There is a clear connection between the level of detail in a model and the generality of the conclusions that can be drawn from it. The appropriate level of generality might be difficult to achieve, e.g. if one tries to construct a model of a human disease, then it should not be specific to a certain human being, and at the same time not as general as to capture the dynamics of any human disease. In each case there is a balance between specificity and generality and where on the spectrum a model is placed often depends on the preferences of the scientist. To highlight this point we will briefly consider an example from theoretical biology, where mathematical models are used in order to study the process of speciation, the evolutionary process that leads to the formation of two species from a single ancestral species.30 The dynamics of this process is a fundamental question in evolutionary biology. When constructing models of speciation minimal assumptions are made about the species in question, since the aim is to understand the general mechanisms driving this process.
By establishing under which conditions (selective pressures, migration rates etc.) speciation occurs within the model, it is possible to compare the results with real closely related species and see if they match the criteria derived from the model. If there is a match then the model provides a possible explanation of the process, if not then the model lacks some crucial element and needs to be revised.
Models that are aimed at more general questions tend, for obvious reasons, to become more abstract. Possibly the most striking example of this comes from the Hungarian mathematician von Neumann. He was interested in one of the most fun- damental questions in biology: What are the mechanisms and logic behind repro-
30Geritz, S.A.H., Kisdi, E., Meszéna, G. and Metz, J.A.J. (1998). Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol. 12:35–57.
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duction?31Instead of studying reproduction in the biological world he reformulated the question into: Is it possible to construct a model in which reproduction occurs?
By a model he meant a formal mathematical system, and to answer the question he formulated a completely new mathematical construct known as a cellular automaton.
A cellular automaton consists of a square lattice, like a chess board, but with the dif- ference that each square can be in a number of different states, usually represented by different colours (von Neumann used 17 states), and the state of each square changes depending on the state of the neighbouring squares. It is the states, together with the rules that govern the transition between different states, that define a cellular automata. von Neumann succeeded in finding a set of rules where a certain structure (configuration of squares in certain states) would over time give rise to an exact copy of itself, and consequently he had shown that reproduction was possible within a mathematical construct. The cellular automaton he devised was not a reflection of any existing real structure, but instead a representation of the basic phenomenon of self-replication. von Neumann’s work was the starting point for a research disci- pline called “artificial life”, which studies general properties of living systems, such as self-organisation, communication and swarming, from an abstract point of view.
Models that are being studied in this field are not models of a particular system, but of abstract properties, but they are still to be considered models.
In contrast to theories, which are often considered as rivalling if they describe the same phenomenon, there is rarely such antagonism between models that overlap.
This is because models often focus on different aspects of a system, and only if they assume a contradictory set of mechanisms do they disagree. This feature of models becomes clearer when we view models as tools: they might look different or be of different kinds (conceptual, symbolic etc.), but still achieve the same goal (describe the same phenomenon); or in other words, models are like different languages that portray reality in different ways. Because of this there is a pluralism amongst models that is not to be found amongst theories.
An example of pluralism among models is two different models that both describe the swarming behaviour of the slime moldDictyostelium discoideum. This amoeba is a single-celled organism that resorts to collective behaviour when conditions are rough, such as during a drought, and this cooperative behaviour is in many ways similar to what is seen in multi-cellular organisms. When the organisms start to dehydrate they secrete a signalling substance that diffuses in the surrounding media.
The substance triggers other cells to also produce it and at the same time move towards the direction of the signal. This gives rise to a spiral pattern of migrating cells, that in turn leads to an aggregation of cells at the centre of the spiral out of which a fruiting body is formed. This looks very much like a stalk and the cells located at the top form spores that can be carried away by the wind to new locations.
The swarming of the cells was first described by the Keller–Segel model, a partial
31von Neumann, J. (1966). The Theory of Self-reproducing Automata, A. Burks, ed., Univ. of Illinois Press, Urbana, IL.
Fig. 6 Real and simulated slime mold. Theupper panel shows aggregation of real slime mold where the fruiting body is starting to form at the centre of the colony (© Rupert Mutzel, Freie Universität Berlin).
Thelower panelshows the result of an agent-based simulation in roughly the same stage of aggregation (Palsson, E. and Othmer, G.H. (2000), A model for individual and collective cell movement in Dictyostelium discoideum, PNAS 97:10448–10453, © (2000) National Academy of Sciences, U.S.A.)
differential equation that describes how the concentration of slime mold cells (in the unit cells/mm2) changes in time and space.32
Since the model only describes the concentration of cells it can be viewed as a relatively coarse-grained description of the phenomenon, and other models that take into account the dynamics of single cells have also been formulated. In those cases the system is described with a so called agent-based model, in which each cell is modelled as a deformable visco-elastic sphere that interacts mechanically with other cells and reacts to external stimuli.33Both models have their pros and cons: the
32Keller, E.F. and Segel, L.A. (1970). Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26.
33Palsson, E. and Othmer, G.H. (2000), A model for individual and collective cell movement in Dictyostelium discoideum, PNAS 97:10448–10453.
52 Structure, Relation and Use
continuous Keller–Segel model can describe a large population of cells, and is also amenable to mathematical analysis, but does not allow for the tracking of individual cells. The agent-based model cannot handle as many cells, and is difficult to analyse, but could be utilised in order to investigate the effects of solitary mutant cells in a population of normal cells (Fig.6).
There are also cases where a single model is able to describe two or more dis- parate phenomena. For example the Keller–Segel model has been applied in order to describe the formation of galaxies. To model this process one represents the stellar material as a self-gravitating Brownian gas, a large collection of particles that move according to a random walk and also attract each other according to Newton’s law of gravity. A mathematical analysis of this situation shows that the density of stel- lar material also obeys the Keller–Segel equation.34 This example shows that there are cases of two disparate systems that are governed by analogous mechanisms and exhibit a kind of universality. A connection can be seen to the mechanical analogies used by 19th-century physicists, where the microscopic world behavedas if it was made of springs and pulleys, but in reality it was not. A healthy degree of cautiousness is therefore recommended: just because two phenomena look alike doesn’t imply that they are governed by the same mechanisms.