Complex dynamical networks can often be analyzed as the interconnection of subsystems: This allows us to considerably simplify the model and better understand the global behavior. Some biological networks can be conveniently analyzed as aggregates of monotone subsystems. Yet, monotonicity is a strong requirement; it relies on the knowledge of the state representation and imposes a severe restriction on the Jacobian (which must be a Metzler matrix). Systems with a monotonic step response (MSR), which include input-output monotone systems as a special case, are a broader class and still have interesting features. The property of having a monotonically increasing step response (or, equivalently, in the linear case, a positive impulse response) can be evinced from experimental data, without an explicit model of the system. We consider networks that can be decomposed as aggregates of MSR subsystems and we provide a structural (parameter-free) classification of oscillatory and multistationary behaviors. The classification is based on the exclusive or concurrent presence of negative and positive cycles in the system aggregate graph, whose nodes are the MSR subsystems. The result is analogous to our earlier classification for aggregates of monotone subsystems. Models of biomolecular networks are discussed to demonstrate the applicability of our classification, which helps build synthetic biomolecular circuits that, by design, are well suited to exhibit the desired dynamics.@en

To simplify the analysis of complex dynamical networks, we have recently proposed an approach that decomposes the overall system into the sign-definite interconnection of subsystems with a Positive Impulse Response (PIR). PIR systems include and significantly generalise input-output monotone systems, and the PIR property (or equivalently, for linear systems, the Monotonic Step Response property) can be evinced from experimental data, without an explicit model of the system. An aggregate of PIR subsystems can be associated with a signed matrix of interaction weights, hence with a signed graph where the nodes represent the subsystems and the arcs represent the interactions among them. In this paper, we prove that stability is structurally ensured (for any choice of the PIR subsystems) if a Metzler matrix depending on the interaction weights is Hurwitz; this condition is non-conservative. We also show how to compute an influence matrix that represents the steady-state effects of the interactions among PIR subsystems.@en

An important challenge in synthetic biology is the construction of periodic circuits with tunable and predictable period. We propose a general architecture, based on the use of recombinase proteins and negative feedback, to build a molecular device for periodic switching between two distinct regimes; the switching rule depends on known concentration thresholds for some circuit components. We analytically characterize the threshold values for which a periodic orbit is guaranteed to exist and attract all trajectories with initial conditions within an invariant set, and we provide expressions for period and amplitude. We describe two distinct biological realizations of the recombinase architecture, and show their capacity to exhibit periodic behaviors via extensive numerical simulations.@en

Recombinases are site-specific proteins found in nature that are capable of rearranging DNA. This function has made them promising gene editing tools in synthetic biology, as well as key elements in complex artificial gene circuits implementing Boolean logic. However, since DNA rearrangement is irreversible, it is still unclear how to use recombinases to build dynamic circuits like oscillators. In addition, this goal is challenging because a few molecules of recombinase are enough for promoter inversion, generating inherent stochasticity at low copy number. Here, we propose six different circuit designs for recombinase-based oscillators operating at a single copy number. We model them in a stochastic setting, leveraging the Gillespie algorithm for extensive simulations, and show that they can yield coherent periodic behaviors. Our results support the experimental realization of recombinase-based oscillators and, more generally, the use of recombinases to generate dynamic behaviors in synthetic biology.@en

Oscillators are essential to fuel autonomous behaviours in molecular systems. Artificial oscillators built with programmable biological molecules such as DNA and RNA are generally easy to build and tune, and can serve as timers for biological computation and regulation. We describe a new artificial nucleic acid biochemical reaction network, andwedemonstrate its capacity to exhibit oscillatory solutions. This network can be built in vitro using nucleic acids and three bacteriophage enzymes, and has the potential to be implemented in cells. Numerical simulations suggest that oscillations occur in a realistic range of reaction rates and concentrations.@en

Biological oscillators are present in nearly all self-regulating systems, from individual cells to entire organisms. In any oscillator structure, a negative feedback loop is necessary, but not sufficient to guarantee the emergence of periodic behaviors. The likelihood of oscillations can be improved by careful tuning of the system time constants and by increasing the loop gain, yet it is unclear whether there is any general relationship between optimal time constants and loop gain. This issue is particularly relevant in genetic oscillators resulting from a chain of different subsequent biochemical events, each with distinct (and uncertain) kinetics. Using two families of genetic oscillators as model examples, we show that the loop gain required for oscillations is minimum when all elements in the loop have the same time constant. On the contrary, we show that homeostasis is ensured if a single element is considerably slower than the others.@en

Molecular titration is emerging as an important biochemical interaction mechanism within synthetic devices built with nucleic acids and the CRISPR/Cas system. We show that molecular titration in the context of feedback circuits is a suitable mechanism to enhance the emergence of oscillations and bistable behaviors. We consider biomolecular modules that can be inhibited or activated by input monomeric regulators; the regulators compete with constitutive titrating species to determine the activity of their target. By tuning the titration rate and the concentration of titrating species, it is possible to modulate the delay and convergence speed of the transient response, and the steepness and dead zone of the stationary response of the modules. These phenomena favor the occurrence of oscillations when modules are interconnected to create a negative feedback loop; bistability is favored in a positive feedback interconnection. Numerical simulations are supported by mathematical analysis showing that the capacity of the closed loop systems to exhibit oscillations or bistability is structural.@en

We consider the problem of identifying structural influences of external inputs on steady-state outputs in a biological network model. We speak of a structural influence if, upon a perturbation due to a constant input, the ensuing variation of the steady-state output value has the same sign as the input (positive influence), the opposite sign (negative influence), or is zero (perfect adaptation), for any feasible choice of the model parameters. All these signs and zeros can constitute a structural influence matrix, whose (i, j) entry indicates the sign of steady-state influence of the jth system variable on the ith variable (the output caused by an external persistent input applied to the jth variable). Each entry is structurally determinate if the sign does not depend on the choice of the parameters, but is indeterminate otherwise. In principle, determining the influence matrix requires exhaustive testing of the system steady-state behaviour in the widest range of parameter values. Here we show that, in a broad class of biological networks, the influence matrix can be evaluated with an algorithm that tests the system steady-state behaviour only at a finite number of points. This algorithm also allows us to assess the structural effect of any perturbation, such as variations of relevant parameters. Our method is applied to nontrivial models of biochemical reaction networks and population dynamics drawn from the literature, providing a parameter-free insight into the system dynamics.@en

Biological systems compute spatial and temporal gradients with a variety of mechanisms, some of which have been shown to include integral feedback. In traditional engineering fields, it is well known that integral components within a negative feedback loop can be used to perform a derivative action. In this paper, we define the concept of a practical differentiator that is inspired by this design principle. We then consider three simple biological circuit examples in which we prove that feedback combined with ultrasensitive, quasi-integral components yields a practical differential network under some assumptions. These examples include phosphory-lation/dephosphorylation cycles, and two networks relying on molecular sequestration.@en

Nonlinear relaxation oscillators in engineering rely on positive feedback to operate. One category of relaxation oscillators is given by astable multivibrators, that include a bistable component at the core of their architecture. Here we describe a molecular network motif that operates as an astable multivibrator, and relies on a bistable switch (the classical Gardner and Collins genetic switch) that is toggled between its stable steady states by a persistent input. We show that oscillations arise in the presence of two negative feedback loops that process the persistent input and influence the production of the molecular species forming the bistable subsystem. We perform a thorough stability analysis of this motif obtaining closed-form practical conditions for the emergence of oscillations, and we examine the sensitivity of the system to parameter variations.@en

Cellular signaling pathways are responsible for decision making that sustains life. Most signaling pathways include post-translational modification cycles, that process multiple inputs and are tightly interconnected. Here we consider a model for phosphorylation/dephosphorylation cycles, and we show that under some assumptions they can operate as molecular neurons or perceptrons, that generate sigmoidal-like activation functions by processing sums of inputs with positive and negative weights. We carry out a steady-state and structural stability analysis for single molecular perceptrons as well as for feedforward interconnections, concluding that interconnected phosphorylation/dephosphorylation cycles may work as multilayer biomolecular neural networks (BNNs) with the capacity to perform a variety of computations. As an application, we design signaling networks that behave as linear and non-linear classifiers.@en

Engineered genetic circuits with tailored functions that mimic how cells process information in changing environments (e.g. cell fate decision, chemotaxis, immune response) have great applications in biomedicine and synthetic biology. Although there is a lot of progress toward the design of gene circuits yielding desired steady states (e.g. logic-based networks), building synthetic circuits for dynamic signal processing (e.g. filters, frequency modulation, and controllers) is still challenging. Here, we provide a model-based approach to build gene networks that can operate as band-pass filters by taking advantage of molecular sequestration. By suitably approximating the dynamics of molecular sequestration, we analyze an Incoherent Feed-Forward Loop (IFFL) and a Negative Feedback (NF) circuit and illustrate how they can achieve band-pass filter behavior. Computational analysis shows that a circuit that incorporates both IFFL and NF motifs improves the filter performance. Our approach facilitates the design of sequestration-based filters, and may support the synthesis of molecular controllers with desired specifications.@en

We describe an approach to stabilize a bistable biological system near its unstable equilibrium using a molecular feedback controller. As a case study we focus on the classical toggle switch by Gardner and Collins. The controller relies on two parallel sequestration motifs, which yield two control species influencing the production rates of the toggle switch proteins. We show that the controller reshapes the equilibrium landscape to a single equilibrium. With numerical simulations we illustrate the effectiveness of our approach in stabilizing the closed-loop system around this unique equilibrium, which falls in a neighborhood of the toggle switch unstable equilibrium, if the controller parameters are properly tuned.@en

We design a new negative feedback molecular oscillator and study its properties analytically and numerically. This oscillator is composed of two synthetic genes interconnected through their RNA outputs. Regulation of the genes activity is achieved by controlling the activity of the enzymes rather than the activity of the promoters. We show that a simple model of this system has the potential to oscillate for appropriate choices of the parameters. Our design can be built experimentally using RNA aptamers.@en