Modelling cell division and endoreduplication in tomato fruit pericarp.

In many developing plant tissues and organs, differentiating cells switch from the classical cell cycle to an alternative partial cycle. This partial cycle bypasses mitosis and allows for multiple rounds of genome duplication without cell division, giving rise to cells with high ploidy numbers. This partial cycle is referred to as endoreduplication. Cell division and endoreduplication are important processes for biomass allocation and yield in tomato. Quantitative trait loci for tomato fruit size or weight are frequently associated with variations in the pericarp cell number, and due to the tight connection between endoreduplication and cell expansion and the prevalence of polyploidy in storage tissues, a functional correlation between nuclear ploidy number and cell growth has also been implicated (karyoplasmic ratio theory). In this paper, we assess the applicability of putative mechanisms for the onset of endoreduplication in tomato pericarp cells via development of a mathematical model for the cell cycle gene regulatory network. We focus on targets for regulation of the transition to endoreduplication by the phytohormone auxin, which is known to play a vital role in the onset of cell expansion and differentiation in developing tomato fruit. We show that several putative mechanisms are capable of inducing the onset of endoreduplication. This redundancy in explanatory mechanisms is explained by analysing system behaviour as a function of their combined action. Namely, when all these routes to endoreduplication are used in a combined fashion, robustness of the regulation of the transition to endoreduplication is greatly improved.

Identifying optimal models to represent biochemical systems.

Biochemical systems involving a high number of components with intricate interactions often lead to complex models containing a large number of parameters. Although a large model could describe in detail the mechanisms that underlie the system, its very large size may hinder us in understanding the key elements of the system. Also in terms of parameter identification, large models are often problematic. Therefore, a reduced model may be preferred to represent the system. Yet, in order to efficaciously replace the large model, the reduced model should have the same ability as the large model to produce reliable predictions for a broad set of testable experimental conditions. We present a novel method to extract an "optimal" reduced model from a large model to represent biochemical systems by combining a reduction method and a model discrimination method. The former assures that the reduced model contains only those components that are important to produce the dynamics observed in given experiments, whereas the latter ensures that the reduced model gives a good prediction for any feasible experimental conditions that are relevant to answer questions at hand. These two techniques are applied iteratively. The method reveals the biological core of a model mathematically, indicating the processes that are likely to be responsible for certain behavior. We demonstrate the algorithm on two realistic model examples. We show that in both cases the core is substantially smaller than the full model.

Exploitation of chemical signaling by parasitoids: impact on host population dynamics.

Chemical information mediates species interactions in a wide range of organisms. Yet, the effect of chemical information on population dynamics is rarely addressed. We designed a spatio-temporal parasitoid--host model to investigate the population dynamics when both the insect host and the parasitic wasp that attacks it can respond to chemical information. The host species, Drosophila melanogaster, uses food odors and aggregation pheromone to find a suitable resource for reproduction. The larval parasitoid, Leptopilina heterotoma, uses these same odors to find its hosts. We show that when parasitoids can respond to food odors, this negatively affects fruit fly population growth. However, extra parasitoid responsiveness to aggregation pheromone does not affect fruit fly population growth. Our results indicate that the use of the aggregation pheromone by D. melanogaster does not lead to an increased risk of parasitism. Moreover, the use of aggregation pheromone by the host enhances its population growth and enables it to persist at higher parasitoid densities.

Complexity reduction preserving dynamical behavior of biochemical networks.

The complexity of biochemical systems, stemming from both the large number of components and the intricate interactions between these components, may hinder us in understanding the behavior of these systems. Therefore, effective methods are required to capture their key components and interactions. Here, we present a novel and efficient reduction method to simplify mathematical models of biochemical systems. Our method is based on the exploration of the so-called admissible region, that is the set of parameters for which the mathematical model yields some required output. From the shape of the admissible region, parameters that are really required in generating the output of the system can be identified and hence retained in the model, whereas the rest is removed. To describe the idea, first the admissible region of a very small artificial network with only three nodes and three parameters is determined. Despite its simplicity, this network reveals all the basic ingredients of our reduction method. The method is then applied to an epidermal growth factor receptor (EGFR) network model. It turns out that only about 34% of the network components are required to yield the correct response to the epidermal growth factor (EGF) that was measured in the experiments, whereas the rest could be considered as redundant for this purpose. Furthermore, it is shown that parameter sensitivity on its own is not a reliable tool for model reduction, because highly sensitive parameters are not always retained, whereas slightly sensitive parameters are not always removable.

Continuous-time modeling of cell fate determination in Arabidopsis flowers.

BACKGROUND: The genetic control of floral organ specification is currently being investigated by various approaches, both experimentally and through modeling. Models and simulations have mostly involved boolean or related methods, and so far a quantitative, continuous-time approach has not been explored. RESULTS: We propose an ordinary differential equation (ODE) model that describes the gene expression dynamics of a gene regulatory network that controls floral organ formation in the model plant Arabidopsis thaliana. In this model, the dimerization of MADS-box transcription factors is incorporated explicitly. The unknown parameters are estimated from (known) experimental expression data. The model is validated by simulation studies of known mutant plants. CONCLUSIONS: The proposed model gives realistic predictions with respect to independent mutation data. A simulation study is carried out to predict the effects of a new type of mutation that has so far not been made in Arabidopsis, but that could be used as a severe test of the validity of the model. According to our predictions, the role of dimers is surprisingly important. Moreover, the functional loss of any dimer leads to one or more phenotypic alterations.

Efficient estimation of the robustness region of biological models with oscillatory behavior.

Robustness is an essential feature of biological systems, and any mathematical model that describes such a system should reflect this feature. Especially, persistence of oscillatory behavior is an important issue. A benchmark model for this phenomenon is the Laub-Loomis model, a nonlinear model for cAMP oscillations in Dictyostelium discoideum. This model captures the most important features of biomolecular networks oscillating at constant frequencies. Nevertheless, the robustness of its oscillatory behavior is not yet fully understood. Given a system that exhibits oscillating behavior for some set of parameters, the central question of robustness is how far the parameters may be changed, such that the qualitative behavior does not change. The determination of such a "robustness region" in parameter space is an intricate task. If the number of parameters is high, it may be also time consuming. In the literature, several methods are proposed that partially tackle this problem. For example, some methods only detect particular bifurcations, or only find a relatively small box-shaped estimate for an irregularly shaped robustness region. Here, we present an approach that is much more general, and is especially designed to be efficient for systems with a large number of parameters. As an illustration, we apply the method first to a well understood low-dimensional system, the Rosenzweig-MacArthur model. This is a predator-prey model featuring satiation of the predator. It has only two parameters and its bifurcation diagram is available in the literature. We find a good agreement with the existing knowledge about this model. When we apply the new method to the high dimensional Laub-Loomis model, we obtain a much larger robustness region than reported earlier in the literature. This clearly demonstrates the power of our method. From the results, we conclude that the biological system underlying is much more robust than was realized until now.

Odor-mediated aggregation enhances the colonization ability of Drosophila melanogaster.

Animal aggregation is a general phenomenon in ecological systems. Aggregations are generally considered as an evolutionary advantageous state in which members derive the benefits of mate choice and protection against natural enemies, balanced by the costs of limiting resources and intraspecific competition. Many insects use chemical information to find conspecifics and to form aggregations. In this study, we describe a spatio-temporal simulation model designed to explore and quantify the effects of the strength of chemical attraction, on the colonization ability of a fruit fly (Drosophila melanogaster) population. We found that the use of infochemicals is crucial for colonizing an area. Fruit flies subject to an Allee effect that are unable to respond to chemical information could not successfully colonize the area and went extinct within four generations. This was mainly caused by very high mortality due to the Allee effect. Even when the Allee effect did not play a role, the random dispersing population had more difficulties in colonizing the area and is doomed to extinction in the long run. When fruit flies had the ability to respond to chemical information, they successfully colonized the orchard. This happened faster, for stronger attraction to chemical information. In addition, more fruit flies were able to find the resources and the settlement on the resources was much higher. This resulted in a reduced mortality due to the Allee effect for fruit flies able to respond to chemical information. Odor-mediated aggregation thus enhances the colonization ability of D. melanogaster. Even a weak attraction to chemical information paved the way to successfully colonize the orchard.

The effect of chemical information on the spatial distribution of fruit flies: II Parameterization, calibration, and sensitivity.

In a companion paper (Lof et al., in Bull. Math. Biol., 2008), we describe a spatio-temporal model for insect behavior. This model includes chemical information for finding resources and conspecifics. As a model species, we used Drosophila melanogaster, because its behavior is documented comparatively well. We divide a population of Drosophila into three states: moving, searching, and settled. Our model describes the number of flies in each state, together with the concentrations of food odor and aggregation pheromone, in time and in two spatial dimensions. Thus, the model consists of 5 spatio-temporal dependent variables, together with their constituting relations. Although we tried to use the simplest submodels for the separate variables, the parameterization of the spatial model turned out to be quite difficult, even for this well-studied species. In the first part of this paper, we discuss the relevant results from the literature, and their possible implications for the parameterization of our model. Here, we focus on three essential aspects of modeling insect behavior. First, there is the fundamental discrepancy between the (lumped) measured behavioral properties (i.e., fruit fly displacements) and the (detailed) properties of the underlying mechanisms (i.e., dispersivity, sensory perception, and state transition) that are adopted as explanation. Detailed quantitative studies on insect behavior when reacting to infochemicals are scarce. Some information on dispersal can be used, but quantitative data on the transition between the three states could not be found. Second, a dose-response relation as used in human perception research is not available for the response of the insects to infochemicals; the behavioral response relations are known mostly in a qualitative manner, and the quantitative information that is available does not depend on infochemical concentration. We show how a commonly used Michaelis-Menten type dose-response relation (incorporating a saturation effect) can be adapted to the use of two different but interrelated stimuli (food odors and aggregation pheromone). Although we use all available information for its parameterization, this model is still overparameterized. Third, the spatio-temporal dispersion of infochemicals is hard to model: Modeling turbulent dispersal on a length scale of 10 m is notoriously difficult. Moreover, we have to reduce this inherently three-dimensional physical process to two dimensions in order to fit in the two-dimensional model for the insects. We investigate the consequences of this dimension reduction, and we demonstrate that it seriously affects the parameterization of the model for the infochemicals. In the second part of this paper, we present the results of a sensitivity analysis. This sensitivity analysis can be used in two manners: firstly, it tells us how general the simulation results are if variations in the parameters are allowed, and secondly, we can use it to infer which parameters need more precise quantification than is available now. It turns out that the short term outcome of our model is most sensitive to the food odor production rate and the fruit fly dispersivity. For the other parameters, the model is quite robust. The dependence of the model outcome with respect to the qualitative model choices cannot be investigated with a parameter sensitivity analysis. We conclude by suggesting some experimental setups that may contribute to answering this question.

The effect of chemical information on the spatial distribution of fruit flies: I Model results.

Animal aggregation is a general phenomenon in ecological systems. Aggregations are generally considered as an evolutionary advantageous state in which members derive the benefits of protection and mate choice, balanced by the costs of limiting resources and competition. In insects, chemical information conveyance plays an important role in finding conspecifics and forming aggregations. In this study, we describe a spatio-temporal simulation model designed to explore and quantify the effects of these infochemicals, i.e., food odors and an aggregation pheromone, on the spatial distribution of a fruit fly (Drosophila melanogaster) population, where the lower and upper limit of local population size are controlled by an Allee effect and competition. We found that during the spatial expansion and strong growth of the population, the use of infochemicals had a positive effect on population size. The positive effects of reduced mortality at low population numbers outweighed the negative effects of increased mortality due to competition. At low resource densities, attraction toward infochemicals also had a positive effect on population size during recolonization of an area after a local population crash, by decreasing the mortality due to the Allee effect. However, when the whole area was colonized and the population was large, the negative effects of competition on population size were larger than the positive effects of the reduction in mortality due to the Allee effect. The use of infochemicals thus has mainly positive effects on population size and population persistence when the population is small and during the colonization of an area.