Tracing the molecular basis of transcriptional dynamics in noisy data by using an experiment-based mathematical model.

Changes in transcription factor levels, epigenetic status, splicing kinetics and mRNA degradation can each contribute to changes in the mRNA dynamics of a gene. We present a novel method to identify which of these processes is changed in cells in response to external signals or as a result of a diseased state. The method employs a mathematical model, for which the kinetics of gene regulation, splicing, elongation and mRNA degradation were estimated from experimental data of transcriptional dynamics. The time-dependent dynamics of several species of adipose differentiation-related protein (ADRP) mRNA were measured in response to ligand activation of the transcription factor peroxisome proliferator-activated receptor δ (PPARδ). We validated the method by monitoring the mRNA dynamics upon gene activation in the presence of a splicing inhibitor. Our mathematical model correctly identifies splicing as the inhibitor target, despite the noise in the data.

Marked heterogeneity in growth characteristics of myoblast clonal cultures and myoblast mixed cultures obtained from the same individual.

BACKGROUND: Sarcopenia is defined as an age-related decrease in skeletal muscle mass and function while adjacent satellite cells are unable to compensate for this loss. However, myoblast cultures can be established even in the presence of sarcopenia. OBJECTIVE: It is yet unknown whether satellite cells from failing muscle in older age are equally affected, as human satellite cells have been assessed using myoblast mixed cultures and not by using myoblast clonal cultures. We questioned to what extent myoblast mixed cultures reflect the in vivo characteristics of single satellite cells from adult skeletal muscle. METHODS: We established a myoblast mixed culture and three myoblast clonal cultures out of the same muscle biopsy and cultured these cells for 100 days. Replicative capacity and oxidative stress resistance were compared. RESULTS: We found marked heterogeneity between the myoblast clonal cultures that all had a significantly lower replicative capacity when compared to the mixed culture. Replicative capacity of the clonal cultures was inversely related to the ß-galactosidase activity after exposure to oxidative stress. Addition of L-carnosine enhanced the remaining replicative capacity in all cultures with a concomitant marginal decrease in ß-galactosidase activity. CONCLUSIONS: It is concluded that myoblast mixed cultures in vitro do not reflect the marked heterogeneity between single isolated satellite cells. The consequences of the heterogeneity on muscle performance remain to be established.

Gene circuit analysis of the terminal gap gene huckebein.

The early embryo of Drosophila melanogaster provides a powerful model system to study the role of genes in pattern formation. The gap gene network constitutes the first zygotic regulatory tier in the hierarchy of the segmentation genes involved in specifying the position of body segments. Here, we use an integrative, systems-level approach to investigate the regulatory effect of the terminal gap gene huckebein (hkb) on gap gene expression. We present quantitative expression data for the Hkb protein, which enable us to include hkb in gap gene circuit models. Gap gene circuits are mathematical models of gene networks used as computational tools to extract regulatory information from spatial expression data. This is achieved by fitting the model to gap gene expression patterns, in order to obtain estimates for regulatory parameters which predict a specific network topology. We show how considering variability in the data combined with analysis of parameter determinability significantly improves the biological relevance and consistency of the approach. Our models are in agreement with earlier results, which they extend in two important respects: First, we show that Hkb is involved in the regulation of the posterior hunchback (hb) domain, but does not have any other essential function. Specifically, Hkb is required for the anterior shift in the posterior border of this domain, which is now reproduced correctly in our models. Second, gap gene circuits presented here are able to reproduce mutants of terminal gap genes, while previously published models were unable to reproduce any null mutants correctly. As a consequence, our models now capture the expression dynamics of all posterior gap genes and some variational properties of the system correctly. This is an important step towards a better, quantitative understanding of the developmental and evolutionary dynamics of the gap gene network.

Rapid flow cytometric method for measuring senescence associated beta-galactosidase activity in human fibroblasts.

Senescence associated-beta-galactosidase (SA-beta-gal) activity is a widely used marker for cellular senenescence. SA-beta-gal activity is routinely detected cytochemically, manually discriminating negative from positive cells. This method is time-consuming, subjective and therefore prone to operator-error. We aimed to optimize a flow cytometric method described by other workers using endothelial cells to better differentiate between populations of fibroblasts in degrees of SA-beta-gal activity. Skin fibroblasts were isolated from young (mean age +/- SD: 25.5 +/- 1.8) and very old (age 90.2 +/- 0.3) subjects. Different pH modulators were tested for toxicity. To induce stress-induced senescence, fibroblasts were exposed to rotenone. Senescence was assessed measuring SA-beta-gal activity by cytochemistry (X-gal) and by flow cytometry (C(12)FDG). The pH modulator Bafilomycin A1 (Baf A1) was found to be least toxic for fibroblasts and to differentiate best between nonstressed and stressed fibroblast populations. Under nonstressed conditions, fibroblasts from very old subjects showed higher SA-beta-gal activity than fibroblasts from young subjects. This difference was found for both the flow cytometric and cytochemical methods (P = 0.013 and P = 0.056 respectively). Under stress-induced conditions the flow cytometric method but not the cytochemical method revealed significant higher SA-beta-gal activity in fibroblasts from very old compared to young subjects (P = 0.004 and P = 0.635 respectively). We found the modified flow cytometric method measuring SA-beta-gal activity superior in discriminating between degrees of senescence in different populations of fibroblasts.

Efficient parameter estimation for spatio-temporal models of pattern formation: case study of Drosophila melanogaster.

MOTIVATION: Diffusable and non-diffusable gene products play a major role in body plan formation. A quantitative understanding of the spatio-temporal patterns formed in body plan formation, by using simulation models is an important addition to experimental observation. The inverse modelling approach consists of describing the body plan formation by a rule-based model, and fitting the model parameters to real observed data. In body plan formation, the data are usually obtained from fluorescent immunohistochemistry or in situ hybridizations. Inferring model parameters by comparing such data to those from simulation is a major computational bottleneck. An important aspect in this process is the choice of method used for parameter estimation. When no information on parameters is available, parameter estimation is mostly done by means of heuristic algorithms. RESULTS: We show that parameter estimation for pattern formation models can be efficiently performed using an evolution strategy (ES). As a case study we use a quantitative spatio-temporal model of the regulatory network for early development in Drosophila melanogaster. In order to estimate the parameters, the simulated results are compared to a time series of gene products involved in the network obtained with immunohistochemistry. We demonstrate that a (mu,lambda)-ES can be used to find good quality solutions in the parameter estimation. We also show that an ES with multiple populations is 5-140 times as fast as parallel simulated annealing for this case study, and that combining ES with a local search results in an efficient parameter estimation method.

Computational methods for diffusion-influenced biochemical reactions.

MOTIVATION: We compare stochastic computational methods accounting for space and discrete nature of reactants in biochemical systems. Implementations based on Brownian dynamics (BD) and the reaction-diffusion master equation are applied to a simplified gene expression model and to a signal transduction pathway in Escherichia coli. RESULTS: In the regime where the number of molecules is small and reactions are diffusion-limited predicted fluctuations in the product number vary between the methods, while the average is the same. Computational approaches at the level of the reaction-diffusion master equation compute the same fluctuations as the reference result obtained from the particle-based method if the size of the sub-volumes is comparable to the diameter of reactants. Using numerical simulations of reversible binding of a pair of molecules we argue that the disagreement in predicted fluctuations is due to different modeling of inter-arrival times between reaction events. Simulations for a more complex biological study show that the different approaches lead to different results due to modeling issues. Finally, we present the physical assumptions behind the mesoscopic models for the reaction-diffusion systems. AVAILABILITY: Input files for the simulations and the source code of GMP can be found under the following address: http://www.cwi.nl/projects/sic/bioinformatics2007/

Modelling genetic regulation of growth and form in a branching sponge.

We present a mathematical model of the genetic regulation controlling skeletogenesis and the influence of the physical environment on a branching sponge with accretive growth (e.g. Haliclona oculata or Lubomirskia baikalensis). From previous work, it is known that high concentrations of silicate induce spicule formation and upregulate the silicatein gene. The upregulation of this gene activates locally the production of spicules in the sponge and the deposition of the skeleton. Furthermore, it is known that the expression of the gene Iroquois induces the formation of an aquiferous system, consisting of exhalant and inhalant pores. We propose a model of the regulatory network controlling the separation in time and space of the skeletogenesis and the formation of the aquiferous system. The regulatory network is closely linked with environmental influences. In building a skeleton, silicate is absorbed from the environment. In our model, silicate is transported by diffusion through the environment and absorbed at the surface of a geometric model of the sponge, resulting in silicate gradients emerging in the neighbourhood of the sponge. Our model simulations predict sponge morphology and the positioning of the exhalant pores over the surface of the sponge.

Spatial stochastic modelling of the phosphoenolpyruvate-dependent phosphotransferase (PTS) pathway in Escherichia coli.

MOTIVATION: Many biochemical networks involve reactions localized on the cell membrane. This can give rise to spatial gradients of the concentration of cytosolic species. Moreover, the number of membrane molecules can be small and stochastic effects can become relevant. Pathways usually consist of a complex interaction network and are characterized by a large set of parameters. The inclusion of spatial and stochastic effects is a major challenge in developing quantitative and dynamic models of pathways. RESULTS: We have developed a particle-based spatial stochastic method (GMP) to simulate biochemical networks in space, including fluctuations from the diffusion of particles and reactions. Gradients emerging from membrane reactions can be resolved. As case studies for the GMP method we used a simple gene expression system and the phosphoenolpyruvate:glucose phosphotransferase system pathway. AVAILABILITY: The source code for the GMP method is available at http://www.science.uva.nl/research/scs/CellMath/GMP.

Uncertainty Propagation in Nerve Impulses Through the Action Potential Mechanism.

We investigate the propagation of probabilistic uncertainty through the action potential mechanism in nerve cells. Using the Hodgkin-Huxley (H-H) model and Stochastic Collocation on Sparse Grids, we obtain an accurate probabilistic interpretation of the deterministic dynamics of the transmembrane potential and gating variables. Using Sobol indices, out of the 11 uncertain parameters in the H-H model, we unravel two main uncertainty sources, which account for more than 90 % of the fluctuations in neuronal responses, and have a direct biophysical interpretation. We discuss how this interesting feature of the H-H model allows one to reduce greatly the probabilistic degrees of freedom in uncertainty quantification analyses, saving CPU time in numerical simulations and opening possibilities for probabilistic generalisation of other deterministic models of great importance in physiology and mathematical neuroscience.

A global sensitivity analysis approach for morphogenesis models.

BACKGROUND: Morphogenesis is a developmental process in which cells organize into shapes and patterns. Complex, non-linear and multi-factorial models with images as output are commonly used to study morphogenesis. It is difficult to understand the relation between the uncertainty in the input and the output of such 'black-box' models, giving rise to the need for sensitivity analysis tools. In this paper, we introduce a workflow for a global sensitivity analysis approach to study the impact of single parameters and the interactions between them on the output of morphogenesis models. RESULTS: To demonstrate the workflow, we used a published, well-studied model of vascular morphogenesis. The parameters of this cellular Potts model (CPM) represent cell properties and behaviors that drive the mechanisms of angiogenic sprouting. The global sensitivity analysis correctly identified the dominant parameters in the model, consistent with previous studies. Additionally, the analysis provided information on the relative impact of single parameters and of interactions between them. This is very relevant because interactions of parameters impede the experimental verification of the predicted effect of single parameters. The parameter interactions, although of low impact, provided also new insights in the mechanisms of in silico sprouting. Finally, the analysis indicated that the model could be reduced by one parameter. CONCLUSIONS: We propose global sensitivity analysis as an alternative approach to study the mechanisms of morphogenesis. Comparison of the ranking of the impact of the model parameters to knowledge derived from experimental data and from manipulation experiments can help to falsify models and to find the operand mechanisms in morphogenesis. The workflow is applicable to all 'black-box' models, including high-throughput in vitro models in which output measures are affected by a set of experimental perturbations.

Systems biology: parameter estimation for biochemical models.

Mathematical models of biological processes have various applications: to assist in understanding the functioning of a system, to simulate experiments before actually performing them, to study situations that cannot be dealt with experimentally, etc. Some parameters in the model can be directly obtained from experiments or from the literature. Others have to be inferred by comparing model results to experiments. In this minireview, we discuss the identifiability of models, both intrinsic to the model and taking into account the available data. Furthermore, we give an overview of the most frequently used approaches to search the parameter space.

Stress-induced responses of human skin fibroblasts in vitro reflect human longevity.

Unlike various model organisms, cellular responses to stress have not been related to human longevity. We investigated cellular responses to stress in skin fibroblasts that were isolated from young and very old subjects, and from offspring of nonagenarian siblings and their partners, representatives of the general population. Fibroblasts were exposed to rotenone and hyperglycemia and assessed for senescence-associated beta-galactosidase (SA-beta-gal) activity by flow cytometry. Apoptosis/cell death was measured with the Annexin-V/PI assay and cell-cycle analysis (Sub-G1 content) and growth potential was determined by the colony formation assay. Compared with fibroblasts from young subjects, baseline SA-beta-gal activity was higher in fibroblasts from old subjects (P = 0.004) as were stress-induced increases (rotenone: P < 0.001, hyperglycemia: P = 0.027). For measures of apoptosis/cell death, fibroblasts from old subjects showed higher baseline levels (Annexin V+/PI+ cells: P = 0.040, Sub-G1: P = 0.014) and lower stress-induced increases (Sub-G1: P = 0.018) than fibroblasts from young subjects. Numbers and total size of colonies under nonstressed conditions were higher for fibroblasts from young subjects (P = 0.017 and 0.006, respectively). Baseline levels of SA-beta-gal activity and apoptosis/cell death were not different between fibroblasts from offspring and partner. Stress-induced increases were lower for SA-beta-gal activity (rotenone: P = 0.064, hyperglycemia: P < 0.001) and higher for apoptosis/cell death (Annexin V+/PI- cells: P = 0.041, Annexin V+/PI+ cells: P = 0.008). Numbers and total size of colonies under nonstressed conditions were higher for fibroblasts from offspring (P = 0.001 and 0.024, respectively) whereas rotenone-induced decreases were lower (P = 0.008 and 0.004, respectively). These data provide strong support for the hypothesis that in vitro cellular responses to stress reflect the propensity for human longevity.

Parameter estimation and determinability analysis applied to Drosophila gap gene circuits.

BACKGROUND: Mathematical modeling of real-life processes often requires the estimation of unknown parameters. Once the parameters are found by means of optimization, it is important to assess the quality of the parameter estimates, especially if parameter values are used to draw biological conclusions from the model. RESULTS: In this paper we describe how the quality of parameter estimates can be analyzed. We apply our methodology to assess parameter determinability for gene circuit models of the gap gene network in early Drosophila embryos. CONCLUSION: Our analysis shows that none of the parameters of the considered model can be determined individually with reasonable accuracy due to correlations between parameters. Therefore, the model cannot be used as a tool to infer quantitative regulatory weights. On the other hand, our results show that it is still possible to draw reliable qualitative conclusions on the regulatory topology of the gene network. Moreover, it improves previous analyses of the same model by allowing us to identify those interactions for which qualitative conclusions are reliable, and those for which they are ambiguous.

Why the phosphotransferase system of Escherichia coli escapes diffusion limitation.

We calculated the implications of diffusion for the phosphoenolpyruvate:glucose phosphotransferase system (glucose-PTS) of Escherichia coli in silicon cells of various magnitudes. For a cell of bacterial size, diffusion limitation of glucose influx was negligible. Nevertheless, a significant concentration gradient for one of the enzyme species, nonphosphorylated IIA(Glc), was found. This should have consequences because the phosphorylation state of IIA(Glc) is an important intracellular signal. For mammalian cell sizes we found significant diffusion limitation, as well as strong concentration gradients in many PTS components, and strong effects on glucose and energy signaling. We calculated that the PTS may sense both extracellular glucose and the intracellular free-energy state. We discuss i), that the effects of diffusion on cell function should prevent this highly effective bacterial system from functioning in eukaryotic cells, ii), that in the larger eukaryotic cell any similar chain of mobile group-transfer proteins can neither sustain the same volumetric flux as in bacteria nor transmit a signal far into the cell, and iii), that systems such as these may exhibit spatial differentiation in their sensitivity to different signals.

Flux control of the bacterial phosphoenolpyruvate:glucose phosphotransferase system and the effect of diffusion.

We analyzed the role of diffusion and cell size on the flux control properties of the glucose-PTS of Escherichia coli, in silicon cells under various metabolic conditions. To our surprise, the influence of the concentration of phosphoryl-donor PEP on the distribution of control was small. We found for cells of bacterial size that PTS-flux control was mainly located in processes taking place in the membrane and that diffusion hardly controlled the flux (< 2.8 %). Enlargement of the cells shifted the control from membrane to cytoplasm and from process rates to diffusion rates, the latter now having a total control of about 38 %. In the presence of glucose, nearly all diffusion flux control resided in the component that links the cytoplasmic processes to those in the membrane.