Linking Discrete and Stochastic Models: The Chemical Master Equation as a Bridge between Process Hitting and Proper Generalized Decomposition

Modeling frameworks bring structure and analysis tools to large and non-intuitive systems but come with certain inherent assumptions and limitations, sometimes to an inhibitive extent. By building bridges in existing models, we can exploit the advantages of each, widening the range of analysis possible for larger, more detailed models of gene regulatory networks. In this paper, we create just such a link between Process Hitting [6,7,8], a recently introduced discrete framework, and the Chemical Master Equation in such a way that allows the application of powerful numerical techniques, namely Proper Generalized Decomposition [1,2,3], to overcome the curse of dimensionality. With these tools in hand, one can exploit the formal analysis of discrete models without sacrificing the ability to obtain a full space state solution, widening the scope of analysis and interpretation possible. As a demonstration of the utility of this methodology, we have applied it here to the p53-mdm2 network [4,5], a widely studied biological regulatory network

A Theoretical Exploration of Birhythmicity in the p53-Mdm2 Network

Experimental observations performed in the p53-Mdm2 network, one of the key protein modules involved in the control of proliferation of abnormal cells in mammals, revealed the existence of two frequencies of oscillations of p53 and Mdm2 in irradiated cells depending on the irradiation dose. These observations raised the question of the existence of birhythmicity, i.e. the coexistence of two oscillatory regimes for the same external conditions, in the p53-Mdm2 network which would be at the origin of these two distinct frequencies. A theoretical answer has been recently suggested by Ouattara, Abou-Jaoudé and Kaufman who proposed a 3-dimensional differential model showing birhythmicity to reproduce the two frequencies experimentally observed. The aim of this work is to analyze the mechanisms at the origin of the birhythmic behavior through a theoretical analysis of this differential model. To do so, we reduced this model, in a first step, into a 3-dimensional piecewise linear differential model where the Hill functions have been approximated by step functions, and, in a second step, into a 2-dimensional piecewise linear differential model by setting one autonomous variable as a constant in each domain of the phase space. We find that two features related to the phase space structure of the system are at the origin of the birhythmic behavior: the existence of two embedded cycles in the transition graph of the reduced models; the presence of a bypass in the orbit of the large amplitude oscillatory regime of low frequency. Based on this analysis, an experimental strategy is proposed to test the existence of birhythmicity in the p53-Mdm2 network. From a methodological point of view, this approach greatly facilitates the computational analysis of complex oscillatory behavior and could represent a valuable tool to explore mathematical models of biological rhythms showing sufficiently steep nonlinearities

Modeling Stochasticity and Variability in Gene Regulatory Networks

Modeling stochasticity in gene regulatory networks is an important and complex problem in molecular systems biology. To elucidate intrinsic noise, several modeling strategies such as the Gillespie algorithm have been used successfully. This paper contributes an approach as an alternative to these classical settings. Within the discrete paradigm, where genes, proteins, and other molecular components of gene regulatory networks are modeled as discrete variables and are assigned as logical rules describing their regulation through interactions with other components. Stochasticity is modeled at the biological function level under the assumption that even if the expression levels of the input nodes of an update rule guarantee activation or degradation there is a probability that the process will not occur due to stochastic effects. This approach allows a finer analysis of discrete models and provides a natural setup for cell population simulations to study cell-to-cell variability. We applied our methods to two of the most studied regulatory networks, the outcome of lambda phage infection of bacteria and the p53-mdm2 complex.Comment: 23 pages, 8 figure

4D synchrotron X-ray tomographic quantification of the transition from cellular to dendrite growth during directional solidification

Solidification morphology directly impacts the mechanical properties of materials; hence many models of the morphological evolution of dendritic structures have been formulated. However, there is a paucity of validation data for directional solidification models, especially the direct observations of metallic alloys, both for cellular and dendritic structures. In this study, we performed 4D synchrotron X-ray tomographic imaging (three spatial directions plus time), to study the transition from cellular to a columnar dendritic morphology and the subsequent growth of columnar dendrite in a temperature gradient stage. The cellular morphology was found to be highly complex, with frequent lateral bridging. Protrusions growing out of the cellular front with the onset of morphological instabilities were captured, together with the subsequent development of these protrusions into established dendrites. Other mechanisms affecting the solidification microstructure, including dendrite fragmentation/pinch-off were also captured and the quantitative results were compared to proposed mechanisms. The results demonstrate that 4D imaging can provide new data to both inform and validate solidification models

Boolean dynamics revisited through feedback interconnections

Boolean models of physical or biological systems describe the global dynamics of the system and their attractors typically represent asymptotic behaviors. In the case of large networks composed of several modules, it may be difficult to identify all the attractors. To explore Boolean dynamics from a novel viewpoint, we will analyse the dynamics emerging from the composition of two known Boolean modules. The state transition graphs and attractors for each of the modules can be combined to construct a new asymptotic graph which will (1) provide a reliable method for attractor computation with partial information; (2) illustrate the differences in dynamical behavior induced by the updating strategy (asynchronous, synchronous, or mixed); and (3) show the inherited organization/structure of the original network’s state transition graph.publishe

The Nucleocapsid Region of HIV-1 Gag Cooperates with the PTAP and LYPXnL Late Domains to Recruit the Cellular Machinery Necessary for Viral Budding

HIV-1 release is mediated through two motifs in the p6 region of Gag, PTAP and LYPXnL, which recruit cellular proteins Tsg101 and Alix, respectively. The Nucleocapsid region of Gag (NC), which binds the Bro1 domain of Alix, also plays an important role in HIV-1 release, but the underlying mechanism remains unclear. Here we show that the first 202 residues of the Bro1 domain (Broi) are sufficient to bind Gag. Broi interferes with HIV-1 release in an NC–dependent manner and arrests viral budding at the plasma membrane. Similar interrupted budding structures are seen following over-expression of a fragment containing Bro1 with the adjacent V domain (Bro1-V). Although only Bro1-V contains binding determinants for CHMP4, both Broi and Bro1-V inhibited release via both the PTAP/Tsg101 and the LYPXnL/Alix pathways, suggesting that they interfere with a key step in HIV-1 release. Remarkably, we found that over-expression of Bro1 rescued the release of HIV-1 lacking both L domains. This rescue required the N-terminal region of the NC domain in Gag and the CHMP4 binding site in Bro1. Interestingly, release defects due to mutations in NC that prevented Bro1 mediated rescue of virus egress were rescued by providing a link to the ESCRT machinery via Nedd4.2s over-expression. Our data support a model in which NC cooperates with PTAP in the recruitment of cellular proteins necessary for its L domain activity and binds the Bro1–CHMP4 complex required for LYPXnL–mediated budding

Linking Discrete and Stochastic Models: The Chemical Master Equation as a Bridge between Process Hitting and Proper Generalized Decomposition

Modeling frameworks bring structure and analysis tools to large and non-intuitive systems but come with certain inherent assumptions and limitations, sometimes to an inhibitive extent. By building bridges in existing models, we can exploit the advantages of each, widening the range of analysis possible for larger, more detailed models of gene regulatory networks. In this paper, we create just such a link between Process Hitting [6,7,8], a recently introduced discrete framework, and the Chemical Master Equation in such a way that allows the application of powerful numerical techniques, namely Proper Generalized Decomposition [1,2,3], to overcome the curse of dimensionality. With these tools in hand, one can exploit the formal analysis of discrete models without sacrificing the ability to obtain a full space state solution, widening the scope of analysis and interpretation possible. As a demonstration of the utility of this methodology, we have applied it here to the p53-mdm2 network [4,5], a widely studied biological regulatory network.Modeling frameworks bring structure and analysis tools to large and non-intuitive systems but come with certain inherent assumptions and limitations, sometimes to an inhibitive extent. By building bridges in existing models, we can exploit the advantages of each, widening the range of analysis possible for larger, more detailed models of gene regulatory networks. In this paper, we create just such a link between Process Hitting [6,7,8], a recently introduced discrete framework, and the Chemical Master Equation in such a way that allows the application of powerful numerical techniques, namely Proper Generalized Decomposition [1,2,3], to overcome the curse of dimensionality. With these tools in hand, one can exploit the formal analysis of discrete models without sacrificing the ability to obtain a full space state solution, widening the scope of analysis and interpretation possible. As a demonstration of the utility of this methodology, we have applied it here to the p53-mdm2 network [4,5], a widely studied biological regulatory network