Regulation strategies for two-output biomolecular networks.

Feedback control theory facilitates the development of self-regulating systems with desired performance which are predictable and insensitive to disturbances. Feedback regulatory topologies are found in many natural systems and have been of key importance in the design of reliable synthetic bio-devices operating in complex biological environments. Here, we study control schemes for biomolecular processes with two outputs of interest, expanding previously described concepts based on single-output systems. Regulation of such processes may unlock new design possibilities but can be challenging due to coupling interactions; also potential disturbances applied on one of the outputs may affect both. We therefore propose architectures for robustly manipulating the ratio/product and linear combinations of the outputs as well as each of the outputs independently. To demonstrate their characteristics, we apply these architectures to a simple process of two mutually activated biomolecular species. We also highlight the potential for experimental implementation by exploring synthetic realizations both in vivo and in vitro. This work presents an important step forward in building bio-devices capable of sophisticated functions.

Biomolecular mechanisms for signal differentiation.

Cells can sense temporal changes of molecular signals, allowing them to predict environmental variations and modulate their behavior. This paper elucidates biomolecular mechanisms of time derivative computation, facilitating the design of reliable synthetic differentiator devices for a variety of applications, ultimately expanding our understanding of cell behavior. In particular, we describe and analyze three alternative biomolecular topologies that are able to work as signal differentiators to input signals around their nominal operation. We propose strategies to preserve their performance even in the presence of high-frequency input signal components which are detrimental to the performance of most differentiators. We find that the core of the proposed topologies appears in natural regulatory networks and we further discuss their biological relevance. The simple structure of our designs makes them promising tools for realizing derivative control action in synthetic biology.

Evolution of opposing regulatory interactions underlies the emergence of eukaryotic cell cycle checkpoints.

In eukaryotes the entry into mitosis is initiated by activation of cyclin-dependent kinases (CDKs), which in turn activate a large number of protein kinases to induce all mitotic processes. The general view is that kinases are active in mitosis and phosphatases turn them off in interphase. Kinases activate each other by cross- and self-phosphorylation, while phosphatases remove these phosphate groups to inactivate kinases. Crucial exceptions to this general rule are the interphase kinase Wee1 and the mitotic phosphatase Cdc25. Together they directly control CDK in an opposite way of the general rule of mitotic phosphorylation and interphase dephosphorylation. Here we investigate why this opposite system emerged and got fixed in almost all eukaryotes. Our results show that this reversed action of a kinase-phosphatase pair, Wee1 and Cdc25, on CDK is particularly suited to establish a stable G2 phase and to add checkpoints to the cell cycle. We show that all these regulators appeared together in LECA (Last Eukaryote Common Ancestor) and co-evolved in eukaryotes, suggesting that this twist in kinase-phosphatase regulation was a crucial step happening at the emergence of eukaryotes.

Exact maximal reduction of stochastic reaction networks by species lumping.

MOTIVATION: Stochastic reaction networks are a widespread model to describe biological systems where the presence of noise is relevant, such as in cell regulatory processes. Unfortunately, in all but simplest models the resulting discrete state-space representation hinders analytical tractability and makes numerical simulations expensive. Reduction methods can lower complexity by computing model projections that preserve dynamics of interest to the user. RESULTS: We present an exact lumping method for stochastic reaction networks with mass-action kinetics. It hinges on an equivalence relation between the species, resulting in a reduced network where the dynamics of each macro-species is stochastically equivalent to the sum of the original species in each equivalence class, for any choice of the initial state of the system. Furthermore, by an appropriate encoding of kinetic parameters as additional species, the method can establish equivalences that do not depend on specific values of the parameters. The method is supported by an efficient algorithm to compute the largest species equivalence, thus the maximal lumping. The effectiveness and scalability of our lumping technique, as well as the physical interpretability of resulting reductions, is demonstrated in several models of signaling pathways and epidemic processes on complex networks. AVAILABILITY AND IMPLEMENTATION: The algorithms for species equivalence have been implemented in the software tool ERODE, freely available for download from https://www.erode.eu. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

The Beacon Calculus: A formal method for the flexible and concise modelling of biological systems.

Biological systems are made up of components that change their actions (and interactions) over time and coordinate with other components nearby. Together with a large state space, the complexity of this behaviour can make it difficult to create concise mathematical models that can be easily extended or modified. This paper introduces the Beacon Calculus, a process algebra designed to simplify the task of modelling interacting biological components. Its breadth is demonstrated by creating models of DNA replication dynamics, the gene expression dynamics in response to DNA methylation damage, and a multisite phosphorylation switch. The flexibility of these models is shown by adapting the DNA replication model to further include two topics of interest from the literature: cooperative origin firing and replication fork barriers. The Beacon Calculus is supported with the open-source simulator bcs (https://github.com/MBoemo/bcs.git) to allow users to develop and simulate their own models.

Coupled membrane transporters reduce noise.

Molecular systems are inherently probabilistic and operate in a noisy environment, yet, despite all these uncertainties, molecular functions are surprisingly reliable and robust. The principles used by natural systems to deal with noise are still not well understood, especially in a nonhomogeneous environment where molecules can diffuse across different compartments. In this paper we show that membrane transport mechanisms have very effective properties of noise reduction. In particular, we show that active transport mechanisms (those that can transport against a gradient of concentration by using energy or by means of the concentration gradient of other substances), such as symporters and antiporters, have surprising efficiency in noise reduction, which outperforms passive diffusion mechanisms and are well below Poisson levels. We link our results to the coupled transport of potassium, sodium, and glucose to show that the noise in internal glucose level can be greatly reduced. Our results show that compartmentalization can be a highly effective mechanism of noise reduction and suggests that membrane transport could give this extra benefit, contributing to the emergence of complex compartmentalization in eukaryotes.

Computing with biological switches and clocks.

The complex dynamics of biological systems is primarily driven by molecular interactions that underpin the regulatory networks of cells. These networks typically contain positive and negative feedback loops, which are responsible for switch-like and oscillatory dynamics, respectively. Many computing systems rely on switches and clocks as computational modules. While the combination of such modules in biological systems leads to a variety of dynamical behaviours, it is also driving development of new computing algorithms. Here we present a historical perspective on computation by biological systems, with a focus on switches and clocks, and discuss parallels between biology and computing. We also outline our vision for the future of biological computing.

Molecular Filters for Noise Reduction.

Living systems are inherently stochastic and operate in a noisy environment, yet despite all these uncertainties, they perform their functions in a surprisingly reliable way. The biochemical mechanisms used by natural systems to tolerate and control noise are still not fully understood, and this issue also limits our capacity to engineer reliable, quantitative synthetic biological circuits. We study how representative models of biochemical systems propagate and attenuate noise, accounting for intrinsic as well as extrinsic noise. We investigate three molecular noise-filtering mechanisms, study their noise-reduction capabilities and limitations, and show that nonlinear dynamics such as complex formation are necessary for efficient noise reduction. We further suggest that the derived molecular filters are widespread in gene expression and regulation and, particularly, that microRNAs can serve as such noise filters. To our knowledge, our results provide new insight into how biochemical networks control noise and could be useful to build robust synthetic circuits.

Single molecules can operate as primitive biological sensors, switches and oscillators.

BACKGROUND: Switch-like and oscillatory dynamical systems are widely observed in biology. We investigate the simplest biological switch that is composed of a single molecule that can be autocatalytically converted between two opposing activity forms. We test how this simple network can keep its switching behaviour under perturbations in the system. RESULTS: We show that this molecule can work as a robust bistable system, even for alterations in the reactions that drive the switching between various conformations. We propose that this single molecule system could work as a primitive biological sensor and show by steady state analysis of a mathematical model of the system that it could switch between possible states for changes in environmental signals. Particularly, we show that a single molecule phosphorylation-dephosphorylation switch could work as a nucleotide or energy sensor. We also notice that a given set of reductions in the reaction network can lead to the emergence of oscillatory behaviour. CONCLUSIONS: We propose that evolution could have converted this switch into a single molecule oscillator, which could have been used as a primitive timekeeper. We discuss how the structure of the simplest known circadian clock regulatory system, found in cyanobacteria, resembles the proposed single molecule oscillator. Besides, we speculate if such minimal systems could have existed in an RNA world.

Chemical reaction network designs for asynchronous logic circuits.

Chemical reaction networks (CRNs) are a versatile language for describing the dynamical behaviour of chemical kinetics, capable of modelling a variety of digital and analogue processes. While CRN designs for synchronous sequential logic circuits have been proposed and their implementation in DNA demonstrated, a physical realisation of these devices is difficult because of their reliance on a clock. Asynchronous sequential logic, on the other hand, does not require a clock, and instead relies on handshaking protocols to ensure the temporal ordering of different phases of the computation. This paper provides novel CRN designs for the construction of asynchronous logic, arithmetic and control flow elements based on a bi-molecular reaction motif with catalytic reactions and uniform reaction rates. We model and validate the designs for the deterministic and stochastic semantics using Microsoft's GEC tool and the probabilistic model checker PRISM, demonstrating their ability to emulate the function of asynchronous components under low molecular count.

Programming discrete distributions with chemical reaction networks.

We explore the range of probabilistic behaviours that can be engineered with Chemical Reaction Networks (CRNs). We give methods to "program" CRNs so that their steady state is chosen from some desired target distribution that has finite support in [Formula: see text], with [Formula: see text]. Moreover, any distribution with countable infinite support can be approximated with arbitrarily small error under the [Formula: see text] norm. We also give optimized schemes for special distributions, including the uniform distribution. Finally, we formulate a calculus to compute on distributions that is complete for finite support distributions, and can be compiled to a restricted class of CRNs that at steady state realize those distributions.

Maximal aggregation of polynomial dynamical systems.

Ordinary differential equations (ODEs) with polynomial derivatives are a fundamental tool for understanding the dynamics of systems across many branches of science, but our ability to gain mechanistic insight and effectively conduct numerical evaluations is critically hindered when dealing with large models. Here we propose an aggregation technique that rests on two notions of equivalence relating ODE variables whenever they have the same solution (backward criterion) or if a self-consistent system can be written for describing the evolution of sums of variables in the same equivalence class (forward criterion). A key feature of our proposal is to encode a polynomial ODE system into a finitary structure akin to a formal chemical reaction network. This enables the development of a discrete algorithm to efficiently compute the largest equivalence, building on approaches rooted in computer science to minimize basic models of computation through iterative partition refinements. The physical interpretability of the aggregation is shown on polynomial ODE systems for biochemical reaction networks, gene regulatory networks, and evolutionary game theory.

Stochastic analysis of Chemical Reaction Networks using Linear Noise Approximation.

Stochastic evolution of Chemical Reactions Networks (CRNs) over time is usually analyzed through solving the Chemical Master Equation (CME) or performing extensive simulations. Analysing stochasticity is often needed, particularly when some molecules occur in low numbers. Unfortunately, both approaches become infeasible if the system is complex and/or it cannot be ensured that initial populations are small. We develop a probabilistic logic for CRNs that enables stochastic analysis of the evolution of populations of molecular species. We present an approximate model checking algorithm based on the Linear Noise Approximation (LNA) of the CME, whose computational complexity is independent of the population size of each species and polynomial in the number of different species. The algorithm requires the solution of first order polynomial differential equations. We prove that our approach is valid for any CRN close enough to the thermodynamical limit. However, we show on four case studies that it can still provide good approximation even for low molecule counts. Our approach enables rigorous analysis of CRNs that are not analyzable by solving the CME, but are far from the deterministic limit. Moreover, it can be used for a fast approximate stochastic characterization of a CRN.

The Formal Language and Design Principles of Autonomous DNA Walker Circuits.

Simple computation can be performed using the interactions between single-stranded molecules of DNA. These interactions are typically toehold-mediated strand displacement reactions in a well-mixed solution. We demonstrate that a DNA circuit with tethered reactants is a distributed system and show how it can be described as a stochastic Petri net. The system can be verified by mapping the Petri net onto a continuous-time Markov chain, which can also be used to find an optimal design for the circuit. This theoretical machinery can be applied to create software that automatically designs a DNA circuit, linking an abstract propositional formula to a physical DNA computation system that is capable of evaluating it. We conclude by introducing example mechanisms that can implement such circuits experimentally and discuss their individual strengths and weaknesses.

Noise Reduction in Complex Biological Switches.

Cells operate in noisy molecular environments via complex regulatory networks. It is possible to understand how molecular counts are related to noise in specific networks, but it is not generally clear how noise relates to network complexity, because different levels of complexity also imply different overall number of molecules. For a fixed function, does increased network complexity reduce noise, beyond the mere increase of overall molecular counts? If so, complexity could provide an advantage counteracting the costs involved in maintaining larger networks. For that purpose, we investigate how noise affects multistable systems, where a small amount of noise could lead to very different outcomes; thus we turn to biochemical switches. Our method for comparing networks of different structure and complexity is to place them in conditions where they produce exactly the same deterministic function. We are then in a good position to compare their noise characteristics relatively to their identical deterministic traces. We show that more complex networks are better at coping with both intrinsic and extrinsic noise. Intrinsic noise tends to decrease with complexity, and extrinsic noise tends to have less impact. Our findings suggest a new role for increased complexity in biological networks, at parity of function.

Unlimited multistability and Boolean logic in microbial signalling.

The ability to map environmental signals onto distinct internal physiological states or programmes is critical for single-celled microbes. A crucial systems dynamics feature underpinning such ability is multistability. While unlimited multistability is known to arise from multi-site phosphorylation seen in the signalling networks of eukaryotic cells, a similarly universal mechanism has not been identified in microbial signalling systems. These systems are generally known as two-component systems comprising histidine kinase (HK) receptors and response regulator proteins engaging in phosphotransfer reactions. We develop a mathematical framework for analysing microbial systems with multi-domain HK receptors known as hybrid and unorthodox HKs. We show that these systems embed a simple core network that exhibits multistability, thereby unveiling a novel biochemical mechanism for multistability. We further prove that sharing of downstream components allows a system with n multi-domain hybrid HKs to attain 3n steady states. We find that such systems, when sensing distinct signals, can readily implement Boolean logic functions on these signals. Using two experimentally studied examples of two-component systems implementing hybrid HKs, we show that bistability and implementation of logic functions are possible under biologically feasible reaction rates. Furthermore, we show that all sequenced microbial genomes contain significant numbers of hybrid and unorthodox HKs, and some genomes have a larger fraction of these proteins compared with regular HKs. Microbial cells are thus theoretically unbounded in mapping distinct environmental signals onto distinct physiological states and perform complex computations on them. These findings facilitate the understanding of natural two-component systems and allow their engineering through synthetic biology.

Gener: a minimal programming module for chemical controllers based on DNA strand displacement.

UNLABELLED: : Gener is a development module for programming chemical controllers based on DNA strand displacement. Gener is developed with the aim of providing a simple interface that minimizes the opportunities for programming errors: Gener allows the user to test the computations of the DNA programs based on a simple two-domain strand displacement algebra, the minimal available so far. The tool allows the user to perform stepwise computations with respect to the rules of the algebra as well as exhaustive search of the computation space with different options for exploration and visualization. Gener can be used in combination with existing tools, and in particular, its programs can be exported to Microsoft Research's DSD tool as well as to LaTeX. AVAILABILITY AND IMPLEMENTATION: Gener is available for download at the Cosbi website at http://www.cosbi.eu/research/prototypes/gener as a windows executable that can be run on Mac OS X and Linux by using Mono. CONTACT: ozan@cosbi.eu.

Programming chemistry in DNA-addressable bioreactors.

We present a formal calculus, termed the chemtainer calculus, able to capture the complexity of compartmentalized reaction systems such as populations of possibly nested vesicular compartments. Compartments contain molecular cargo as well as surface markers in the form of DNA single strands. These markers serve as compartment addresses and allow for their targeted transport and fusion, thereby enabling reactions of previously separated chemicals. The overall system organization allows for the set-up of programmable chemistry in microfluidic or other automated environments. We introduce a simple sequential programming language whose instructions are motivated by state-of-the-art microfluidic technology. Our approach integrates electronic control, chemical computing and material production in a unified formal framework that is able to mimic the integrated computational and constructive capabilities of the subcellular matrix. We provide a non-deterministic semantics of our programming language that enables us to analytically derive the computational and constructive power of our machinery. This semantics is used to derive the sets of all constructable chemicals and supermolecular structures that emerge from different underlying instruction sets. Because our proofs are constructive, they can be used to automatically infer control programs for the construction of target structures from a limited set of resource molecules. Finally, we present an example of our framework from the area of oligosaccharide synthesis.

Morphisms of reaction networks that couple structure to function.

BACKGROUND: The mechanisms underlying complex biological systems are routinely represented as networks. Network kinetics is widely studied, and so is the connection between network structure and behavior. However, similarity of mechanism is better revealed by relationships between network structures. RESULTS: We define morphisms (mappings) between reaction networks that establish structural connections between them. Some morphisms imply kinetic similarity, and yet their properties can be checked statically on the structure of the networks. In particular we can determine statically that a complex network will emulate a simpler network: it will reproduce its kinetics for all corresponding choices of reaction rates and initial conditions. We use this property to relate the kinetics of many common biological networks of different sizes, also relating them to a fundamental population algorithm. CONCLUSIONS: Structural similarity between reaction networks can be revealed by network morphisms, elucidating mechanistic and functional aspects of complex networks in terms of simpler networks.