Bistable Toggle Research Articles

Editorial for the special issue “Control‐theoretic approaches for systems in the life sciences”

Editorial for the special issue “Control<scp>‐theoretic</scp> approaches for systems in the life sciences”

Data-Driven Operator Theoretic Methods for Phase Space Learning and Analysis

This paper uses data-driven operator theoretic approaches to explore the global phase space of a dynamical system. We defined conditions for discovering new invariant subsets in the state space of a dynamical system starting from an invariant subset based on the spectral properties of the Koopman operator. When the system evolution is known locally in several invariant subsets in the state space of a dynamical system, a phase space stitching result is derived that yields the global Koopman operator. Additionally, in the case of equivariant systems, a phase space stitching result is developed to identify the global Koopman operator using the symmetry properties between the invariant subsets of the dynamical system and time-series data from any one of the invariant subsets. Finally, these results are extended to topologically conjugate dynamical systems; in particular, the relation between the Koopman tuple of topologically conjugate systems is established. The proposed results are demonstrated on several second-order nonlinear dynamical systems including a bistable toggle switch. Our method elucidates a strategy for designing discovery experiments: experiment execution can be done in many steps, and models from different invariant subsets can be combined to approximate the global Koopman operator.

Bi-stable toggle switching in magnetic tunnel junctions using sub-nanosecond Joule heat pulses

In this study, bi-stable toggle magnetization switching in magnetic tunnel junctions induced by electrically injected sub-nanosecond unipolar heat pulses was demonstrated. The switching probability of magnetization between bi-stable states was estimated by applying 0.5 ns voltage pulses and measuring the perpendicular component of the magnetization direction. The maximum switching probability was approximately 70%, suggesting that bi-stable toggle magnetization switching was induced by the torque created by the Joule-heat-induced magnetic anisotropy change. Joule-heat-driven magnetization switching has the potential to become a fundamental technology for fast spin control.

An Automated Tabletop Continuous Culturing System with Multicolor Fluorescence Monitoring for Microbial Gene Expression and Long-Term Population Dynamics

Real-time monitoring of gene expression dynamics and population levels in a multispecies microbial community could enable the study of the role of changing gene expression patterns on eco-evolutionary outcomes. Here we report the design and validation of a unique experimental platform with an in situ fluorescence measurement system that has high dynamic range and temporal resolution and is capable of monitoring multiple fluorophores for long-term gene expression and population dynamics experiments. We demonstrate the capability of our system to capture gene expression dynamics in response to external perturbations in two synthetic genetic systems: a simple inducible genetic circuit and a bistable toggle switch. Finally, in exploring the population dynamics of a two species microbial community, we show that our system can capture the switch between competitive exclusion and long-term coexistence in response to different nutrient conditions.

Multicellular Feedback Control of a Genetic Toggle-Switch in Microbial Consortia

We describe a multicellular approach to control a target cell population endowed with a bistable toggle-switch. The idea is to engineer a synthetic microbial consortium consisting of three different cell populations. In such a consortium, two populations, the Togglers, responding to some reference input, can induce the switch of a bistable memory mechanism in a third population, the Targets, so as to activate or deactivate some additional functionalities in the cells. Communication among the three populations is established by orthogonal quorum sensing molecules that are used to close a feedback control loop across the populations. The control design is validated via in-silico experiments in BSim, a realistic agent-based simulator of bacterial populations.

Sharing Resources Can Lead to Monostability in a Network of Bistable Toggle Switches

The vision of synthetic biology is to engineer complex cellular systems in a modular fashion. Unfortunately, multiple factors hinder such modular design, thus the scalability of rationally engineering synthetic gene circuits. One major barrier emerges due to the limited availability of shared cellular resources, intertwining the behavior of otherwise disconnected components. It was recently demonstrated that these unwanted and pervasive coupling effects can be accurately predicted both in vitro and in vivo . Here, we not only analyze how these effects influence the collective behavior of multiple toggle switches but we also illustrate how to leverage this predominantly disadvantageous coupling. In particular, we prove that by turning on/off a load module, the network of toggle switches can be switched between monostable and multistable modes. Importantly, this can be achieved without any direct regulatory connection among the toggle switches or between the load module and any of the toggle switches, thus presenting a major experimental advantage over relying on traditional regulatory linkages. This idea of exploiting and leveraging resource competition is illustrated through the design of a cellular random number generator.

Balancing a genetic toggle switch by real-time feedback control and periodic forcing

Cybergenetics is a novel field of research aiming at remotely pilot cellular processes in real-time with to leverage the biotechnological potential of synthetic biology. Yet, the control of only a small number of genetic circuits has been tested so far. Here we investigate the control of multistable gene regulatory networks, which are ubiquitously found in nature and play critical roles in cell differentiation and decision-making. Using an in silico feedback control loop, we demonstrate that a bistable genetic toggle switch can be dynamically maintained near its unstable equilibrium position for extended periods of time. Importantly, we show that a direct method based on dual periodic forcing is sufficient to simultaneously maintain many cells in this undecided state. These findings pave the way for the control of more complex cell decision-making systems at both the single cell and the population levels, with vast fundamental and biotechnological applications.

Cell-cycle transitions: a common role for stoichiometric inhibitors.

The cell division cycle is the process by which eukaryotic cells replicate their chromosomes and partition them to two daughter cells. To maintain the integrity of the genome, proliferating cells must be able to block progression through the division cycle at key transition points (called "checkpoints") if there have been problems in the replication of the chromosomes or their biorientation on the mitotic spindle. These checkpoints are governed by protein-interaction networks, composed of phase-specific cell-cycle activators and inhibitors. Examples include Cdk1:Clb5 and its inhibitor Sic1 at the G1/S checkpoint in budding yeast, APC:Cdc20 and its inhibitor MCC at the mitotic checkpoint, and PP2A:B55 and its inhibitor, alpha-endosulfine, at the mitotic-exit checkpoint. Each of these inhibitors is a substrate as well as a stoichiometric inhibitor of the cell-cycle activator. Because the production of each inhibitor is promoted by a regulatory protein that is itself inhibited by the cell-cycle activator, their interaction network presents a regulatory motif characteristic of a "feedback-amplified domineering substrate" (FADS). We describe how the FADS motif responds to signals in the manner of a bistable toggle switch, and then we discuss how this toggle switch accounts for the abrupt and irreversible nature of three specific cell-cycle checkpoints.

The Switch in a Genetic Toggle System with Lévy Noise.

A bistable toggle switch is a paradigmatic model in the field of biology. The dynamics of the system induced by Gaussian noise has been intensively investigated, but Gaussian noise cannot incorporate large bursts typically occurring in real experiments. This paper aims to examine effects of variations from one protein imposed by a non-Gaussian Lévy noise, which is able to describe even large jumps, on the coherent switch and the on/off switch via the steady-state probability density, the joint steady-state probability density, and the mean first passage time. We find that a large burst of one protein due to the Lévy noises can induce coherent switches even with small noise intensities in contrast to the Gaussian case which requires large intensities for this. The influences of the stability index, skewness parameter and noise intensity on the on/off switch are analyzed, leading to an adjustment of the concentrations of both proteins and a decision which stable point to stay most. The mean first passage times show complex effects under Lévy noise, especially the stability index and skewness parameter. Our results also imply that the presence of non-Gaussian Lévy noises has fundamentally changed the escape mechanism in such a system compared with Gaussian noise.

Creating Single-Copy Genetic Circuits

Synthetic biology is increasingly used to develop sophisticated living devices for basic and applied research. Many of these genetic devices are engineered using multi-copy plasmids, but as the field progresses from proof-of-principle demonstrations to practical applications, it is important to develop single-copy synthetic modules that minimize consumption of cellular resources and can be stably maintained as genomic integrants. Here we use empirical design, mathematical modeling, and iterative construction and testing to build single-copy, bistable toggle switches with improved performanceand reduced metabolic load that can be stably integrated into the host genome. Deterministic and stochastic models led us to focus on basal transcription to optimize circuit performance and helped to explain the resulting circuit robustness across alarge range of component expression levels. The design parameters developed here provide important guidance for future efforts to convert functional multi-copy gene circuits into optimized single-copy circuits for practical, real-world use.

Solution of the chemical master equation by radial basis functions approximation with interface tracking.

BackgroundThe chemical master equation is the fundamental equation of stochastic chemical kinetics. This differential-difference equation describes temporal evolution of the probability density function for states of a chemical system. A state of the system, usually encoded as a vector, represents the number of entities or copy numbers of interacting species, which are changing according to a list of possible reactions. It is often the case, especially when the state vector is high-dimensional, that the number of possible states the system may occupy is too large to be handled computationally. One way to get around this problem is to consider only those states that are associated with probabilities that are greater than a certain threshold level.ResultsWe introduce an algorithm that significantly reduces computational resources and is especially powerful when dealing with multi-modal distributions. The algorithm is built according to two key principles. Firstly, when performing time integration, the algorithm keeps track of the subset of states with significant probabilities (essential support). Secondly, the probability distribution that solves the equation is parametrised with a small number of coefficients using collocation on Gaussian radial basis functions. The system of basis functions is chosen in such a way that the solution is approximated only on the essential support instead of the whole state space.DiscussionIn order to demonstrate the effectiveness of the method, we consider four application examples: a) the self-regulating gene model, b) the 2-dimensional bistable toggle switch, c) a generalisation of the bistable switch to a 3-dimensional tristable problem, and d) a 3-dimensional cell differentiation model that, depending on parameter values, may operate in bistable or tristable modes. In all multidimensional examples the manifold containing the system states with significant probabilities undergoes drastic transformations over time. This fact makes the examples especially challenging for numerical methods.ConclusionsThe proposed method is a new numerical approach permitting to approximately solve a wide range of problems that have been hard to tackle until now. A full representation of multi-dimensional distributions is recovered. The method is especially attractive when dealing with models that yield solutions of a complex structure, for instance, featuring multi-stability.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-015-0210-y) contains supplementary material, which is available to authorized users.

DNA-Binding Kinetics Determines the Mechanism of Noise-Induced Switching in Gene Networks

Gene regulatory networks are multistable dynamical systems in which attractor states represent cell phenotypes. Spontaneous, noise-induced transitions between these states are thought to underlie critical cellular processes, including cell developmental fate decisions, phenotypic plasticity in fluctuating environments, and carcinogenesis. As such, there is increasing interest in the development of theoretical and computational approaches that can shed light on the dynamics of these stochastic state transitions in multistable gene networks. We applied a numerical rare-event sampling algorithm to study transition paths of spontaneous noise-induced switching for a ubiquitous gene regulatory network motif, the bistable toggle switch, in which two mutually repressive genes compete for dominant expression. We find that the method can efficiently uncover detailed switching mechanisms that involve fluctuations both in occupancies of DNA regulatory sites and copy numbers of protein products. In addition, we show that the rate parameters governing binding and unbinding of regulatory proteins to DNA strongly influence the switching mechanism. In a regime of slow DNA-binding/unbinding kinetics, spontaneous switching occurs relatively frequently and is driven primarily by fluctuations in DNA-site occupancies. In contrast, in a regime of fast DNA-binding/unbinding kinetics, switching occurs rarely and is driven by fluctuations in levels of expressed protein. Our results demonstrate how spontaneous cell phenotype transitions involve collective behavior of both regulatory proteins and DNA. Computational approaches capable of simulating dynamics over many system variables are thus well suited to exploring dynamic mechanisms in gene networks.

A termination criterion for parameter estimation in stochastic models in systems biology

Parameter estimation procedures are a central aspect of modeling approaches in systems biology. They are often computationally expensive, especially when the models take stochasticity into account. Typically parameter estimation involves the iterative optimization of an objective function that describes how well the model fits some measured data with a certain set of parameter values. In order to limit the computational expenses it is therefore important to apply an adequate stopping criterion for the optimization process, so that the optimization continues at least until a reasonable fit is obtained, but not much longer. In the case of stochastic modeling, at least some parameter estimation schemes involve an objective function that is itself a random variable. This means that plain convergence tests are not a priori suitable as stopping criteria.This article suggests a termination criterion suited to optimization problems in parameter estimation arising from stochastic models in systems biology. The termination criterion is developed for optimization algorithms that involve populations of parameter sets, such as particle swarm or evolutionary algorithms. It is based on comparing the variance of the objective function over the whole population of parameter sets with the variance of repeated evaluations of the objective function at the best parameter set. The performance is demonstrated for several different algorithms. To test the termination criterion we choose polynomial test functions as well as systems biology models such as an Immigration-Death model and a bistable genetic toggle switch. The genetic toggle switch is an especially challenging test case as it shows a stochastic switching between two steady states which is qualitatively different from the model behavior in a deterministic model.

Using noise to control heterogeneity of isogenic populations in homogenous environments

We explore the extent to which the phenotypes of individual, genetically identical cells can be controlled independently from each other using only a single homogeneous environmental input. We show that such control is theoretically impossible if restricted to a deterministic setting, but it can be achieved readily if one exploits heterogeneities introduced at the single-cell level due to stochastic fluctuations in gene regulation. Using stochastic analyses of a bistable genetic toggle switch, we develop a control strategy that maximizes the chances that a chosen cell will express one phenotype, while the rest express another. The control mechanism uses UV radiation to enhance identically protein degradation in all cells. Control of individual cells is made possible only by monitoring stochastic protein fluctuations and applying UV control at favorable times and levels. For two identical cells, our stochastic control law can drive protein expression of a chosen cell above its neighbor with a better than 99% success rate. In a population of 30 identical cells, we can drive a given cell to remain consistently within the top 20%. Although cellular noise typically impairs predictability for biological responses, our results show that it can also simultaneously improve controllability for those same responses.

Control of gene expression using a red- and far-red light–responsive bi-stable toggle switch

Light-triggered gene expression systems offer an unprecedented spatiotemporal resolution that cannot be achieved with classical chemically inducible genetic tools. Here we describe a protocol for red light-responsive gene expression in mammalian cells. This system can be toggled between stable ON and OFF states by short pulses of red and far-red light, respectively. In the protocol, CHO-K1 cells are transfected to allow red light-inducible expression of the secreted alkaline phosphatase (SEAP) reporter, and gene expression is tuned by illumination with light of increasing wavelengths. As a starting point for elaborate red light-responsive gene expression, we outline the reversible activation of gene expression and describe how a spatial pattern can be 'printed' on a monolayer of cells by using a photomask. The core protocol requires only 4 d from seeding of the cells to reporter quantification, and other than light-emitting diode (LED) illumination boxes no elaborate equipment is required.

Stability of bacterial toggle switches is enhanced by cell-cycle lengthening by several orders of magnitude

Bistable regulatory elements are important for nongenetic inheritance, increase of cell-to-cell heterogeneity allowing adaptation, and robust responses at the population level. Here, we study computationally the bistable genetic toggle switch-a small regulatory network consisting of a pair of mutual repressors-in growing and dividing bacteria. We show that as cells with an inhibited growth exhibit high stability of toggle states, cell growth and divisions lead to a dramatic increase of toggling rates. The toggling rates were found to increase with rate of cell growth, and can be up to six orders of magnitude larger for fast growing cells than for cells with the inhibited growth. The effect is caused mainly by the increase of protein and mRNA burst sizes associated with the faster growth. The observation that fast growth dramatically destabilizes toggle states implies that rapidly growing cells may vigorously explore the epigenetic landscape enabling nongenetic evolution, while cells with inhibited growth adhere to the local optima. This can be a clever population strategy that allows the slow growing (but stress resistant) cells to survive long periods of unfavorable conditions. Simultaneously, at favorable conditions, this stress resistant (but slowly growing-or not growing) subpopulation may be replenished due to a high switching rate from the fast growing population.

Multi-Finite Buffer Method for Direct Solution of Discrete Chemical Master Equation

Many biological reaction networks are intrinsically stochastic due to random thermal fluctuations. Stochasticity is significant when the copy number of participating molecular species are small. The discrete Chemical Master Equation (dCME) provides a general framework to study the underlying stochastic processes of biological networks. Although the direct solution of dCME is advantageous over approximation methods such as the Langevin and the Fokker-Planck equations, it is challenging to obtain exact solution to the dCME. The Finite Buffer dCME Method provides an optimal algorithm to enumerate the underlying state space, and has been used to compute the exact solutions of dCME for several problems. In this study, we extend the finite buffer method by introducing multiple buffer queues for more effective construction of the state space and for quantitative control of errors, when buffer sizes are limited. By introducing the concept of open Independent Birth-Death (IBD) units, which are non-intersecting sets of reactions grouped by common synthesis and degradation reactions, we can enumerate the state space optimally and assess errors for each open IBD from the probability of buffer depletion, when the buffer size is limited. We also describe theoretical estimation of the error bound for any given buffer size of an IBD, so its buffer size can be optimized. We demonstrate the effectiveness of our approach in computing time-evolving and steady state probability landscapes, as well as first passage time distribution using the birth-death process, the bistable Schlogl model, the bistable toggle switch model, and the phage lambda lysogenic-lytic switching model as examples. We also compare our results with those using other methods.

General Applicability of Synthetic Gene-Overexpression for Cell-Type Ratio Control via Reprogramming

Control of the cell-type ratio in multistable systems requires wide-range control of the initial states of cells. Here, using a synthetic circuit in E. coli, we describe the use of a simple gene-overexpression system combined with a bistable toggle switch, for the purposes of enabling the wide-range control of cellular states and thus generating arbitrary cell-type ratios. Theoretically, overexpression induction temporarily alters the bistable system to a monostable system, in which the location of the single steady state of cells can be manipulated over a wide range by regulating the overexpression levels. This induced cellular state becomes the initial state of the basal bistable system upon overexpression cessation, which restores the original bistable system. We experimentally demonstrated that the overexpression induced a monomodal cell distribution, and subsequent overexpression withdrawal generated a bimodal distribution. Furthermore, as designed theoretically, regulating the overexpression levels by adjusting the concentrations of small molecules generated arbitrary cell-type ratios.

A general strategy for cellular reprogramming: The importance of transcription factor cross-repression

Transcription factor cross-repression is an important concept in cellular differentiation. A bistable toggle switch constitutes a molecular mechanism that determines cellular commitment and provides stability to transcriptional programs of binary cell fate choices. Experiments support that perturbations of these toggle switches can interconvert these binary cell fate choices, suggesting potential reprogramming strategies. However, more complex types of cellular transitions could involve perturbations of combinations of different types of multistable motifs. Here, we introduce a method that generalizes the concept of transcription factor cross-repression to systematically predict sets of genes, whose perturbations induce cellular transitions between any given pair of cell types. Furthermore, to our knowledge, this is the first method that systematically makes these predictions without prior knowledge of potential candidate genes and pathways involved, providing guidance on systems where little is known. Given the increasing interest of cellular reprogramming in medicine and basic research, our method represents a useful computational methodology to assist researchers in the field in designing experimental strategies.

A red/far-red light-responsive bi-stable toggle switch to control gene expression in mammalian cells

Growth and differentiation of multicellular systems is orchestrated by spatially restricted gene expression programs in specialized subpopulations. The targeted manipulation of such processes by synthetic tools with high-spatiotemporal resolution could, therefore, enable a deepened understanding of developmental processes and open new opportunities in tissue engineering. Here, we describe the first red/far-red light-triggered gene switch for mammalian cells for achieving gene expression control in time and space. We show that the system can reversibly be toggled between stable on- and off-states using short light pulses at 660 or 740 nm. Red light-induced gene expression was shown to correlate with the applied photon number and was compatible with different mammalian cell lines, including human primary cells. The light-induced expression kinetics were quantitatively analyzed by a mathematical model. We apply the system for the spatially controlled engineering of angiogenesis in chicken embryos. The system’s performance combined with cell- and tissue-compatible regulating red light will enable unprecedented spatiotemporally controlled molecular interventions in mammalian cells, tissues and organisms.