Over the three centuries that have elapsed since its invention, calculus has become the lingua franca for describing dynamic systems mathematically. Across a vast array of applications from the behavior of a suspension bridge under load to the orbit of a communications satellite, calculus has often been the intellectual foundation upon which our science and technology have advanced. It was only natural therefore that researchers in the relatively new field of biology would eagerly embrace an approach that has yielded such transformative advances in other fields. The application of calculus to the deterministic modeling of biological systems however, can be problematic for a number of different reasons.
The distinction typically drawn between biology and fields such as chemistry and physics is that biology is the study of living systems whereas the objects of interest to the chemist and the physicist are “dead” matter. This distinction is far from a clear one to say the least, especially when you consider that much of modern biological research is very much concerned with these “dead” objects from which living systems are constructed. In this light then, the term “molecular biology” could almost be considered an oxymoron.
From the perspective of the modern biologist, what demands to be understood are the emergent properties of the myriad interactions of all this “dead” matter, at that higher level of abstraction that can truly be called “biology”. Under the hood as it were, molecules diffuse, collide and react in a dance whose choreography can be described by the rules of physics and chemistry – but the living systems that are composed of these molecules, adapt, reproduce and respond to their ever changing environments in staggeringly subtle, complex and orchestrated ways, many of which we have barely even begun to understand.
The dilemma for the biologist is that the kind of deterministic models applied to such great effect in other fields, are often a very poor description of the biological system being studied, particularly when it is “biological” insights that are being sought. On the other hand, after three centuries of application across almost every conceivable domain of dynamic analysis, calculus is a tried and tested tool whose analytical power is hard to ignore.
One of the challenges confronting biologists in their attempts to model the dynamic behaviors of biological systems is having too many moving parts to describe. This problem arises from the combinatorial explosion of species and states that confounds the mathematical biologist’s attempts to use differential equations to model anything but the simplest of cell signaling or metabolic networks. The two equally unappealing options available under these circumstances are to build a very descriptive model of one tiny corner of the system being studied, or to build a very low resolution model of the larger system, replete with simplifying assumptions, approximations and even wholesale omissions. I have previously characterized this situation as an Uncertainty Principle for Traditional Mathematical Approaches to Biological Modeling in which you are able to have scope or resolution, but not both at the same time.
There is a potentially even greater danger in this scenario that poses a threat to the scientific integrity of the modeling study itself, since decisions about what to simplify, aggregate or omit from the model to make its assembly and execution tractable, require a set of a priori hypotheses about what are and what are not, the important features of the system i.e. it requires a decision about what subset of components of the model will determine its overall behavior before the model has ever been run.
Interestingly, there is also a potential upside to this rather glum scenario. The behavior of a model with such omissions and/or simplifications may fail even to approximate our observations in the laboratory. This could be evidence that the features of the model we omitted or simplified might actually be far more important to its behavior than we initially suspected. Of course, they might not be and the disconnect between the model and the observations may be entirely due to other flaws in the model, in its underlying assumptions,or even in the observations themselves. The point worth noting here however, and one that many researchers with minimal exposure to modeling often fail to appreciate, is that even incomplete or “wrong” models can be very useful and illuminating.
The role of stochasticity and noise in the function of biological systems is an area that is only just starting to be widely recognized and explored, and it is an aspect of biology that is not captured using the kind of modeling approaches based upon the bulk properties of the system, that we are discussing here. A certain degree of noise is a characteristic of almost any complex system and there is a tendency to think of it only in terms of its nuisance value – like the kind of unwanted noise that reduces the fidelity of an audio reproduction or a cellphone conversation. There is however evidence that biological noise might even be useful to organisms under certain circumstances – even something that can be exploited to confer an evolutionary advantage.
The low copy number of certain genes for example, will yield noisy expression patterns in which the fluctuations in the rates of gene expression are significant relative to the overall gene expression “signal”. Certain microorganisms under conditions of biological stress, can exploit the stochasticity inherent in the expression of stress-related genes with low copy numbers, to essentially subdivide into smaller, micro-populations differentiated by their stress responses. This is a form of hedging strategy that spreads and thereby mitigates risk, not unlike the kind of strategy used in the world of finance to reduce the risk of major losses in an investment portfolio.
In spite of all these shortcomings, I certainly don’t want to leave anybody with the impression that deterministic modeling is a poor approach, or that it has no place in computational biology. In the right context, it is an extremely powerful and useful approach to understanding the dynamic behaviors of systems. It is however important to recognize that many of the emergent properties (like adaptation for example) of the physico-chemical systems that organisms are at a fundamental level – where on the conceptual spectrum from physics through chemistry to biology we are positioned much further towards the biology end – are often ill-suited to analysis using the deterministic modeling approach. Given the nature of this intellectual roadblock, it may well be time for computational biologists to consider looking beyond differential equations as their modeling tool of choice, and developing new approaches for biological modeling, better suited to the task at hand.
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