Javier Buceta

Noise-induced spatial patterns

By modifying the spatial coupling a la Swift-Hohenberg in the model introduced in Phys. Rev. Lett. 73 (1994) 3395, we obtain a system that displays noise-induced spatial patterns. We present a mean field theory of this phenomenon and verify some of its predictions by numerical simulations.

Extinction in population dynamics.

We study a generic reaction-diffusion model for single-species population dynamics that includes reproduction, death, and competition. The population is assumed to be confined in a refuge beyond which conditions are so harsh that they lead to certain extinction. Standard continuum mean field models in one dimension yield a critical refuge length L(c) such that a population in a refuge larger than this is assured survival. Herein we extend the model to take into account the discreteness and finiteness of the population, which leads us to a stochastic description. We present a particular critical criterion for likely extinction, namely, that the standard deviation of the population be equal to the mean. According to this criterion, we find that while survival can no longer be guaranteed for any refuge size, for sufficiently weak competition one can make the refuge large enough (certainly larger than L(c)) to cause extinction to be unlikely. However, beyond a certain value of the competition rate parameter it is no longer possible to escape a likelihood of extinction even in an infinite refuge. These unavoidable fluctuations therefore have a severe impact on refuge design issues.

Stationary and oscillatory spatial patterns induced by global periodic switching.

We propose a new mechanism for pattern formation based on the global alternation of two dynamics, neither of which exhibits patterns. When driven by either one of the separate dynamics, the system goes to a spatially homogeneous state associated with that dynamics. However, when the two dynamics are globally alternated sufficiently rapidly, the system exhibits stationary spatial patterns. Somewhat slower switching leads to oscillatory patterns. We support our findings by numerical simulations and discuss the results in terms of the symmetries of the system and the ratio of two relevant characteristic times, the switching period and the relaxation time to a homogeneous state in each separate dynamics.

Robustness and Stability of the Gene Regulatory Network Involved in DV Boundary Formation in the Drosophila Wing

Gene regulatory networks have been conserved during evolution. The Drosophila wing and the vertebrate hindbrain share the gene network involved in the establishment of the boundary between dorsal and ventral compartments in the wing and adjacent rhombomeres in the hindbrain. A positive feedback-loop between boundary and non-boundary cells and mediated by the activities of Notch and Wingless/Wnt-1 leads to the establishment of a Notch dependent organizer at the boundary. By means of a Systems Biology approach that combines mathematical modeling and both in silico and in vivo experiments in the Drosophila wing primordium, we modeled and tested this regulatory network and present evidence that a novel property, namely refractoriness to the Wingless signaling molecule, is required in boundary cells for the formation of a stable dorsal-ventral boundary. This new property has been validated in vivo, promotes mutually exclusive domains of Notch and Wingless activities and confers stability to the dorsal-ventral boundary. A robustness analysis of the regulatory network complements our results and ensures its biological plausibility.

Dynamics and mechanical stability of the developing dorsoventral organizer of the wing imaginal disc.

Shaping the primordia during development relies on forces and mechanisms able to control cell segregation. In the imaginal discs of Drosophila the cellular populations that will give rise to the dorsal and ventral parts on the wing blade are segregated and do not intermingle. A cellular population that becomes specified by the boundary of the dorsal and ventral cellular domains, the so-called organizer, controls this process. In this paper we study the dynamics and stability of the dorsal-ventral organizer of the wing imaginal disc of Drosophila as cell proliferation advances. Our approach is based on a vertex model to perform in silico experiments that are fully dynamical and take into account the available experimental data such as: cell packing properties, orientation of the cellular divisions, response upon membrane ablation, and robustness to mechanical perturbations induced by fast growing clones. Our results shed light on the complex interplay between the cytoskeleton mechanics, the cell cycle, the cell growth, and the cellular interactions in order to shape the dorsal-ventral organizer as a robust source of positional information and a lineage controller. Specifically, we elucidate the necessary and sufficient ingredients that enforce its functionality: distinctive mechanical properties, including increased tension, longer cell cycle duration, and a cleavage criterion that satisfies the Hertwig rule. Our results provide novel insights into the developmental mechanisms that drive the dynamics of the DV organizer and set a definition of the so-called Notch fence model in quantitative terms.

Interplay of cytoskeletal activity and lipid phase stability in dynamic protein recruitment and clustering

Recent experiments have revealed that some membrane proteins aggregate to form clusters. This type of process has been proven to be dynamic and to be actively maintained by external kinetics. Additionally, this dynamic recruiting is cholesterol- and actin-dependent, suggesting that raft organization and cytoskeleton rearrangement play a crucial role. In the present study, we propose a simple model that provides a general framework to describe the dynamical behavior of lipid-protein assemblies. Our results suggest that lipid-mediated interactions and cytoskeleton-anchored proteins contribute to the modulation of such behavior. In particular, we find a resonant condition between the membrane protein and cytoskeleton dynamics that results in the invariance of the ratio of clustered proteins that is found in in vivo experimental observations.

Comprehensive study of phase transitions in relaxational systems with field-dependent coefficients

We present a comprehensive study of phase transitions in single-field systems that relax to a nonequilibrium global steady state. The mechanism we focus on is not the so-called Stratonovich drift combined with collective effects, but is instead similar to the one associated with noise-induced transitions in the manner of Horsthemke-Lefever in zero-dimensional systems. As a consequence, the noise interpretation (e.g., Ito vs Stratonovich) merely shifts the phase boundaries. With the help of a mean-field approximation, we present a broad qualitative picture of the various phase diagrams that can be found in these systems. To complement the theoretical analysis we present numerical simulations that confirm the findings of the mean-field theory.

Stochastic Stabilization of Phenotypic States: The Genetic Bistable Switch as a Case Study

We study by means of analytical calculation and stochastic simulations how intrinsic noise modifies the bifurcation diagram of gene regulatory processes that can be effectively described by the Langevin formalism. In a general context, our study raises the intriguing question of how biochemical fluctuations redesign the epigenetic landscape in differentiation processes. We have applied our findings to a general class of regulatory processes that includes the simplest case that displays a bistable behavior and hence phenotypic variability: the genetic auto-activating switch. Thus, we explain why and how the noise promotes the stability of the low-state phenotype of the switch and show that the bistable region is extended when increasing the intensity of the fluctuations. This phenomenology is found in a simple one-dimensional model of the genetic switch as well as in a more detailed model that takes into account the binding of the protein to the promoter region. Altogether, we prescribe the analytical means to understand and quantify the noise-induced modifications of the bifurcation points for a general class of regulatory processes where the genetic bistable switch is included.

Spatial patterns induced purely by dichotomous disorder.

We study conditions under which spatially extended systems with coupling in the manner of Swift and Hohenberg exhibit spatial patterns induced purely by the presence of quenched dichotomous disorder. Complementing the theoretical results based on a generalized mean-field approximation, we also present numerical simulations of particular dynamical systems that exhibit the proposed phenomenology.

Patterns in reaction–diffusion systems generated by global alternation of dynamics

We recently proposed a mechanism for inducing a Turing instability by alternation of reaction–diffusion dynamics each of which is pattern-free. Herein we shed light on the question of which particular switching schemes produce that instability and which do not. We also consider a particular reaction–diffusion model to illustrate the mechanism.