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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.

We investigate toxin interference competition between S. cerevisiae colonies grown on a solid medium. In vivo experiments show that the outcome of this competition depends strongly on nutrient availability and cell densities. Here we present a new model for S. cerevisiae colonies, calculating the local height and composition of the colonies. The model simulates yeast colonies that show a good fit to experimental data. Simulations of colonies that start out with a homogeneous mixture of toxin producing and toxin sensitive cells can display remarkable pattern formation, depending on the initial ratio of the strains. Simulations in which the toxin producing and toxin sensitive species start at nearby positions clearly show that toxin production is advantageous.

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.

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. Electronic Supplementary Material The online version of this article (10.1007/s11538-008-9327-0) contains supplementary material, which is available to authorized users.

In this study, we consider a herbivore-predator metapopulation model, consisting of two patches. Only the herbivores are allowed to migrate between the patches. Furthermore, the intrinsic growth rates of the herbivores depend on climatic fluctuations. Our objective is the conservation of the herbivores. We are interested in finding the parameter perturbations that, when applied, will reduce the extinction risk of one of the subpopulations the most. The risk of extinction is measured in terms of the fifth percentiles of the subpopulations. This is the value below which the subpopulation is found 5 out of 100 times in a series of values taken at fixed time intervals. We make use of the short-term behavior of the model by formulating the tangent linear equations. In the neighborhood of a short interval of a reference orbit the linear error growth can then be calculated. The parameter perturbation that causes the largest error growth, the so-called first singular vector, can be computed with the use of adjoint equations. The adjoint system acts as a backward integration. It turns out that at certain moments in time such a singular vector has a direction for which a parameter perturbation is likely effective in changing the dynamics in a long model simulation. The selection is based on a specific local behavior of the error growth. This adjoint method is compared with a method in which parameter perturbations are randomly chosen. Here we carry out a test for a model with only few parameters. However, it can be applied to models with a very large number of parameters, making the adjoint method an interesting alternative for the random method as then the required number of runs cannot be realized within a realistic computing time. A same objection holds for the use of a systematic method of finding an optimum, e.g. by computing the gradient in the fifth percentile with respect to the parameter values.

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.

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.

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.

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.

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. Keywords Parameterizati

The stochastic dynamics of a chemostat with three trophic levels, substrate-bacterium-worm, is analyzed. It is assumed that the worm population is perturbed by environmental stochastic noise causing extinction in finite time. A diffusion model of the process is formulated. With singular perturbation methods applied to the corresponding Fokker-Planck equation an estimate of the expected extinction time is derived. This chemostat can be seen as an experimental sewage-treatment system in which the worm population facilitates the reduction of remaining sludge

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.

An algorithm is developed to determine the optimum sustainable yield subject to the condition that the resulting solution has a stable limit cycle. Periodic harvesting is considered and the algorithm is applied to a logistic population and a prey-predator system.

By solving asymptotically the Dirichlet problem for the backward Kolmogorov equation describing the random walk of a particle in a dispersive flow it is computed at what rate contaminated particles arrive at a well. Moreover, an estimate of the average travel time is given. In the mathematical analysis adsorption and first order decay can be included.

It is well known that the solutions of functional differential equations have jump-discontinuities in their derivatives, unless some rather restrictive compatibility conditions are imposed upon the initial function. In this paper a method for the calculation of the position and the value of these jumps is presented, and conditions that are sufficient for the solution to be smooth between the jump-discontinuities are given.

If an ordinary differential equation is solved by using a linear multistep method, then it is well-known that the asymptotic behaviour of the error is determined by the characteristic roots of the discretized differential equation. In particular the so-called essential roots (the roots on or close to the unit circle) play an important role. If a functional differential equation is solved by means of a linear multistep method, then the number of characteristic roots of the discretized equation is inversely proportional to the stepsize used. In this paper it is shown that only a fixed number of these roots are strictly inside the unit circle, and all the others are “essential”.

Let $u^ * $ be the solution of the delay differential equation \[ \begin{gathered} u'(t) = f(t,u(t),u(t - \tau ))\quad {\text{for}}\,t > 0, \hfill \\ u(t) = \varphi (t)\qquad{\text{for}} - \tau \leqq t \leqq 0. \hfill \\ \end{gathered} \]For $t_j = jh$, let $u_h (t_j )$ be the approximation for $u^ * (t_j )$ obtained by the explicit midpoint rule, initialized by one Euler step. It is shown that if f and $\varphi $ are sufficiently differentiable and ${\tau / {h \in \mathbb{N}}}$, there are functions $e_k $ such that for j even, \[u_h \left( {t_j } \right) - u^ * \left( {t_j } \right) = \sum _{k = 1}^r {h^{2k} e_k } \left( {t_j } \right) + O\left( {h^{2r + 2} } \right).\] This expansion of the global discretization error exists in spite of the fact that the solution $u^ * $ usually has jump discontinuities in its derivatives at the points $t = n\tau $, $n \in \mathbb{N}$. Thus repeated extrapolation to the limit (Richardson extrapolation) may be applied to improve the approximation. Furthermore, it is show...

A new way to define linear multistep methods for functional differential equations is presented, and some of their properties are analyzed. The asymptotic behavior of the global discretization error is investigated. Finally, Milne's device is generalized to functional differen- tial equations. The effect of the nonsmoothness of the exact solution is taken into account. 1. Introduction. Consider the functional differential equation (FDE) (1) x'(t) = F(t, xt) (to < t < T), xto = O, where x(t) E RI, 0 E C((-T,0), Rn), FE C((to,T) x C((-T,0), Rn), RI) and xt: 9 -* x(t + 9) for 0 E ( - T, 0), t E (to, T). It is well known that the solution x of (1) is usually not smooth, that is, even if F and 4 are C? functions, x may have jump discontinuities in its derivatives. The occurrence of these jump discontinuities may lead to order-breakdown for numerical methods if no special provisions are made. It seems that all available techniques require information about the location of the jumps. If the delay is not state-dependent, then these locations may be known a priori. If not, then they may be calculated numerically (see Feldstein and Neves (6)). When the location of a jump discontinuity is known, two different techniques are available: (i) Make the location of the jump a grid point, and restart. In practice, this

One of the main obstacles in systems biology is complexity, a feature that is inherent to living systems. This complexity stems both from the large number of components involved and from the intricate interactions between these components. When the system is described by a mathematical model, we frequently end up with a large nonlinear set of mathematical equation that contains many parameters. Such a large model usually has a number of undesirable properties, e.g., its dynamical behavior is hard to understand, its parameters are difficult to identify, and its simulation requires a very long computing time. In this thesis, we present several strategies that may help to overcome these problems. On the level of method development, we focus on two issues: a) method development to analyze robustness, and b) method development to reduce model complexity. On the level of practical systems biology, we develop and analyze a model for the cell cycle in tomato fruit pericarp. Robustness, that is the ability of a system to preserve biological functionality in spite of internal and external perturbations, is an essential feature of a biological system. Any mathematical model that describes this system should reflect this property. This implies the needs of a mathematical method to evaluate the robustness of mathematical models for biological processes. However, assessing robustness of a complex non-linear model that contains many parameters is not straightforward. In this thesis, we present a novel method to evaluate the robustness of mathematical models efficiently. This method enables us to find which parameter combinations in a model are responsible for its robustness. In this way, we get more insight into the underlying mechanisms that govern the robustness of the biological system. The advantage of our method is that the effort to apply the method scales linearly with the number of parameters. It is therefore very efficient when it is applied on mathematical models that contain a large number of parameters. The complexity in a model can be brought down by simplifying the model. In this thesis, we also present a novel reduction method to simplify mathematical representations of biological models. In this method, biological components and parameters that do not contribute to the observed dynamics are considered redundant and hence are removed from the model. This results in a simpler model with less components and parameters, without losing predictive capabilities for any testable experimental condition. Since the reduced model contains less parameters, parameter identification can be carried out more efficaciously. In the last part of this thesis we show how modeling can help us in understanding the cell cycle in tomato fruit pericarp. The cell cycle in this system is quite unique since the classical cell cycle, in which the cell division takes place, after some periods turns into a partial cycle where the cell keeps replicating its DNA but skips the division. Several mechanisms that are putatively responsible for this transition have been proposed. With modeling, we show that although each of these putative mechanisms could lead on its own the cell cycle to this transition, also their combination could lead to the same result. We also show that the mechanisms that yield the transition are very robust.

The study of the role of chemical information in species interactions has been mostly restricted to studies at the level of individual organisms. The central question in this thesis is how intraspecific chemical information conveyance and exploitation thereof by a natural enemy affects the spatial population dynamics of a species. To answer this question, I developed a spatio-temporal model where both host and parasitoid can respond to infochemicals. Our model system consists of the fruit fly Drosophila melanogaster, and its natural enemy, the larval parasitoid Leptopilina heterotoma. D. melanogaster uses its aggregation pheromone in combination with odours from fermenting fruits to localise suitable resources for reproduction. L. heterotoma uses these same odours to localise its host. For D. melanogaster, aggregation on a resource can be beneficial when a population is small and has to overcome negative effects associated with low population densities. Such negative effects, known as the Allee effect, can for instance be caused by difficulties in resource exploitation or in finding a mate. Aggregation also involves costs. Individuals within an aggregation frequently experience more severe competition for food, space and mates than they would experience when being on their own. Furthermore, I investigated which behavioural decisions enhance the ability to find - and distinguish between – odour sources that differ in their suitability for reproduction. On the individual level, this research showed that, like real fruit flies, the modelled fruit flies need to have a preference for the presence of both aggregation pheromone and food odours, over food odours alone, to be able to distinguish between the two types of odour sources. The results show that this stronger preference does not have to be innate. As long as fruit flies are able to remember and adjust their current preference based on the odour concentrations that they perceive, more fruit flies find the more attractive odour source. On a population level, this thesis shows that the use of chemical information by D. melanogaster affects its population dynamics. In the absence of its natural enemy, and when the Drosphila population is small, the use of food odours and aggregation pheromone has a positive effect on population growth and enhances the fruit fly’s colonization ability. When the population becomes larger, however, the negative effects of larval competition are stronger than the positive effects of reduced mortality due to the Allee effect. The use of chemical information was crucial to colonize an area from the boundaries. A fruit fly population that was unable to use chemical information could not colonize the area and went extinct. When parasitoids can use chemical information, parasitism rates are higher, resulting in a slower population growth of their host. No difference was recorded in fruit fly population size and in larval mortality due to parasitism, when parasitoids exploited the aggregation pheromone of the fruit fly adults as compared with the simulations where the parasitoids could only respond to chemicals emitted by the host habitat. In contrast, the use of chemical information by the host enhanced its population growth and enabled it to survive, even at higher parasitoid densities. This research showed that mortality when the population was small had a greater impact on population size than mortality due to competition or parsitism. Food patches are not always abundant in nature. Thus, the reproductive success of fruit flies is mainly determined by their opportunities of producing clutches (i.e. locating patches) rather than by preventing over-aggregation or parasitism. As a result, the use of chemical information has a net positive effect on fruit fly population dynamics, despite the fact that L. heterotoma is able to exploit it.