One-hit stochastic decline in a mechanochemical model of cytoskeleton-induced neuron death I: cell-fate arrival times

Much experimental evidence shows that the cytoskeleton is a downstream target and effector during cell death in numerous neurodegenerative diseases, including Parkinson's, Huntington's, and Alzheimer's diseases. However, recent evidence indicates that cytoskeletal dysfunction can also trigger neuronal death, by mechanisms as yet poorly understood. This is the first of two papers in which we study a mathematical model of cytoskeleton-induced neuron death. In our model, assembly control of the neuronal cytoskeleton interacts with both cellular stress levels and cytosolic free radical concentrations to trigger neurodegeneration. This trigger mechanism is further modulated by the presence of cell interactions in the form of a diffusible toxic factor released by dying neurons. We find that, consistent with empirical observations, our model produces one-hit exponential and sigmoid patterns of cell dropout. In all cases, cell dropout is exponential-tailed and described accurately by a gamma distribution. The transition between exponential and sigmoidal is gradual, and determined by a synergetic interaction between the magnitude of fluctuations in cytoskeleton assembly control and by the degree of cell coupling. We conclude that a single mechanism involving neuron interactions and fluctuations in cytoskeleton assembly control is compatible with the experimentally observed range of neuronal attrition kinetics.     

One-hit stochastic decline in a mechanochemical model of cytoskeleton-induced neuron death III: diffusion pulse death zones

This is the third of three papers in which we study a mathematical model of cytoskeleton-induced neuron death. In the first two papers of this suite [Lomasko, T., Clarke, G., Lumsden, C., 2007a. One-hit stochastic decline in a mechanochemical model of cytoskeleton-induced neuron death I: cell fate arrival times. J. Theor. Biol. 249, 1-17, doi:10.1016/j.jtbi.2007.05.031; Lomasko, T., Clarke, G., Lumsden, C., 2007b. One-hit stochastic decline in a mechanochemical model of cytoskeleton-induced neuron death II: transition state metastability. J. Theor. Biol. 249, 18-28, doi:10.1016/j.jtbi.2007.05.032], we established that the mean-field limit of our model relates the known patterns of neuron decline to specific scales of cytoskeleton reorganization and cell-cell interaction by diffusible death factors. In the mean-field limit, the spatially variable concentration of diffusing death factor is replaced by a constant average value. Recent empirical advances now permit the actual diffusion of such factors to be followed in intact neuropil. In this paper we therefore extend the model beyond the mean-field limit, to include the diffusion dynamics of death factor bursts released from dying neurons. A range of novel tissue degeneration patterns is observed, for which we confirm and extend the mean-field prediction that sigmoidal patterns of neuron population decay are a principal hallmark of cell death in the presence of death factor release.     

Perturbation avalanches and criticality in gene regulatory networks

Boolean networks are simplified models of gene regulatory networks. We derive an approximation of the size distribution of perturbation avalanches in Boolean networks based on known results in the theory of branching processes. We show numerically that the approximation works well for different kinds of Boolean networks. It has been suggested that gene regulatory networks may be dynamically critical. To study this, as an application of the presented theory we present a novel method for estimating an order parameter from microarray data. According to the available data and our method, we find that gene regulatory networks appear to be stable and reside near the phase transition between order and chaos.     

Neuron firing in driven nonlinear integrate-and-fire models

Statistical properties of neuron firing are studied in the framework of a nonlinear leaky integrate-and-fire model that is driven by a slow periodic subthreshold signal. The firing events are characterized by first passage time densities. The experimentally better accessible interspike interval density generally depends on the sojourn times in a refractory state of the neuron. This aspect is not part of the integrate-and-fire model and must be modelled additionally. For a large class of refractory dynamics, a general expression for the interspike interval density is given and further evaluated for the two cases with an instantaneous resetting (i.e. no refractory state) and a refractory state possessing a deterministic lifetime. First passage time densities and interspike interval densities following from the proposed theory compare favorably with precise numerical simulations.     

Spatial distribution of VEGF isoforms and chemotactic signals in the vicinity of a tumor

We propose a mathematical model that describes the formation of gradients of different isoforms of vascular endothelial growth factor (VEGF). VEGF is crucial in the process of tumor-induced angiogenesis, and recent experiments strongly suggest that the molecule is most potent when bound to the extracellular matrix (ECM). Using a system of reaction-diffusion equations, we study diffusion of VEGF, binding of VEGF to the ECM, and cleavage of VEGF from the ECM by matrix metalloproteases (MMPs). We find that spontaneous gradients of matrix-bound VEGF are possible for an isoform that binds weakly to the ECM (i.e. VEGF₁₆₅), but cleavage by MMPs is required to form long-range gradients of isoforms that bind rapidly to the ECM (i.e. VEGF₁₈₉). We also find that gradient strengths and ranges are regulated by MMPs. Finally, we find that VEGF molecules cleaved from the ECM may be distributed in patterns that are not conducive to chemotactic migration toward a tumor, depending on the spatial distribution of MMP molecules. Our model elegantly explains a number of in vivo observations concerning the significance of different VEGF isoforms, points to VEGF₁₆₅ as an especially significant therapeutic target and indicator of a tumor's angiogenic potential, and enables predictions that are subject to testing with in vitro experiments.     

Comparative evaluation of reverse engineering gene regulatory networks with relevance networks, graphical gaussian models and bayesian networks

MOTIVATION: An important problem in systems biology is the inference of biochemical pathways and regulatory networks from postgenomic data. Various reverse engineering methods have been proposed in the literature, and it is important to understand their relative merits and shortcomings. In the present paper, we compare the accuracy of reconstructing gene regulatory networks with three different modelling and inference paradigms: (1) Relevance networks (RNs): pairwise association scores independent of the remaining network; (2) graphical Gaussian models (GGMs): undirected graphical models with constraint-based inference, and (3) Bayesian networks (BNs): directed graphical models with score-based inference. The evaluation is carried out on the Raf pathway, a cellular signalling network describing the interaction of 11 phosphorylated proteins and phospholipids in human immune system cells. We use both laboratory data from cytometry experiments as well as data simulated from the gold-standard network. We also compare passive observations with active interventions. RESULTS: On Gaussian observational data, BNs and GGMs were found to outperform RNs. The difference in performance was not significant for the non-linear simulated data and the cytoflow data, though. Also, we did not observe a significant difference between BNs and GGMs on observational data in general. However, for interventional data, BNs outperform GGMs and RNs, especially when taking the edge directions rather than just the skeletons of the graphs into account. This suggests that the higher computational costs of inference with BNs over GGMs and RNs are not justified when using only passive observations, but that active interventions in the form of gene knockouts and over-expressions are required to exploit the full potential of BNs. AVAILABILITY: Data, software and supplementary material are available from http://www.bioss.sari.ac.uk/staff/adriano/research.html     

Multivariate analysis of noise in genetic regulatory networks

Stochasticity is an intrinsic property of genetic regulatory networks due to the low copy numbers of the major molecular species, such as, DNA, mRNA, and regulatory proteins. Therefore, investigation of the mechanisms that reduce the stochastic noise is essential in understanding the reproducible behaviors of real organisms and is also a key to design synthetic genetic regulatory networks that can reliably work. We use an analytical and systematic method, the linear noise approximation of the chemical master equation along with the decoupling of a stoichiometric matrix. In the analysis of fluctuations of multiple molecular species, the covariance is an important measure of noise. However, usually the representation of a covariance matrix in the natural coordinate system, i.e. the copy numbers of the molecular species, is intractably complicated because reactions change copy numbers of more than one molecular species simultaneously. Decoupling of a stoichiometric matrix, which is a transformation of variables, significantly simplifies the representation of a covariance matrix and elucidates the mechanisms behind the observed fluctuations in the copy numbers. We apply our method to three types of fundamental genetic regulatory networks, that is, a single-gene autoregulatory network, a two-gene autoregulatory network, and a mutually repressive network. We have found that there are multiple noise components differently originating. Each noise component produces fluctuation in the characteristic direction. The resulting fluctuations in the copy numbers of the molecular species are the sum of these fluctuations. In the examples, the limitation of the negative feedback in noise reduction and the trade-off of fluctuations in multiple molecular species are clearly explained. The analytical representations show the full parameter dependence. Additionally, the validity of our method is tested by stochastic simulations.     

The incorporation of epigenetics in artificial gene regulatory networks

Artificial gene regulatory networks are computational models that draw inspiration from biological networks of gene regulation. Since their inception they have been used to infer knowledge about gene regulation and as methods of computation. These computational models have been shown to possess properties typically found in the biological world, such as robustness and self organisation. Recently, it has become apparent that epigenetic mechanisms play an important role in gene regulation. This paper describes a new model, the Artificial Epigenetic Regulatory Network (AERN) which builds upon existing models by adding an epigenetic control layer. Our results demonstrate that AERNs are more adept at controlling multiple opposing trajectories when applied to a chaos control task within a conservative dynamical system, suggesting that AERNs are an interesting area for further investigation.     

On the dynamics of random Boolean networks subject to noise: Attractors, ergodic sets and cell types

The asymptotic dynamics of random Boolean networks subject to random fluctuations is investigated. Under the influence of noise, the system can escape from the attractors of the deterministic model, and a thorough study of these transitions is presented. We show that the dynamics is more properly described by sets of attractors rather than single ones. We generalize here a previous notion of ergodic sets, and we show that the Threshold Ergodic Sets so defined are robust with respect to noise and, at the same time, that they do not suffer from a major drawback of ergodic sets. The system jumps from one attractor to another of the same Threshold Ergodic Set under the influence of noise, never leaving it. By interpreting random Boolean networks as models of genetic regulatory networks, we also propose to associate cell types to Threshold Ergodic Sets rather than to deterministic attractors or to ergodic sets, as it had been previously suggested. We also propose to associate cell differentiation to the process whereby a Threshold Ergodic Set composed by several attractors gives rise to another one composed by a smaller number of attractors. We show that this approach accounts for several interesting experimental facts about cell differentiation, including the possibility to obtain an induced pluripotent stem cell from a fully differentiated one by overexpressing some of its genes.     

Shape-dependent control of cell growth, differentiation, and apoptosis: switching between attractors in cell regulatory networks

Development of characteristic tissue patterns requires that individual cells be switched locally between different phenotypes or "fates;" while one cell may proliferate, its neighbors may differentiate or die. Recent studies have revealed that local switching between these different gene programs is controlled through interplay between soluble growth factors, insoluble extracellular matrix molecules, and mechanical forces which produce cell shape distortion. Although the precise molecular basis remains unknown, shape-dependent control of cell growth and function appears to be mediated by tension-dependent changes in the actin cytoskeleton. However, the question remains: how can a generalized physical stimulus, such as cell distortion, activate the same set of genes and signaling proteins that are triggered by molecules which bind to specific cell surface receptors. In this article, we use computer simulations based on dynamic Boolean networks to show that the different cell fates that a particular cell can exhibit may represent a preprogrammed set of common end programs or "attractors" which self-organize within the cell's regulatory networks. In this type of dynamic network model of information processing, generalized stimuli (e.g., mechanical forces) and specific molecular cues elicit signals which follow different trajectories, but eventually converge onto one of a small set of common end programs (growth, quiescence, differentiation, apoptosis, etc.). In other words, if cells use this type of information processing system, then control of cell function would involve selection of preexisting (latent) behavioral modes of the cell, rather than instruction by specific binding molecules. Importantly, the results of the computer simulation closely mimic experimental data obtained with living endothelial cells. The major implication of this finding is that current methods used for analysis of cell function that rely on characterization of linear signaling pathways or clusters of genes with common activity profiles may overlook the most critical features of cellular information processing which normally determine how signal specificity is established and maintained in living cells.     

Robustness and state-space structure of Boolean gene regulatory models

Robustness to perturbation is an important characteristic of genetic regulatory systems, but the relationship between robustness and model dynamics has not been clearly quantified. We propose a method for quantifying both robustness and dynamics in terms of state-space structures, for Boolean models of genetic regulatory systems. By investigating existing models of the Drosophila melanogaster segment polarity network and the Saccharomyces cerevisiae cell-cycle network, we show that the structure of attractor basins can yield insight into the underlying decision making required of the system, and also the way in which the system maximises its robustness. In particular, gene networks implementing decisions based on a few genes have simple state-space structures, and their attractors are robust by virtue of their simplicity. Gene networks with decisions that involve many interacting genes have correspondingly more complicated state-space structures, and robustness cannot be achieved through the structure of the attractor basins, but is achieved by larger attractor basins that dominate the state space. These different types of robustness are demonstrated by the two models: the D. melanogaster segment polarity network is robust due to simple attractor basins that implement decisions based on spatial signals; the S. cerevisiae cell-cycle network has a complicated state-space structure, and is robust only due to a giant attractor basin that dominates the state space.     

Structure of regulatory networks and diversity of gene expression patterns

Complexity of gene regulatory network has been considered to be responsible for diversity of cells. Different types of cells, characterized by the expression patterns of genes, are produced in early development through the dynamics of gene activities based on the regulatory network. However, very little is known about relationship between the structure of regulatory networks and the dynamics of gene activities. In this paper, I introduce new idea of “steady-state compatibility” by which the diversity of possible gene activities can be determined from the topological structure of gene regulatory networks. The basic premise is very simple: the activity of a gene should be a function of the controlling genes. Thus, a gene should always show unique expression activity if the activities of the controlling genes are unique. Based on this, the maximum possible diversity of steady states is determined using only information regarding regulatory linkages without knowing the regulatory functions of genes. By extending this idea, some general properties were derived. For example, multiple loop structures in regulatory networks are necessary for increasing the diversity of gene activity. On the other hand, connected multiple loops sharing the same genes do not increase the diversity. The method was applied to a gene regulatory network responsible for early development in a sea urchin species. A set of important genes responsible for generating diversities of gene activities was derived based on the concept of compatibility of steady states.     

Evolutionary modelling of feed forward loops in gene regulatory networks

Feed forward loops (FFLs) are gene regulatory network motifs. They exist in different types, defined by the signs of the effects of genes in the motif on one another. We examine 36 feed forward loops in Escherichia coli, using evolutionary simulations to predict the forms of FFL expected to evolve to generate the pattern of expression of the output gene. These predictions are tested using likelihood ratios, comparing likelihoods of the observed FFL structures with their likelihoods under null models. The very high likelihood ratios generated, of over 10(11), suggest that evolutionary simulation is a valuable component in the explanation of FFL structure.     

A continuous optimization approach for inferring parameters in mathematical models of regulatory networks

Background

The advances of systems biology have raised a large number of sophisticated mathematical models for describing the dynamic property of complex biological systems. One of the major steps in developing mathematical models is to estimate unknown parameters of the model based on experimentally measured quantities. However, experimental conditions limit the amount of data that is available for mathematical modelling. The number of unknown parameters in mathematical models may be larger than the number of observation data. The imbalance between the number of experimental data and number of unknown parameters makes reverse-engineering problems particularly challenging.

Results

To address the issue of inadequate experimental data, we propose a continuous optimization approach for making reliable inference of model parameters. This approach first uses a spline interpolation to generate continuous functions of system dynamics as well as the first and second order derivatives of continuous functions. The expanded dataset is the basis to infer unknown model parameters using various continuous optimization criteria, including the error of simulation only, error of both simulation and the first derivative, or error of simulation as well as the first and second derivatives. We use three case studies to demonstrate the accuracy and reliability of the proposed new approach. Compared with the corresponding discrete criteria using experimental data at the measurement time points only, numerical results of the ERK kinase activation module show that the continuous absolute-error criteria using both function and high order derivatives generate estimates with better accuracy. This result is also supported by the second and third case studies for the G1/S transition network and the MAP kinase pathway, respectively. This suggests that the continuous absolute-error criteria lead to more accurate estimates than the corresponding discrete criteria. We also study the robustness property of these three models to examine the reliability of estimates. Simulation results show that the models with estimated parameters using continuous fitness functions have better robustness properties than those using the corresponding discrete fitness functions.

Conclusions

The inference studies and robustness analysis suggest that the proposed continuous optimization criteria are effective and robust for estimating unknown parameters in mathematical models.

Electronic supplementary material

The online version of this article (doi:10.1186/1471-2105-15-256) contains supplementary material, which is available to authorized users.     

Preservation of dynamic properties in qualitative modeling frameworks for gene regulatory networks

Mathematical modeling often helps to provide a systems perspective on gene regulatory networks. In particular, qualitative approaches are useful when detailed kinetic information is lacking. Multiple methods have been developed that implement qualitative information in different ways, e.g., in purely discrete or hybrid discrete/continuous models. In this paper, we compare the discrete asynchronous logical modeling formalism for gene regulatory networks due to R. Thomas with piecewise affine differential equation models. We provide a local characterization of the qualitative dynamics of a piecewise affine differential equation model using the discrete dynamics of a corresponding Thomas model. Based on this result, we investigate the consistency of higher-level dynamical properties such as attractor characteristics and reachability. We show that although the two approaches are based on equivalent information, the resulting qualitative dynamics are different. In particular, the dynamics of the piecewise affine differential equation model is not a simple refinement of the dynamics of the Thomas model