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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.

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.

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.

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.

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”.

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