Computing with competition in biochemical networks

Exploring Emergent Properties in Enzymatic Reaction Networks: Design and Control of Dynamic Functional Systems

The intricate and complex features of enzymatic reaction networks (ERNs) play a key role in the emergence and sustenance of life. Constructing such networks in vitro enables stepwise build up in complexity and introduces the opportunity to control enzymatic activity using physicochemical stimuli. Rational design and modulation of network motifs enable the engineering of artificial systems with emergent functionalities. Such functional systems are useful for a variety of reasons such as creating new-to-nature dynamic materials, producing value-added chemicals, constructing metabolic modules for synthetic cells, and even enabling molecular computation. In this review, we offer insights into the chemical characteristics of ERNs while also delving into their potential applications and associated challenges.

Pattern recognition in the nucleation kinetics of non-equilibrium self-assembly

Inspired by biology's most sophisticated computer, the brain, neural networks constitute a profound reformulation of computational principles1-3. Analogous high-dimensional, highly interconnected computational architectures also arise within information-processing molecular systems inside living cells, such as signal transduction cascades and genetic regulatory networks4-7. Might collective modes analogous to neural computation be found more broadly in other physical and chemical processes, even those that ostensibly play non-information-processing roles? Here we examine nucleation during self-assembly of multicomponent structures, showing that high-dimensional patterns of concentrations can be discriminated and classified in a manner similar to neural network computation. Specifically, we design a set of 917 DNA tiles that can self-assemble in three alternative ways such that competitive nucleation depends sensitively on the extent of colocalization of high-concentration tiles within the three structures. The system was trained in silico to classify a set of 18 grayscale 30 × 30 pixel images into three categories. Experimentally, fluorescence and atomic force microscopy measurements during and after a 150 hour anneal established that all trained images were correctly classified, whereas a test set of image variations probed the robustness of the results. Although slow compared to previous biochemical neural networks, our approach is compact, robust and scalable. Our findings suggest that ubiquitous physical phenomena, such as nucleation, may hold powerful information-processing capabilities when they occur within high-dimensional multicomponent systems.

Computing Arithmetic Functions Using Immobilised Enzymatic Reaction Networks

Living systems use enzymatic reaction networks to process biochemical information and make decisions in response to external or internal stimuli. Herein, we present a modular and reusable platform for molecular information processing using enzymes immobilised in hydrogel beads and compartmentalised in a continuous stirred tank reactor. We demonstrate how this setup allows us to perform simple arithmetic operations, such as addition, subtraction and multiplication, using various concentrations of substrates or inhibitors as inputs and the production of a fluorescent molecule as the readout.

Neural extraction of multiscale essential structure for network dismantling

Diverse real world systems can be abstracted as complex networks consisting of nodes and edges as functional components. Percolation theory has shown that the failure of a few of nodes could lead to the collapse of a whole network, which brings up the network dismantling problem: How to select the least number of nodes to decompose a network into disconnected components each smaller than a predefined threshold? For its NP-hardness, many heuristic approaches have been proposed to measure and rank each node according to its importance to network structural stability. However, these measures are from a uniscale viewpoint by regarding one complex network as a flatted topology. In this article, we argue that nodes' structural importance can be measured in different scales of network topologies. Built upon recent deep learning techniques, we propose a self-supervised learning based network dismantling framework (NEES), which can hierarchically merge some compact substructures to convert a network into a coarser one with fewer nodes and edges. During the merging process, we design neural models to extract essential structures and utilize self-attention mechanisms to learn nodes' importance hierarchy in each scale. Experiments on real world networks and synthetic model networks show that the proposed NEES outperforms the state-of-the-art schemes in most cases in terms of removing the least number of target nodes to dismantle a network. The dismantling effectiveness of our neural extraction framework also highlights the emerging role of multi-scale essential structures.

A small-molecule chemical interface for molecular programs

In vitro molecular circuits, based on DNA-programmable chemistries, can perform an increasing range of high-level functions, such as molecular level computation, image or chemical pattern recognition and pattern generation. Most reported demonstrations, however, can only accept nucleic acids as input signals. Real-world applications of these programmable chemistries critically depend on strategies to interface them with a variety of non-DNA inputs, in particular small biologically relevant chemicals. We introduce here a general strategy to interface DNA-based circuits with non-DNA signals, based on input-translating modules. These translating modules contain a DNA response part and an allosteric protein sensing part, and use a simple design that renders them fully tunable and modular. They can be repurposed to either transmit or invert the response associated with the presence of a given input. By combining these translating-modules with robust and leak-free amplification motifs, we build sensing circuits that provide a fluorescent quantitative time-response to the concentration of their small-molecule input, with good specificity and sensitivity. The programmability of the DNA layer can be leveraged to perform DNA based signal processing operations, which we demonstrate here with logical inversion, signal modulation and a classification task on two inputs. The DNA circuits are also compatible with standard biochemical conditions, and we show the one-pot detection of an enzyme through its native metabolic activity. We anticipate that this sensitive small-molecule-to-DNA conversion strategy will play a critical role in the future applications of molecular-level circuitry.

Using Models to (Re-)Design Synthetic Circuits

Mathematical models play an important role in the design of synthetic gene circuits, by guiding the choice of biological components and their assembly into novel gene networks. Here, we present a guide for biologists to build and utilize models of gene networks (synthetic or natural) to analyze dynamical properties of these networks while considering the low numbers of molecules inside cells that results in stochastic gene expression. We start by describing how to write down a model and discussing the level of details to include. We then briefly demonstrate how to simulate a network's dynamics using deterministic differential equations that assume high numbers of molecules. To consider the role of stochastic gene expression in single cells, we provide a detailed tutorial on running stochastic Gillespie simulations of a network, including instructions on coding the Gillespie algorithm with example code. Finally, we illustrate how using a combination of quantitative experimental characterization of a synthetic circuit and mathematical modeling can guide the iterative redesign of a synthetic circuit to achieve the desired properties. This is shown using a classic synthetic oscillator, the repressilator, which we recently redesigned into the most precise and robust synthetic oscillator to date. We thus provide a toolkit for synthetic biologists to build more precise and robust synthetic circuits, which should lead to a deeper understanding of the dynamics of gene regulatory networks.

Multiplex Digital MicroRNA Detection Using Cross-Inhibitory DNA Circuits

Ubiquitous post-transcriptional regulators in eukaryotes, microRNAs are currently emerging as promising biomarkers of physiological and pathological processes. Multiplex and digital detection of microRNAs represents a major challenge toward the use of microRNA signatures in clinical settings. The classical reverse transcription polymerase chain reaction quantification approach has important limitations because of the need for thermocycling and a reverse transcription step. Simpler, isothermal alternatives have been proposed, yet none could be adapted in both a digital and multiplex format. This is either because of a lack of sensitivity that forbids single molecule detection or molecular cross-talk reactions that are responsible for nonspecific amplification. Building on an ultrasensitive isothermal amplification mechanism, we present a strategy to suppress cross-talk reactions, allowing for robust isothermal and multiplex detection of microRNA targets. Our approach relies on target-specific DNA circuits interconnected with DNA-encoded inhibitors that repress nonspecific signal amplification. We demonstrate the one-step, isothermal, digital, and simultaneous quantification of various pairs of important microRNA targets.

On the importance of reaction networks for synthetic living systems

The goal of creating a synthetic cell necessitates the development of reaction networks which will underlie all of its behaviours. Recent developments in in vitro systems, based upon both DNA and enzymes, have created networks capable of a range of behaviours e.g. information processing, adaptation and diffusive signalling. These networks are based upon reaction motifs that when combined together produce more complex behaviour. We highlight why it is inevitable that networks, based on enzymes or enzyme-like catalysts, will be required for the construction of a synthetic cell. We outline several of the challenges, including (a) timing, (b) regulation and (c) energy distribution, that must be overcome in order to transition from the simple networks we have today to much more complex networks capable of a variety of behaviours and which could find application one day within a synthetic cell.

Neuronal message passing using Mean-field, Bethe, and Marginal approximations

Neuronal computations rely upon local interactions across synapses. For a neuronal network to perform inference, it must integrate information from locally computed messages that are propagated among elements of that network. We review the form of two popular (Bayesian) message passing schemes and consider their plausibility as descriptions of inference in biological networks. These are variational message passing and belief propagation - each of which is derived from a free energy functional that relies upon different approximations (mean-field and Bethe respectively). We begin with an overview of these schemes and illustrate the form of the messages required to perform inference using Hidden Markov Models as generative models. Throughout, we use factor graphs to show the form of the generative models and of the messages they entail. We consider how these messages might manifest neuronally and simulate the inferences they perform. While variational message passing offers a simple and neuronally plausible architecture, it falls short of the inferential performance of belief propagation. In contrast, belief propagation allows exact computation of marginal posteriors at the expense of the architectural simplicity of variational message passing. As a compromise between these two extremes, we offer a third approach - marginal message passing - that features a simple architecture, while approximating the performance of belief propagation. Finally, we link formal considerations to accounts of neurological and psychiatric syndromes in terms of aberrant message passing.

Model-driven engineering of supramolecular buffering by multivalency

The ability of chemists to regulate the concentration of molecules is extremely important. However, as reactions are slowly superseded by more complex reaction networks, new ways of regulating molecular concentrations are needed. Recently, we described a system in which the concentration of a monovalent molecule with catalytic activity was buffered over a wide concentration range by its binding to a divalent molecule. Guided by model predictions, we are able to experimentally optimize the system by increasing the valency of the buffer, with even-numbered valencies displaying superior buffering capabilities. These results allow us to understand and gain more control over the activities of molecules in complex molecular systems, thereby obtaining insights into natural systems as well as creating adaptive artificial systems.

A supramolecular system in which the concentration of a molecule is buffered over several orders of magnitude is presented. Molecular buffering is achieved as a result of competition in a ring–chain equilibrium of multivalent ureidopyrimidinone monomers and a monovalent naphthyridine molecule which acts as an end-capper. While we previously only considered divalent ureidopyrimidinone monomers we now present a model-driven engineering approach to improve molecular buffering using multivalent ring–chain systems. Our theoretical models reveal an odd–even effect where even-valent molecules show superior buffering capabilities. Furthermore, we predict that supramolecular buffering can be significantly improved using a tetravalent instead of a divalent molecule, since the tetravalent molecule can form two intramolecular rings with different “stabilities” due to statistical effects. Our model predictions are validated against experimental ¹H NMR data, demonstrating that model-driven engineering has considerable potential in supramolecular chemistry.

Resource Competition Shapes the Response of Genetic Circuits

A common approach to design genetic circuits is to compose gene expression cassettes together. While appealing, this modular approach is challenged by the fact that expression of each gene depends on the availability of transcriptional/translational resources, which is in turn determined by the presence of other genes in the circuit. This raises the question of how competition for resources by different genes affects a circuit's behavior. Here, we create a library of genetic activation cascades in E. coli bacteria, where we explicitly tune the resource demand by each gene. We develop a general Hill-function-based model that incorporates resource competition effects through resource demand coefficients. These coefficients lead to nonregulatory interactions among genes that reshape the circuit's behavior. For the activation cascade, such interactions result in surprising biphasic or monotonically decreasing responses. Finally, we use resource demand coefficients to guide the choice of ribosome binding site and DNA copy number to restore the cascade's intended monotonically increasing response. Our results demonstrate how unintended circuit's behavior arises from resource competition and provide a model-guided methodology to minimize the resulting effects.

Materials learning from life: concepts for active, adaptive and autonomous molecular systems

A broad overview of functional aspects in biological and synthetic out-of-equilibrium systems.

Boosting functionality of synthetic DNA circuits with tailored deactivation

Molecular programming takes advantage of synthetic nucleic acid biochemistry to assemble networks of reactions, in vitro, with the double goal of better understanding cellular regulation and providing information-processing capabilities to man-made chemical systems. The function of molecular circuits is deeply related to their topological structure, but dynamical features (rate laws) also play a critical role. Here we introduce a mechanism to tune the nonlinearities associated with individual nodes of a synthetic network. This mechanism is based on programming deactivation laws using dedicated saturable pathways. We demonstrate this approach through the conversion of a single-node homoeostatic network into a bistable and reversible switch. Furthermore, we prove its generality by adding new functions to the library of reported man-made molecular devices: a system with three addressable bits of memory, and the first DNA-encoded excitable circuit. Specific saturable deactivation pathways thus greatly enrich the functional capability of a given circuit topology.

Nonlinearity in synthetic molecular circuits is usually achieved by manipulation of network topology or of production kinetics. Here, the authors achieve bistability and other nonlinear behaviours by manipulating the individual degradation rate laws of circuit components using saturable pathways.

High-resolution mapping of bifurcations in nonlinear biochemical circuits

Analog molecular circuits can exploit the nonlinear nature of biochemical reaction networks to compute low-precision outputs with fewer resources than digital circuits. This analog computation is similar to that employed by gene-regulation networks. Although digital systems have a tractable link between structure and function, the nonlinear and continuous nature of analog circuits yields an intricate functional landscape, which makes their design counter-intuitive, their characterization laborious and their analysis delicate. Here, using droplet-based microfluidics, we map with high resolution and dimensionality the bifurcation diagrams of two synthetic, out-of-equilibrium and nonlinear programs: a bistable DNA switch and a predator-prey DNA oscillator. The diagrams delineate where function is optimal, dynamics bifurcates and models fail. Inverse problem solving on these large-scale data sets indicates interference from enzymatic coupling. Additionally, data mining exposes the presence of rare, stochastically bursting oscillators near deterministic bifurcations.

Proteolytic crosstalk in multi-protease networks

Processive proteases, such as ClpXP in E. coli, are conserved enzyme assemblies that can recognize and rapidly degrade proteins. These proteases are used for a number of purposes, including degrading mistranslated proteins and controlling cellular stress response. However, proteolytic machinery within the cell is limited in capacity and can lead to a bottleneck in protein degradation, whereby many proteins compete ('queue') for proteolytic resources. Previous work has demonstrated that such queueing can lead to pronounced statistical relationships between different protein counts when proteins compete for a single common protease. However, real cells contain many different proteases, e.g. ClpXP, ClpAP, and Lon in E. coli, and it is not clear how competition between proteins for multiple classes of protease would influence the dynamics of cellular networks. In the present work, we theoretically demonstrate that a multi-protease proteolytic bottleneck can substantially couple the dynamics for both simple and complex (oscillatory) networks, even between substrates with substantially different affinities for protease. For these networks, queueing often leads to strong positive correlations between protein counts, and these correlations are strongest near the queueing theoretic point of balance. Furthermore, we find that the qualitative behavior of these networks depends on the relative size of the absolute affinity of substrate to protease compared to the cross affinity of substrate to protease, leading in certain regimes to priority queue statistics.

Molecular ping-pong Game of Life on a two-dimensional DNA origami array

We propose a design for programmed molecular interactions that continuously change molecular arrangements in a predesigned manner. We introduce a model where environmental control through laser illumination allows platform attachment/detachment oscillations between two floating molecular species. The platform is a two-dimensional DNA origami array of tiles decorated with strands that provide both, the floating molecular tiles to attach and to pass communicating signals to neighbouring array tiles. In particular, we show how algorithmic molecular interactions can control cyclic molecular arrangements by exhibiting a system that can simulate the dynamics similar to two-dimensional cellular automata on a DNA origami array platform.

A model for aryl hydrocarbon receptor-activated gene expression shows potency and efficacy changes and predicts squelching due to competition for transcription co-activators

A stochastic model of nuclear receptor-mediated transcription was developed based on activation of the aryl hydrocarbon receptor (AHR) by 2,3,7,8-tetrachlorodibenzodioxin (TCDD) and subsequent binding the activated AHR to xenobiotic response elements (XREs) on DNA. The model was based on effects observed in cells lines commonly used as in vitro experimental systems. Following ligand binding, the AHR moves into the cell nucleus and forms a heterodimer with the aryl hydrocarbon nuclear translocator (ARNT). In the model, a requirement for binding to DNA is that a generic coregulatory protein is subsequently bound to the AHR-ARNT dimer. Varying the amount of coregulator available within the nucleus altered both the potency and efficacy of TCDD for inducing for transcription of CYP1A1 mRNA, a commonly used marker for activation of the AHR. Lowering the amount of available cofactor slightly increased the EC50 for the transcriptional response without changing the efficacy or maximal response. Further reduction in the amount of cofactor reduced the efficacy and produced non-monotonic dose-response curves (NMDRCs) at higher ligand concentrations. The shapes of these NMDRCs were reminiscent of the phenomenon of squelching. Resource limitations for transcriptional machinery are becoming apparent in eukaryotic cells. Within single cells, nuclear receptor-mediated gene expression appears to be a stochastic process; however, intercellular communication and other aspects of tissue coordination may represent a compensatory process to maintain an organism’s ability to respond on a phenotypic level to various stimuli within an inconstant environment.

Computation in dynamically bounded asymmetric systems

Previous explanations of computations performed by recurrent networks have focused on symmetrically connected saturating neurons and their convergence toward attractors. Here we analyze the behavior of asymmetrical connected networks of linear threshold neurons, whose positive response is unbounded. We show that, for a wide range of parameters, this asymmetry brings interesting and computationally useful dynamical properties. When driven by input, the network explores potential solutions through highly unstable 'expansion' dynamics. This expansion is steered and constrained by negative divergence of the dynamics, which ensures that the dimensionality of the solution space continues to reduce until an acceptable solution manifold is reached. Then the system contracts stably on this manifold towards its final solution trajectory. The unstable positive feedback and cross inhibition that underlie expansion and divergence are common motifs in molecular and neuronal networks. Therefore we propose that very simple organizational constraints that combine these motifs can lead to spontaneous computation and so to the spontaneous modification of entropy that is characteristic of living systems.

Dynamic DNA-toolbox reaction circuits: a walkthrough

In living organisms, the integration of signals from the environment and the molecular computing leading to a cellular response are orchestrated by Gene Regulatory Networks (GRN). However, the molecular complexity of in vivo genetic regulation makes it next to impossible to describe in a quantitative manner. Reproducing, in vitro, reaction networks that could mimic the architecture and behavior of in vivo networks, yet lend themselves to mathematical modeling, represents a useful strategy to understand, and even predict, the function of GRN. In this paper, we define a set of in vitro, DNA-based molecular transformations that can be linked to each other in such a way that the product of one transformation can activate or inhibit the production of one or several other DNA compounds. Therefore, these reactions can be wired in arbitrary networks. This approach provides an experimental way to reproduce the dynamic features of genetic regulation in a test tube. We introduce the rules to design the necessary DNA species, a guide to implement the chemical reactions and ways to optimize the experimental conditions. We finally show how this framework, or "DNA toolbox", can be used to generate an inversion module, though many other behaviors, including oscillators and bistable switches, can be implemented.

Computer-assisted design for scaling up systems based on DNA reaction networks

In the past few years, there have been many exciting advances in the field of molecular programming, reaching a point where implementation of non-trivial systems, such as neural networks or switchable bistable networks, is a reality. Such systems require nonlinearity, be it through signal amplification, digitalization or the generation of autonomous dynamics such as oscillations. The biochemistry of DNA systems provides such mechanisms, but assembling them in a constructive manner is still a difficult and sometimes counterintuitive process. Moreover, realistic prediction of the actual evolution of concentrations over time requires a number of side reactions, such as leaks, cross-talks or competitive interactions, to be taken into account. In this case, the design of a system targeting a given function takes much trial and error before the correct architecture can be found. To speed up this process, we have created DNA Artificial Circuits Computer-Assisted Design (DACCAD), a computer-assisted design software that supports the construction of systems for the DNA toolbox. DACCAD is ultimately aimed to design actual in vitro implementations, which is made possible by building on the experimental knowledge available on the DNA toolbox. We illustrate its effectiveness by designing various systems, from Montagne et al.'s Oligator or Padirac et al.'s bistable system to new and complex networks, including a two-bit counter or a frequency divider as well as an example of very large system encoding the game Mastermind. In the process, we highlight a variety of behaviours, such as enzymatic saturation and load effect, which would be hard to handle or even predict with a simpler model. We also show that those mechanisms, while generally seen as detrimental, can be used in a positive way, as functional part of a design. Additionally, the number of parameters included in these simulations can be large, especially in the case of complex systems. For this reason, we included the possibility to use CMA-ES, a state-of-the-art optimization algorithm that will automatically evolve parameters chosen by the user to try to match a specified behaviour. Finally, because all possible functionality cannot be captured by a single software, DACCAD includes the possibility to export a system in the synthetic biology markup language, a widely used language for describing biological reaction systems. DACCAD can be downloaded online at http://www.yannick-rondelez.com/downloads/.

Noise in biology

Noise permeates biology on all levels, from the most basic molecular, sub-cellular processes to the dynamics of tissues, organs, organisms and populations. The functional roles of noise in biological processes can vary greatly. Along with standard, entropy-increasing effects of producing random mutations, diversifying phenotypes in isogenic populations, limiting information capacity of signaling relays, it occasionally plays more surprising constructive roles by accelerating the pace of evolution, providing selective advantage in dynamic environments, enhancing intracellular transport of biomolecules and increasing information capacity of signaling pathways. This short review covers the recent progress in understanding mechanisms and effects of fluctuations in biological systems of different scales and the basic approaches to their mathematical modeling.

Translational cross talk in gene networks

It has been shown experimentally that competition for limited translational resources by upstream mRNAs can lead to an anticorrelation between protein counts. Here, we investigate a stochastic model for this phenomenon, in which gene transcripts of different types compete for a finite pool of ribosomes. Throughout, we utilize concepts from the theory of multiclass queues to describe a qualitative shift in protein count statistics as the system transitions from being underloaded (ribosomes exceed transcripts in number) to being overloaded (transcripts exceed ribosomes in number). The exact analytical solution of a simplified stochastic model, in which the numbers of competing mRNAs and ribosomes are fixed, exhibits weak positive correlations between steady-state protein counts when total transcript count slightly exceeds ribosome count, whereas the solution can exhibit strong negative correlations when total transcript count significantly exceeds ribosome count. Extending this analysis, we find approximate but reasonably accurate solutions for a more realistic model, in which abundances of mRNAs and ribosomes are allowed to fluctuate randomly. Here, ribosomal fluctuations contribute positively and mRNA fluctuations contribute negatively to correlations, and when mRNA fluctuations dominate ribosomal fluctuations, a strong anticorrelation extremum reliably occurs near the transition from the underloaded to the overloaded regime.

Scaling down DNA circuits with competitive neural networks

DNA has proved to be an exquisite substrate to compute at the molecular scale. However, nonlinear computations (such as amplification, comparison or restoration of signals) remain costly in term of strands and are prone to leak. Kim et al. showed how competition for an enzymatic resource could be exploited in hybrid DNA/enzyme circuits to compute a powerful nonlinear primitive: the winner-take-all (WTA) effect. Here, we first show theoretically how the nonlinearity of the WTA effect allows the robust and compact classification of four patterns with only 16 strands and three enzymes. We then generalize this WTA effect to DNA-only circuits and demonstrate similar classification capabilities with only 23 strands.

In vitro regulatory models for systems biology

The reductionist approach has revolutionized biology in the past 50 years. Yet its limits are being felt as the complexity of cellular interactions is gradually revealed by high-throughput technology. In order to make sense of the deluge of "omic data", a hypothesis-driven view is needed to understand how biomolecular interactions shape cellular networks. We review recent efforts aimed at building in vitro biochemical networks that reproduce the flow of genetic regulation. We highlight how those efforts have culminated in the rational construction of biochemical oscillators and bistable memories in test tubes. We also recapitulate the lessons learned about in vivo biochemical circuits such as the importance of delays and competition, the links between topology and kinetics, as well as the intriguing resemblance between cellular reaction networks and ecosystems.

Predator-prey molecular ecosystems

Biological organisms use intricate networks of chemical reactions to control molecular processes and spatiotemporal organization. In turn, these living systems are embedded in self-organized structures of larger scales, for example, ecosystems. Synthetic in vitro efforts have reproduced the architectures and behaviors of simple cellular circuits. However, because all these systems share the same dynamic foundations, a generalized molecular programming strategy should also support complex collective behaviors, as seen, for example, in animal populations. We report here the bottom-up assembly of chemical systems that reproduce in vitro the specific dynamics of ecological communities. We experimentally observed unprecedented molecular behaviors, including predator-prey oscillations, competition-induced chaos, and symbiotic synchronization. These synthetic systems are tailored through a novel, compact, and versatile design strategy, leveraging the programmability of DNA interactions under the precise control of enzymatic catalysis. Such self-organizing assemblies will foster a better appreciation of the molecular origins of biological complexity and may also serve to orchestrate complex collective operations of molecular agents in technological applications.