Words to plants

Fatima Cvrèková 

A review of the book Plants to ecosystems: advances in computational life sciences. Marek T. Michalewicz, ed., vii + 134 pp., CSIRO, Colingwood, Australia 1997, ISBN 0-643-05942-3(v.1), Folia Geobotanica 33(3):374-376, 1998.

What could a biologist expect from a mathematician interested in modelling life? An illustration of the range of possibilities that mathematics and computing provide to a botanist can be found in the presented book, which is the first of two volumes summarizing a selection of papers presented at a Symposium on Computational Challenges in Life Sciences held at  Melbourne, Australia. In editor´s words, the purpose of this symposium was to summarize „the conceptual and methodological view of plant modelling as it is currently developing“. An average biologist with a platonic attraction towards mathematical modelling would expect a balanced coverage of topics related to computer models of plants, plant communities, populations, landscapes and ecosystems including both plants and other organisms. This is, however, not the case: despite the title, the contents of this volume is mainly focused on the  „individual plant“ or „individual plant + its immediate environment“ level.

There are two exceptions among the 7 chapters. The article by D.G. Green is devoted to modelling plant populations in landscapes, especially using the methodology of cellular automata, familiar to many of us in the form of Conway´s „Life Game“. This approach is used here to build models of such phenomena as spreading of pests, of fires or of novel genotypes through multi-species plant communities. The second exception is the last chapter by J.A. Kaandorp and P.M.A. Sloot, dealing with models of growth of sessile marine animals, such as corals and sponges. While such a topic might seem exotic in a „botanical“ book, its rationale becomes obvious when we realize that the suspension-feeding marine animals also represent an example of organisms whose morphogenesis is directed by availability of nutrients delivered by diffusion. Root systems of higher plants might be viewed as a more sofisticated case of such an environment-dependent morphogenesis (and, indeed, a chapter by A.J. Diggle presents examples of modelling the topology of root systems depending on soil properties). Moreover, as soon as we realize how complex mathematics and how powerful computer technology was required to model the shaping of a „simple“ sponge, we stop wondering why the scope of the volume is limited the way it is.

The main focus of the book lies in discussion of mathematical models of individual plant development. The article by P. Prusinkiewicz, J. Hanan, M. Hammel and R. Mìch gives an inspiring introduction into the principles of construction of so-called Lindenmayer system (L-system)-based models of plant morphology and morphogenesis (or virtual plants, as the authors prefer to call them). L-system applications, integration of virtual plants into crop-level models including the environment, and building of virtual plant models from „real-life“ data are the subject of two more chapters by J.Hanan, P.M. Room, W.R. Remphrey and P. Prusinkiewicz. The remaining chapter  by C. Godin, Y. Guédon, E. Costes and Y. Caglio is also dedicated to the „interface“ between mathematical models and experimental measurements, providing an alternative to the L-system approach to reconstructing plant shapes from measured 3-dimensional coordinates of living plants. The figures in the book provide nice examples, although they are limited by plain impossibility to present dynamic, 3-dimensional models on a piece of paper, regardless of how good the quality of print is (and it is excellent!). Even the static figures are more than sufficient to tempt the reader into trying some of the cited internet links and to search for more. I would highly recommend visiting e.g. Laurens Lapre´s site  and an on-line book by P. Prusinkiewicz (available also here, in a slightly different format), where you will find, among other things, an almost startlingly life-like animated model of ash tree shoot expansion (be patient, the movie is fairly large).

What is the principle of L-system-based plant models? A summary of a hypothetical example (by and for a mathematically naive biologist) would go as follows: Assume that a plant body can be divided into a finite number of modules, structural subunits of few well-defined types (such as terminal buds, internodes with leaves and axillar buds, flowers, fruit). Imagine that the plant develops in discrete time-steps, corresponding e.g. to the plastochron. Define what happens to each type of module in each time-step: e.g. terminal buds produce internodes and new terminal buds or flowers, depending on the context, flowers produce fruit, some axillar buds give rise to internodes and terminal buds, others remain dormant - again in a context-dependent manner. Describe the starting structure of the plant and the rules for module transformation in a formal language, first developed by Aristid Lindenmeyer in the sixties (hence L-systems). The module structure translates into „parametric words“, with more-less empirical parameters, and the rules determining module fate take on the form of „productions“ or „rewriting rules“ for defining the context and replacing ancestor modules with their descendants according to this context. Let the model „develop“, i.e. compute the module structure for many discrete time steps, and, last but not least, feed the formal descriptions of  the results from each step into a program that produces a graphical representation. As soon as the mathematical description is formulated, the parameters (and numerical constants contained in the rewriting rules) are by no means restricted to empirically derived values. Exploration of the space of available parameter values reveals that the same model with different sets of constants can lead to structures reminiscent of a cherry tree, a plum tree or a fern leaf. Indeed, a substantial part of the book is not about modelling at the levels between plants and ecosystems, but about „creating“ virtual plants from (parametric) words. And here is an interesting point hidden, a point tightly connected with the very nature of mathematical modelling in biology.

Mathematical models have been applied for purposes as diverse as description of single enzyme reaction kinetics and for predictions of ecosystem behaviour, for interpolation of physiological parameters from measured real-life data and for generation of „data“ from virtual, computer-simulated experiments. In some cases the models, though based on observations or measured data, apparently are not much concerned with the underlying processes of life. A good example would be the classical description of population size development by a logistic curve, with numerical parameters determined empirically and lacking obvious physiological correlates. Other models have originated in the understanding and subsequent formalization of the underlying physiology. Monod´s description of population development in a bacterial culture, based on the assumption that culture growth is limited by a single rate-limiting enzyme reaction that follows the well-known standard Michaelis-Menten kinetics, is an example of this kind of models. Mathematical models in biology could be viewed as covering a wide field between these two extremes. However, the lack of obvious physiological correlates to the model´s parameters in the first type of models usually tends to be perceived as a drawback, and biologists often consider the second type of models more valuable. We apparently have an urge to seek for „real“ things behind model parameters, a tendency to convert abstract concepts into entities (S.J. Gould) or to believe that whatever received a name must be an entity or being, having an independent existence of its own (J.S. Mill). A model that resists such reification of parameters (e.g. the logistic growth curve), is viewed as less satisfactory than one that invites it, as does the Michaelis-Monod growth curve.

The L-system-based plant models present a special challenge for those who insist on reification of all parameters. We have accepted the notion that one can get a cherry tree or a fern from a single cell by succession of physiological processes governed by the interplay of gene activities; we do even accept the  possibility that the genes regulating the development of the cherry tree and the fern may not be as different as the two plants are - what makes the difference between them may well be different rules of regulation rather than point mutations in the genes. Finding the differences that make the differences is one of the greatest challenges to contemporary biology, and much of the ongoing effort in genomics and developmental biology is driven by the hope that a satisfactory description of ontogeny on the molecular biology level will provide means to identify the „switches“ that gave rise to the emergence of novel morphogenetic patterns throughout evolution. However, how should we face the finding that the morphological difference between our fern and the cherry tree may be as well (and at present much better!) described in terms of differences of formal parameters of a L-system model, whose „real-life“ correlates are far from obvious?

We could learn to live with two or more alternative and complementary views of morphogenesis, as projections of the complex, multi-dimensional reality onto two planes that never touch each other. We could describe the developmental mechanics using the tools and concepts of molecular biology and leave the richness of inter-related shapes of non-model organisms to the taxonomists and to those who will use the whole elaborate (L-system or other) modelling methodology to reach a consistent, self-confined description of natural shapes. That would require a kind of resignation on both sides - giving up the aim of explaining all aspects of morphogenesis by the approach of experimental biology, and renouncing the possibility that the mathematician´s effort might ever bring anything more than a crude description of the formal logic of shape transformations. Whatsoever this logic is, we would be discouraged to ask questions concerning its molecular, physiological, biological nature. Instead, we would be at best left with an image of two necessarily incomplete, non-overlaping wiews of the world, while the exponents of each of them could only comfort themselves by pointing out equal imperfection of their opponents´ approach. We now realize that real-life bacteria actively control their entry into the stationary phase rather than allowing the environment to manipulate them into it, and that common metabolic enzymes are hardly suited for the task of the limiting factor from Monod´s growth curve. Of course, there could be a protein sensing the concentration of some critical environmental compound, whose kinetics follows the model, but that is not what the original model said. With a bit of cynism we could therefore conclude that a „successful reification“ of a „physiological“model may be rather a succesful wording of an aethiological myth, that „physiological“, reifiable models are no better than abstract ones, and that there is therefore nothing wrong with models devoid of physiologically explainable parameters.

But there may be another possibility. There may be the right time to start asking a different kind of questions, a time to move the focus of plant modelling from the „plants to ecosystems“ to the „cells to plants“ level. What biological processes could be reflected in the model parameters? For instance, how would a change in the gradients of regulatory molecules shift the parameters of a corresponding L-system model? Could we model differences in plant tissue sensitivity to a particular phytohormone?  Could we try to set up a physiological model to derive parameters for our L-systems? I believe that we have the choice to start taking the models for what they are - crude but useful tools for description of phenomena we know little about - rather than blaming them for what they are not. If we are lucky, on our way towards sensible physiological interpretation of abstract mathematical models we may arrive to new insights into the biological nature of (plant) morphogenesis. It will not change much on the fact that there are the two views - that of physiology and that of abstract modelling. Just the emphasis will shift on the fact that these are views of one world.

It is beyond the scope of this review to solve big mysteries on the spot; we are here merely to make a few notes about a book. Although endowed with a somewhat misleading title, the volume edited by Marek T. Michalewicz represents a highly inspiring reading for a generally interested biologist with at least a passive understanding of the formal language of mathematics and the internal workings of a computer. It is clearly not for those who by default skip equations interspersed in the text. On the other hand it could provide a rich source of ideas for a mathematician or a programmer interested in a biological application of his skills and techniques. While for specialized technical information (from either the „botanical“ or the „computational“ side of the topic) one would have to refer to the abundant cited literature, the reviewed volume presents a succesful example of communicating the aims and ways of biologists to programmers and vice versa. To my opinion, this is something still rather rare and always desirable, at least if we hope to extend the usefulness of mathematical modeling to the whole spectrum of phenomena between the plants and ecosystems, and maybe even beyond.