable.

So, when computational or numerical modeling comes into play? This approach is handy at least in too scenarios. One situation could be when the biological process is too complicated to be fully defined mathematically in a small set of equations. The second situation would be when the mathematical equations are rather simple but there are too many equations. For instance, let's imagine that we are trying to model ant colony behavior. The behavior of a single ant is perhaps easy to model but colonies have thousands of ants. Imagine modeling potential germ carriers in a city of millions.

Also in this category there is a biological process that we address in further detail in this work: microtubules. We model these dynamical biopolymers individually in relatively simple terms using tools such as random numbers, Monte Carlo and Markov chains. But we are not interested in modeling only single microtubules but also their collective properties. Cells may have several hundred microtubules and then the computational approach suites well.

In general, the choice of underlying mathematical tool is made based on the nature of the biological process and questions being asked. For example, if we are studying protein folding and we are only interested in the most stable structure, a Monte Carlo approach may be the most appropriate tool. Another example could be the study of the molecular evolution of genes. In this case, random nucleotide mutations obeying laws of a Markov process would be the best choice. The Ising and Potts models on their part have been successfully used to understand patterning and cell migration respectively and a variety of other biological puzzels.

In summary, the purpose of this work is to introduce these mathematical tools and encourage its application to understand biological problems.

Figure 2: Structure of out online tutorials

In he tutorial section of this site we present yet another example of a natural process that can be modeled mathematically: microtubule dynamics. We see in this example that this network of biological polymers can be meaningfully modeled using simple mathematical tools such as random number generators, Monte Carlo computations and Markov chains.

Future work

In these tutorials, we start with the very basic mathematics you will need to build biological models. Statistics and probability rule here (much as they do in quantum mechanics in physics). After a brief introduction to random numbers and walks, Monte Carlo techniques, and Markov chains, you will study a few of the basic modeling techniques used. From there, you will proceed into new and largely unexplored territory in mathematical biology.

In the future, you can expect tutorials on a number of applications currently under study by biologists, mathematicians, physicists, and computer scientists. These include models of termite behavior, flocking and swarming, cell aggregation, myxobacteria, and even how the leopard got his spots (or, perhaps, how the zebra got her stripes).