[show abstract][hide abstract] ABSTRACT: Recent computational studies indicate that the molecular noise of a cellular process may be a rich source of information about process dynamics and parameters. However, accessing this source requires stochastic models that are usually difficult to analyze. Therefore, parameter estimation for stochastic systems using distribution measurements, as provided for instance by flow cytometry, currently remains limited to very small and simple systems. Here we propose a new method that makes use of low-order moments of the measured distribution and thereby keeps the essential parts of the provided information, while still staying applicable to systems of realistic size. We demonstrate how cell-to-cell variability can be incorporated into the analysis obviating the need for the ubiquitous assumption that the measurements stem from a homogeneous cell population. We demonstrate the method for a simple example of gene expression using synthetic data generated by stochastic simulation. Subsequently, we use time-lapsed flow cytometry data for the osmo-stress induced transcriptional response in budding yeast to calibrate a stochastic model, which is then used as a basis for predictions. Our results show that measurements of the mean and the variance can be enough to determine the model parameters, even if the measured distributions are not well-characterized by low-order moments only--e.g., if they are bimodal.
Proceedings of the National Academy of Sciences 05/2012; 109(21):8340-5. · 9.74 Impact Factor
[show abstract][hide abstract] ABSTRACT: A model predictive control law is given by the solution to a parametric optimization problem that can be pre-computed offline and provides an explicit map from state to control input. In this paper, an algorithm is introduced based on wavelet multiresolution analysis that returns a low complexity explicit model predictive control law built on a hierarchy of second-order interpolets. The resulting interpolation is shown to be everywhere feasible and continuous. Further, tests to confirm stability and to compute a bound on the performance loss are introduced. Since the controller approximation is built on a grid hierarchy, convergence to a stabilizing control law is guaranteed and the evaluation of the control law in real-time systems is naturally fast and runs in a bounded logarithmic time. Two examples are provided; A two-dimensional example with an evaluation speed of 31 ns and a four-dimensional example with an evaluation speed of 119 ns.
IEEE Transactions on Automatic Control 12/2011; · 2.72 Impact Factor
[show abstract][hide abstract] ABSTRACT: In stochastic models of chemically reacting systems that contain bimolecular reactions, the dynamics of the moments of order up to n of the species populations do not form a closed system, in the sense that their time-derivatives depend on moments of order n + 1. To close the dynamics, the moments of order n + 1 are generally approximated by nonlinear functions of the lower order moments. If the molecule counts of some of the species have a high probability of becoming zero, such approximations may lead to imprecise results and stochastic simulation is the only viable alternative for system analysis. Stochastic simulation can produce exact realizations of chemically reacting systems, but tends to become computationally expensive, especially for stiff systems that involve reactions at different time scales. Further, in some systems, important stochastic events can be very rare and many simulations are necessary to obtain accurate estimates. The computational cost of stochastic simulation can then be prohibitively large. In this paper, we propose a novel method for estimating the moments of chemically reacting systems. The method is based on closing the moment dynamics by replacing the moments of order n + 1 by estimates calculated from a small number of stochastic simulation runs. The resulting stochastic system is then used in an extended Kalman filter, where estimates of the moments of order up to n, obtained from the same simulation, serve as outputs of the system. While the initial motivation for the method was improving over the performance of stochastic simulation and moment closure methods, we also demonstrate that it can be used in an experimental setting to estimate moments of species that cannot be measured directly from time course measurements of the moments of other species.
The Journal of chemical physics 10/2011; 135(16):165102. · 3.09 Impact Factor
[show abstract][hide abstract] ABSTRACT: DNA replication is an important process in the life of a cell. It has to be completed with extreme accuracy in a specific phase of the cell cycle, known as the S phase. Eukaryotic DNA replication is a rather complex and uncertain process. Several mathematical models have been recently proposed in the literature to interpret experimental data from various organisms. A common concern of many of these models is the so-called random gap problem, the observation that eukaryotic DNA replication should last longer than experimental evidence suggests due to its stochastic nature. One of the biological hypotheses proposed for resolving the random gap problem postulates the presence of a limiting factor regulating the rate with which DNA replication initiates. We show how this hypothesis can be captured in the Piecewise Deterministic Markov Process modeling framework. Monte Carlo simulations allow us to analyze the proposed model and compare model predictions with independent experimental data.
[show abstract][hide abstract] ABSTRACT: We introduce bounds on the finite-time performance of Markov chain Monte Carlo (MCMC) algorithms in solving global stochastic optimization problems defined over continuous domains. It is shown that MCMC algorithms with finite-time guarantees can be developed with a proper choice of the target distribution and by studying their convergence in total variation norm. This work is inspired by the concept of finite-time learning with known accuracy and confidence developed in statistical learning theory.
IEEE Transactions on Automatic Control 01/2011; · 2.72 Impact Factor
[show abstract][hide abstract] ABSTRACT: We introduce a methodological framework based on the concepts of safety and optimality to interpret organismal strategies that are intrinsically related to survival behaviors. We focus on the production of the antibiotic subtilin by the bacterium Bacillus subtilis, which is one among a set of possible responses to environmental stress that are elicited by the bacterium, and we investigate the activation strategies over the genes involved in the process. We argue that these activation strategies can be synthesized as the outcome of an optimal control problem that yields a survival probability. This optimization procedure is generated from a probabilistic safety problem, which is formally related to the survival probability. We claim that a proper choice of the value function for the optimization problem that encodes the survival analysis can be related to the activation mechanisms for subtilin production.
[show abstract][hide abstract] ABSTRACT: DNA replication is one of the most fundamental processes in the life of every cell. In earlier work a model to capture the mechanics of the DNA replication process was developed in the stochastic hybrid systems framework. Monte Carlo simulations of the model allowed us to make novel predictions regarding the mechanisms behind DNA replication based on experimental data for the fission yeast. Here the stochastic hybrid model is adopted to the Hybrid Input/Output Automaton formalism. We then verify that the model captures the mechanisms of DNA replication process by induction proofs. Our results demonstrate that the model is indeed a faithful representation of the physical reality and lend theoretical support for the predictions of the model.
Proceedings of the 13th ACM International Conference on Hybrid Systems: Computation and Control, HSCC 2010, Stockholm, Sweden, April 12-15, 2010; 01/2010
[show abstract][hide abstract] ABSTRACT: Under the so called Subliminal Control concept, an automated system, commanding minor speed adjustments imperceptible by the Air Traffic Controller (ATC), tries to keep the Air Traffic Controller’s risk perception low, emulating a “lucky traffic”. In this paper we outline such a concept and investigate several implementation considerations of subliminal control. A proposed subliminal controller is tested against several encounter geometries for level flights in simulations using a stochastic environment that comprises wind forecast uncertainties. The results demonstrate that subliminal control has the potential to reduce the workload of the ATC in several cases.
Transportation Research Part C: Emerging Technologies. 01/2010;
[show abstract][hide abstract] ABSTRACT: We introduce bounds on the finite-time performance of Markov chain Monte Carlo algorithms in approaching the global solution of stochastic optimization problems over continuous domains. A comparison with other state-of-the-art methods having finite-time guarantees for solving stochastic programming problems is included. Comment: 29 pages, 6 figures. Revised version based on referees report
[show abstract][hide abstract] ABSTRACT: This paper addresses the computational overhead involved in probabilistic reachability computations for a general class of controlled stochastic hybrid systems. An approximate dynamic programming approach is proposed to mitigate the curse of dimensionality issue arising in the solution to the stochastic optimal control reformulation of the probabilistic reachability problem. An algorithm tailored to this problem is introduced and compared with the standard numerical solution to dynamic programming on a benchmark example.
[show abstract][hide abstract] ABSTRACT: DNA replication is one of the most fundamental processes in the life of every cell. In earlier work a model to capture the mechanics of the DNA replication process was developed. The model allowed us to make novel predictions regarding the mechanisms behind DNA replication based on experimental data for the fission yeast. One of the difficulties we had to overcome in the process was tuning of the model parameters based on experimental data, which, for lack of better methods had to be done manually. Here we propose a methodology for systematizing this process, inspired by techniques for multi-objective optimization.
BioInformatics and BioEngineering, 2008. BIBE 2008. 8th IEEE International Conference on; 11/2008
[show abstract][hide abstract] ABSTRACT: DNA replication in eukaryotic cells initiates from hundreds of origins along their genomes, leading to complete duplication of genetic information before cell division. The large number of potential origins, coupled with system uncertainty, dictates the need for new analytical tools to capture spatial and temporal patterns of DNA replication genome-wide. We have developed a stochastic hybrid model that reproduces DNA replication throughout a complete genome. The model can capture different modes of DNA replication and is applicable to various organisms. Using genome-wide data on the location and firing efficiencies of origins in the fission yeast, we show how the DNA replication process evolves during S-phase in the presence of stochastic origin firing. Simulations reveal small regions of the genome that extend S-phase to three times its reported duration. The low levels of late replication predicted by the model are below the detection limit of techniques used to measure S-phase length. Parameter sensitivity analysis shows that increased replication fork speeds genome-wide, or additional origins are not sufficient to reduce S-phase to its reported length. We model the redistribution of a limiting initiation factor during S-phase and show that it could shorten S-phase to the reported duration. Alternatively, S-phase may be extended, and what has traditionally been defined as G2 may be occupied by low levels of DNA synthesis with the onset of mitosis delayed by activation of the G2/M checkpoint.
Proceedings of the National Academy of Sciences 09/2008; 105(34):12295-300. · 9.74 Impact Factor
[show abstract][hide abstract] ABSTRACT: The term stochastic hybrid systems defines a class of dynamical and control systems that involve the interaction of continuous dynamics, discrete dynamics and probabilistic uncertainty. Over the last decade stochastic hybrid systems have emerged as a powerful modeling paradigm in a wide range of application areas. The talk will provide an overview of recent developments in this rapidly evolving field of research. We will introduce some of the theoretical foundations and challenges of stochastic hybrid systems and outline computational methods that can be used to analyze and control such systems, based primarily on randomized algorithms. The discussion will be motivated by several applied modeling, analysis and control problems from the areas of systems biology and air traffic management. This extended abstract provides some motivation for the talk, as well as an extensive list of references that can serve as a starting point for further reading.
Control and Decision Conference, 2008. CCDC 2008. Chinese; 08/2008
[show abstract][hide abstract] 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.
IEEE Transactions on Automatic Control 02/2008; · 2.72 Impact Factor
[show abstract][hide abstract] ABSTRACT: This paper concerns parameter identification for the class of jump Markov linear systems. We present recursive formulae for the computation of the second-order statistics of a jump Markov process and discuss their application to the estimation of the model parameters given data from a single or repeated experiments. Simulation results are provided to show the effectiveness of our approach.
Decision and Control, 2007 46th IEEE Conference on; 01/2008
[show abstract][hide abstract] ABSTRACT: In this paper, we give new characterizations of the stochastic reachability problem for stochastic hybrid systems in the language of different theories that can be employed in studying stochastic processes (Markov processes, potential theory, optimal control). These characterizations are further used to obtain the probabilities involved in the context of stochastic reachability as viscosity solutions of some variational inequalities.
Decision and Control, 2007 46th IEEE Conference on; 01/2008
[show abstract][hide abstract] ABSTRACT: Based on a model of subtilin production by Bacillus subtilis, in this paper we discuss the parameter identification of stochastic hybrid dynamics that are typically found in biological regulatory networks. In accordance with the structure of the 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 profiles. Techniques for parameter estimation from sparse and irregularly sampled observations are developed and applied to simulated data. Numerical results are provided to show the effectiveness of our methods.
Decision and Control, 2007 46th IEEE Conference on; 01/2008
[show abstract][hide abstract] ABSTRACT: We present a novel control scheme for multiple non-holonomic vehicles under uncertainty, which can guarantee collision avoidance while complying with constraints imposed on the vehicles. Dipolar navigation functions are used for decentralized conflict-free control, while model predictive control is used in a centralized manner in order to ensure that the resulting trajectories remain feasible with respect to the constraints present and to optimize the performance objectives. The model used is chosen to resemble air traffic control problems, with some uncertainty introduced in the system. The efficiency of the control strategy is demonstrated by realistic simulations.
Proceedings of the 47th IEEE Conference on Decision and Control, CDC 2008, December 9-11, 2008, Cancún, México; 01/2008
[show abstract][hide abstract] ABSTRACT: Simulated annealing is a popular method for approaching the solution of a global optimization problem. Existing results on its performance apply to discrete combinatorial optimization where the optimization variables can assume only a finite set of possible values. We introduce a new general formulation of simulated annealing which allows one to guarantee finite-time performance in the optimization of functions of continuous variables. The results hold universally for any optimization problem on a bounded domain and establish a connection between simulated annealing and up-to-date theory of convergence of Markov chain Monte Carlo methods on continuous domains. This work is inspired by the concept of finite-time learning with known accuracy and confidence developed in statistical learning theory.