Mathematical models in biology, value and limits are linked to the impossibility of a reductionist method; the importance of a multilevel approach in the study of biological phenomena. The use of mathematical models in biology involves several conditions and limitations. A first limitation is the need to simplify mathematical models a lot. Otherwise, the complexity of calculation would make their use unnecessary; the second consists of the fact that the models describe only a part of the properties of the biological system.
Often it is necessary to limit you to "empirical models" ad hoc relationships that describe the behavior of the system. In other words, a reductionist attitude is not possible; the biological properties that are found at a certain level of complexity of a system are not simply deducible from those that govern the system at lower levels. A multilevel approach is therefore needed. In particular, it is useful to study the biological system as a whole without breaking it down into pieces.
In biological research, the use of mathematical models is now widespread. Therefore, when it comes to approaches of this type, it is first of all necessary to clarify what the intentions of the experimenters are in making use of these tools. As a premise, it is good to specify that mathematical models applied to biological systems can be empirical or mechanistic. The empirical ones are generally based on simple observations and describe the system without a real understanding of the laws that govern it; the opposite is true for mechanistic ones, which are based (or presume to be based) on actual knowledge of the laws that govern the phenomena investigated.
It is clear that mechanistic models are the ones that most interest, and in some sense attract, experimenters both because they should allow more faithful predictions of the system's behavior and because their success is, in turn, diagnostic of a real understanding of its properties. In fact, the intent of each modeling is to understand the properties of the system as thoroughly as possible and ultimately, if possible, to define the essential aspects of the life phenomenon.
We first introduce a brief description of the method and the purposes with which the mathematical tool is used for the development of a mechanistic model) we start from a collection of experimental data relating to an object (such as a protein or a cellular system) and starting they develop models, i.e., equations that represent the chemical-physical laws that govern the system) measurements are subsequently carried out on the system. In particular, it should be able to predict the response of the system to environmental variations, starting from its elementary properties.
So the method consists of an experimental part (data collection) and a theoretical part (reworking to predict emerging properties). There is no doubt that in some cases, this approach has worked (and when it works, it is always a happy event). But as a rule, successes are usually partial.
In principle, the application of a model to a given system can lead to a better understanding of the same in two ways) verifying whether the mathematical model correctly preaches, or not, the behavior of the system (and in both cases the understanding of the system is enriched) or also obtaining the quantitative parameters inherent in the model, which are therefore descriptive of the properties of the system (such as, for example, the speed constants in the case of chemical kinetic models).
It is also appropriate to specify that, especially if applied to biological systems, the models have virtually two main limitations in each case, which are at methodologically distinct levels almost always they introduce more or less substantial simplifications with respect to the properties of the system they should represent. This is required either by the impossibility of developing equations that faithfully describe the behavior of the system or by the complexity of the developed equations, which makes them unusable on a practical level. No less important, the models always describe a part of all system properties. Typical in this regard is the fact that there are usually distinct, descriptive systems for treating the functional and structural properties of protein molecules, and as a matter of fact, complete integration of the two systems is clearly far from being feasible.
The examples we have produced show very clearly that in biological systems, modeling encounters considerable difficulties already at relatively limited levels of complexity. But the question we can ask ourselves, on the basis of this initial reflection, is whether all the problems related to modeling are attributable to an insufficiency of the mathematical tool. Many examples suggest that the problem is more complex instead.
We can elaborate this concept by examining proteins, biological macromolecules of primary importance (together with nucleic acids), which support virtually every biological function.
In terms of the chemical constitution, these are linear polymers deriving from the condensation of a repertoire of 20 amino acids.
At the time of the synthesis, they are filiform, and even before the synthesis are completed, or immediately afterward, they curl up reaching a very precise three-dimensional structure, called the native, that is the only one associated with the function that the protein has to perform naturally. Today we know with certainty that the information that leads to this three-dimensional structure is contained entirely in the amino acid sequence (therefore in the protein itself).
This theoretical framework of reference is well established and already raised two fundamental questions at the start of the investigations on proteins (around the fifties-sixties of the twentieth century) how can we predict the structure from the sequence?
Since the structure that the molecule actually assumes is the most thermodynamically stable (i.e., with a minimum of energy), one of the practical ways in principle to predict the structure from the sequence is computational. For example, starting from the largest conformation of the protein, increasingly stable conformations are calculated that form from the extended one. In practice, the protein folding process is simulated in silico (i.e., with the computer), that is, the curling of the protein that takes place within the cells to give increasingly stable and compact conformations.
In fact, the computing power that is required for this operation requires clusters of several powerful computers that work several days in order to reproduce a phenomenon that occurs on the time scale of seconds or fractions of a second. Despite the grandeur of the means required, these bioinformatics methods have a good chance of success only with proteins whose size does not exceed 100 amino acids or slightly more (keep in mind that a medium-sized protein typically consists of 300-400 amino acids). Therefore in most cases, this methodology is not applicable. For medium-sized proteins, instead, methods that are based on the comparison of the structure to be determined with those of proteins with structures already determined by structural investigation methods must be used.
In any case, whatever the way a protein structure is predicted computationally, it is essential to subject it to a subsequent validation. It is necessary to obtain experimental evidence that confirms the correctness of the prediction. For example, if it is discovered that a given amino acid has an important role in the stability of the molecule, using molecular biology methods, it will be necessary to replace it with another that for its properties cannot have the same role, and evaluate whether the protein has lost stability at after the replacement.