Alfredo Braunstein

The formation of polarized signaling domains on cell membranes is a fundamental example of biological pattern formation. While such patterns resemble structures from equilibrium phase separation, they are intrinsically nonequilibrium, driven by energy-consuming enzymatic cycles that switch molecules like phosphoinositides or small GTPases between distinct states. Here, we develop a minimal model of this enzyme-driven phase-ordering process. Starting from microscopic reaction kinetics, we derive a mesoscopic theory that belongs to the class of active model A with a global constraint. This framework yields an explicit mean-field phase diagram and closed-form expressions for key observables, such as interfacial tension, domain fractions, and phase-coexistence boundaries, in terms of kinetic rates. In this context, phase coexistence is controlled by nonequilibrium parameters such as- catalytic rates and enzymatic asymmetry, rather than equilibrium parameters such as saturation concentrations. The resulting phase-separated domains rapidly exchange material with their surroundings. Their maintenance requires a continuous power input determined by enzymatic kinetics. The predicted phenomenology is consistent with experimental observations on reconstituted systems of phosphoinositide and Rab5 membrane patterning. We further study how metastable uniform states decay via nucleation of minority-phase domains and subsequent coarsening, driven by an effective interfacial tension. Using largedeviation theory, we derive the critical nucleation radius under the action of the intrinsic, multiplicative chemical noise. The analytical results are quantitatively confirmed by stochastic simulations of the process. Our work provides a theoretical framework identifying key biochemical parameters controlling active phase separation on membrane scaffolds, offering testable predictions for experiments.

We propose an analytic approach for the steady-state dynamics of Markov processes on locally tree-like graphs. It is based on time-translation invariant probability distributions for edge trajectories, which we encode in terms of infinite matrix products. For homogeneous ensembles on regular graphs, the distribution is parametrized by a single d× d× r^2 d × d × r 2 tensor, where r r is the number of states per variable, and d d is the matrix-product bond dimension. While the method becomes exact in the large- d d limit, it typically provides highly accurate results even for small bond dimensions d d . The d^2r^2 d 2 r 2 parameters are determined by solving a fixed point equation, for which we provide an efficient belief-propagation procedure. We apply this approach to a variety of models, including Ising-Glauber dynamics with symmetric and asymmetric couplings, as well as the SIS model. Even for small d d , the results are compatible with Monte Carlo estimates and accurately reproduce known exact solutions. The method provides access to precise temporal correlations, which, in some regimes, would be virtually impossible to estimate by sampling.

In Bayesian inference, computing the posterior distribution from the data is typically a nontrivial problem, which usually requires approximations such as mean-field approaches or numerical methods, like the Monte Carlo Markov chain. Being a high-dimensional distribution over a set of correlated variables, the posterior distribution can undergo the notorious replica symmetry-breaking transition. When that happens, several mean-field methods and virtually every Monte Carlo scheme cannot provide a reasonable approximation to the posterior and its marginals. Replica symmetry is believed to be guaranteed whenever the data are generated with known prior and likelihood distributions, namely under the so-called Nishimori conditions. In this paper, we break this belief by providing a counterexample showing that under the Nishimori conditions, replica symmetry breaking arises. Introducing a simple, geometrical model that can be thought of as a patient-zero retrieval problem in a highly infectious regime of the epidemic Susceptible-Infectious model, we show that under the Nishimori conditions there is evidence of replica symmetry breaking. We achieve this result by computing the instability of the replica symmetric cavity method toward the one-step replica symmetry-broken phase. The origin of this phenomenon-replica symmetry breaking under the Nishimori conditions-is likely due to the correlated disorder appearing in the epidemic models.

A novel generalized mean-field approximation, called the small-coupling dynamic cavity (SCDC) method, for epidemic inference and risk assessment is presented. The method is developed within a fully Bayesian framework and accounts for noncausal effects generated by the presence of observations. It is based on a graphical model representation of the epidemic stochastic process and utilizes dynamic cavity equations to derive a set of self-consistent equations for probability marginals defined on the edges of the contact graph. By performing a small-coupling expansion, a pair of time-dependent cavity messages is obtained, which capture the probability of individual infection and the conditioning power of observations. In its efficient formulation, the computational cost per iteration of the SCDC algorithm is linear in the duration of the epidemic dynamics. While the method is derived for the susceptible-infected (SI) model, it is straightforwardly applicable to many other Markovian epidemic processes, including recurrent ones. This linear complexity is particularly advantageous for recurrent epidemic processes, where inference methods are typically exponentially complex in the duration of the epidemic dynamics. The method exhibits high accuracy in assessing individual risk, as demonstrated by tests on the SI model applied to various classes of synthetic contact networks, where it performs on par with belief propagation techniques and generally exceeds the performance of individual-based mean-field methods. Additionally, the method was applied to recurrent epidemic models, where it showed interesting performance even for relatively large values of the infection probability, highlighting its versatility and effectiveness in challenging scenarios. Although convergence issues may arise due to long-range correlations in contact graphs, the estimated marginal probabilities remain sufficiently accurate for reliable risk estimation. Future work includes extending the method to non-Markovian recurrent epidemic models and investigating the role of second-order terms in the small-coupling expansion of the observation-reweighted dynamic cavity equations.

The recent COVID-19 pandemic underscores the significance of early stage nonpharmacological intervention strategies. The widespread use of masks and the systematic implementation of contact tracing strategies provide a potentially equally effective and socially less impactful alternative to more conventional approaches, such as large-scale mobility restrictions. However, manual contact tracing faces strong limitations in accessing the network of contacts, and the scalability of currently implemented protocols for smartphone-based digital contact tracing becomes impractical during the rapid expansion phases of the outbreaks, due to the surge in exposure notifications and associated tests. A substantial improvement in digital contact tracing can be obtained through the integration of probabilistic techniques for risk assessment that can more effectively guide the allocation of diagnostic tests. In this study, we first quantitatively analyze the diagnostic and social costs associated with these containment measures based on contact tracing, employing three state-of-the-art models of SARS-CoV-2 spreading. Our results suggest that probabilistic techniques allow for more effective mitigation at a lower cost. Secondly, our findings reveal a remarkable efficacy of probabilistic contact-tracing techniques in performing backward and multistep tracing and capturing superspreading events.

We investigate the information-theoretical limits of inference tasks in epidemic spreading on graphs in the thermodynamic limit. The typical inference tasks consist in computing observables of the posterior distribution of the epidemic model given observations taken from a ground-truth (sometimes called planted) random trajectory. We can identify two main sources of quenched disorder: the graph ensemble and the planted trajectory. The epidemic dynamics however induces nontrivial long-range correlations among individuals' states on the latter. This results in nonlocal correlated quenched disorder which unfortunately is typically hard to handle. To overcome this difficulty, we divide the dynamical process into two sets of variables: a set of stochastic independent variables (representing transmission delays), plus a set of correlated variables (the infection times) that depend deterministically on the first. Treating the former as quenched variables and the latter as dynamic ones, computing disorder average becomes feasible by means of the replica-symmetric cavity method. We give theoretical predictions on the posterior probability distribution of the trajectory of each individual, conditioned to observations on the state of individuals at given times, focusing on the susceptible infectious (SI) model. In the Bayes-optimal condition, i.e., when true dynamic parameters are known, the inference task is expected to fall in the replica-symmetric regime. We indeed provide predictions for the information theoretic limits of various inference tasks, in form of phase diagrams. We also identify a region, in the Bayes-optimal setting, with strong hints of replica-symmetry breaking. When true parameters are unknown, we show how a maximum-likelihood procedure is able to recover them with mostly unaffected performance.

Computing observables from conditioned dynamics is typically computationally hard, because, although obtaining independent samples efficiently from the unconditioned dynamics is usually feasible, generally most of the samples must be discarded (in a form of importance sampling) because they do not satisfy the imposed conditions. Sampling directly from the conditioned distribution is non-trivial, as conditioning breaks the causal properties of the dynamics which ultimately renders the sampling procedure efficient. One standard way of achieving it is through a Metropolis Monte-Carlo procedure, but this procedure is normally slow and a very large number of Monte-Carlo steps is needed to obtain a small number of statistically independent samples. In this work, we propose an alternative method to produce independent samples from a conditioned distribution. The method learns the parameters of a generalized dynamical model that optimally describe the conditioned distribution in a variational sense. The outcome is an effective, unconditioned, dynamical model, from which one can trivially obtain independent samples, effectively restoring causality of the conditioned distribution. The consequences are twofold: on the one hand, it allows us to efficiently compute observables from the conditioned dynamics by simply averaging over independent samples. On the other hand, the method gives an effective unconditioned distribution which is easier to interpret. The method is flexible and can be applied virtually to any dynamics. We discuss an important application of the method, namely the problem of epidemic risk assessment from (imperfect) clinical tests, for a large family of time-continuous epidemic models endowed with a Gillespie-like sampler. We show that the method compares favorably against the state of the art, including the soft-margin approach and mean-field methods.

Reconstructing missing information in epidemic spreading on contact networks can be essential in prevention and containment strategies. For instance, identifying and warning infective but asymptomatic individuals (e.g., manual contact tracing) helped contain outbreaks in the COVID-19 pandemic. The number of possible epidemic cascades typically grows exponentially with the number of individuals involved. The challenge posed by inference problems in the epidemics processes originates from the difficulty of identifying the almost negligible subset of those compatible with the evidence (for instance, medical tests). Here we present a new generative neural networks framework that can sample the most probable infection cascades compatible with observations. Moreover, the framework can infer the parameters governing the spreading of infections. The proposed method obtains better or comparable results with existing methods on the patient zero problem, risk assessment, and inference of infectious parameters in synthetic and real case scenarios like spreading infections in workplaces and hospitals.

Significance: Contact tracing mobile applications are clear candidates for enabling us to slow down an epidemic and keep society running while holding the health risks down. Currently used mobile applications aim to notify individuals who were recently in significant contact with an individual who tested COVID-19 positive. In our work, we aim to quantify the epidemiological gain one would obtain if, additionally, individuals who were recently in contact could exchange messages of information. With such a message-passing addition, the risk of contracting COVID-19 could be estimated with much better accuracy than simple contact tracing. We conclude that probabilistic risk estimation is capable of enhancing performance of digital contact tracing and should be considered in the mobile tracing applications.