POSITION      

POSITIONS SERVED           

EDUCATION

• National Scientific Qualification as Associate Professor, February 2014
  

• Ph.D. in Electronic Engineering, Faculty of Engineering, University of L'Aquila, February 10, 2000. Supervisor: Prof. A. Germani. The thesis (in Italian) deals with the control of undamped flexible structure, by using a functional approach. Most important results concerns the implementation of a finite dimensional control law, whose performances converge to the ones of the infinite dimensional optimal control law, as the approximation index increases to infinity. Taking into account the stability properties of the proposed control law, they are robust with respect to the system physical parameters.

• Qualification to the profession of Engineer, 120/120, University of L'Aquila, December 1995.

• Laurea Degree in Electronic Engineering, 110/110 cum laude, University of L'Aquila, Engineering Faculty, July 20, 1995. The thesis (in Italian) deals with the optimal polynomial filtering for linear discrete-time non Gaussian systems. Supervisor: Prof. A. Germani.

• "G. Galilei" High School Scientific Degree, 60/60, Pescara, July 1989.

TEACHING ACTIVITY
UNIVERSITY OF L'AQUILA,
A.A. 2018/19

MESSAGE: STUDENTS CAN CONTACT PROF. P. PALUMBO IN ROME AT:
06 49937134     (IASI - Via dei Taurini, 19)

• Visiting Professor of "Systems Biology", Master of Science in Mathematical Modeling in Engineering (lessons  in English)

• Teaching Assistant in:
  •  "Teoria dei Sistemi", Prof. C. Manes, first level laurea degree in Information and Control, Electronic, Telecommunication Engineering
  • "Teoria dei Sistemi", Prof. P. Pepe, first level laurea degree in Magement Engineering

• At the University of L'Aquila, Pasquale Palumbo has been Contract Professor of:
  • "Systems Biology" (6CFU, lessons in English), master laurea degree in Mathematical Engineering, since A.A. 2010/11;
  • "Intensive Programme on Mathematical Models in Life Sciences" (3CFU, lessons in English), master laurea degree in Mathematical Engineering, since A.A. 2010/11;
    
  • "Systems Biology" (9CFU, lessons in English), master laurea degree in Mathematical Engineering, A.A. 2008/09 - 2009/10;
    
  • "Teoria dei Sistemi II" (6CFU), laurea degree in Information and Control Engineering, A.A. 2003/04 - 2006/07;
  • "Controlli Automatici II", laurea degree in Information and Control Engineering, A.A. 2002/03;
  • "Calcolo delle Probabilità e Statistica", laurea degree in Civil Engineering, A.A. 2001/02;
  • "Calcolo delle Probabilità", laurea degree in Civil and Environmental Engineering, A.A. 2000/01

• "ESERCIZI DI TEORIA DEI SISTEMI - PARTE I", Libreria Universitaria Benedetti, L'Aquila, September 2003, ISBN 88-87182-14-0
• ESERCIZI DI TEORIA DEI SISTEMI - II EDIZIONE", Libreria Universitaria Benedetti, L'Aquila, Settembre 2008, ISBN 978-88-87182-31-6

More than 100 exercises all endowed with solutions above time-invariant finite-dimensional linear systems:
1) continuous and discrete time evolution: time and frequency domain approach (Laplace and Z transforms)
2) representation of transfer functions: Bode and polar
 diagrams
3) time-invariant linear systems stability; continuous-time systems (Routh criterion), discrete-time systems (Jury criterion), feedback systems (Nyquist criterion)
NEW PARAGRAPH on nonlinear systems stability
4) NEW CHAPTER on
controllability, observability and Kalman decomposition

• CONTROL OF FLEXIBLE SYSTEMS: A FUNCTIONAL APPRACH. Flexible structures can be accurately described by continuum models, i.e. dynamic systems with infinite-dimensional state space. Therefore, any finite-dimensional model necessarily neglects the higher order vibration modes. On the other hand, a control law can be implemented only if it admits a finite-dimensional representation. Two main approaches can be followed to obtain finite-dimensional control laws for infinite-dimensional systems, such as flexible structures: according to the classical approach, a finite-dimensional control law is designed on the basis of a finite-dimensional approximation of the flexible structure (finite elements, Galerkin projections, or other approximation schemes); however, such a classical approach neglects the higher order modes of vibration of the structure, so that in many cases the unmodeled modes can be excited by the control law itself, so inducing undesired and unexpected vibrations in the structure, phenomenon known as spillover. Because of the intrinsic infinite-dimensional nature of the mechanical structure, any control law designed on the basis of a finite-dimensional model may produce this undesired effect. On the other hand, according to a modern approach, a finite-dimensional control law is designed on the basis of a distributed parameter model of the flexible structure. Such a modern approach allows to overcome the drawbacks of the classical one, in that it provides feedback control laws suitably designed exploiting the infinite-dimensional model, where all the modes of vibration of the system are considered. With this approach, the spillover effects are avoided by ensuring the closed loop stability for the original flexible structure. When the regulator is achieved without taking its physical realizability into account, the resulting feedback control law may well be defined as an operator acting on an infinite-dimensional state space. In these cases, a finite-dimensional approximation scheme is needed in order to implement the control law. In order to make effective such a methodology, for any approximation index, the corresponding regulator has to ensure the modal stability of the system, in order to overcome the spillover. The paper R[7] (taken from the Ph.D. dissertation) deals with the LQG control of an undamped Euler-Bernoulli cantilever beam with a tip mass at the free end. The optimal solution to such a problem is available in the literature, and is achieved in an infinite-dimensional setting; also an approximate implementable transfer function is provided. However it is not clear how far the proposed approximate compensator is from the optimal solution. The idea of the paper is to present a sequence of approximate, physically implementable regulators converging to the optimal LQG solution. It is proved that, for any given finite time horizon, the evolution of the system state driven by the approximate control input converges in L2-norm to the evolution of the system state driven by the optimal control law, as the order of the approximation scheme increases. An interesting feature is that the transfer function of the approximated compensator monotonically approaches the shape of the transfer function of the optimal compensator, as it can be appreciated in the magnitude/phase Bode representation. Preliminary results have been presented in C[2]. In R[7] it is proved also that, for each order of the approximation, the closed loop system is strongly stable, and, moreover, the proposed compensator guarantees modal stability of the closed loop system also in the presence of stiffness/inertia uncertainties. Such performances are indeed achieved by means of a Galerkin approximation scheme, based on the complete set of the generalized eigenfunctions of the structure instead of the usual splines. In C[3], an implementable Luenberger-like observer has been proposed for a similar undamped flexible structure, whose measurements are acquired with a not negligible delay.

• POLYNOMIAL FILTERING OF NON GAUSSIAN LINEAR SYSTEMS. It is well known that the optimal solution to the minimum variance filtering problem is given by the expectation value of the state conditioned by all the measurements up to the current time, that is the projection onto the linear space of all the Borel functions of the measurements. In the linear Gaussian case the optimal filter is given by the Kalman filter, which is a linear transformation of the measurements. Unfortunately, in the non Gaussian case, there is not a simple characterization of the conditional expectation, so that it is worthwhile to consider suboptimal estimates which have a simpler mathematical structure. The simplest suboptimal estimate is the optimal affine one. It consists in projecting the state onto the subspace of all the linear transformations of the output. For linear systems the optimal affine estimate is still achieved by the Kalman filter. Suboptimal estimates comprised between the optimal linear and the conditional expectation can be considered by projecting onto subspaces of polynomial transformations of the measurements. A quadratic filter has been implemented in C[1], for a digital linear system, whose measurements are affected by quantization noise.
  

• Singular (descriptor) systems. The optimal polynomial approach has been proposed for the filtering of linear singular systems (also known as descriptor systems). Pubblications in this field use the measurements in order to overcome the lack of knowledge coming from the singular model. According to non restrictive hypotheses, a regular system is associated to the singular model, whose state, filtered by using a polynomial estimate, univocally identify the descriptor vector. In R[2], the polynomial filter has been proposed in the discrete-time framework (a preliminar version has been presented in C[12]). It has been proved that the linear version C[4], gives back the maximum likelihood estimate, in the Gaussian case. In C[5] a Kalman-Bucy-like filter is achieved for a singular stochastic differential system, in the Ito formulation.

• FILTERING AND IDENTIFICATION OF UNCERTAIN SYSTEMS.
  
• Linear variable structure systems. These are switching linear systems, that is systems whose matrices switch according to an unknown parameter. Taking into account the case of a Markovian switching parameter, a suitable state space realization has been proposed. It has been proved that the minimum variance linear approach C[7] (or the others available in literature) is not enough to simultaneously estimate both the state and the unknown parameter, unless a known input suitably excites the system. On the contrary, a polynomial approach C[10] is able also to identify the system even in complete absence of a known input. The case concerning only switching measurements is very interesting, since it models a digital telecommunication system: by applying the procedure presented in C[10] a real time algorithm is implemented providing in real-time both system identification and channel estimation R[14] (in C[27] a preliminary version of R[14] has been presented). In R[4], the case of absence of any stochastic or deterministic characterization of the switching parameter has been investigated. Such a loss of knowledge has been modeled by using an extended singular system, driven by bounded covariance extended noises, whose second order moments are not completely known. Then the minimax criterion is adopted (a preliminary version of the paper has been presented in C[6]). Simulations reported in R[4] show the effectiveness of the proposed approach, compared with other robust filters, such has the ones based on LMIs, or on the maximum entropy criterion. Moreover, in R[4], the asymptotic properties of the filter have been investigated. 

• Linear interval systems. In C[11, 13] the uncertainties of the model deal with one or more unknown constant (or even slowly time-varying) parameters, taking values in a suitably defined discrete C[11] or continuous C[13] range. An extended state is defined, containing among its components the uncertain parameters and, then, is filtered by using the polynomial approach. The same problem of simultaneous parameter identification and filtering has been investigated in C[18], for a generic class of interval nonlinear systems. Analogously to the linear case, the approach is based on a suitably defined extended state, including the unknown parameters among its components: the extended state estimate is then achieved by means of the Polynomial EKF (PEKF), R[3].

• Discrete-time linear systems forced by unknown nonlinearities. In the discrete time framework, the case of stochastic linear systems driven by unknown nonlinearities has been investigated in C[8]: an extended singular system is considered, in order to model the uncertainties, and the state is filtered by using the minimum variance criterion.

• Fault distance estimation of an electric power line. In R[21] and C[37] a filtering approach has been applied to detect the fault distance in a medium-voltage power line, based on distance protection concepts and synchronization of the voltage and current samples coming from the two terminals of the faulted MV line segment
  

• FILTERING AND CONTROL OF STOCHASTIC NONLINEAR SYSTEMS.
• Bilinear systems. Taking into account the field of continuous time systems, the state estimate problem has been investigated in R[1] for a bilinear stochastic differential system, driven by unknown inputs. According to the Ito formulation, the measurements overcome the loss of knowledge concerning the unknown input, so that the optimal linear state estimate is achieved. 
  

• Carleman approximation based methodologies. In R[3] the polynomial version of the Extended Kalman Filter (EKF), so widely used in many engineering frameworks, has been developed. According to the linear approximation of the system, the EKF performs well if the initial estimation error and the disturbing noises are small enough. On the other hand, the Polynomial-EKF (PEKF) is based on the Carleman stochastic bilinearization of the system, whose extended state is filtered by means of the minimum variance estimate; a preliminary version of R[3] has been presented in C[9]. A double index version of the PEKF has been proposed in C[15], where the first index is related to the degree of approximation of the nonlinear system, while the other is related to the order of the polynomial estimate adopted. The Carleman approximation has been also used in R[10], in order to estimate the state of a nonlinear stochastic differential system: the algorithm consists of the optimal linear filter applied to the bilinear differential system approximating the original model (a preliminary version had been presented in C[19]). The same nonlinear stochastic framework has been investigated in R[27], where a novel algorithm is proposed, denoted as Observer Follower Filter (OFF), based on a two-steps, mixed approach. The first step makes use of a high-gain observer-based estimator for nonlinear systems, applied to the system equations in order to provide the trajectory around which a ν-degree Carleman approximation of the stochastic differential system is achieved, second step. According to the high-gain estimator exploited in R[27], a bounded mean square error for the observer is achieved. Numerical simulations show the effectiveness of the proposed methodology, and the improvements of the OFF with respect to the standard Extended Kalman–Bucy Filter (EKBF) obtained by increasing the order of the Carleman approximation. Preliminary results have been presented in C[39], C[40]. In C[49] the OFF methodology has been applied to the case of stochastic differential systems with sampled measurements. The Carleman approximation technique has been also applied to solve optimal control problems for linear stochastic systems affected by disturbances generated by a nonlinear stochastic exosystem. The discrete-time case has been investigated in R[18] (preliminary results have been presented in C[25]), the continuous-time case has been presented in C[28].
  

• Log-Rice signals filtering. In C[23] a polynomial filtering approach has been used in a telecommunication framework: a recursive filtering algorithm is proposed for log-Rice signals generated by the envelope of noisy narrow band Gaussian signals through noisy logarithmic amplifiers. A class of filters is considered, and within this class the filter providing the optimal estimate is computed.
  

• Mobile robot localization. A polynomial filtering approach has been used also in a robotic framework C[24], where a new analytical algorithm performs the localization of a mobile robot using odometry and laser readings. The algorithm provides the optimal affine filter in a class of suitably defined estimators. The comparison with the standard Extended Kalman Filter shows the efficiency of the proposed filter also in critical situations where the system nonlinearities cause a bad behavior of the EKF.

• Roll-motion control for a sea-surface vehicle. In C[29] and C[31] a control system is proposed for the regulation problem of the roll-motion of a manned sea-surface vehicle: a stochastic bilinear model is exploited to properly take into account the wind/sea-wave uncertainties. A special focus concerns the parameter identification process.
  

• POSITIVE SYSTEMS: STATE SPACE REALIZATION BY MEANS OF INTERNAL POSITIVE SYSTEMS. In some technological frame-works, such as Charge Routing Networks (CRNs), or fiber-optic filters, only positive state–space realizations of digital signal processing algorithms, such as filters or control laws, can be implemented. On the other hand, the imposition of an a priori positivity constraint on the processing algorithm is a too strong design limitation. For this reason, some authors studied the problem of state–space realization of generic stationary filters through combination of positive systems, in the discrete-time framework. The single-input/single-output (SISO) case has been widely investigated and important results are available in the literature. On the contrary, no specific theoretical results exist for multi-input/multi-output (MIMO) systems. In R[17], the problem of the Internal Positive Realization (IPR) of MIMO systems and filters is formulated and a straightforward method for the construction of IPRs is proposed. The stability properties of the resulting positive realization are also investigated. The method is illustrated on two engineering applications. Preliminary results have been presented in C[26].

• OPTIMAL CONTROL FOR TIME-DELAY SYSTEMS. The optimal control problem for a time-invariant linear systems with an arbitrary constant time-delay in the input channel is investigated in R[30]. A state feedback is provided for the infinite horizon case with a quadratic cost function. The solution is memoryless, except at an initial time interval of measure equal to the time-delay. If the initial input is set equal to zero, then the optimal feedback control law is memoryless from the beginning. Stability results are established for the closed loop system, in the scalar case. Preliminary results have been presented in C[35], C[42]. This paper has been awarded with the Editor's Award of the Kybernetyka Journal for the year 2013, http://www.kybernetika.cz/award.html
  

• HIGHER ORDER METHODS FOR THE SOLUTION OF NONLINEAR EQUATIONS. It is well known that, looking for the solution of nonlinear equations, the Newton's method has local quadratic convergence, according to a set of non restrictive convergence conditions. In R[6] a new iterative method for the scalar case is proposed, based on a suitable polynomial model derived from the Taylor series expansion. Providing the same Newton's convergence conditions, it achieves a convergence rate increasing with the approximation degree. Moreover, an easy implementable scheme is obtained, according to a specific matrix factorization, achieved in a closed form (i.e. no further numerical computation), by using the Jordan decomposition.

• STATE OBSERVERS FOR NONLINEAR DETERMINISTIC SYSTEMS. The problem of asymptotic state reconstruction for a class of continuous-time systems characterized by linear input-state dynamics and polynomial state-output function is investigated in C[14]. It is shown that the dynamics of systems in this class can be embedded into the dynamics of systems of higher dimension, with time-varying linear state dynamics and linear state-output map. An asymptotic state observer for the original system is presented, whose design is based on the equations of the extended system. The interest in this observer is in its capability of state reconstruction also in cases in which the original system is not drift-observable (observable for zero input) nor uniformly observable (observable for any input). In C[22] the design of asymptotic state observers for systems characterized by output functions that are ratios of polynomials in the state is investigated. The case of linear and bilinear input-state dynamics is considered, and conditions for exponential error decay are provided. The first step towards the construction of the observer is to show that the dynamics of a system in the considered class can be embedded into the dynamics of a system of higher dimension, with time-varying linear state dynamics and linear output map.
  

• MODELING AND CONTROL OF THE GLUCOSE/INSULIN HOMEOSTASIS. The homeostasis of glucose, involving the secretion of its controlling hormone insulin by pancreas is the object of several mathematical models over the past thirty years. One of the goals of this modeling effort is the measurement of the degree to which a given subject is able to accomodate a load of glucose (for instance administred as an intra-venous bolus, the well knwon Intra-Venous Glucose Tolerance Test, IVGTT). The fact that several attempts at modeling the glucose-insulin system have been made in the past points to the actual difficulty in constructing a model which is at the same time mathematically coherent, statistically robust and physiologically meaningful. In other words, the model should exhibit satisfactory properties of the solutions, its parameters should be statistically estimable with sufficient precision from data sets obtained from standard experimental procedures, and it should conform to established physiological concepts. So far, indeed, no single model exhibiting simultaneously all of these features has been proposed. In R[8] a family of delay-differential models of the glucose-insulin system is presented, whose members represent adequately the Intra-Venous Glucose Tolerance Test and allied experimental procedures of diabetological interest. Different models of the family have been identified from IVGTT data R[9], providing excellent results in terms of parameter estimation. These models, then, appear statistically and physiologically promising. In R[8] the mathematical analysis has been performed. All the models in the family admit positive bounded unique solutions for any positive initial condition and are persistent. The models agree with the physics underlying the experiments, and they all present a unique positive equilibrium point. Local stability is investigated in a pair of interesting member models: one, a discrete-delays differential system (a preliminar version of the model is presented in C[20]); the other, a distributed-delay system reducing to an ordinary differential system evolving on a suitably defined extended state space. In both cases conditions are given on the physical parameters in order to ensure the local asymptotic stability of the equilibrium point. These conditions are always satisfied, given the actual parameter estimates obtained experimentally. A study of the global stability properties is also performed. A further analysis of the global stability has been proposed in R[12], R[23]. A study to assess the pancreatic insulin delivery rate from insulin and C-peptide concentrations has been proposed in R[28], C[44]. A review on the glucose-insulin mathematical models has been presented in R[29], with special focus on multi-scale features.
  

• THE ARTIFICIAL PANCREAS. For Artificial Pancreas is meant a feedback devise able to automatically regulate the plasma glucose concentration for all the cases when the pancreatic insulin release is impaired or missing. The research activity in this framework has been developed with the goal of tracking a reference glucose profile in both the cases of intravenous or subcutaneous insulin administration, according to a model-based algorithm that suitably exploits the nonlinear Delay Differential Equations (DDE) model published in R[8], R[9]. DDE systems are naturally suited to model the pancreatic insulin response, not negligible in Type II diabetic patients: as a matter of fact, they reveal to be particularly useful to design model-based control laws in Type II diabetes therapies. As far as the intravenous administration, main results are published in R[15] and R[24]. In the former paper, a disturbance is added to the insulin kinetics in order to model uncertainties concerning both the insulin delivery rate and the mechanism actuating the insulin pump, and a feedback control law which yields input-to-state stability of the closed loop error system with respect to the disturbance is provided (a preliminar version has been presented in C[30]). In R[24] an observer-based algorithm is presented, which properly exploits only glucose measurements, by means of a state observer designed for nonlinear DDE systems, B[5]. In the spirit of the separation theorem, a nonlinear control law is proposed, based on the exact input/output feedback linearization, which makes use of the observer estimates instead of the full state measurements. The local convergence of the tracking error to zero is theoretically proved. In C[45] it has been shown that the closed-loop system satisfies the local Input-to-State Stability (ISS) property with respect to the unknown disturbance.  Such a control law has been tested on a set of virtual patients R[32], modeled by a different model, recently approved by the Food and Drug Administration as an alternative to clinical trials. Simulations are performed, taking into account the standard technology concerning blood glucose sensors and insulin delivery devices. Numerical results show the robustness of the proposed approach with respect to the uncertainties of the model parameters, as well as to the glucose measurement errors and insulin pump malfunctioning.  Preliminary results have been presented in B[1], C[16], C[21], C[30], C[32], C[41]). A digital version of the algorithm has been presented in C[34], based on the DDE model discretization. In C[57], the same DDE model has been exploited to synthesize the LMI feedback control. As far as what concerns subcutaneously administered insulin therapies, the DDE model has been endowed with a subcutaneous compartment modeling the absorption from the subcutaneous depot. According to the observer-based feedback linearization theory for time-delay systems, plasma glycemia  is controlled to track a desired glucose profile according to only glucose measurements C[38], C[43], C[48], B[2]. These model-based approaches can be applied also to experimental frameworks such as the clamp, a standard perturbation procedure, widely adopted to estimate the insulin sensitivity of a healthy subject. In this case the control input is the exogenous glucose administration, released to balance the experimental insulin perturbation. Some results have been presented in C[51].
  
  

  • Modeling the subcutaneous insulin absorption. The modeling of insulin absorption from a subcutaneous infusion has been investigated in R[5], where a pair of time-varying periodic models have been proposed, both taking into account the fact that different absorption rates are associated with different times of a 24 hour period (a preliminary version of the paper has been presented in C[17]).

• MODELING THE PANCREATIC INSULIN SECRETION. An islet population model is proposed in R[16] for pancreatic insulin secretion. Without detailing the chain of biochemical events giving rise to the delivery of insulin packets, the effect of the islets' bursting response to varying glucose concentration is described by a simple second order nonlinear model, of the same functional form for all islets, but with a random distribution of parameter values over the one million islets considered. The islet equations are coupled to a traditional model of the glucose/insulin dynamics to complete a description of the feed-back control of the glucose/insulin system. The model is thus based upon the completely random cooperation of a large number of independent controllers, all reacting to the same prevailing plasma glucose concentrations, but with distributed reaction characteristics. It is shown that the proposed model is able to replicate in silico different observed phenomena such as low frequency glycemia-insulinemia oscillations (ultradian oscillations, with a period between 50 and 150 minutes, amplified by constant glucose administration and entrained by an oscillating exogenous glucose infusion), as well as concordant induction of high-frequency insulin oscillations by a rapid periodic pulsatile glucose infusion. In order to reproduce by simulation all of the above observed phenomena, a single set of (hyper-)parameters has been used throughout, showing that it is indeed possible that a single model may explain the results of several published experimental protocols. A slightly modified version of the model has been exploited in R[33] to the end of replicating also Grodky (C[50]) in vitro experiments and in vivo IVGTT.
  

• DIABETES PROGRESSION. Few attempts have been made to model mathematically the progression of Type 2 diabetes. A realistic representation of the disease process is preliminary to the possibility of inferring the effectiveness of therapeutic protocols. However, writing a good model for diabetes progression is difficult, because of the long time-span of the disease, which makes experimental verification extremely awkward. In this context, it is of primary importance that the assumptions underlying the model equations properly reflect established physiology and that the mathematical formulation of the model not give rise to physically unlikely behavior of the solutions. In R[13], a model of diabetes progression is formulated, its assumptions are discussed, its qualitative characteristics are established and its performance over a lengthy time-span is simulated under different disease and treatment scenarios. The differences with two previously proposed models of diabetes progression are discussed. The integration of this model of diabetes progression with simple models for the evolution of insulin sensitivity and for the effectiveness of treatment is proposed.
  

• PHARMACOKINETICS AND PHARMACODYNAMICS.
  
• The apparent permeability index. The apparent permeability coefficient is widely used for the screening of the absorption characteristics (absorption is larger than permeation, and includes disintegration, dissolution, etc.) of new candidate drugs, and is routinely obtained from in vitro or ex vivo experiments. A classical example, widely used in the pharmaceutical industry is the in vitro Caco-2 cell culture model. The index is defined as the initial flux of compound through the membrane (normalized by membrane surface area and donor concentration) and is typically computed by adapting a straight line to the initial portion of the recorded amounts in the receiver compartment, possibly disregarding the first few points when lagging of the transfer process through the membrane is evident. Modelling the transfer process via a two-compartmental system yields an immediate analogue of the common Papp, as the initial slope of the receiver quantity, but the two-compartment model is often inappropriate for reproducing observations. A three-compartment model, describing the cellular layer as well as donor and receiver compartments, represents the kinetics far better, but has the disadvantage of having always zero initial flow rate to the receiver compartment: in these circumstances the direct analogue of the Papp index may not be computed. In R[11] an alternative definition of an apparent permeability index (PPapp) is proposed for three-compartment models, and is shown to reduce to the classical formulation as the cellular layer's volume tends towards zero. This index characterizes the intrinsic permeability of the membrane to the compound under investigation, can be directly computed in a completely observer-independent fashion from the model parameter values, and reduces to the usual Papp when the linear two-compartment description is sufficient to describe the kinetics.
• Graft rejection model. R[25] proposes an Ordinary Differential Equations model accounting for immune cell proliferation in response to the sudden entry of graft antigens, through different activation mechanisms. The model considers the effect of a single immunosuppressive medication (e.g. cyclosporine), subject to first-order linear kinetics and acting by modifying, in a saturable concentration-dependent fashion, the proliferation coefficient. The proposed model substantially simplifies the chain of events potentially leading to organ rejection. It is however able to simulate quantitatively the time course of graft-related antigen and competent immunoreactive cell populations, showing the long-term alternative outcomes of rejection, tolerance or tolerance at a reduced functional tissue mass. The introduced model is mathematically consistent with known physiology and can reproduce variations in immune status and allograft survival after transplantation. The model can be adapted to represent different therapeutic schemes and may offer useful indications for the optimization of therapy protocols in the transplanted patient.

• GROWTH AND CYCLE MODELS OF THE BUDDING YEAST SACCHAROMYCES CEREVISIAE. Large-scale "omics" data are often represented as networks of interacting components, but such representation is inherently static and, as such, cannot provide a realistic picture of the temporal dynamics of complex cellular functions. These difficulties suggest moving to a modeling strategy that explicitly takes into account both the wiring of the components and the task they perform. From an engineering perspective, this problem resembles that of "circuit analysis". In R[20], we focus on a limited but relevant biological circuit, the G1 to S transition in yeast cell cycle, and investigate both the network representation and the corresponding circuit described by a mathematical model, by means of a wide range of numerical simulation analysis. Reliable predictions of system-level properties are achieved and the parameters that mostly affect these properties are found out. In R[35] we present a novel mathematical model of the basic molecular mechanism controlling the G1/S transition, whose major regulatory feature is multisite phosphorylation of nuclear Whi5. Cln3–Cdk1, whose nuclear amount is proportional to cell size, and then Cln1,2–Cdk1, randomly phosphorylate both decoy and functional Whi5 sites. Full phosphorylation of functional sites releases Whi5 inhibitory activity, activating G1/S transcription. Simulation analysis shows that this mechanism ensures coherent release of Whi5 inhibitory action and accounts for many experimentally observed properties of mitotically growing or conjugating G1 cells. Cell cycle progression and transcriptional analyses of a Whi5 phosphomimetic mutant verify the model prediction that coherent transcription of the G1/S regulon and ensuing G1/S transition requires full phosphorylation of Whi5 functional sites. The publication has been advertised on the CNR website, and it has been reported on over 30 websites on science and information, included "Le Scienze", "Focus", "Panorama", "Il Tempo". In R[22] we summarize the major connections between cell growth and cell cycle in the model eukaryote Saccharomyces cerevisiae.

• MATHEMATICAL MODELS FOR TRANSCRIPTION NETWORKS. The modern systems biology approach to the study of molecular cellular biology, consists in the development of computational tools to support the formulation of new hypotheses on the molecular mechanisms underlying the observed cell behavior. Recent biotechnologies are able to provide precise measures of gene expression time courses in response to a large variety of internal and environmental perturbations.

• Network motif validation. In R[19], we propose a simple algorithm for the selection of the best regulatory network motif among a number of alternatives, using the expression time course of the genes which are the final targets of the activated signalling pathway. To this aim, we considered the Hill nonlinear ODEs model to simulate the behavior of two ubiquitous motifs: the single input motif and the multi output feed-forward loop motif.

• Protein estimation from transcript measurements. Protein estimates from transcript measurements. In R[26] a mathematical tool is presented, to estimate unknown variables of transcription networks, according to a set of measurements of the transcriptional activity of promoters. The approach is based on the use of the mathematical model of the network under investigation and on the use of a state-observer for nonlinear systems. Particular attention is devoted to the application of the proposed methodology to the FFL network motiv. Preliminary results have been presented in C[33]. 

• Discrete-time models for transcription networks. In C[36] discrete-time dynamical models have been presented for most common network motifs, with a special focus on the discussion of peculiar dynamical features, with respect to different discretization times.
  

• CHEMICAL MASTER EQUATIONS (CME). The Chemical Master Equation (CME) is a well known tool for studying (bio)chemical processes involving few copies of the species involved, because it is a framework able to capture random behaviors that are neglected by deterministic approaches based on the concentration dynamics. In R[34], we investigate some structural properties of CMEs and their solutions. We introduce a generalized notion of one-step process, which results in a sparse dynamical matrix describing the collection of the scalar CMEs, also showing a recursive block-tridiagonal structure. Further properties are inferred by means of a graph-theoretical interpretation of the reaction network. Preliminary results have been presented in C[46]. In C[47] the CME approach has been properly exploited to model multisite phosphorylation/dephosphorylation cycles, by using the quasi-steady state approximation of enzymatic kinetics. The CME dynamics is written from the coefficients of the deterministic reaction-rate equations and the stationary distribution is computed explicitly, according to a recently developed realization scheme. Further investigation on the topic has been carried out in C[56], with a special focus on the double time-scale raising in enzymatic reactions.
  

• SYNTHETIC BIOLOGY. Synthetic Biology combines different branches of biology and engineering (spanning from biotechnologies to mathematical modeling abstractions), aiming at properly designing synthetic biological circuits, able to replicate emergent properties potentially useful for biotechnology industry, human health and environment. In R[36], we investigate the role of negative feedback in noise propagation for a basic (though rather general) enzymatic reaction scheme. Two distinct feedback control schemes on enzyme expression are here considered: one from the final product of the pathway activity, the other from the enzyme accumulation (negative autoregulation). Both the feedback schemes are designed to provide the same steady-state average values of the involved players, in order to evaluate the feedback performances according to the same working mode. Computations are carried out numerically (by means of the Stochastic Simulation Algorithm) and analytically (via Stochastic Hybrid System modeling), the latter allowing to infer information on which model parameter setting leads to a more efficient noise attenuation, according to the chosen feedback scheme. In addition to highlighting the clear role of the feedback in providing a substantial noise reduction, our investigation concludes that the effect of the feedback is enhanced by increasing the promoter sensitivity for both the feedback schemes. A further interesting biological insight is that an increase in the promoter sensitivity provides more benefits to the feedback from the product with respect to the feedback from the enzyme, in terms of enlarging the parameter design space. Preliminary results have been presented in C[58], and a variant concerning stochastic product clearance will be presented in C[62].

• TUMOR GROWTH CONTROL: a closed-loop control is presented in C[59] with the aim of tumor volume reduction by means of anti-angiogenic administration. To this end a basic (though widely accepted) model of tumor growth is considered, and the output-feedback linearization theory is exploited, with the feedback designed on the basis of a state observer for nonlinear systems.