2026

Biological networks are customarily described as structurally robust. This means that they often function extremely well under large forms of perturbations affecting both the concentrations and the kinetic parameters. In order to explain this property, various mathematical notions have been proposed in the literature. In this paper, we propose the notion of structural contractivity, building on the previous work of the authors. That previous work characterized the long-term dynamics of classes of biological interaction networks, based on ``rate-dependent Lyapunov functions."" Here, we show that stronger notions of convergence can be established by proving structural contractivity with respect to nonstandard polyhedral ℓ ∞ -norms. In particular, we show that such networks are nonexpansive. With additional verifiable conditions, we show that they are strictly contractive over arbitrary positive compact sets. In addition, we show that such networks entrain to periodic inputs. We illustrate our theory with examples drawn from the modeling of intracellular signaling pathways.

Metastatic melanoma presents a formidable challenge in oncology due to its high invasiveness and resistance to current treatments. Central to its ability to metastasize is the Notch signaling pathway, which, when activated through direct cell-cell interactions, propels cells into a metastatic state through mechanisms akin to the epithelial-mesenchymal transition (EMT). While the upregulation of miR-222 has been identified as a critical step in this metastatic progression, the mechanism through which this upregulation persists in the absence of active Notch signaling remains unclear. Here we introduce a dynamical system model that integrates miR-222 gene regulation with histone feedback mechanisms. Through computational analysis, we delineate the non-linear decision boundaries that govern melanoma cell fate transitions, taking into account the dynamics of Notch signaling and the role of epigenetic modifications. Our approach highlights the critical interplay between Notch signaling pathways and epigenetic regulation in dictating the fate of melanoma cells.

This paper studies gradient dynamics subject to additive stochastic noise, which may arise from sources such as stochastic gradient estimation, measurement noise, or stochastic sampling errors. To analyze the robustness of such stochastic gradient systems, the concept of small-covariance noise-to-state stability (NSS) is introduced, along with a Lyapunov-based characterization. Furthermore, the classical Polyak–Lojasiewicz (PL) condition on the objective function is generalized to the K-PL condition via comparison functions, thereby extending its applicability to a broader class of optimization problems. It is shown that the stochastic gradient dynamics exhibit small-covariance NSS if the objective function satisfies the K-PL condition and possesses a globally Lipschitz continuous gradient. This result implies that the trajectories of stochastic gradient dynamics converge to a neighborhood of the optimum with high probability, with the size of the neighborhood determined by the noise covariance. Moreover, if the K-PL condition is strengthened to a K_∞-PL condition, the dynamics are NSS; whereas if it is weakened to a general positive-definite-PL condition, the dynamics exhibit integral NSS. The results further extend to objectives without globally Lipschitz gradients through appropriate step-size tuning. The proposed framework is further applied to the robustness analysis of policy optimization for the linear quadratic regulator (LQR) and logistic regression.

This work explores the integration of machine learning (ML) and mechanistic models (MM). While ML has demonstrated remarkable success in data-driven modeling across engineering, biology, and other scientific fields, MM remain essential for their interpretability and capacity to extrapolate beyond observed conditions based on established principles such as chemical kinetics and physiological processes. However, MM can be labor-intensive to construct and often rely on simplifying assumptions that may not fully capture real-world complexity. It is thus desirable to combine MM and ML approaches so as to enable more robust predictions, enhanced system insights, and improved handling of sparse or noisy data. A key challenge when doing so is ensuring that ML components do not disregard mechanistic information, potentially leading to overfitting or reduced interpretability. To address that challenge, this paper introduces the idea of Partially Uncertain Model Structures (PUMS) and investigates conditions that discourage the ML components from ignoring mechanistic constraints. This work also introduces the concept of embedded Physics-Informed Neural Networks (ePINNs), which consist of two loss-sharing neural networks that seamlessly blend ML and MM components. This work arose in the study of the context problem in synthetic biology. Engineered genetic circuits may exhibit unexpected behavior in living cells due to resource sharing. To illustrate the advantages of the ePINNs approach, this paper applies the framework to a gene network model subject to resource competition, demonstrating the effectiveness of this hybrid modeling approach in capturing complex system interactions while maintaining physical consistency.

In this paper we study feedback interconnections of positive nonlinear SISO systems and introduce two complementary properties: persistent-input/persistent-output (PIPO) for the plant and persistent-input/transient-output (PITO) for the controller. Assuming forward completeness of the closed loop and a one-sided affine growth bound on the controller output, we show that PIPO and PITO jointly imply boundedness of the control signal. The result is input-output in nature and does not require linearity or monotonicity of the interconnected subsystems, although it does not in general guarantee boundedness of the full controller state. We provide a structural PIPO condition for positive monotone plants with a class-K_infinity steady-state characteristic, establish PITO for the antithetic integral controller with explicit gains, and illustrate the framework on a nonlinear integral-feedback motif with multiplicative controller growth.

Cancer therapies often fail when intolerable toxicity or drug-resistant cancer cells undermine otherwise effective treatment strategies. Over the past decade, adaptive therapy has emerged as a promising approach to postpone emergence of resistance by altering dose timing based on tumor burden thresholds. Despite encouraging results, these protocols often overlook the crucial role of toxicity-induced treatment breaks, which may permit tumor regrowth. Herein, we explore the following question: would toxicity feedback improve or hinder the efficacy of adaptive therapy? To address this question, we propose a mathematical framework for incorporating toxic feedback into treatment design. We find that the degree of competition between sensitive and resistant populations, along with the growth rate of resistant cells, critically modulates the impact of toxicity feedback on time to progression. Further, our model identifies circumstances where strategic treatment breaks, which may be based on either tumor size or toxicity, can mitigate overtreatment and extend time to progression, both at the baseline parameterization and across a heterogeneous virtual population. Taken together, these findings highlight the importance of integrating toxicity considerations into the design of adaptive therapy.

Understanding how opposing regulatory factors shape gene expression is essential for understanding complex biological systems. A motivating observation, drawn from cancer epigenetics, is that removing an activating factor can sometimes lead to higher, not lower, expression of a gene that is also subject to a repressing factor. Prior theoretical work explained this counterintuitive behavior by competition of repressors and activators for genomic binding sites. However, it has been difficult to test this directly in natural systems, where layers of regulation obscure causal relationships. This paper introduces a fully synthetic, tunable genetic platform in a prokaryotic model system that reconstitutes this competition mechanism in a controlled and isolated setting. The genetic platform contains a target gene with binding sites for both an activator and a repressor, together with separate overlapping decoy binding sites for the same regulators. Activator and repressor functions are implemented using CRISPRa and CRISPRi, which permit independent control of regulator expression levels, design of the binding sites, and modulation of the binding affinities. Using this minimal system, we demonstrate that increasing activator expression level can reduce expression of the target gene when both regulators are present, consistent with the hypothesis that additional activator molecules displace the repressor from decoy sites, which becomes available to repress the target. By demonstrating how competition for genomic binding sites can invert expected regulatory responses, this synthetic framework provides a system for understanding similar paradoxical behaviors in natural regulatory networks and establishes a foundation for future studies in more complex mammalian contexts.

We study the problem of designing a controller that satisfies an arbitrary number of affine inequalities at every point in the state space. This is motivated by the use of guardrails in autonomous systems. Indeed, a variety of key control objectives, such as stability, safety, and input saturation, are guaranteed by closed-loop systems whose controllers satisfy such inequalities. Many works in the literature design such controllers as the solution to a state-dependent quadratic program (QP) whose constraints are precisely the inequalities. When the input dimension and number of constraints are high, computing a solution of this QP in real time can become computationally burdensome. Additionally, the solution of such optimization problems is not smooth in general, which can degrade the performance of the system. This paper provides a novel method to design a smooth controller that satisfies an arbitrary number of affine constraints. This why we refer to it as a universal formula for control. The controller is given at every state as the minimizer of a strictly convex function. To avoid computing the minimizer of such function in real time, we introduce a method based on neural networks (NN) to approximate the controller. Remarkably, this NN can be used to solve the controller design problem for any task with less than a fixed input dimension and number of affine constraints, and is completely independent of the state dimension. Additionally, we show that the NN-based controller only needs to be trained with datapoints from a compact set in the state space, which significantly simplifies the training process. Various simulations showcase the performance of the proposed solution, and also show that the NN-based controller can be used to warmstart an optimization scheme that refines the approximation of the true controller in real time, significantly reducing the computational cost compared to a generic initialization.

The Lojasiewicz inequality characterizes objective-value convergence along gradient flows and, in special cases, yields exponential decay of the cost. However, such results do not directly imply convergence of the state. In this paper, we use contraction theory to derive state-space guarantees for gradient systems satisfying generalized Lojasiewicz inequalities. We first show that, when the objective has a unique strongly convex minimizer, the generalized Lojasiewicz inequality implies semi-global exponential stability; on arbitrary compact subsets, this yields exponential stability. We then give two curvature-based sufficient conditions, together with constraints on the Lojasiewicz rate, under which the nonconvex gradient flow is globally incrementally exponentially stable, a property strictly stronger than global exponential stability. A few examples are presented at the end of the paper to validate the proposed theory.

This note establishes a theoretical framework for finding (potentially overparameterized) approximations of a function on a compact set with a-priori bounds for the generalization error. The approximation method considered is to choose, among all functions that (approximately) interpolate a given data set, one with a minimal Lipschitz constant. The paper establishes rigorous generalization bounds over practically relevant classes of approximators, including deep neural networks. It also presents a neural network implementation based on Lipschitz-bounded network layers and an augmented Lagrangian method. The results are illustrated for a problem of learning the dynamics of an input-to-state stable system with certified bounds on simulation error.

This paper investigates how the compositional structure of neural networks shapes their optimization landscape and training dynamics. We analyze the gradient flow associated with overparameterized optimization problems, which can be interpreted as training a neural network with linear activations. Remarkably, we show that the global convergence properties can be derived for any cost function that is proper and real analytic. We then specialize the analysis to scalar cost functions, where the geometry of the landscape can be fully characterized. In this setting, we demonstrate that key structural features – such as the location and stability of saddle points – are universal across all admissible costs, depending solely on the overparameterized representation rather than on problem-specific details. Moreover, we show that convergence can be arbitrarily accelerated depending on the initialization, as measured by an imbalance metric introduced in this work. Finally, we discuss how these insights may generalize to neural networks with sigmoidal activations, showing through a simple example that certain geometric and dynamical properties persist beyond the linear case.

Boumal, Criscitiello and Rebjock (BCR) proved that if M is a contractible, connected and complete Riemannian manifold, then every smooth function f: M→ R satisfying the global Polyak–Łojasiewicz inequality (PŁI) is necessarily of the form f = f^* + \|φ\|^2 with φ a submersion. Informally, minimizing such a function amounts to solving a nonlinear least-squares problem in new coordinates. The global PŁI hypothesis fails, however, in many problems of interest, among them continuous-time LQR policy optimization in optimal control and a standard formulation of logistic regression. A hierarchy of weakened PŁ inequalities has been introduced in order to cover such problems, and more generally to study the effect of noise and adversarial perturbations on gradient flows. This note shows that, with minor modifications, the same reduction to a nonlinear least-squares problem holds under a substantially weaker hypothesis, ``semiglobal'' PŁI, which is satisfied in both of the examples just mentioned. That condition asks that f satisfy an estimate \|∇ f(x)\| ≥ α(f(x)-f^*) for all x, with α merely positive definite and bounded below by a positive multiple of s for small s>0.

It is known that cooperative systems, and more generally systems monotone with respect to cones, behave under appropriate irreducibility conditions like one-dimensional systems, as evidenced by results such as Hirsch's generic convergence theorem. It is also known, following work of Sanchez and others, that two-cooperative systems, those preserving a generally nonconvex 2-dimensional cone (tested through the diminishing of sign variations), behave like two-dimensional systems, as evidenced by Poincare–Bendixson-type theorems. In these notes I attempt to give some geometric intuition for these dimensionality reductions, based on Birkhoff–Hilbert contractions of the projective metric on a positive cone.

This paper considers systems in which two or more ligands bind independently to distinct sites in a common scaffold. Such systems arise in a range of applications, including immunotherapy and synthetic biology. We show that each stoichiometric compatibility class contains a unique steady state, and that this steady state is asymptotically stable. The main result gives a rigorous proof that the steady-state concentration of the fully bound complex, viewed as a function of the total scaffold concentration, has a unique maximum. This biphasic dose response behavior is a characteristic feature of scaffolding systems and, in the special case of two ligands, plays an important role in the design and analysis of bispecific antibody drugs.

Antithetic feedback controllers have become a key experimental and theoretical tool in synthetic biology. Introduced by Khammash and collaborators about 10 years ago, they are employed in order to achieve the practical regulation of protein expression, including tracking and robust disturbance rejection. In closed-loop, there are unique equilibria which, depending on parameter values, can be unstable. It had been shown, however, that this instability is not arbitrary: any bounded trajectory that stays away from the equilibrium must converge to a periodic orbit. This motivated a long-standing open question: is every trajectory bounded? In other words, even if the equilibrium is unstable, can nonlinear effects prevent unbounded excursions in the state space? This paper provides an affirmative answer, establishing the boundedness of all solutions. Previous attempts to prove this fact using Lyapunov functions had no success. Instead, this paper takes a completely different approach, specific to antithetic configurations, in which the key idea is to think of the controller as providing a ``persistently negative feedback'' which acts far away from the equilibrium in such a way so as to keep trajectories from diverging. This new approach, although tailored to the antithetic controller, might be useful in other applications as well.

We study the monotonicity of the cumulative dose response (cDR) for a class of incoherent feedforward motif (IFFMs) systems with linear intermediate dynamics and nonlinear output dynamics. While the instantaneous dose response (DR) may be nonmonotone with respect to the input, the cDR can still be monotone. To analyze this phenomenon, we derive an integral representation of the sensitivity of cDR with respect to the input and establish general sufficient conditions for both monotonicity and non-monotonicity. These results reduce the problem to verifying qualitative sign properties along system trajectories. We apply this framework to four canonical IFFM systems and obtain a complete characterization of their behavior. In particular, IFFM1 and IFFM3 exhibit monotone cDR despite potentially non-monotone DR, while IFFM2 is monotone already at the level of DR, which implies monotonicity of cDR. In contrast, IFFM4 violates these conditions, leading to a loss of monotonicity. Numerical simulations indicate that these properties persist beyond the structured initial conditions used in the analysis. Overall, our results provide a unified framework for understanding how network structure governs monotonicity in cumulative input–output responses.

In this work we study the convergence of gradient methods for nonconvex optimization problems – specifically the effect of the problem formulation to the convergence behavior of the solution of a gradient flow. We show through a simple example that, surprisingly, the gradient flow solution can be exponentially or asymptotically convergent, depending on how the problem is formulated. We then deepen the analysis and show that a policy optimization strategy for the continuous-time linear quadratic regulator (LQR) (which is known to present only asymptotic convergence globally) presents almost global exponential convergence if the problem is overparameterized through a linear feed-forward neural network (LFFNN). We prove this qualitative improvement always happens for a simplified version of the LQR problem and derive explicit convergence rates for the gradient flow. Finally, we show that both the qualitative improvement and the quantitative rate gains persist in the general LQR through numerical simulations.

In several applications, including in synthetic biology, one often has input/output data on a system composed of many modules, and although the modules’ input/output functions and signals may be unknown, knowledge of the composition architecture can allow to significantly reduce the amount of training data required to learn the system’s input/output mapping. Learning the modules’ input/output functions is also necessary for designing new systems from different composition architectures. Here, we propose a modular learning framework, which incorporates prior knowledge of the system’s compositional structure to (a) identify the composing modules’ input/output functions from the system’s input/output data and (b) achieve this by using a reduced amount of data compared to what would be required without knowledge of the compositional structure. To achieve this, we introduce the notion of modular identifiability, which allows to recover the modules’ input/output functions from a subset of the system’s input/output data, and provide theoretical guarantees on a class of systems motivated by genetic circuits. We illustrate the theory through computational studies, showing that a neural network (NNET) that accounts for the compositional structure is able to learn the composing modules’ input/output functions and to predict the system’s output on inputs which lie outside of the training set. By reducing the need for experimental data, and allowing modules’ identification, this framework offers the potential to ease the design of synthetic biological circuits and of multi-module systems more generally.