Jaap Molenaar

Redefining plant systems biology: from cell to ecosystem

Molecular biologists typically restrict systems biology to cellular levels. By contrast, ecologists define biological systems as communities of interacting individuals at different trophic levels that process energy, nutrient and information flows. Modern plant breeding needs to increase agricultural productivity while decreasing the ecological footprint. This requires a holistic systems biology approach that couples different aggregation levels while considering the variables that affect these biological systems from cell to community. The challenge is to generate accurate experimental data that can be used together with modelling concepts and techniques that allow experimentally verifying in silico predictions. The coupling of aggregation levels in plant sciences, termed Integral Quantification of Biological Organization (IQ(BiO)), might enhance our abilities to generate new desired plant phenotypes.

Simulation of Organ Patterning on the Floral Meristem Using a Polar Auxin Transport Model

An intriguing phenomenon in plant development is the timing and positioning of lateral organ initiation, which is a fundamental aspect of plant architecture. Although important progress has been made in elucidating the role of auxin transport in the vegetative shoot to explain the phyllotaxis of leaf formation in a spiral fashion, a model study of the role of auxin transport in whorled organ patterning in the expanding floral meristem is not available yet. We present an initial simulation approach to study the mechanisms that are expected to play an important role. Starting point is a confocal imaging study of Arabidopsis floral meristems at consecutive time points during flower development. These images reveal auxin accumulation patterns at the positions of the organs, which strongly suggests that the role of auxin in the floral meristem is similar to the role it plays in the shoot apical meristem. This is the basis for a simulation study of auxin transport through a growing floral meristem, which may answer the question whether auxin transport can in itself be responsible for the typical whorled floral pattern. We combined a cellular growth model for the meristem with a polar auxin transport model. The model predicts that sepals are initiated by auxin maxima arising early during meristem outgrowth. These form a pre-pattern relative to which a series of smaller auxin maxima are positioned, which partially overlap with the anlagen of petals, stamens, and carpels. We adjusted the model parameters corresponding to properties of floral mutants and found that the model predictions agree with the observed mutant patterns. The predicted timing of the primordia outgrowth and the timing and positioning of the sepal primordia show remarkable similarities with a developing flower in nature.

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.

A mathematical model for BRI1 mediated signaling in root growth and hypocotyl elongation

Brassinosteroid (BR) signaling is essential for plant growth and development. In Arabidopsis (Arabidopsis thaliana), BRs are perceived by the BRASSINOSTEROID INSENSITIVE1 (BRI1) receptor. Root growth and hypocotyl elongation are convenient downstream physiological outputs of BR signaling. A computational approach was employed to predict root growth solely on the basis of BRI1 receptor activity. The developed mathematical model predicts that during normal root growth, few receptors are occupied with ligand. The model faithfully predicts root growth, as observed in bri1 loss-of-function mutants. For roots, it incorporates one stimulatory and two inhibitory modules, while for hypocotyls, a single inhibitory module is sufficient. Root growth as observed when BRI1 is overexpressed can only be predicted assuming that a decrease occurred in the BRI1 half-maximum response values. Root growth appears highly sensitive to variation in BR concentration and much less to reduction in BRI1 receptor level, suggesting that regulation occurs primarily by ligand availability and biochemical activity.

A quantitative and dynamic model of the Arabidopsis flowering time gene regulatory network.

Various environmental signals integrate into a network of floral regulatory genes leading to the final decision on when to flower. Although a wealth of qualitative knowledge is available on how flowering time genes regulate each other, only a few studies incorporated this knowledge into predictive models. Such models are invaluable as they enable to investigate how various types of inputs are combined to give a quantitative readout. To investigate the effect of gene expression disturbances on flowering time, we developed a dynamic model for the regulation of flowering time in Arabidopsis thaliana. Model parameters were estimated based on expression time-courses for relevant genes, and a consistent set of flowering times for plants of various genetic backgrounds. Validation was performed by predicting changes in expression level in mutant backgrounds and comparing these predictions with independent expression data, and by comparison of predicted and experimental flowering times for several double mutants. Remarkably, the model predicts that a disturbance in a particular gene has not necessarily the largest impact on directly connected genes. For example, the model predicts that SUPPRESSOR OF OVEREXPRESSION OF CONSTANS (SOC1) mutation has a larger impact on APETALA1 (AP1), which is not directly regulated by SOC1, compared to its effect on LEAFY (LFY) which is under direct control of SOC1. This was confirmed by expression data. Another model prediction involves the importance of cooperativity in the regulation of APETALA1 (AP1) by LFY, a prediction supported by experimental evidence. Concluding, our model for flowering time gene regulation enables to address how different quantitative inputs are combined into one quantitative output, flowering time.

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.

Continuous-time modeling of cell fate determination in Arabidopsis flowers

BackgroundThe 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.ResultsWe 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.ConclusionsThe 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.

A fast algorithm to assess local structural identifiability

The paper presents a novel method for assessing the local structural identifiability question for a general non-linear state-space model. The method is a combination of (i) the application of a singular value decomposition to a parametric output sensitivity matrix that is created by simply integrating the model once and, (ii) a symbolic computation for a reduced model that is guided by the SVD results and allows a confirmation of the conclusions regarding identifiability obtained in the first step. In case there is a lack of identifiability, the symbolic computation quickly results in determination of the exact structure of the nullspace and a suitable re-parametrisation. The method is discussed in detail and applied to three case studies, of which the last two are considerably large, containing 22 and 43 parameters to be identified, respectively.

Crop management impacts the efficiency of quantitative trait loci (QTL) detection and use: case study of fruit load×QTL interactions

Mapping studies using populations with introgressed marker-defined genomic regions are continuously increasing knowledge about quantitative trait loci (QTL) that correlate with variation in important crop traits. This knowledge is useful for plant breeding, although combining desired traits in one genotype might be complicated by the mode of inheritance and co-localization of QTL with antagonistic effects, and by physiological trade-offs, and feed-back or feed-forward mechanisms. Therefore, integrating advances at the genetic level with insight into influences of environment and crop management on crop performance remains difficult. Whereas mapping studies can pinpoint correlations between QTL and phenotypic traits for specific conditions, ignoring or overlooking the importance of environment or crop management can jeopardize the relevance of such assessments. Here, we focus on fruit load (a measure determining competition among fruits on one plant) and its strong modulation of QTL effects on fruit size and composition. Following an integral approach, we show which fruit traits are affected by fruit load, to which underlying processes these traits can be linked, and which processes at lower and higher integration levels are affected by fruit load (and subsequently influence fruit traits). This opinion paper (i) argues that a mechanistic framework to interpret interactions between fruit load and QTL effects is needed, (ii) pleads for consideration of the context of agronomic management when detecting QTL, (iii) makes a case for incorporating interacting factors in the experimental set-up of QTL mapping studies, and (iv) provides recommendations to improve efficiency in QTL detection and use, with particular focus on model-based marker-assisted breeding.

Prediction uncertainty assessment of a systems biology model requires a sample of the full probability distribution of its parameters

Multi-parameter models in systems biology are typically ‘sloppy’: some parameters or combinations of parameters may be hard to estimate from data, whereas others are not. One might expect that parameter uncertainty automatically leads to uncertain predictions, but this is not the case. We illustrate this by showing that the prediction uncertainty of each of six sloppy models varies enormously among different predictions. Statistical approximations of parameter uncertainty may lead to dramatic errors in prediction uncertainty estimation. We argue that prediction uncertainty assessment must therefore be performed on a per-prediction basis using a full computational uncertainty analysis. In practice this is feasible by providing a model with a sample or ensemble representing the distribution of its parameters. Within a Bayesian framework, such a sample may be generated by a Markov Chain Monte Carlo (MCMC) algorithm that infers the parameter distribution based on experimental data. Matlab code for generating the sample (with the Differential Evolution Markov Chain sampler) and the subsequent uncertainty analysis using such a sample, is supplied as Supplemental Information.