### Modeling & Computation

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The research lines of the Modeling & Computation area are listed below:

- Complex Flows & Non Linear Time Series Analysis
- Computational Biology
- Photonics in Random Media
- Scientific Computing & Big Datas
- Statistical Mechanics
- Turbulence in Plasmas

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The experimental study of nonlinear and complex systems relies on the design of specific advanced data analysis tools, which need to be able to identify properties of the dynamics which are often subtle, hidden in chaotic motions, or simply by the noise. In such cases, it is necessary to use appropriate techniques that also need to be benchmarked for a quantitative reliability. These techniques, often based on advanced statistical and mathematical tools, can be then applied on suitable systems to provide important validation or input for theoretical models and numerical simulations.

Solar flares and the Sun-Earth connection. In recent years, the study of solar variability and its influence on Earth has increased, both because of a more compelling need for a good protection against solar storms that arise from the massive use of the modern technology, and for the exponential quantitative and qualitative increase of measurements available. Our approach is based on the exploitation of in-situ and remote measurements of the Sun-Earth system, including remote solar imaging, magnetic field, and energetic particles, and solar wind and magnetospheric in situ and remote measurements. The main research topics concern the dynamics of the solar active regions and their relationship with flaring activity through the analysis of the complexity of the magnetic field configuration in flaring active regions [Sorriso-Valvo et al., 2015]. Other studies concern the interpretation in terms of coupling models is also studied from data, for example through the analysis of proper modes [Vecchio et al., 2005].

Geophysical time series analysis. Several geophysical systems can be studied in the framework of complex systems and nonlinear dynamics. Examples are: earthquakes [Carbone et al., 2005], geomagnetic field reversals [Carbone et al., 2006], geomagnetic activity. These phenomena often can be observed as time series, which can be thus studied using, e.g., statistical tools. Their description is useful to validate theoretical models, to advance the general knowledge of the process, and to some extent to help improving the predictability of catastrophic events. Time series of various phenomena are studied and their statistical properties assessed.

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Theoretical methods and mathematical modeling can be applied to investigate complex physical systems that are found in biological organisms. Computer simulations are a common tool to analyze the interactions of biological structures at the molecular level, or the connections among different subsystems in large-scale networks, which are crucial to understand the basic mechanisms in the proteomics and metabolomics fields.

Ligand binding. Receptor-ligand interactions are responsible of molecular recognition, binding, transport and release. These mechanisms are fundamental in many processes that are vital in any living organism, and can be exploited in several applications of nanotechnology. Molecular dynamics simulations and docking calculations are theoretical tools that allow us to determine a model of interaction between a receptor and a variety of ligands. These computational techniques are useful to predict structural and dynamic properties that determine the functionality of a molecular complex between transport proteins and small compounds, such as ligands of pharmaceutical interest.

Protein folding. Proteins are molecular ‘nanomachines’ that acquire their tridimensional structure through a spontaneous process of folding. This phenomenon is central for some of the basic mechanisms of life. Furthermore, their competitive processes (unfolding and misfolding) lead to a loss of function and are correlated to some serious pathologies. Protein folding is an intrinsically complex molecular process that needs to be investigated with a combined theoretical and experimental approach. Molecular dynamics simulations and other computational methods are able to reveal crucial details of the folding reaction.
Regulatory RNA. Overcoming the decades’ old view according to which they are mere intermediates between DNA and proteins in the central dogma of molecular biology, RNAs are increasingly recognized for the remarkable variety of functional roles they play in eukaryotic gene expression. Non-coding RNAs (ncRNAs), in particular, are deeply involved in developmental programs and disease, to the point of having become targets of novel therapeutic approaches that aim at either inhibiting or enhancing their functionality. Importantly, most ncRNAs act by protein-mediated binding to other nucleic acids, leading to complex inter-dependencies be- tween coding and ncRNAs, RNA-binding proteins and DNA. Their central role in regulation is perhaps best exemplified by microRNAs (miRNAs), small ncRNAs of 20–25 nucleotides (nt) that mediate post-transcriptional regulation (PTR) via gene expression silencing in animals. miRNAs are believed to affect the expression of about two-thirds of protein-coding genes in humans. Their apparent ubiquity, together with the strong topological heterogeneities that characterize the PTR network that maps out the known interactions between miRNAs and other RNA species (such as mRNAs), has lead to the idea that competition for shared miRNAs can cause an effective positive interaction between different transcripts, currently referred to as the ‘ceRNA effect’. In this context, we are interested in quantifying the role and effectiveness of miRNA-mediated RNA cross-talk, with the goal of clarifying (i) under which conditions it outperforms other regulatory mechanisms, for instance in processing gene expression noise; (b) whether it can carry a significant systemic role; and (c) its relevance in specific biological cases of differentiation and disease.

Cell growth and its biosynthetic costs. The coupling between the physiology of cell growth and cellular composition has been actively investigated since the 1940s. In exponentially growing bacteria, such interdependence is best expressed in a quantitative way by the bacterial ‘growth laws’ that directly relate the protein, DNA and RNA content of a cell to the growth rate. Many such laws have been experimentally characterized and many more are currently being probed at increasingly high resolution. The emerging scenario suggests that proteome organization in bacteria is actively regulated in response to the growth conditions. Several phenomenological models explain the origin of different growth laws at coarse-grained levels. By contrast, genome-scale approaches probing such relationships at the molecular level are far less developed. We have developed a mathematical modeling scheme called Constrained Allocation Flux Balance Analysis or CAFBA, in which the costs of gene expression are accounted for effectively through a single global constraint on metabolic fluxes, encodeing for the relative adjustment of proteome sectors at different growth rates. Using bacteria as the initial model organisms, we are interested in quantifying the trade-off between cell growth and its associated biosynthetic costs, generating testable predictions about the way in which the usage of metabolic pathways and protein expression levels are modulated by the growth conditions.

Cell-to-cell variability in exponentially growing bacteria. Current experimental techniques (see e.g. the ‘mother machine’) can probe physiological variability by characterizing e.g. growth rate distributions for bacterial populations at single cell resolution. These distributions reflect noise at various levels, from intracellular stochasticity in gene expression and metabolite levels to fluctuations in the extracellular medium. However, upon controlling the latter, they provide a window to analyze the role of noise in the genotype-phenotype relationship. Straightforward sam- pling of the feasible space predicted by mathematical models, however, does not explain the observed statistics. We are therefore interested in identifying a physical or biological principle that drives the selection of observed growth states and hence shed light on the origin of the observed phenotypic diversity.

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In recent years, random lasing materials (e.g. powders, porous media, precipitates in solution, or photonic crystals with impurities) have been extensively studied experimentally. Pumping energy into these systems causes them to re-emit multi-mode coherent light, with a spectrum displaying randomly arranged peaks in frequency. Starting from the structure and geometry of the atoms and molecules that scatter the light waves, one would eventually want a theory that predicts the onset, the nature and the features of the light modes and answering the following questions.
What shape and size do light modes display in space ?
In which dimension and under which conditions do they localize because of disorder?
On which frequencies do light modes emit in cavity-less media?
Can there be a random laser pulse in time?
Do competing random laser modes phase-lock as in multimode standard lasers?
How strong is the coupling magnitude and how is it related to the coupling modes spatial overlap and the etherogeneous optical susceptibility?
The latter two questions are connected to the coupling property of depending on the spatial overlap of the electromagnetic fields of the interacting modes. This feature ascribes to the problem of assessing the structure of an interacting network of light-modes in a statistical mechanics representation. Indeed, a set of modes can interact only if their electromagnetic fields overlap in space and, in the lasing regime, non-linear amplification occurs only if the frequencies of the modes satisfy some kind of mode-locking condition. These rules strongly influence the set of feasible interactions in which each mode is viewed as a network node. A key challenge that we address is the characterization of the structure of this network of wave-modes, including the strengths and signs of the relevant random interactions, as is required, e.g., in order to distinguish apart physical regimes of laser stationary behaviour. To this aim a Hamiltonian theory has been derived and investigated in systems with different kinds of bond-disorder, ranging from standard ordered multimode mode-locking lasers to recently introduced glassy random lasers.

Glassy Random Laser and Experimental Measurement of Replica Symmetry Breaking
The investigation of the glassy behaviour of light in the framework of our theory is made possible by means of a newly introduced overlap parameter, the Intensity Fluctuation Overlap (IFO) measuring the correlation between intensity fluctuations of waves in random media. This order parameter allows to identify the laser transition in arbitrary physical regimes, with varying amount of disorder and non-linearity. In particular, in random media it allows for the identification of the glassy nature of some kind of random laser, in terms of emission spectra data, the only data so far accessible in random laser measurements. The model devised from first principles in whose framework the parameter is defined is the nonlinear phasor statistical mechanical model. This is a generalised complex spherical spin-glass model solvable in the mean-field approximation by Replica Symmetry Breaking theory. IFO measurements are possible in real experiments, recently leading to a validation of the RSB theory and a new characterisation of lasers in terms of spectral intensity fluctuations.

Interference of Coupling of Waves in Random Media
The light modes interaction network has to be inferred starting from data acquired in measurements, of spectra and correlations of phases and amplitudes of the light modes, and this inference problem is closely analogous to those in our other areas of application of statistical inference. Starting with the analysis of the inverse problem in statistical mechanical systems with continuous variables, like XY and complex phasors, our inference project is concerned with the bottom-up approach for studying statistical models for application to wave and optics. The parameters describing a given model system, like active links in the network system and external field affecting the system, are inferred using the data set which is made available by experimental or numerical measurements.
We adopt various inference techniques to reconstruct the interaction networks and to estimate the coupling values: mean-field approach, Pseudo Likelihood Maximization (PLM) with L1 and L2 regularizations and PLM with decimation. Such inverse problems for network reconstruction are considered on graphs of different kinds, from 2D and 3D nearest-neighbour lattices, Bethe and Erdos-Renyi sparse random graph to dense random graphs.

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We exploit parallel computing on single and multi Graphic Processing Units (GPU’s) for several problems. In particular, we develop optimised parallel codes for continuous variables Monte Carlo dynamics, Population dynamics for Belief propagation and Cavity method in random graphs, and Pseudolikelihood maximisation.

Algorithms. We are interested in the development of efficient computational techniques for the study of inference and optimization problems in large experimental data bases, mostly for complex biological systems. Among the key application domains are the analysis of gene expression at single cell resolution, the study of kinetic and/or thermodynamical conservation laws in cellular metabolic networks, the analysis of evolutionary variability in protein sequences, the characterization of cell-to-cell variability in microbial populations (at both the physiological and the molecular level), and the inference of complex interaction networks (protein-protein, protein-DNA, RNA-RNA) from genomic and/or thermodynamic data. In addition, we work on the development of multi-scale models for metabolic engineering of unicellular organisms and large-scale simulation of human tissues.

Biophysical simulations: Molecular competition on receptors. Many biological processes are based on the interaction between a receptor and various partner molecules that can bind it. To clarify these interactions, ‘in vitro’ experiments usually analyze the receptor in the presence of another single molecule. Nevertheless, ‘in vivo’ mechanisms are much more complex, and there can be competition phenomena among different molecular partners for the same receptor, or among different molecules of the same type that could associate to distinct regions of the same receptors through various binding modes. Computer simulations can help in a systematical mapping of the various possible combinations.

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We study disordered and frustrated systems such as spin-glasses, structural glasses and random photonics models by means of advanced methods in statistical mechanical of disordered and complex systems, namely Replica Symmetry Breaking theory, Cavity method, Belief and Survey propagation, Supersymmetric path integral formulation of the dynamics (à la Martin-Siggia-Rose), Renormalization group approaches (on hierarchical lattice, on finite dimensional squared and cubic cells, on hierarchical models) and enhanced Monte Carlo methods for the numerical simulation of the dynamics both at and off-equilibrium.

By means of the above mentioned techniques we investigate phase transitions and states organisation in complex systems both at high dimensionality, where the mean-field approximation is exact) and in low dimension, where a phase transition is still present but it belongs to a different universally class. Phenomena studied in the recent years are the spin-glass transition and the low temperature replica symmetry breaking, the structural glass transition and of the organisation of the stable and metastable glassy states below the dynamic arrest transition, the random field Ising model transition, the Anderson localization.

Glassy and slowly relaxing systems. A glass can be viewed as a liquid in which a huge slowing down of the diffusive motion of the particles has destroyed its ability to flow on experimental timescales. The slowing down is expressed through the relaxation time, that is, generally speaking, the characteristic time at which the slowest measurable processes relax to equilibrium. Cooling down from the liquid phase, the slow degrees of freedom of the glass former are no longer accessible and the viscosity of the undercooled melt grows several orders of magnitude in a relatively small temperature interval. As a result, in the cooling process, from some point on, the time effectively spent at a certain temperature is not enough to attain equilibrium: the system is said to have fallen out of equilibrium. Nature and characterization of this non-equilibrium glassy regime and of the glass transition are a challenging issue that stimulates deep theoretical work concerning frustrated systems in diverse representations. We work on the theoretical representation of the behavior of viscous liquids, structural glasses and spin-glasses, on the critical slowing down occurring near-by the dynamic arrest, on the aging dynamics, on the extension of glass theories beyond the limit of validity of mean-field approximation.

Inverse problem in statistical mechanics. Given a data set and a model with some unknown parameters, the inverse problem aims to find the values of the model parameters that best fit the data. We focus on systems of interacting elements, in which the inverse problem concerns the statistical inference of the underling interaction network and of its coupling coefficients from observed data on the dynamics of the system. Versions of this problem are encountered in physics, biology, social sciences and finance, neuroscience (just to cite a few), and are becoming more and more important due to the increase in the amount of data available from these fields. A standard approach used in statistical inference is to predict the interaction couplings by maximizing the likelihood function. This technique, however, requires the evaluation of the partition function that, in the most general case, concerns a number of computations scaling exponentially with the system size. Boltzmann machine learning approach uses Monte Carlo sampling to compute the gradients of the Log-likelihood looking for stationary points but this method is computationally manageable only for small systems. A series of faster approximations, such as naive mean-field, independent-pair approximation inversion of Thouless-Anderson-Palmer equations, small correlations expansion, adaptive TAP, adaptive cluster expansion or Bethe approximations have been developed in the last 15 years. These techniques take as input means and correlations of observed variables and most of them assume a fully connected graph as underlying connectivity network, or expand around it by perturbative dilution. In most cases, network reconstruction turns out to be not accurate for small data sizes and/or when couplings are strong or, else, if the original interaction network is sparse. A further method, substantially improving performances for small data, is the so-called Pseudo-Likelyhood Method (PLM), implemented with regularization or with decimation. We work on the analysis of the performances of the various inference methods, on their improvement and on their application to new problems.

Disordered protein states. The ordered structure of proteins is one of the basic paradigms of classical biology, and it provides an explanation for many aspects of their functioning. Nevertheless, in many cases proteins operate in environments far from equilibrium, or possess labile conformations that convert towards order only under particular conditions. Examples include protein folding/unfolding in the presence of temperature and pressure variations, or configuration reorganizations induced by ligand binding in intrinsically disordered proteins. The statistical properties of these ensembles of structures can be studied with sampling techniques based on classical molecular dynamics simulations.

Molecular networks. We are interested in characterizing emergent properties of large networks of interacting molecules of biological significance, e.g. proteins or nucleic acids, using equilibrium and non-equilibrium statistical mechanics methods. Our central goal is to understand what makes these networks optimal and in which precise sense, how the laws of physics limit their performance in such tasks as noise or information processing, and whether they can sustain collective effects similar to those that characterize more traditional systems studied in statistical physics. In turn, our hope is to gain insight about the evolution of the large-scale organization of the known molecular networks that govern cellular and multi-cellular activities.

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Turbulence is an ubiquitous phenomenon that can be observed on a huge range of scales, from galaxy clusters down to micro- and nano-fluidics. It is observed mostly in neutral flows, but also in charged, magnetized flows such as astrophysical plasmas. The study of turbulence requires a multiple approach: theoretical, experimental, based on data analysis and on numerical simulations. All these aspects are exploited here, with particular focus on space and laboratory turbulent plasmas. Most of the visible matter in the universe is in the state of plasma. Often times, astrophysical plasmas have highly turbulent dynamics, resulting in a large number of interesting processes such as: energy dissipation, particle acceleration, excitation of electromagnetic waves, particle heating, magnetic reconnection, formation of shocks. All these phenomena can be studied in-situ only in space plasma, where instruments on-board scientific space missions can take measurements. Data can be studied using specific diagnostic tools, which allow the validation of theories and models. A substantial use of numerical simulations is also necessary. The study of turbulence in the interplanetary space is therefore of broad interest for the understanding of the dynamics of astrophysical plasmas, but also for its implications on laboratory plasmas and for the Sun-Earth interaction. The study of space plasmas turbulence is based on three main approaches: the analysis of data provided by the scientific mission; the development of theoretical models and novel data analysis techniques; the use of numerical simulations (massive computational resources are often required; these are provided by large facilities for high performance computing, such as CINECA, or the UNICAL HPCC. Examples are the full characterization of intermittency in solar wind turbulence [Sorriso-Valvo et al. 1999; 2015], and the validation of the theoretical prediction for the scaling law of the energy flux in solar wind turbulence [Sorriso-Valvo et al., 2007].