Ruben Perez-Carrasco

A theoretical framework for the regulation of Shh morphogen-controlled gene expression

How morphogen gradients govern the pattern of gene expression in developing tissues is not well understood. Here, we describe a statistical thermodynamic model of gene regulation that combines the activity of a morphogen with the transcriptional network it controls. Using Sonic hedgehog (Shh) patterning of the ventral neural tube as an example, we show that the framework can be used together with the principled parameter selection technique of approximate Bayesian computation to obtain a dynamical model that accurately predicts tissue patterning. The analysis indicates that, for each target gene regulated by Gli, which is the transcriptional effector of Shh signalling, there is a neutral point in the gradient, either side of which altering the Gli binding affinity has opposite effects on gene expression. This explains recent counterintuitive experimental observations. The approach is broadly applicable and provides a unifying framework to explain the temporospatial pattern of morphogen-regulated gene expression.

A Statistical Approach Reveals Designs for the Most Robust Stochastic Gene Oscillators.

The engineering of transcriptional networks presents many challenges due to the inherent uncertainty in the system structure, changing cellular context, and stochasticity in the governing dynamics. One approach to address these problems is to design and build systems that can function across a range of conditions; that is they are robust to uncertainty in their constituent components. Here we examine the parametric robustness landscape of transcriptional oscillators, which underlie many important processes such as circadian rhythms and the cell cycle, plus also serve as a model for the engineering of complex and emergent phenomena. The central questions that we address are: Can we build genetic oscillators that are more robust than those already constructed? Can we make genetic oscillators arbitrarily robust? These questions are technically challenging due to the large model and parameter spaces that must be efficiently explored. Here we use a measure of robustness that coincides with the Bayesian model evidence, combined with an efficient Monte Carlo method to traverse model space and concentrate on regions of high robustness, which enables the accurate evaluation of the relative robustness of gene network models governed by stochastic dynamics. We report the most robust two and three gene oscillator systems, plus examine how the number of interactions, the presence of autoregulation, and degradation of mRNA and protein affects the frequency, amplitude, and robustness of transcriptional oscillators. We also find that there is a limit to parametric robustness, beyond which there is nothing to be gained by adding additional feedback. Importantly, we provide predictions on new oscillator systems that can be constructed to verify the theory and advance design and modeling approaches to systems and synthetic biology.

Intrinsic Noise Profoundly Alters the Dynamics and Steady State of Morphogen-Controlled Bistable Genetic Switches

During tissue development, patterns of gene expression determine the spatial arrangement of cell types. In many cases, gradients of secreted signalling molecules—morphogens—guide this process by controlling downstream transcriptional networks. A mechanism commonly used in these networks to convert the continuous information provided by the gradient into discrete transitions between adjacent cell types is the genetic toggle switch, composed of cross-repressing transcriptional determinants. Previous analyses have emphasised the steady state output of these mechanisms. Here, we explore the dynamics of the toggle switch and use exact numerical simulations of the kinetic reactions, the corresponding Chemical Langevin Equation, and Minimum Action Path theory to establish a framework for studying the effect of gene expression noise on patterning time and boundary position. This provides insight into the time scale, gene expression trajectories and directionality of stochastic switching events between cell states. Taking gene expression noise into account predicts that the final boundary position of a morphogen-induced toggle switch, although robust to changes in the details of the noise, is distinct from that of the deterministic system. Moreover, the dramatic increase in patterning time close to the boundary predicted from the deterministic case is substantially reduced. The resulting stochastic switching introduces differences in patterning time along the morphogen gradient that result in a patterning wave propagating away from the morphogen source with a velocity determined by the intrinsic noise. The wave sharpens and slows as it advances and may never reach steady state in a biologically relevant time. This could explain experimentally observed dynamics of pattern formation. Together the analysis reveals the importance of dynamical transients for understanding morphogen-driven transcriptional networks and indicates that gene expression noise can qualitatively alter developmental patterning.

Catch bond drives stator mechanosensitivity in the bacterial flagellar motor

Significance The bacterial flagellar motor (BFM) is the rotary motor powering swimming of many motile bacteria. Many of the components of this molecular machine are dynamic, a property which allows the cell to optimize its behavior in accordance with the surrounding environment. A prime example is the stator unit, a membrane-bound ion channel that is responsible for applying torque to the rotor. The stator units are mechanosensitive, with the number of engaged units dependent on the viscous load on the motor. We measure the kinetics of the stators as a function of the viscous load and find that the mechanosensitivity of the BFM is governed by a catch bond: a counterintuitive type of bond that becomes stronger under force. The bacterial flagellar motor (BFM) is the rotary motor that rotates each bacterial flagellum, powering the swimming and swarming of many motile bacteria. The torque is provided by stator units, ion motive force-powered ion channels known to assemble and disassemble dynamically in the BFM. This turnover is mechanosensitive, with the number of engaged units dependent on the viscous load experienced by the motor through the flagellum. However, the molecular mechanism driving BFM mechanosensitivity is unknown. Here, we directly measure the kinetics of arrival and departure of the stator units in individual motors via analysis of high-resolution recordings of motor speed, while dynamically varying the load on the motor via external magnetic torque. The kinetic rates obtained, robust with respect to the details of the applied adsorption model, indicate that the lifetime of an assembled stator unit increases when a higher force is applied to its anchoring point in the cell wall. This provides strong evidence that a catch bond (a bond strengthened instead of weakened by force) drives mechanosensitivity of the flagellar motor complex. These results add the BFM to a short, but growing, list of systems demonstrating catch bonds, suggesting that this “molecular strategy” is a widespread mechanism to sense and respond to mechanical stress. We propose that force-enhanced stator adhesion allows the cell to adapt to a heterogeneous environmental viscosity and may ultimately play a role in surface-sensing during swarming and biofilm formation.

Combining a Toggle Switch and a Repressilator within the AC-DC Circuit Generates Distinct Dynamical Behaviors

Summary Although the structure of a genetically encoded regulatory circuit is an important determinant of its function, the relationship between circuit topology and the dynamical behaviors it can exhibit is not well understood. Here, we explore the range of behaviors available to the AC-DC circuit. This circuit consists of three genes connected as a combination of a toggle switch and a repressilator. Using dynamical systems theory, we show that the AC-DC circuit exhibits both oscillations and bistability within the same region of parameter space; this generates emergent behaviors not available to either the toggle switch or the repressilator alone. The AC-DC circuit can switch on oscillations via two distinct mechanisms, one of which induces coherence into ensembles of oscillators. In addition, we show that in the presence of noise, the AC-DC circuit can behave as an excitable system capable of spatial signal propagation or coherence resonance. Together, these results demonstrate how combinations of simple motifs can exhibit multiple complex behaviors.

Minimum Action Path theory reveals the details of stochastic biochemical transitions out of oscillatory cellular states

Cell state determination is the outcome of intrinsically stochastic biochemical reactions. Transitions between such states are studied as noise-driven escape problems in the chemical species space. Escape can occur via multiple possible multidimensional paths, with probabilities depending nonlocally on the noise. Here we characterize the escape from an oscillatory biochemical state by minimizing the Freidlin-Wentzell action, deriving from it the stochastic spiral exit path from the limit cycle. We also use the minimized action to infer the escape time probability density function.

Memory functions reveal structural properties of gene regulatory networks

Gene regulatory networks (GRNs) control cellular function and decision making during tissue development and homeostasis. Mathematical tools based on dynamical systems theory are often used to model these networks. The size and complexity of these models, however, mean that their behaviour is not always intuitive and the underlying mechanisms can be difficult to decipher. For this reason, methods that simplify and aid exploration of complex networks are necessary. To this end we develop a broadly applicable form of the Zwanzig-Mori projection. By first converting a thermodynamic state ensemble model of gene regulation into mass action reactions we derive a general method that produces a set of time evolution equations for a subset of components of a network. The influence of the rest of the network, the bulk, is captured by memory functions that describe how the subnetwork reacts to its own past state via components in the bulk. We demonstrate that the method captures the full dynamics of a simple cross-repressive transcriptional motif while simplifying analysis of the subnetwork and providing a natural interpretation of the memory functions. We then apply the approach to a GRN from the vertebrate neural tube, a well characterised developmental transcriptional network composed of four interacting transcription factors. This reveals the function of specific links within the neural tube network and identifies features of the regulatory structure that increase the robustness of the network to initial conditions. Taken together, the study provides evidence that Zwanzig-Mori projections offer powerful and effective tools for simplifying and exploring the behaviour of GRNs.Gene regulatory networks (GRNs) control cellular function and decision making during tissue development and homeostasis. Mathematical tools based on dynamical systems theory are often used to model these networks, but the size and complexity of these models mean that their behaviour is not always intuitive and the underlying mechanisms can be difficult to decipher. For this reason, methods that simplify and aid exploration of complex networks are necessary. To this end we develop a broadly applicable form of the Zwanzig-Mori projection. By first converting a thermodynamic state ensemble model of gene regulation into mass action reactions we derive a general method that produces a set of time evolution equations for a subset of components of a network. The influence of the rest of the network, the bulk, is captured by memory functions that describe how the subnetwork reacts to its own past state via components in the bulk. These memory functions provide probes of near-steady state dynamics, revealing information not easily accessible otherwise. We illustrate the method on a simple cross-repressive transcriptional motif to show that memory functions not only simplify the analysis of the subnetwork but also have a natural interpretation. We then apply the approach to a GRN from the vertebrate neural tube, a well characterised developmental transcriptional network composed of four interacting transcription factors. The memory functions reveal the function of specific links within the neural tube network and identify features of the regulatory structure that specifically increase the robustness of the network to initial conditions. Taken together, the study provides evidence that Zwanzig-Mori projections offer powerful and effective tools for simplifying and exploring the behaviour of GRNs.

Deterministic and Stochastic Study for a Microscopic Angiogenesis Model: Applications to the Lewis Lung Carcinoma.

Angiogenesis modelling is an important tool to understand the underlying mechanisms yielding tumour growth. Nevertheless, there is usually a gap between models and experimental data. We propose a model based on the intrinsic microscopic reactions defining the angiogenesis process to link experimental data with previous macroscopic models. The microscopic characterisation can describe the macroscopic behaviour of the tumour, which stability analysis reveals a set of predicted tumour states involving different morphologies. Additionally, the microscopic description also gives a framework to study the intrinsic stochasticity of the reactive system through the resulting Langevin equation. To follow the goal of the paper, we use available experimental information on the Lewis lung carcinoma to infer meaningful parameters for the model that are able to describe the different stages of the tumour growth. Finally we explore the predictive capabilities of the fitted model by showing that fluctuations are determinant for the survival of the tumour during the first week and that available treatments can give raise to new stable tumour dormant states with a reduced vascular network.

A multiscale model of epigenetic heterogeneity reveals the kinetic routes of pathological cell fate reprogramming

The inherent capacity of somatic cells to switch their phenotypic status in response to damage stimuli in vivo might have a pivotal role in ageing and cancer. However, how the entry-exit mechanisms of phenotype reprogramming are established remains poorly understood. In an attempt to elucidate such mechanisms, we herein introduce a stochastic model of combined epigenetic regulation (ER)-gene regulatory network (GRN) to study the plastic phenotypic behaviours driven by ER heterogeneity. Furthermore, based on the existence of multiple scales, we formulate a method for stochastic model reduction, from which we derive an efficient hybrid simulation scheme that allows us to deal with such complex systems. Our analysis of the coupled system reveals a regime of tristability in which pluripotent stem-like and differentiated steady-states coexist with a third indecisive state. Crucially, ER heterogeneity of differentiation genes is for the most part responsible for conferring abnormal robustness to pluripotent stem-like states. We then formulate epigenetic heterogeneity-based strategies capable of unlocking and facilitating the transit from differentiation-refractory (pluripotent stem-like) to differentiation-primed epistates. The application of the hybrid numerical method validated the likelihood of such switching involving solely kinetic changes in epigenetic factors. Our results suggest that epigenetic heterogeneity regulates the mechanisms and kinetics of phenotypic robustness of cell fate reprogramming. The occurrence of tunable switches capable of modifying the nature of cell fate reprogramming from pathological to physiological might pave the way for new therapeutic strategies to regulate reparative reprogramming in ageing and cancer. Author summary Certain modifications of the structure and functioning of the protein/DNA complex called chromatin can allow adult, fully differentiated cells to adopt a stem cell-like pluripotent state in a purely epigenetic manner, not involving changes in the underlying DNA sequence. Such reprogramming-like phenomena may constitute an innate reparative route through which human tissues respond to injury and could also serve as a novel regenerative strategy in human pathological situations in which tissue or organ repair is impaired. However, it should be noted that in vivo reprogramming would be capable of maintaining tissue homeostasis provided the acquisition of pluripotency features is strictly transient and accompanied by an accurate replenishment of the specific cell types being lost. Crucially, an excessive reprogramming to pluripotency in the absence of controlled re-differentiation would impair the repair or the replacement of damaged cells, thereby promoting pathological alterations of cell fate. A mechanistic understanding of how the degree of chromatin plasticity dictates the reparative versus pathological behaviour of in vivo reprogramming to rejuvenate aged tissues while preventing tumorigenesis is urgently needed, including especially the intrinsic epigenetic heterogeneity of the tissue resident cells being reprogrammed. We here introduce a novel method that mathematically captures how epigenetic heterogeneity is actually the driving force that governs the routes and kinetics to entry into and exit from a pathological pluripotent-like state. Moreover, our approach computationally validates the likelihood of unlocking chronic, unrestrained pluripotent states and drive their differentiation down the correct path by solely manipulating the intensity and direction of few epigenetic control switches. Our approach could inspire new therapeutic approaches based on in vivo cell reprogramming for efficient tissue regeneration and rejuvenation and cancer treatment.

Degradation rate uniformity determines success of oscillations in repressive feedback regulatory networks

Ring oscillators are biochemical circuits consisting of a ring of interactions capable of sustained oscillations. The nonlinear interactions between genes hinder the analytical insight into their function, usually requiring computational exploration. Here, we show that, despite the apparent complexity, the stability of the unique steady state in an incoherent feedback ring depends only on the degradation rates and a single parameter summarizing the feedback of the circuit. Concretely, we show that the range of regulatory parameters that yield oscillatory behaviour is maximized when the degradation rates are equal. Strikingly, this result holds independently of the regulatory functions used or number of genes. We also derive properties of the oscillations as a function of the degradation rates and number of nodes forming the ring. Finally, we explore the role of mRNA dynamics by applying the generic results to the specific case with two naturally different degradation timescales.