An Optogenetic Platform for Real-Time, Single-Cell Interrogation of Stochastic Transcriptional Regulation.

Transcription is a highly regulated and inherently stochastic process. The complexity of signal transduction and gene regulation makes it challenging to analyze how the dynamic activity of transcriptional regulators affects stochastic transcription. By combining a fast-acting, photo-regulatable transcription factor with nascent RNA quantification in live cells and an experimental setup for precise spatiotemporal delivery of light inputs, we constructed a platform for the real-time, single-cell interrogation of transcription in Saccharomyces cerevisiae. We show that transcriptional activation and deactivation are fast and memoryless. By analyzing the temporal activity of individual cells, we found that transcription occurs in bursts, whose duration and timing are modulated by transcription factor activity. Using our platform, we regulated transcription via light-driven feedback loops at the single-cell level. Feedback markedly reduced cell-to-cell variability and led to qualitative differences in cellular transcriptional dynamics. Our platform establishes a flexible method for studying transcriptional dynamics in single cells.

A universal biomolecular integral feedback controller for robust perfect adaptation.

Homeostasis is a recurring theme in biology that ensures that regulated variables robustly-and in some systems, completely-adapt to environmental perturbations. This robust perfect adaptation feature is achieved in natural circuits by using integral control, a negative feedback strategy that performs mathematical integration to achieve structurally robust regulation1,2. Despite its benefits, the synthetic realization of integral feedback in living cells has remained elusive owing to the complexity of the required biological computations. Here we prove mathematically that there is a single fundamental biomolecular controller topology3 that realizes integral feedback and achieves robust perfect adaptation in arbitrary intracellular networks with noisy dynamics. This adaptation property is guaranteed both for the population-average and for the time-average of single cells. On the basis of this concept, we genetically engineer a synthetic integral feedback controller in living cells4 and demonstrate its tunability and adaptation properties. A growth-rate control application in Escherichia coli shows the intrinsic capacity of our integral controller to deliver robustness and highlights its potential use as a versatile controller for regulation of biological variables in uncertain networks. Our results provide conceptual and practical tools in the area of cybergenetics3,5, for engineering synthetic controllers that steer the dynamics of living systems3-9.

Universal structural requirements for maximal robust perfect adaptation in biomolecular networks.

Adaptation is a running theme in biology. It allows a living system to survive and thrive in the face of unpredictable environments by maintaining key physiological variables at their desired levels through tight regulation. When one such variable is maintained at a certain value at the steady state despite perturbations to a single input, this property is called robust perfect adaptation (RPA). Here we address and solve the fundamental problem of maximal RPA (maxRPA), whereby, for a designated output variable, RPA is achieved with respect to perturbations in virtually all network parameters. In particular, we show that the maxRPA property imposes certain structural constraints on the network. We then prove that these constraints are fully characterized by simple linear algebraic stoichiometric conditions which differ between deterministic and stochastic descriptions of the dynamics. We use our results to derive a new internal model principle (IMP) for biomolecular maxRPA networks, akin to the celebrated IMP in control theory. We exemplify our results through several known biological examples of robustly adapting networks and construct examples of such networks with the aid of our linear algebraic characterization. Our results reveal the universal requirements for maxRPA in all biological systems, and establish a foundation for studying adaptation in general biomolecular networks, with important implications for both systems and synthetic biology.

A genetic mammalian proportional-integral feedback control circuit for robust and precise gene regulation.

The processes that keep a cell alive are constantly challenged by unpredictable changes in its environment. Cells manage to counteract these changes by employing sophisticated regulatory strategies that maintain a steady internal milieu. Recently, the antithetic integral feedback motif has been demonstrated to be a minimal and universal biological regulatory strategy that can guarantee robust perfect adaptation for noisy gene regulatory networks in Escherichia coli. Here, we present a realization of the antithetic integral feedback motif in a synthetic gene circuit in mammalian cells. We show that the motif robustly maintains the expression of a synthetic transcription factor at tunable levels even when it is perturbed by increased degradation or its interaction network structure is perturbed by a negative feedback loop with an RNA-binding protein. We further demonstrate an improved regulatory strategy by augmenting the antithetic integral motif with additional negative feedback to realize antithetic proportional-integral control. We show that this motif produces robust perfect adaptation while also reducing the variance of the regulated synthetic transcription factor. We demonstrate that the integral and proportional-integral feedback motifs can mitigate the impact of gene expression burden, and we computationally explore their use in cell therapy. We believe that the engineering of precise and robust perfect adaptation will enable substantial advances in industrial biotechnology and cell-based therapeutics.

Digital Quantification of Proteins and mRNA in Single Mammalian Cells.

Absolute quantification of macromolecules in single cells is critical for understanding and modeling biological systems that feature cellular heterogeneity. Here we show extremely sensitive and absolute quantification of both proteins and mRNA in single mammalian cells by a very practical workflow that combines proximity ligation assay (PLA) and digital PCR. This digital PLA method has femtomolar sensitivity, which enables the quantification of very small protein concentration changes over its entire 3-log dynamic range, a quality necessary for accounting for single-cell heterogeneity. We counted both endogenous (CD147) and exogenously expressed (GFP-p65) proteins from hundreds of single cells and determined the correlation between CD147 mRNA and the protein it encodes. Using our data, a stochastic two-state model of the central dogma was constructed and verified using joint mRNA/protein distributions, allowing us to estimate transcription burst sizes and extrinsic noise strength and calculate the transcription and translation rate constants in single mammalian cells.

Optogenetic closed-loop feedback control of the unfolded protein response optimizes protein production.

In biotechnological protein production processes, the onset of protein unfolding at high gene expression levels leads to diminishing production yields and reduced efficiency. Here we show that in silico closed-loop optogenetic feedback control of the unfolded protein response (UPR) in S. cerevisiae clamps gene expression rates at intermediate near-optimal values, leading to significantly improved product titers. Specifically, in a fully-automated custom-built 1L-photobioreactor, we used a cybergenetic control system to steer the level of UPR in yeast to a desired set-point by optogenetically modulating the expression of α-amylase, a hard-to-fold protein, based on real-time feedback measurements of the UPR, resulting in 60% higher product titers. This proof-of-concept study paves the way for advanced optimal biotechnology production strategies that diverge from and complement current strategies employing constitutive overexpression or genetically hardwired circuits.

Optogenetic actuator - ERK biosensor circuits identify MAPK network nodes that shape ERK dynamics.

Combining single-cell measurements of ERK activity dynamics with perturbations provides insights into the MAPK network topology. We built circuits consisting of an optogenetic actuator to activate MAPK signaling and an ERK biosensor to measure single-cell ERK dynamics. This allowed us to conduct RNAi screens to investigate the role of 50 MAPK proteins in ERK dynamics. We found that the MAPK network is robust against most node perturbations. We observed that the ERK-RAF and the ERK-RSK2-SOS negative feedback operate simultaneously to regulate ERK dynamics. Bypassing the RSK2-mediated feedback, either by direct optogenetic activation of RAS, or by RSK2 perturbation, sensitized ERK dynamics to further perturbations. Similarly, targeting this feedback in a human ErbB2-dependent oncogenic signaling model increased the efficiency of a MEK inhibitor. The RSK2-mediated feedback is thus important for the ability of the MAPK network to produce consistent ERK outputs, and its perturbation can enhance the efficiency of MAPK inhibitors.

Engineering AraC to make it responsive to light instead of arabinose.

The L-arabinose-responsive AraC and its cognate PBAD promoter underlie one of the most often used chemically inducible prokaryotic gene expression systems in microbiology and synthetic biology. Here, we change the sensing capability of AraC from L-arabinose to blue light, making its dimerization and the resulting PBAD activation light-inducible. We engineer an entire family of blue light-inducible AraC dimers in Escherichia coli (BLADE) to control gene expression in space and time. We show that BLADE can be used with pre-existing L-arabinose-responsive plasmids and strains, enabling optogenetic experiments without the need to clone. Furthermore, we apply BLADE to control, with light, the catabolism of L-arabinose, thus externally steering bacterial growth with a simple transformation step. Our work establishes BLADE as a highly practical and effective optogenetic tool with plug-and-play functionality-features that we hope will accelerate the broader adoption of optogenetics and the realization of its vast potential in microbiology, synthetic biology and biotechnology.

DeepCME: A deep learning framework for computing solution statistics of the chemical master equation.

Stochastic models of biomolecular reaction networks are commonly employed in systems and synthetic biology to study the effects of stochastic fluctuations emanating from reactions involving species with low copy-numbers. For such models, the Kolmogorov's forward equation is called the chemical master equation (CME), and it is a fundamental system of linear ordinary differential equations (ODEs) that describes the evolution of the probability distribution of the random state-vector representing the copy-numbers of all the reacting species. The size of this system is given by the number of states that are accessible by the chemical system, and for most examples of interest this number is either very large or infinite. Moreover, approximations that reduce the size of the system by retaining only a finite number of important chemical states (e.g. those with non-negligible probability) result in high-dimensional ODE systems, even when the number of reacting species is small. Consequently, accurate numerical solution of the CME is very challenging, despite the linear nature of the underlying ODEs. One often resorts to estimating the solutions via computationally intensive stochastic simulations. The goal of the present paper is to develop a novel deep-learning approach for computing solution statistics of high-dimensional CMEs by reformulating the stochastic dynamics using Kolmogorov's backward equation. The proposed method leverages superior approximation properties of Deep Neural Networks (DNNs) to reliably estimate expectations under the CME solution for several user-defined functions of the state-vector. This method is algorithmically based on reinforcement learning and it only requires a moderate number of stochastic simulations (in comparison to typical simulation-based approaches) to train the "policy function". This allows not just the numerical approximation of various expectations for the CME solution but also of its sensitivities with respect to all the reaction network parameters (e.g. rate constants). We provide four examples to illustrate our methodology and provide several directions for future research.

Likelihood-free nested sampling for parameter inference of biochemical reaction networks.

The development of mechanistic models of biological systems is a central part of Systems Biology. One major challenge in developing these models is the accurate inference of model parameters. In recent years, nested sampling methods have gained increased attention in the Systems Biology community due to the fact that they are parallelizable and provide error estimates with no additional computations. One drawback that severely limits the usability of these methods, however, is that they require the likelihood function to be available, and thus cannot be applied to systems with intractable likelihoods, such as stochastic models. Here we present a likelihood-free nested sampling method for parameter inference which overcomes these drawbacks. This method gives an unbiased estimator of the Bayesian evidence as well as samples from the posterior. We derive a lower bound on the estimators variance which we use to formulate a novel termination criterion for nested sampling. The presented method enables not only the reliable inference of the posterior of parameters for stochastic systems of a size and complexity that is challenging for traditional methods, but it also provides an estimate of the obtained variance. We illustrate our approach by applying it to several realistically sized models with simulated data as well as recently published biological data. We also compare our developed method with the two most popular other likeliood-free approaches: pMCMC and ABC-SMC. The C++ code of the proposed methods, together with test data, is available at the github web page https://github.com/Mijan/LFNS_paper.

Temporal perturbation of ERK dynamics reveals network architecture of FGF2/MAPK signaling.

Stimulation of PC-12 cells with epidermal (EGF) versus nerve (NGF) growth factors (GFs) biases the distribution between transient and sustained single-cell ERK activity states, and between proliferation and differentiation fates within a cell population. We report that fibroblast GF (FGF2) evokes a distinct behavior that consists of a gradually changing population distribution of transient/sustained ERK signaling states in response to increasing inputs in a dose response. Temporally controlled GF perturbations of MAPK signaling dynamics applied using microfluidics reveal that this wider mix of ERK states emerges through the combination of an intracellular feedback, and competition of FGF2 binding to FGF receptors (FGFRs) and heparan sulfate proteoglycan (HSPG) co-receptors. We show that the latter experimental modality is instructive for model selection using a Bayesian parameter inference. Our results provide novel insights into how different receptor tyrosine kinase (RTK) systems differentially wire the MAPK network to fine-tune fate decisions at the cell population level.

The probability distribution of the reconstructed phylogenetic tree with occurrence data.

We study the problem of computing the probability distribution of phylogenetic trees that commonly arise in areas ranging from epidemiology to macroevolution. We focus on homogeneous birth death trees with incomplete sampling and consider observations from three distinct sampling schemes. First, individuals can be sampled and removed, through time, and included in the tree. Second, they can be occurrences which are sampled and removed through time and not included in the tree. Third, extant individuals can be sampled and included in the tree. The outcome of the process is thus composed of the reconstructed phylogenetic tree spanning all individuals sampled and included in the tree, and a timeline of occurrence events which are not placed along the tree. We derive a formula for computing the joint probability density of this outcome, which can readily be used to perform maximum likelihood or Bayesian estimation of the parameters of the model. In the context of epidemiology, our probability density enables the estimation of transmission rates through a joint analysis of epidemiological case count data and phylogenetic trees reconstructed from pathogen sequences. Within macroevolution, our equations form the basis for incorporating fossil occurrences from paleontological databases together with extant species phylogenies for estimating speciation and extinction rates. This work provides the theoretical framework for bridging not only the gap between phylogenetics and epidemiology, but also that between phylogenetics and paleontology.

Synthetic control systems for high performance gene expression in mammalian cells.

Tunable induction of gene expression is an essential tool in biology and biotechnology. In spite of that, current induction systems often exhibit unpredictable behavior and performance shortcomings, including high sensitivity to transactivator dosage and plasmid take-up variation, and excessive consumption of cellular resources. To mitigate these limitations, we introduce here a novel family of gene expression control systems of varying complexity with significantly enhanced performance. These include: (i) an incoherent feedforward circuit that exhibits output tunability and robustness to plasmid take-up variation; (ii) a negative feedback circuit that reduces burden and provides robustness to transactivator dosage variability; and (iii) a new hybrid circuit integrating negative feedback and incoherent feedforward that combines the benefits of both. As with endogenous circuits, the complexity of our genetic controllers is not gratuitous, but is the necessary outcome of more stringent performance requirements. We demonstrate the benefits of these controllers in two applications. In a culture of CHO cells for protein manufacturing, the circuits result in up to a 2.6-fold yield improvement over a standard system. In human-induced pluripotent stem cells they enable precisely regulated expression of an otherwise poorly tolerated gene of interest, resulting in a significant increase in the viability of the transfected cells.

Sensitivity Analysis for Multiscale Stochastic Reaction Networks Using Hybrid Approximations.

We consider the problem of estimating parameter sensitivities for stochastic models of multiscale reaction networks. These sensitivity values are important for model analysis, and the methods that currently exist for sensitivity estimation mostly rely on simulations of the stochastic dynamics. This is problematic because these simulations become computationally infeasible for multiscale networks due to reactions firing at several different timescales. However it is often possible to exploit the multiscale property to derive a "model reduction" and approximate the dynamics as a Piecewise deterministic Markov process, which is a hybrid process consisting of both discrete and continuous components. The aim of this paper is to show that such PDMP approximations can be used to accurately and efficiently estimate the parameter sensitivity for the original multiscale stochastic model. We prove the convergence of the original sensitivity to the corresponding PDMP sensitivity, in the limit where the PDMP approximation becomes exact. Moreover, we establish a representation of the PDMP parameter sensitivity that separates the contributions of discrete and continuous components in the dynamics and allows one to efficiently estimate both contributions.

A finite state projection method for steady-state sensitivity analysis of stochastic reaction networks.

Consider the standard stochastic reaction network model where the dynamics is given by a continuous-time Markov chain over a discrete lattice. For such models, estimation of parameter sensitivities is an important problem, but the existing computational approaches to solve this problem usually require time-consuming Monte Carlo simulations of the reaction dynamics. Therefore, these simulation-based approaches can only be expected to work over finite time-intervals, while it is often of interest in applications to examine the sensitivity values at the steady-state after the Markov chain has relaxed to its stationary distribution. The aim of this paper is to present a computational method for the estimation of steady-state parameter sensitivities, which instead of using simulations relies on the recently developed stationary finite state projection algorithm [Gupta et al., J. Chem. Phys. 147, 154101 (2017)] that provides an accurate estimate of the stationary distribution at a fixed set of parameters. We show that sensitivity values at these parameters can be estimated from the solution of a Poisson equation associated with the infinitesimal generator of the Markov chain. We develop an approach to numerically solve the Poisson equation, and this yields an efficient estimator for steady-state parameter sensitivities. We illustrate this method using several examples.

Molecular circuits for dynamic noise filtering.

The invention of the Kalman filter is a crowning achievement of filtering theory-one that has revolutionized technology in countless ways. By dealing effectively with noise, the Kalman filter has enabled various applications in positioning, navigation, control, and telecommunications. In the emerging field of synthetic biology, noise and context dependency are among the key challenges facing the successful implementation of reliable, complex, and scalable synthetic circuits. Although substantial further advancement in the field may very well rely on effectively addressing these issues, a principled protocol to deal with noise-as provided by the Kalman filter-remains completely missing. Here we develop an optimal filtering theory that is suitable for noisy biochemical networks. We show how the resulting filters can be implemented at the molecular level and provide various simulations related to estimation, system identification, and noise cancellation problems. We demonstrate our approach in vitro using DNA strand displacement cascades as well as in vivo using flow cytometry measurements of a light-inducible circuit in Escherichia coli.

Iterative experiment design guides the characterization of a light-inducible gene expression circuit.

Systems biology rests on the idea that biological complexity can be better unraveled through the interplay of modeling and experimentation. However, the success of this approach depends critically on the informativeness of the chosen experiments, which is usually unknown a priori. Here, we propose a systematic scheme based on iterations of optimal experiment design, flow cytometry experiments, and Bayesian parameter inference to guide the discovery process in the case of stochastic biochemical reaction networks. To illustrate the benefit of our methodology, we apply it to the characterization of an engineered light-inducible gene expression circuit in yeast and compare the performance of the resulting model with models identified from nonoptimal experiments. In particular, we compare the parameter posterior distributions and the precision to which the outcome of future experiments can be predicted. Moreover, we illustrate how the identified stochastic model can be used to determine light induction patterns that make either the average amount of protein or the variability in a population of cells follow a desired profile. Our results show that optimal experiment design allows one to derive models that are accurate enough to precisely predict and regulate the protein expression in heterogeneous cell populations over extended periods of time.

Implementation Considerations, Not Topological Differences, Are the Main Determinants of Noise Suppression Properties in Feedback and Incoherent Feedforward Circuits.

Biological systems use a variety of mechanisms to deal with the uncertain nature of their external and internal environments. Two of the most common motifs employed for this purpose are the incoherent feedforward (IFF) and feedback (FB) topologies. Many theoretical and experimental studies suggest that these circuits play very different roles in providing robustness to uncertainty in the cellular environment. Here, we use a control theoretic approach to analyze two common FB and IFF architectures that make use of an intermediary species to achieve regulation. We show the equivalence of both circuits topologies in suppressing static cell-to-cell variations. While both circuits can suppress variations due to input noise, they are ineffective in suppressing inherent chemical reaction stochasticity. Indeed, these circuits realize comparable improvements limited to a modest 25% variance reduction in best case scenarios. Such limitations are attributed to the use of intermediary species in regulation, and as such, they persist even for circuit architectures that combine both IFF and FB features. Intriguingly, while the FB circuits are better suited in dealing with dynamic input variability, the most significant difference between the two topologies lies not in the structural features of the circuits, but in their practical implementation considerations.

A finite state projection algorithm for the stationary solution of the chemical master equation.

The chemical master equation (CME) is frequently used in systems biology to quantify the effects of stochastic fluctuations that arise due to biomolecular species with low copy numbers. The CME is a system of ordinary differential equations that describes the evolution of probability density for each population vector in the state-space of the stochastic reaction dynamics. For many examples of interest, this state-space is infinite, making it difficult to obtain exact solutions of the CME. To deal with this problem, the Finite State Projection (FSP) algorithm was developed by Munsky and Khammash [J. Chem. Phys. 124(4), 044104 (2006)], to provide approximate solutions to the CME by truncating the state-space. The FSP works well for finite time-periods but it cannot be used for estimating the stationary solutions of CMEs, which are often of interest in systems biology. The aim of this paper is to develop a version of FSP which we refer to as the stationary FSP (sFSP) that allows one to obtain accurate approximations of the stationary solutions of a CME by solving a finite linear-algebraic system that yields the stationary distribution of a continuous-time Markov chain over the truncated state-space. We derive bounds for the approximation error incurred by sFSP and we establish that under certain stability conditions, these errors can be made arbitrarily small by appropriately expanding the truncated state-space. We provide several examples to illustrate our sFSP method and demonstrate its efficiency in estimating the stationary distributions. In particular, we show that using a quantized tensor-train implementation of our sFSP method, problems admitting more than 100 × 106 states can be efficiently solved.

An engineering viewpoint on biological robustness.

In his splendid article "Can a biologist fix a radio?--or, what I learned while studying apoptosis," Y. Lazebnik argues that when one uses the right tools, similarity between a biological system, like a signal transduction pathway, and an engineered system, like a radio, may not seem so superficial. Here I advance this idea by focusing on the notion of robustness as a unifying lens through which to view complexity in biological and engineered systems. I show that electronic amplifiers and gene expression circuits share remarkable similarities in their dynamics and robustness properties. I explore robustness features and limitations in biology and engineering and highlight the role of negative feedback in shaping both.