VOLUME 1, ISSUE 6, DECEMBER 2007 PUBLICATION NEWS Joint Special Issue on Systems Biology in the Transactions on Circuits and Systems: Part-I and the Transactions on Automatic Control, January 2008 From a scientific perspective this century has been heralded as the beginning of the age of biology. It is probably safer to say that biology is beginning to make more tangible contacts with engineering, physical and information sciences - an interaction from which all parties have only to benefit. Slogans such as these are often interpreted by various communities of researchers in many different ways. The need to extract "meaning" - or better yet "knowledge" - from the profusion of data generated primarily by advances in instrumentation is a domain where much attention have been directed recently (biology is not the only discipline befuddled with the availability of much data - there are other examples e.g., astronomy). The data-driven domains in biology that include attempts at gene sequencing, understanding the functions and structure of large biological molecules such as the proteins (or more generally the domain of "omics" research) etc. characterize what has been recently dubbed as the field of Bioinformatics. For sure there are other areas of biology of interest to engineering and physical sciences. One that easily comes to mind is the fundamental issue of evolution of life on earth (and possibly elsewhere in the universe!) from its basic ingredients at the atomic and molecular level. No matter how far fetched it may appear, we are aware of serious attempts to understand such issues based on genuinely engineering principles at all levels of biology - beginning from molecular to the cellular level including all the way up to the evolution of organisms and species. A yet different and perhaps a more pragmatic domain of engineering research in this context is the biomedical domain. Since much has been said and is already being written in the Circuits & Systems Society by members of its two Technical Committees (Bio-CAS and LISSA) in its new Publication "IEEE Transactions on Biomedical Circuits and Systems", I may refrain from any further comments here. Systems Biology is another such sub-discipline with strong relevance to engineering methodology and principles - and is especially so for members of the CAS community. Despite this, the area has probably not received adequate coverage in IEEE, in general, and Circuits & Systems in particular. A somewhat naive and simple description of Systems Biology could be that it deals with interactions of systems of a large number of biological ingredients - be that molecules, cells, organisms or species. Familiar issues of modeling, control, communication, prediction - all become relevant in this context. Research goals can range from understanding of fundamental principles to applications in drug design and healthcare. Since most of these systems are networked in some sense, are large in terms of number of its components, and interactions often nonlinear, the area has the smack of what has been touted as "Complex Systems" research in many circles in recent years (the fact that it is much different from Bioinformatics should be clear at this point). One thing is for sure true, that the field is genuinely cross-disciplinary, and cannot be pursued within the traditional confines of disciplinary boundaries. Against this backdrop, Dr. M. Vidyasagar, one of the AEs in the editorial board of TCAS-I, and I started a discussion of the possibility of a special issue on Systems Biology as soon as my term as the EiC of TCAS-I started in January 2006. It turns out that at the same time Prof. Christos Cassandras - the Editor-in-Chief of IEEE Trans. on Automatic Control (T-AC) - was independently considering a special issue on the very same topic with Prof. Mustafa Khammash and Prof. Claire Tomlin as guest editors. This was a remarkable opportunity for us to fulfill the desire of the CAS Executive Committee back in 2005, that TCAS-I should focus on interdisciplinary research, one way of doing which could be to publish joint special issues with other IEEE transactions or even non-IEEE publications. While Prof. Vidyasagar needs no introduction to the Circuits & Systems Society, the team of three guest editors (Dr. Khammash is a professor at UC Santa Barbara, and Prof. Tomlin is a recent MacArthur award winner from Stanford U) became natural candidates for editing a joint special issue, for which an open call for papers was soon announced. Some of the readers of the CAS transactions may find the papers in the special issue to be somewhat tilted towards the systems (as opposed to circuits) aspects of the field. I would like to remind those of us in CAS that the fields of Circuit Theory and Control Systems were historically closer than it appears to be today. I am also reminded of the period during the sixties when many giants in control theory were, in fact, active members of the CAS community. For example, R.E. Kalman was an Associate Editor of TCAS, when Dante C. Youla was the Editor (in-Chief). After all, they knew that circuits are nothing but systems with concrete examples. Perhaps it is now time for a renewed synthesis of ideas from the two fields. Systems Biology can very well provide the context for such a synthesis. The present special issue is a result of an effort that spanned almost two years. Prof. Christos Cassandras, the EiC for T-AC, and I needed to jump many bureaucratic hoops of IEEE to make this unusual special issue happen with the help of IEEE publications staff beginning from Prof. Saifur Rahman to Fran Zappula, Mona Mittra and many others. In the process, we discovered that in some sense, it is one of a kind special issue that was never done before within IEEE - not at least after electronic publishing (e.g., IEEE Xplore) came of age. It is a good time and place to thank all those involved - Christos, in particular, who has always been very cooperative. Apart from crossing bureaucratic boundaries in collaborative publishing, we again learned a few aspects of the cultural differences in publishing in the two fields of engineering and biosciences. Serious questions were raised by potential authors from the biosciences community as to why they should wait for years for their paper to appear in journals that are not even listed in PubMed (most IEEE journals are not). For TCAS-I, this was not its first encounter with the issue. The question had already surfaced earlier in 2006 when TCAS-I published a special issue on "Advances on Life Sciences Systems and Applications". The lightning speed with which the biosciences community handles review and publication of scientific facts is daunting. One can conjecture the reasons for such cultural differences based on the style in which scientific work is carried out and flavor of the results reported in the two fields, but if IEEE has to seriously get involved in publishing of bioscience papers then it needs to come to terms with this dilemma. The special issue consists of 19 papers grouped into 9 subtopics. A list of titles, authors and abstracts follows. Happy reading... Sankar Basu EiC, TCAS-I Foreword - Guest editorial STOCHASTIC MODELLING Title: Structural Alignment of RNAs Using Profile-csHMMs and Its Application to RNA Homology Search: Overview and New Results Authors: Byung-Jun Yoon and P. P. Vaidyanathan ABSTRACT: Systematic research on noncoding RNAs (ncRNAs) has revealed that many ncRNAs are actively involved in various biological networks. Therefore, in order to fully understand the mechanisms of these networks, it is crucial to understand the roles of ncRNAs. Unfortunately, the annotation of ncRNA genes that give rise to functional RNA molecules has begun only recently, and it is far from being complete. Considering the huge amount of genome sequence data, we need efficient computational methods for finding ncRNA genes. One effective way of finding ncRNA genes is to look for regions that are similar to known ncRNA genes. As many ncRNAs have well-conserved secondary structures, we need statistical models that can represent such structures for this purpose. In this paper, we propose a new method for representing RNA sequence profiles and finding structural alignment of RNAs, based on profile context-sensitive HMMs (profile-csHMMs). Unlike existing models, the proposed approach can handle any kind of RNA secondary structures, including pseudoknots. We show that profile-csHMMs can provide an effective framework for the computational analysis of RNAs and the identification of ncRNA genes. Title: The 4M (Mixed Memory Markov Model) Algorithm for Finding Genes in Prokaryotic Genomes Authors: M. Vidyasagar, Sharmila S. Mande, C. V. Siva Kumar Reddy, V. Raja Rao ABSTRACT: In this paper, we present a new algorithm called 4M (Mixed Memory Markov Model) for finding genes from the genomes of prokaryotes. This is achieved by modelling the known coding regions of the genome as a set of sample paths of a multi-step Markov chain (call it C), and the known noncoding regions as a set of sample paths of another multi-step Markov chain (call it NC). The new feature of the 4M algorithm is that different states are allowed to have different memory length, in contrast to a fixed multi-step Markov model used in GeneMark in its various versions. At the same time, compared to an algorithm like Glimmer3 that uses an interpolation of Markov models of different memory lengths, the statistical significance of the conclusions drawn from the 4M algorithm is quite easy to quantify. Thus when a whole genome annotation is carried out and several new genes are predicted, it is extremely easy to rank these predictions in terms of the confidence one has in the predictions. The basis of the 4M algorithm is a simple rank condition satisfied by the matrix of frequencies associated with a Markov chain. The 4M algorithm is validated by applying it to 75 organisms belonging to practically all known families of bacteria and archae. The performance of the 4M algorithm is compared to those of Glimmer3, GeneMark2.5d, and GeneMarkHMM2.6g. It is found that in a vast majority of cases, the 4M algorithm finds many more genes than it misses, compared to any of the other three algorithms. Next, the 4M algorithm is used to carry out whole genome annotation of thirteen organisms, by using 50% of the known genes as the training input for the coding model, and 20% of the known non-genes as the training input for the non-coding model. After this, all the open reading frames are classified. It is found that the 4M algorithm is highly specific, in that it picks out virtually all of the known genes, while predicting that only a small number of the ORFs whose status is unknown to be genes. Title: Subtilin Production by Bacillus Subtilis: Stochastic Hybrid Models and Parameter Identification Authors: Eugenio Cinquemani, Riccardo Porreca, Giancarlo Ferrari-Trecate, and John Lygeros ABSTRACT: This paper presents methods for the parameter identification of a model of subtilin production by Bacillus subtilis. Based on a stochastic hybrid model, identification is split in two subproblems: estimation of the genetic network regulating subtilin production from gene expression data, and estimation of population dynamics based on nutrient and population level data. Techniques for identification of switching dynamics from sparse and irregularly sampled observations are developed and applied to simulated data. Numerical results are provided to show the effectiveness of our methods. Title: Stochastic modeling and control of biological systems: the lactose regulation system of Escherichia coli Authors: A. Agung Julius, A. M. Halasz, M. Selman Sakar, H. Rubin, V. Kumar, George J. Pappas ABSTRACT: In this paper we present a comprehensive framework for stochastic modeling, model abstraction, and controller design for a biological system. The first half of the paper concerns modeling and model abstraction of the system. Most models in systems biology are deterministic models with ordinary differential equations in the concentration variables. We present a stochastic hybrid model of the lactose regulation system of E. coli bacteria that capture important phenomena which cannot be described by continuous deterministic models. We then show that the resulting stochastic hybrid model can be abstracted into a much simpler model, a two-state continuous time Markov chain. The second half of the paper discusses controller design for a specific architecture. The architecture consists of measurement of a global quantity in a colony of bacteria as an output feedback, and manipulation of global environmental variables as control actuation. We show that controller design can be performed on the abstracted (Markov chain) model and implementation on the real model yields the desired result. STABILTY AND STABILIZATION Title: Further Results on Stabilization of Periodic Trajectories for a Chemostat with Two Species Authors: Frederic Mazenc, Michael Malisoff, and Jerome Harmand ABSTRACT: We discuss an important class of problems involving the tracking of prescribed trajectories in the chemostat model. We provide new tracking results for chemostats with two species and one limiting substrate, based on Lyapunov function methods. In particular, we use a linear feedback control of the dilution rate and an appropriate time-varying substrate input concentration to produce a locally exponentially stable oscillatory behavior. This means that all trajectories for the nutrient and corresponding species concentrations in the closed loop chemostat that stay near the oscillatory reference trajectory are attracted to the reference trajectory exponentially fast. We also obtain a globally stable oscillatory reference trajectory for the species concentrations, using a nonlinear feedback control depending on the dilution rate and the substrate input concentration. This guarantees that all trajectories for the closed loop chemostat dynamics are attracted to the reference trajectory. Finally, we construct an explicit Lyapunov function for the corresponding global error dynamics. We demonstrate the efficacy of our method in a simulation. Title: A passivity-based approach to stability of spatially distributed systems with a cyclic interconnection structure Authors: Mihailo R. Jovanovic, Murat Arcak, and Eduardo D. Sontag ABSTRACT: A class of distributed systems with a cyclic interconnection structure is considered. These systems arise in several biochemical applications and they can undergo diffusion driven instability which leads to a formation of spatially heterogeneous patterns. In this paper, a class of cyclic systems in which addition of diffusion does not have a destabilizing effect is identified. For these systems global stability results hold if the ``secant'' criterion is satisfied. In the linear case, it is shown that the secant condition is necessary and sufficient for the existence of a decoupled quadratic Lyapunov function, which extends a recent diagonal stability result to partial differential equations. For reaction-diffusion equations with non-decreasing coupling nonlinearities global asymptotic stability of the origin is established. All of the derived results remain true for both linear and nonlinear positive diffusion terms. Similar results are shown for compartmental systems. Title: Bistable biological systems: a characterization through local compact input-to-state stability Authors: Madalena Chaves, Thomas Eissing and Frank Allgower ABSTRACT: Many biological systems have the capacity to operate in two distinct modes, in a stable manner. Typically, the system can switch from one stable mode to the other in response to a specific external input. Mathematically, these bistable systems are usually described by models that exhibit (at least) two distinct stable steady states. On the other hand, to capture biological variability, it seems more natural to associate to each stable mode of operation an appropriate invariant set in the state space rather than a single fixed point. A general formulation is proposed in this paper, which allows freedom in the form of kinetic interactions, and is suitable for establishing conditions on the existence of one or more disjoint forward-invariant sets for the given system. Stability with respect to each set is studied in terms of a local notion of input-to-state stability with respect to compact sets. Two well known systems that exhibit bi-stability are analyzed in this framework: the lac operon and an apoptosis network. For the first example, the question of designing an input that drives the system to switch between modes is also considered. IDENTIFICATION Title: Model for Photosynthesis and Photoinhibition: Parameter Identification Based on the Harmonic Irradiation O_2 Response Measurement Authors: Sergej Celikovsky, Stepan Papacek, Branislav Rehak ABSTRACT: A method for parameter identification of a model describing the growth of the algae is presented. The method is based on the description in the form of the so-called photosynthetic factory. The experimental data are gained by measuring the steady state photosynthetic production when the input of the photosynthetic factory (being the light intensity) is a harmonic signal. Estimation of parameters is based on a sufficient number of experiments compared with simulated data via the least-squares technique. As the input signal is harmonic and the dynamics of the unforced system is exponentially stable, the resulting asymptotical steady state trajectory of the photosynthetic factory is also periodic and can be computed via determining appropriate center manifold graph by solving the corresponding first order PDE. The latter is performed by the finite element method (FEM). The application of the proposed method is demonstrated on an example using real experimental data. Title: An Adjoint-based Parameter Identification Algorithm applied to Planar Cell Polarity Signaling Authors: Robin L. Raffard, Keith Amonlirdviman, Jeffrey D. Axelrod, Claire J. Tomlin ABSTRACT: This paper presents an adjoint-based algorithm for performing automatic parameter identification on differential equation models of biological systems. The algorithm locally solves an optimization problem, in which the cost reflects the deviation between the observed data and the output of the parameterized mathematical model, and the constraints are the governing parameterized equations. The tractability and the speed of convergence (to local minima) of the algorithm are strongly favorable to numerical parameter search algorithms which do not make use of the adjoint. Furthermore, initializing the algorithm with different instantiations of the parameters allows one to effectively search the parameter space. Results of the application of this algorithm to a previously presented mathematical model of Planar Cell Polarity (PCP) signaling in the wings of Drosophila melanogaster are presented, and some new insights into the PCP mechanism that are enabled by the algorithm are described. MODELLING AND CONTROL Title: Passivity and Optimal Control of Descriptor Biological Complexity Systems Authors: Peiyong Liu, Qingling Zhang, Xiaoguang Yang, Li Yang ABSTRACT: Abstract-Accurate mathematical models, to describe and analyze complex systems, cannot be established. This is especially true for biological complex systems because of the complexity of interaction among constituent units inside the whole system. Descriptor biological complex systems are a new research field in descriptor systems; there has been little research about passivity and optimal control of descriptor biological complex systems. Passivity analysis and feedback controller design of the system offers an important basis for the research of descriptor system theory applied to biological complex systems. In this paper, a poly-chamber model of the endocrine disruptor - Diethylstibestrol (DES) - moving in a human body is developed based on physiological rules. Passivity of this model is described and proved systematically. A feedback controller for this descriptor biological complex system is designed under the station of strict passivity, and an example of the controller is given for a particular instantiation of the model. Title: Systems analysis of regulatory processes underlying eukaryotic gradient perception Authors: J. Krishnan and Pablo Iglesias ABSTRACT: This paper analyzes regulatory processes involved in the biological process of directed cell locomotion --- known as chemotaxis. We focus on the nature of regulation involved in a subprocess of chemotaxis called gradient perception. We examine two different models from a dynamics/control perspective to gain insight into the working mechanisms of these models. One model is a minimal model which reconciles gradient perception to the property of adaptation. The second model is a biochemical model of a lipid network and its regulation by enzymes. In both cases we focus on the extent of regulation of the modules/networks by inputs, and examine various limiting cases. Both the models and the resulting insights have broader applicability than the context in which they were developed. PIECEWISE AFFINE & MONOTONE SYSTEMS Title: Controllability Analysis of Biosystems Based on Piecewise Affine Systems Approach Authors: Shun-ichi Azuma, Eriko Yanagisawa, and Jun-ichi Imura ABSTRACT: This paper discusses the controllability problem of biosystems based on the piecewise affine system model representation. First, we consider what kind of controllability problems are useful for analyzing and controlling biosystems, and then the controllable set problem, that is, a problem of finding a state set in which each state can be driven to a given target state, is formulated. It is shown that this problem will be very useful for many kinds of problems on control of biosystems such as the input allocation problem and the stabilization problem. Next, based on our previous probabilistic controllability analysis technique for hybrid dynamical systems, a more sophisticated method for solving the above complex problem in an approximated and suitable way for biosystem analysis is proposed. Finally, the proposed framework is applied to the quorum sensing system of the pathogen shape Pseudomonas aeruginosa for explaining how it is formulated and what solutions are obtained. Title: The Switching Threshold Reconstruction Problem for Piecewise Affine Models of Genetic Regulatory Networks Authors: Samuel Drulhe, Giancarlo Ferrari-Trecate, Hidde de Jong ABSTRACT: This paper discusses the controllability problem of biosystems based on the piecewise affine system model representation. First, we consider what kind of controllability problems are useful for analyzing and controlling biosystems, and then the controllable set problem, that is, a problem of finding a state set in which each state can be driven to a given target state, is formulated. It is shown that this problem will be very useful for many kinds of problems on control of biosystems such as the input allocation problem and the stabilization problem. Next, based on our previous probabilistic controllability analysis technique for hybrid dynamical systems, a more sophisticated method for solving the above complex problem in an approximated and suitable way for biosystem analysis is proposed. Finally, the proposed framework is applied to the quorum sensing system of the pathogen shape Pseudomonas aeruginosa for explaining how it is formulated and what solutions are obtained. OSCILLATIONS Title: Oscillations in I/O Monotone Systems Authors: David Angeli and Eduardo D. Sontag ABSTRACT: Oscillatory behavior is a key property of many biological systems. The Small-Gain Theorem (SGT) for input/output monotone systems provides a sufficient condition for global asymptotic stability of an equilibrium and hence its violation is a necessary condition for the existence of periodic solutions. One advantage of the use of the monotone SGT technique is its robustness with respect to all perturbations that preserve monotonicity and stability properties of a very low-dimensional (in many interesting examples, just one-dimensional) model reduction. This robustness makes the technique useful in the analysis of molecular biological models in which there is large uncertainty regarding the values of kinetic and other parameters. However, verifying the conditions needed in order to apply the SGT is not always easy. This paper provides an approach to the verification of the needed properties, and illustrates the approach through an application to a classical model of circadian oscillations, as a nontrivial ``case study,'' and also provides a theorem in the converse direction of predicting oscillations when the SGT conditions fail.. Title: Sensitivity Measures for Oscillating Systems: Application to Mammalian Circadian Gene Network Authors: Stephanie R. Taylor, Rudiyanto Gunawan, Linda R. Petzold, Francis J. Doyle III Vital physiological behaviors exhibited daily by bacteria, plants, and animals are governed by endogenous oscillators called circadian clocks. The most salient feature of the circadian clock is its ability to change its internal time (phase) to match that of the external environment. The circadian clock, like many oscillators in nature, is regulated at the cellular level by a complex network of interacting components. As a complementary approach to traditional biological investigation, we utilize mathematical models and systems theoretic tools to elucidate these mechanisms. The models are systems of ordinary differential equations exhibiting stable limit cycle behavior. To study the robustness of circadian phase behavior, we use sensitivity analysis. As the standard set of sensitivity tools are not suitable for the study of phase behavior, we introduce a novel tool, the parametric impulse phase response curve (pIPRC). NOISE Title: Noise in gene regulatory networks Authors: Ioannis Lestas, Johan Paulsson, Nicholas E Ross, Glenn Vinnicombe ABSTRACT: Life processes in single cells and at the molecular level are inherently stochastic. Quantifying the noise is, however, far from trivial, as a major contribution comes from intrinsic fluctuations, arising from the randomness in the times between discrete jumps. It is shown in the paper how a noise filtering setup with an operator theoretic interpretation can be relevant for analyzing the intrinsic stochasticity in jump processes described by master equations. Such interpretation naturally exists in linear noise approximations, but it also provides an exact description of the jump process when the transition rates are linear. As an important example, it is shown in the paper how by addressing the proximity of the underlying dynamics in an appropriate topology, a sequence of coupled birth death processes, which can be relevant in gene expression, tends to a pure delay; this implies important limitations in noise suppression capabilities. Despite the exactness, in a linear regime, of the analysis of noise in conjunction with the network dynamics, the paper emphasizes the importance of also analyzing dynamic behaviour when transition rates are highly nonlinear; otherwise, steady state solutions can be misinterpreted. The examples are taken from systems with macroscopic models leading to bistability or limit cycles. It is discussed that bistability in the deterministic mass action kinetics and bimodality in the steady state solution of the master equation, neither always imply one another, nor do they necessarily lead to efficient switching behaviours: the underlying dynamics need to be taken into account. The lac operon is finally discussed as an example, where, despite the bistability in the deterministic model, there is no obvious bimodality in the probability distribution, with experimentally reported bimodal cell populations being due to other factors, such as growth rate differences. Title: The Finite State Projection Approach for the Analysis of Stochastic Noise in Gene Networks Authors: Brian Munsky and Mustafa Khammash ABSTRACT: In order to capture important subcellular dynamics, researchers in computational biology have begun to turn to mesoscopic models in which molecular interactions at the gene level behave as discrete stochastic events. While the trajectories of such models cannot be described with deterministic expressions, the probability distributions of these trajectories can be described by the set of linear ordinary differential equations known as the chemical master equation (CME). Until recently, it has been believed that the CME could only be solved analytically in the most trivial of problems, and the CME has been analyzed almost exclusively with Kinetic Monte Carlo (KMC) algorithms. However, concepts from linear systems theory have enabled the Finite State Projection (FSP) approach and have significantly enhanced our ability to solve the CME without resorting to KMC simulations. In this paper we review the FSP approach and introduce a variety of systems theory based modifications and enhancements to the FSP algorithm. Notions such as observability, controllability and minimal realizations enable large reductions and increase efficiency with little to no loss in accuracy. Model reduction techniques based upon linear perturbation theory allow for the systematic projection of multiple time scale dynamics onto a slowly varying manifold of much smaller dimension. We also present a powerful new reduction approach, in which we perform computations on a small subset of configuration grid points and then interpolate to find the distribution on the full set. The power of the FSP and its various reduction approaches is illustrated on few important models of genetic regulatory networks. NETWORKS Title: Temporal logic analysis of gene networks under parameter uncertainty Authors: Gregory Batt and Calin Belta and Ron Weiss ABSTRACT: The lack of precise numerical information for the values of biological parameters severely limits the development and analysis of models of genetic regulatory networks. To deal with this problem, we propose a method for the analysis of genetic regulatory networks under parameter uncertainty. We consider models based on piecewise-multiaffine differential equations, dynamical properties expressed in temporal logic, and intervals for the values of uncertain parameters. The problem is then either to guarantee that the system satisfies the expected properties for every possible parameter value -- the corresponding parameter set is then called valid -- or to find valid subsets of a given parameter set. The proposed method uses discrete abstractions and model checking, and allows for efficient search of the parameter space. However, the abstraction process creates spurious behaviors in the abstract systems, along which time does not progress. Consequently, the verification of liveness properties, expressing that something will eventually happen, and implicitly assuming progress of time, often fails. A solution to this second problem is proposed using the notion of transient regions. This approach has been implemented in a tool for robust verification of gene networks (RoVerGeNe) and applied to the tuning of a synthetic network built in E. coli. HIV Title: Clinical tests of therapeutical failures based on mathematical modeling of the HIV infection Authors: Djomangan Adama Ouattara, Marie-Jose' Mhawej, Claude H. Moog ABSTRACT: Clinical tests of therapeutical failures based on mathematical modeling of the HIV infection Authors: Djomangan Adama Ouattara, Marie-Jose' Mhawej, Claude H. Moog ABSTRACT: Clinical tests which are displayed are based on a system theoretic approach for an early diagnosis of the immunological and virological failure of HIV patients. Mathematical characterizations of therapeutical failures are presented in this paper. Mathematical modeling is used for individual patients to help for an early diagnosis of the evolution of the infection. The feasibility of the method is depicted on some patients who start HAART (Highly Active AntiRetroviral Therapy). The identifiability of the continuous-time models which are used, is proved and it is shown to be invariant under discretization. Sankar Basu, EiC, IEEE Transactions on Circuits and Systems Part I