Current research on DEB theory for metabolic organisation

Bas Kooijman

Marseille, 2007/01/17; Lisboa, 2007/02/07

DEB theory for metabolic organisation * and its applications is becoming increasingly popular, with the consequence that I no longer oversee in detail who is doing what. Therefore, I restrict myself to research in which our department of Theoretical Biology of the Vrije Universiteit in Amsterdam is involved in one way or another, while emphasizing that this work was only possible due to contributions of many others worldwide.

After a description of the historical roots, this text describes how the research program unfolded and matured into a program on the metabolic organisation of life in the scale-spectrum from molecules to system earth. Some concrete intriguing open DEB problems are listed separately.

Historical roots

The study of evolutionary perspectives * * * * of metabolic organisation in the last 5 years has been very productive for the development of DEB theory. Inspired by this observation, I first briefly sketch the historical perspectives of DEB research since its start in august 1979. These historical roots help to judge the present situation.

Two questions. It all started with my attempt to find answers to two given simple concrete questions:

  1. How should we quantify the effects of chemical compounds on the reproduction of daphnids as revealed from standardized bioassays?
  2. What harm does a small reduction of this reproduction by toxic stress do for the environment?
The questions turned out to be related, and much more fundamental and much less simple than they seem to be at first sight. The answer to the first question has led to the insight that reproduction cannot be studied in itself; it is intimately linked to the ontogeny of the full energy budget, which can be affected by chemical compounds in different direct and indirect ways involving several time scales. The answer to the second question has led to the notion that it concerns ecosystem dynamics and relates to population dynamics and so to properties of individuals. When DEB research started in 1979, existing models for population dynamics did not relate to properties of individuals. They treated all individuals of a species identically. Yet chemical compounds, just like evolution at another time scale, change the properties of individuals, and consequently the properties of populations and ecosystems. Differences between individuals (of the same species) are key to evolution as well as to the (complex) links between levels or organisation and are fundamental to biology.

First principles. Since the two questions had an environmental risk assessment context, the ability to make predictions for potential effects of toxicants was essential for situations that have not been studied experimentally. This means that the standard empirical methods were useless, since they do not allow such extrapolations beyond the range of observed conditions. This constrained the quest for answers to methods that were based on first principles only.

Generalisations. It turned out that knowledge about energy budgets of individuals is essential to answer both questions, and that the way chemical compounds affect the budgets of individuals is vital for the evaluation of the consequences at higher levels of organisation. It appeared that many aspects of these energy budgets are not species-specific at all and apply to all species (micro-organisms, plants, animals). It also turned out that chemical compounds are just one of many factors (temperature, parasites) that modify properties of individuals * , and that there exists a unifying way to deal with such perturbations. Dynamic systems theory provided the proper context to capture how individuals cycle their way through life by changes in state variables, and how factors modify this by changes of individual-specific parameter values.

Extrapolations. The problem of specifying the model structure seems to be totally different from that of determining the values of its parameters. However, one of the most remarkable properties of the DEB theory is that the model structure has far-reaching implications for the co-variation of parameter values among species, especially if they belong to different species * . Similar rules also appeared to hold for the one-compartment model for toxico-kinetics and its modifications * * . These powerful rules are very useful for extrapolations of model predictions from one species of organism to another and from one chemical compound to another. The DEB theory may be unique in this respect.

The DEB program unfolds

Since its historical roots, DEB research broadened its scope and extended to the sub-organismic (organs, cells, organelles, molecules) as well as supra-organismic (populations, ecosystems, system earth) levels of organisation. This broadening came with a shift in emphasis to links between the metabolic organisation at the molecular and system earth scales * This shift was quite natural given its fundamental setup and its consistency, even within the context to the two original questions.

Levels of organisation. By increasing the range of time-space scales that the theory covers, it became evident that each process has its characteristic scale at which it is most effective. In combination with the strategy to keep the models at the various time-space scales simple and consistent, a new research emphasis emerged of how to phase out particular processes and include other processes when walking up and down along the time-space scales. At both ends of the scale-spectrum hardly anything of the original model for individuals is visible, while it still plays a role in the background since individuals are the unit of selection in evolution and the survival machines of life.

Mathematical methodology. This quantitative research program required new mathematical and philosophical methodologies, and quite a few have been developed since its start, while many are still under development. Although the practical need for such methods has been the motivation to initiate their development, once brought into the beautiful abstract world of mathematics, they start an independent life in their own right and might get very different applications in the future.

Current research at the individual level

Some of the work at sub- and supra-organismic levels required a more detailed and extended treatment of the individual as a dynamic system. A lot of attention has been given to create a formal structure for the theory that allows to "click in" particular modules when necessary, but to refrain from this if possible. Each application comes with its one requirements for particular details and complex models hardly contribute to gaining insight, while this insight is the aim of research of this type. We don't want to have a lot of extra modules simultaneously. This strategy, however, poses constraints on the structure of the theory and puts further emphasis on its internal consistency. Another important research strategy is to develop the theory in close interaction with practical applications. Such applications are not only essential for testing the theory, but also for providing inspiration to structure the application-specific modules and the interface with the core theory. At the same time we use the theory to solve practical problems. The value of any theory is in its applications.

Standard model. The simple situation of an isomorph with three life stages (embryo, juvenile, adult), one type of reserve, one type of structure that feeds on one type of food/resource is called the standard DEB model. Although its structure is now well established, recent conceptual progress has been made in finding a mechanism for the reserve dynamics * , which drives metabolism. We also found a simpler derivation of these kinetics and we can now obtain it as a consequence of the weak homeostasis assumption * . A formal reconstruction of the theory is now developed * , which will facilitate future extensions footed on a solid basis and improve the accessibility of the theory by people with other backgrounds. Last but but least, we developed a new non-destructive method to determine the entropy of living biomass *, , i.e. reserve and structure. This will enable us to test popular but wild claims as "life tends to maximise entropy production".

Auxiliary theory. Apart from progress in the core theory, we made progress * in auxiliary theory that links quantities that are easy to measure to DEB parameters. Most of the quantities that are easy to measure, such as body weight and respiration rates, have contributions from several processes, and are, therefore, not variables in explanatory models, such as the DEB model. Its variables and parameters, however, cannot be measured directly, which generates the need for auxiliary theory. Testing all possible aspects of the DEB theory to experimental data has always been a core topic. Since many popular empirical models turn out to be special cases of the DEB theory for particular situations, DEB theory has a substantial empirical support. However, the need for continued testing can never be completely satisfied.

Extended models for individuals. Meanwhile many extensions of the standard model have been developed, and some of them have matured as well. This includes

Current research at the sub-individual levels

Many extensions of the theory at the individual level make already excursions to the sub-individual levels; they primarily rest on generalisations of the kappa-rule for allocation of utilized reserves to the various end points, and on the concept of synthesizing units, which boils down to an extension of classic enzyme kinetics to complex chemical transformations that deal with fluxes, rather than concentrations of metabolites. This generalisation is meant to deal with spatially heterogeneous environments, such as the cell, and with allocation of substrates to specified endpoints. Classic enzyme kinetics can hardly deal with such concepts.

Classification. The newly developed theory for synthesizing units * classifies chemical compounds as substitutable or complementary, and transformations as sequential or parallel. Mixtures of these four classes turn out to be required to understand evolutionary aspects of symbiogenesis * and other complex transformations, such as co-metabolism in the biodegradation of mixtures of organic compounds * * .

Inhibition. By allowing binding probabilities to depend on what is already bound to synthesizing units, or replacing bounded substrates by newly arriving substrates, inhibition phenomena can be captured for application in adaptation and co-metabolism. These phenomena are of importance for e.g. remineralisation, the degradation of toxic compounds and the competition between substrates to partake in a transformation (think for instance of the use of reserve and structure for maintenance purposes * * ).

Handshaking. Within the context of DEB theory it does not make sense to model cellular metabolism directly in terms of interactions of all chemical compounds within the cell for the simple reason that that are too many of them. The strategy is to insert more layers of organisation first by delineating a limited number of metabolic modules (such as the tricarboxylic acid (TCA) cycle, or the glycolysis), with limited interactions. What happens within these modules is then linked to the communication between the modules via handshaking protocols. It turned out that a simple handshaking protocol for the communication of the enzymes of the TCA cycle is sufficient to cover exactly the varying needs of the cell for building-blocks-providing intermediary metabolites and energy-providing ATP * .

Animal behaviour. From an abstract point of view, enzyme-mediated transformation rates of chemical compounds are dominated by the time budget of enzyme molecules and their ability to parallelize several types of behaviour, such as binding various substrates simultaneously. This problem has abstract similarities with modelling the behaviour of organisms, which offers the possibility to use the theory of synthesizing units for animal behaviour, and study e.g. how social interactions affect feeding * . It is no coincidence that the Michaelis-Menten expression for simple enzyme-mediated transformations is identical to the Holling type II functional response for animal feeding. A richer behavioural repertoire modifies these simple expressions. Several of these extensions have been developed, and many more become available in the near future.

Stochasticity. Behaviour is an important source of stochasticity of biological systems, which combines nicely with the setup of theory for synthesizing units in terms of stochastic processes * . This is especially important for small numbers * . The density of both molecules of any particular chemical species in a cell, and individuals of any particular type (species, age, gender) is typically low, which calls for the inclusion of stochasticity. We only started to explore this line of reasoning. It typically requires a complete reformulation of existing stochastic models, because they are inconsistent with DEB theory, and e.g. ignore energy and mass conservation, or dependences between difference processes that complicate model properties.

Current research at the supra-individual levels

Many extensions of the theory at the individual level make already excursions to the supra-individual levels because individuals in field situations interact with others, e.g. via the setting of resource levels. Since population dynamics is key to the original two questions, a lot of work has been done in the field of DEB-structured population dynamics * * , food webs * * * * * * * and simple ecosystems * * * * * . Nutrient recycling * , stoichiometric constraints, * * * * * * * * syntrophic interactions * * , spatial structure * , transport, * * * behaviour, bio-diversity * are keywords for the present developments. Some of these directions represent additions of more "details", others do the opposite and bring in more chemistry and physics. The link in all these developments, and the reason why they are part of the DEB research program, is the explicit attempt to be fully consistent with the DEB theory, which poses constraints on the model structure; it must be possible to take modules out such that the system reduces to previously studied simpler situations to evaluate the effect of that module. This is necessary to learn whether or not we need such modules in particular situations; we want to reduce model complexity to the absolute minimum, without making it too simple for any particular application.

System earth. The ultimate goal of the DEB research program is to link phenomena at system earth level to that at the molecular level via a long chain of intermediate levels of organisation. Climate change provides some motivation to study this level of organisation, but it is also of fundamental interest in itself. Since the dynamics of green-house gasses relate to the water and carbon cycles * * , and these cycles are linked to the various nutrient cycles, this calls for the development of an integrated biogeochemical climate modelling, where transport processes play an important role. The role of calcification * * , the rock cycle and continental drift in nutrient recycling brings in geological and biological components in climate models at an evolutionary time scale. Although the DEB model of individuals plays no direct role at this level, it still does do indirectly by specifying the activity of organisms in a varying environment, and in the coupling of the various cycles. Moreover, quite a few of its basic principles, such as mass and energy conservation and surface area-volume interactions equally apply at system earth level. This makes it interesting to study how the DEB theory can contribute to this level of organisation.

Evolution. The increase in spatial scale to the system earth level comes with an increase of time scale to that at which evolutionary change is operative. Adaptive dynamics provides a useful context to study speciation and self-organisation at the ecosystem level * * . The art of science here is to simplify DEB-based model formulations enough and/or to develop the methodology of adaptive dynamics enough to link these complementary two lines of reasoning. Spatial structure turned out to play a crucial role here * * * not by providing barriers in genetic exchange, however, but by creating differences in local environmental conditions. Many traditionally studied phenomena, such at the optimisation of life history traits * * * , the evolutionary increase of body size in animals, and the link between body size and geographical distribution * ask for a planetary context. I expect further progress in the contribution of DEB theory in this field.


Many applications of DEB theory have stimulated its development in the past. Since they are as different as the optimisation of sewage treatment * * * and cancer research * * , many persons have difficulty to recognize what these applications have in common, especially because abstract ideas about metabolic organisation are not yet widely spread. I here mention three active fields in more detail.

Ecotoxicology. Applications in ecotoxicity remained on the agenda since the start of the DEB program; they all rest on the idea that effects relate to internal concentrations of compounds (so we have to consider availability, toxico-kinetic and transformation modules * * ), and the DEB model specifies the parameters that can potentially be affected by the compound * . The analysis of results of standardised bioassays is a basic result * * * * , but where the applications really become powerful is in the evaluation of effects when concentrations of toxicants vary in time * * , when compounds interact with the physiology of the organism * * , when we have mixtures of compounds * or want to make the step from effects on individuals to that on populations * * * or other types of extrapolation * . We started to study effects at the ecosystem level * . The availability of the required experimental data is here a limiting factor in the further development.

Fisheries/aquaculture. A very nice recent development in the application of DEB theory is by a group of people that is active in fisheries * * * * * and aquaculture: Aquadeb. Their primary focus is on the eco-physiology of fish and molluscs, both in aquaculture and in natural environments. This includes fundamental as well as ecotoxicological aspects. The group produced a special issue 2 of J. Sea Res. 56 (2006) on DEB applications, and plans for other special issue in 2009. They made remarkable progress in testing the applicability of DEB theory, estimate parameter values, evaluate factors that modify performance in field (e.g. seasonal changes), work out species-specific rules for the handling of the reproduction buffer (which is linked to seasonal effects on body composition). The systematic build-up of parameter estimates for a variety of species will stimulate further theory on parameter values. E.g. do species that live in fresh-water have higher maintenance costs than those in seawater, because of the osmotic work they have to do?

Economics/sustainable developments. Economic systems share many features with biological systems, and quite a few DEB concepts turned out to be applicable in this context * . The similarities includes the interaction between the various levels of organisation, from firms and households, to national and supra-national economies. This development potentially leads to mechanistic models for economic phenomena. Agricultural applications, and optimisation of bio-production include more direct application of DEB theory, which links up to some applications in aquaculture. From a theoretical perspective it is nice to see how problems of optimizing bio-production have many similarities with that of minimizing sludge-production in sewage treatment plants, for instance.

Future developments

Acceptance of DEB theory has been slow in the past for reasons that are quite understandable. The combination of advanced biology and mathematics is confusing for specialists at both ends. A bewildering number of interacting factors hamper a systematic approach; DEB theory is built on a list of coherent and consistent assumptions, rather than just a single crucial idea. The popularity of extreme forms of specialisation in scientific research, the long learning curve, and the sometimes unusual interpretation of well-known facts certainly contributed to the acceptance being slow. Yet, the scientific results are frequently remarkable, and DEB theory continues where many less fundamental approaches have to give up. It is difficult to predict the rate of further development, but I am quite confident that the results of the theory will motivate an increasing number of people to invest in it. After 28 years of systematic work in the context of DEB theory, I can tell that it always remained stimulating and exiting, and scientific problems that asked for solutions kept changing, almost on a daily basis. Being not very patient by nature, I would not have believed it myself, three decades ago.

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