Giulia Giordano

Stability analysis of an artificial biomolecular oscillator with non-cooperative regulatory interactions

ABSTRACT 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, and we demonstrate 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.

Model-Free Plant Tuning

Given a static plant described by a differentiable input-output function, which is completely unknown, but whose Jacobian takes values in a known polytope in the matrix space, this paper considers the problem of tuning (i.e., driving to a desired value) the output, by suitably choosing the input. It is shown that, if the polytope is robustly nonsingular (or has full rank, in the nonsquare case), then a suitable tuning scheme drives the output to the desired point. The proof exploits a Lyapunov-like function and applies a well-known game-theoretic result, concerning the existence of a saddle point for a min-max zero-sum game. When the plant output is represented in an implicit form, it is shown that the same result can be obtained, resorting to a different Lyapunov-like function. The case in which proper input or output constraints must be enforced during the transient is considered as well. Some application examples are proposed to show the effectiveness of the approach.

The Smallest Eigenvalue of the Generalized Laplacian Matrix, with Application to Network-Decentralized Estimation for Homogeneous Systems

The problem of synthesizing network-decentralized observers arises when several agents, corresponding to the nodes of a network, exchange information about local measurements to asymptotically estimate their own state. The network topology is unknown to the nodes, which can rely on information about their neighboring nodes only. For homogeneous systems, composed of identical agents, we show that a network-decentralized observer can be designed by starting from local observers (typically, optimal filters) and then adapting the gain to ensure overall stability. The smallest eigenvalue of the so-called generalized Laplacian matrix is crucial: stability is guaranteed if the gain is greater than the inverse of this eigenvalue, which is strictly positive if the graph is externally connected. To deal with uncertain topologies, we characterize the worst-case smallest eigenvalue of the generalized Laplacian matrix for externally connected graphs, and we prove that the worst-case graph is a chain. This general result provides a bound for the observer gain that ensures robustness of the network-decentralized observer even under arbitrary, possibly switching, configurations, and in the presence of noise.

Flow-Inducing Networks

We consider flow-inducing networks, a class of models that are well-suited to describe important biochemical systems, including the MAPK pathway and the interactions at the trans-Golgi network. A flow-inducing network is given by the interconnection of subsystems (modules), each associated with a stochastic state matrix whose entries depend on the state variables of other modules. This results in an overall nonlinear system; when the interactions are modeled as mass action kinetics, the overall system is bilinear. We provide preliminary results concerning the existence of single or multiple equilibria and their positivity. We also show that instability phenomena are possible and that entropy is not a suitable Lyapunov function. The simplest non-trivial module is the duet, a second order system whose variables represent the concentrations of a species in its activated and inhibited state: under mass action kinetics assumptions, we prove that: 1) a negative loop of duets has a unique equilibrium that is unconditionally stable and 2) a positive loop of duets has either a unique stable equilibrium on the boundary or two equilibria, of which one is unstable on the boundary and one is strictly positive and stable; both properties 1) and 2) hold regardless of the number of duets in the loop.

Biometric Palmprint Verification: A Dynamical System Approach

Most of the existing techniques for palmprint recognition rely on metrics, typically based on static functions, which evaluate the distance between a pair of features. In this paper, we propose a new technique for palmprint verification based on a dynamical system approach for principal palm lines matching. The proposed dynamic algorithm is recursive and involves a positive linear dynamical system, whose evolution depends on the matching level between the two input images. In a preprocessing phase, the procedure iteratively erodes both of the images to be compared, by eliminating points in each image that do not have enough close neighboring points both in the image itself and the comparison image. As a result of the iterations, only the points that have enough neighboring points in both the image itself and in the comparison image can survive. Thus, the output of the dynamical system converges either to zero, when a deep mismatch exists between the two images, or to a high value, when a good matching is observed. The results, in terms of verification, are in line with the state-of-the-art results in the current literature. The main advantage of the approach is its robustness when dealing with low-resolution and noisy images. The impact of noise (e.g., salt and pepper noise) is effectively reduced: images corrupted with such noise are easily recognized, while a randomly generated image is rejected even when compared with itself.

A convex programming approach to the inverse kinematics problem for manipulators under constraints

We propose a novel approach to the problem of inverse kinematics for possibly redundant planar manipulators. We show that, by considering the joints as point masses in a fictitious gravity field, and by adding proper constraints to take into account the length of the links, the kinematic inversion may be cast as a convex programming problem. Convex constraints in the decision variables (in particular, linear constraints in the workspace) are easily managed with the proposed approach. We also show how to exploit the idea for avoiding obstacles while tracking a reference end-effector trajectory and discuss how to extend the results to some kinds of non-planar manipulators. Simulation results are reported, showing the effectiveness of the approach.

Polyhedral Lyapunov functions structurally ensure global asymptotic stability of dynamical networks iff the Jacobian is non-singular

For a vast class of dynamical networks, including chemical reaction networks (CRNs) with monotonic reaction rates, the existence of a polyhedral Lyapunov function (PLF) implies structural (i.e., parameter-free) local stability. Global structural stability is ensured under the additional assumption that each of the variables (chemical species concentrations in CRNs) is subject to a spontaneous infinitesimal dissipation. This paper solves the open problem of global structural stability in the absence of the infinitesimal dissipation, showing that the existence of a PLF structurally ensures global convergence if and only if the system Jacobian passes a structural non-singularity test. It is also shown that, if the Jacobian is structurally non-singular, under positivity assumptions for the system partial derivatives, the existence of an equilibrium is guaranteed. For systems subject to positivity constraints, it is shown that, if the system admits a PLF, under structural non-singularity assumptions, global convergence within the positive orthant is structurally ensured, while the existence of an equilibrium can be proven by means of a linear programming test and the computation of a piecewise-linear-in-rate Lyapunov function.

Ceramide-transfer protein-mediated ceramide transfer is a structurally tunable flow-inducing mechanism with structural feed-forward loops

This paper considers two models of ceramide-transfer protein (CERT)-mediated ceramide transfer at the trans-Golgi network proposed in the literature, short distance shuttle and neck swinging, and seeks structural (parameter-free) features of the two models, which rely exclusively on the peculiar interaction network and not on specific parameter values. In particular, it is shown that both models can be seen as flow-inducing systems, where the flows between pairs of species are tuned by the concentrations of other species, and suitable external inputs can structurally regulate ceramide transfer. In the short distance shuttle model, the amount of transferred ceramide is structurally tuned by active protein kinase D (PKD), both directly and indirectly, in a coherent feed-forward loop motif. In the neck-swinging model, the amount of transferred ceramide is structurally tuned by active PI4KIIIβ, while active PKD has an ambivalent effect, due to the presence of an incoherent feed-forward loop motif that directly inhibits ceramide transfer and indirectly promotes it; the structural role of active PKD is to favour CERT mobility in the cytosol. It is also shown that the influences among key variables often have structurally determined steady-state signs, which can help falsify the models against experimental traces.

Aggregates of Monotonic Step Response Systems: A Structural Classification

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.

A Bounded Complementary Sensitivity Function Ensures Topology-Independent Stability of Homogeneous Dynamical Networks

This paper investigates the topology-independent stability of homogeneous dynamical networks, composed of interconnected equal systems. Precisely, dynamical systems with identical nominal transfer function <inline-formula> <tex-math notation=LaTeX>