Parameter estimation in tree graph metabolic networks.

We study the glycosylation processes that convert initially toxic substrates to nutritionally valuable metabolites in the flavonoid biosynthesis pathway of tomato (Solanum lycopersicum) seedlings. To estimate the reaction rates we use ordinary differential equations (ODEs) to model the enzyme kinetics. A popular choice is to use a system of linear ODEs with constant kinetic rates or to use Michaelis-Menten kinetics. In reality, the catalytic rates, which are affected among other factors by kinetic constants and enzyme concentrations, are changing in time and with the approaches just mentioned, this phenomenon cannot be described. Another problem is that, in general these kinetic coefficients are not always identifiable. A third problem is that, it is not precisely known which enzymes are catalyzing the observed glycosylation processes. With several hundred potential gene candidates, experimental validation using purified target proteins is expensive and time consuming. We aim at reducing this task via mathematical modeling to allow for the pre-selection of most potential gene candidates. In this article we discuss a fast and relatively simple approach to estimate time varying kinetic rates, with three favorable properties: firstly, it allows for identifiable estimation of time dependent parameters in networks with a tree-like structure. Secondly, it is relatively fast compared to usually applied methods that estimate the model derivatives together with the network parameters. Thirdly, by combining the metabolite concentration data with a corresponding microarray data, it can help in detecting the genes related to the enzymatic processes. By comparing the estimated time dynamics of the catalytic rates with time series gene expression data we may assess potential candidate genes behind enzymatic reactions. As an example, we show how to apply this method to select prominent glycosyltransferase genes in tomato seedlings.

The flavonoid pathway in tomato seedlings: transcript abundance and the modeling of metabolite dynamics.

Flavonoids are secondary metabolites present in all terrestrial plants. The flavonoid pathway has been extensively studied, and many of the involved genes and metabolites have been described in the literature. Despite this extensive knowledge, the functioning of the pathway in vivo is still poorly understood. Here, we study the flavonoid pathway using both experiments and mathematical models. We measured flavonoid metabolite dynamics in two tissues, hypocotyls and cotyledons, during tomato seedling development. Interestingly, the same backbone of interactions leads to very different accumulation patterns in the different tissues. Initially, we developed a mathematical model with constant enzyme concentrations that described the metabolic networks separately in both tissues. This model was unable to fit the measured flavonoid dynamics in the hypocotyls, even if we allowed unrealistic parameter values. This suggested us to investigate the effect of transcript abundance on flavonoid accumulation. We found that the expression of candidate flavonoid genes varies considerably with time. Variation in transcript abundance results in enzymatic variation, which could have a large effect on metabolite accumulation. Candidate transcript abundance was included in the mathematical model as representative for enzyme concentration. We fitted the resulting model to the flavonoid dynamics in the cotyledons, and tested it by applying it to the data from hypocotyls. When transcript abundance is included, we are indeed able to explain flavonoid dynamics in both tissues. Importantly, this is possible under the biologically relevant restriction that the enzymatic properties estimated by the model are conserved between the tissues.

Modelling the dynamics of viral suppressors of RNA silencing.

Virus infection in plants is limited by RNA silencing. In turn, viruses can counter RNA silencing with silencing suppressors. Viral suppressors of RNA silencing have been shown to play a role in symptom development in plants. We here study four different strategies employed by silencing suppressors: small interfering RNA (siRNA) binding, double-strand RNA (dsRNA) binding and degrading or inactivating Argonaute. We study the effect of the suppressors on viral accumulation within the cell as well as its spread on a tissue with mathematical and computational models. We find that suppressors which target Argonaute are very effective in a single cell, but that targeting dsRNA or siRNA is much more effective at the tissue level. Although targeting Argonaute can be beneficial for viral spread, it can also cause hindrance in some cases owing to raised levels of siRNAs that can spread to other cells.

RNA silencing can explain chlorotic infection patterns on plant leaves.

BACKGROUND: RNA silencing has been implicated in virus symptom development in plants. One common infection symptom in plants is the formation of chlorotic tissue in leaves. Chlorotic and healthy tissue co-occur on a single leaf and form patterns. It has been shown that virus levels in chlorotic tissue are high, while they are low in healthy tissue. Additionally, the presence of siRNAs is confined to the chlorotic spots and the boundaries between healthy and infected tissue. These results strongly indicate that the interaction between virus growth and RNA silencing plays a role in the formation of infection patterns on leaves. However, how RNA silencing leads to the intricate patterns is not known. RESULTS: Here we elucidate the mechanisms leading to infection patterns and the conditions which lead to the various patterns observed. We present a modeling approach in which we combine intra- and inter-cellular dynamics of RNA silencing and viral growth. We observe that, due to the spread of viruses and the RNA silencing response, parts of the tissue become infected while other parts remain healthy. As is observed in experiments high virus levels coincide with high levels of siRNAs, and siRNAs are also present in the boundaries between infected and healthy tissue. We study how single- and double-stranded cleavage by Dicer and amplification by RNA-dependent RNA polymerase can affect the patterns formed. CONCLUSION: This work shows that RNA silencing and virus growth within a cell, and the local spread of virions and siRNAs between cells can explain the heterogeneous spread of virus in leaf tissue, and therewith the observed infection patterns in plants.

The dynamics and efficacy of antiviral RNA silencing: A model study.

BACKGROUND: Mathematical modeling is important to provide insight in the complicated pathway of RNA silencing. RNA silencing is an RNA based mechanism that is widely used by eukaryotes to fight viruses, and to control gene expression. RESULTS: We here present the first mathematical model that combines viral growth with RNA silencing. The model involves a plus-strand RNA virus that replicates through a double-strand RNA intermediate. The model of the RNA silencing pathway consists of cleavage of viral RNA into siRNA by Dicer, target cleavage of viral RNA via the RISC complex, and a secondary response. We found that, depending on the strength of the silencing response, different viral growth patterns can occur. Silencing can decrease viral growth, cause oscillations, or clear the virus completely. Our model can explain various observed phenomena, even when they seem contradictory at first: the diverse responses to the removal of RNA dependent RNA polymerase; different viral growth curves; and the great diversity in observed siRNA ratios. CONCLUSION: The model presented here is an important step in the understanding of the natural functioning of RNA silencing in viral infections.

The RNA silencing pathway: the bits and pieces that matter.

Cellular pathways are generally proposed on the basis of available experimental knowledge. The proposed pathways, however, may be inadequate to describe the phenomena they are supposed to explain. For instance, by means of concise mathematical models we are able to reveal shortcomings in the current description of the pathway of RNA silencing. The silencing pathway operates by cleaving siRNAs from dsRNA. siRNAs can associate with RISC, leading to the degradation of the target mRNA. We propose and analyze a few small extensions to the pathway: a siRNA degrading RNase, primed amplification of aberrant RNA pieces, and cooperation between aberrant RNA to trigger amplification. These extensions allow for a consistent explanation for various types of silencing phenomena, such as virus induced silencing, transgene and transposon induced silencing, and avoidance of self-reactivity, as well as for differences found between species groups.