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.

Correlative analysis of molecular markers with phenotypic signatures is the simplest model for hypothesis generation. In this paper, a panel of 24 breast cell lines was grown in 3D culture, their morphology was imaged through phase contrast microscopy, and computational methods were developed to segment and represent each colony at multiple dimensions. Subsequently, subpopulations from these morphological responses were identified through consensus clustering to reveal three clusters of round, grape-like, and stellate phenotypes. In some cases, cell lines with particular pathobiological phenotypes clustered together (e.g., ERBB2 amplified cell lines sharing the same morphometric properties as the grape-like phenotype). Next, associations with molecular features were realized through (i) differential analysis within each morphological cluster, and (ii) regression analysis across the entire panel of cell lines. In both cases, the dominant genes that are predictive of the morphological signatures were identified. Specifically, PPARγ has been associated with the invasive stellate morphological phenotype, which corresponds to triple-negative pathobiology. PPARγ has been validated through two supporting biological assays.

Genetic toggle switches are widespread in gene regulatory networks (GRN). Bistability, namely the ability to choose among two different stable states, is an essential feature of switching and memory devices. Cells have many regulatory circuits able to provide bistability that endow a cell with efficient and reliable switching between different physiological modes of operation. It is often assumed that negative feedbacks with cooperative binding (i.e. the formation of dimers or multimers) are a prerequisite for bistability. Here we analyze the relation between bistability in GRN under monomeric regulation and the role of autoloops under a deterministic setting. Using a simple geometric argument, we show analytically that bistability can also emerge without multimeric regulation, provided that at least one regulatory autoloop is present.

Many bacteria exhibit multicellular behaviour, with individuals within a colony coordinating their actions for communal benefit. One example of complex multicellular phenotypes is myxobacterial fruiting body formation, where thousands of cells aggregate into large three-dimensional structures, within which sporulation occurs. Here we describe a novel theoretical model, which uses Monte Carlo dynamics to simulate and explain multicellular development. The model captures multiple behaviours observed during fruiting, including the spontaneous formation of aggregation centres and the formation and dissolution of fruiting bodies. We show that a small number of physical properties in the model is sufficient to explain the most frequently documented population-level behaviours observed during development in Myxococcus xanthus .

We quantify the influence of the topology of a transcriptional regulatory network on its ability to process environmental signals. By posing the problem in terms of information theory, we do this without specifying the function performed by the network. Specifically, we study the maximum mutual information between the input (chemical) signal and the output (genetic) response attainable by the network in the context of an analytic model of particle number fluctuations. We perform this analysis for all biochemical circuits, including various feedback loops, that can be built out of 3 chemical species, each under the control of one regulator. We find that a generic network, constrained to low molecule numbers and reasonable response times, can transduce more information than a simple binary switch and, in fact, manages to achieve close to the optimal information transmission fidelity. These high-information solutions are robust to tenfold changes in most of the networks' biochemical parameters; moreover they are easier to achieve in networks containing cycles with an odd number of negative regulators (overall negative feedback) due to their decreased molecular noise (a result which we derive analytically). Finally, we demonstrate that a single circuit can support multiple high-information solutions. These findings suggest a potential resolution of the “cross-talk” phenomenon as well as the previously unexplained observation that transcription factors that undergo proteolysis are more likely to be auto-repressive.

It is system dynamics that determines the function of cells, tissues and organisms. To develop mathematical models and estimate their parameters are an essential issue for studying dynamic behaviors of biological systems which include metabolic networks, genetic regulatory networks and signal transduction pathways, under perturbation of external stimuli. In general, biological dynamic systems are partially observed. Therefore, a natural way to model dynamic biological systems is to employ nonlinear state-space equations. Although statistical methods for parameter estimation of linear models in biological dynamic systems have been developed intensively in the recent years, the estimation of both states and parameters of nonlinear dynamic systems remains a challenging task. In this report, we apply extended Kalman Filter (EKF) to the estimation of both states and parameters of nonlinear state-space models. To evaluate the performance of the EKF for parameter estimation, we apply the EKF to a simulation dataset and two real datasets: JAK-STAT signal transduction pathway and Ras/Raf/MEK/ERK signaling transduction pathways datasets. The preliminary results show that EKF can accurately estimate the parameters and predict states in nonlinear state-space equations for modeling dynamic biochemical networks.

Nucleic acid sensor elements are proving increasingly useful in biotechnology and biomedical applications. A number of ligand-sensing, conformational-switching ribozymes (also known as allosteric ribozymes or aptazymes) have been generated by some combination of directed evolution or rational design. Such sensor elements typically fuse a molecular recognition domain (aptamer) with a catalytic signal generator (ribozyme). Although the rational design of aptazymes has begun to be explored, the relationships between the thermodynamics of aptazyme conformational changes and aptazyme performance in vitro and in vivo have not been examined in a quantitative framework. We have therefore developed a quantitative and predictive model for aptazymes as biosensors in vitro and as riboswitches in vivo . In the process, we have identified key relationships (or dimensionless parameters) that dictate aptazyme performance, and in consequence, established equations for precisely engineering aptazyme function. In particular, our analysis quantifies the intrinsic trade-off between ligand sensitivity and the dynamic range of activity. We were also able to determine how in vivo parameters, such as mRNA degradation rates, impact the design and function of aptazymes when used as riboswitches. Using this theoretical framework we were able to achieve quantitative agreement between our models and published data. In consequence, we are able to suggest experimental guidelines for quantitatively predicting the performance of aptazyme-based riboswitches. By identifying factors that limit the performance of previously published systems we were able to generate immediately testable hypotheses for their improvement. The robust theoretical framework and identified optimization parameters should now enable the precision design of aptazymes for biotechnological and clinical applications.

Time-delays are common in many physical and biological systems and they give rise to complex dynamic phenomena. The elementary processes involved in template biopolymerization, such as mRNA and protein synthesis, introduce significant time delays. However, there is not currently a systematic mapping between the individual mechanistic parameters and the time delays in these networks. We present here the development of mathematical, time-delay models for protein translation, based on PDE models, which in turn are derived through systematic approximations of first-principles mechanistic models. Theoretical analysis suggests that the key features that determine the time-delays and the agreement between the time-delay and the mechanistic models are ribosome density and distribution, i.e., the number of ribosomes on the mRNA chain relative to their maximum and their distribution along the mRNA chain. Based on analytical considerations and on computational studies, we show that the steady-state and dynamic responses of the time-delay models are in excellent agreement with the detailed mechanistic models, under physiological conditions that correspond to uniform ribosome distribution and for ribosome density up to 70%. The methodology presented here can be used for the development of reduced time-delay models of mRNA synthesis and large genetic networks. The good agreement between the time-delay and the mechanistic models will allow us to use the reduced model and advanced computational methods from nonlinear dynamics in order to perform studies that are not practical using the large-scale mechanistic models.