Mass fluctuation kinetics : analysis and computation of equilibria and local dynamics

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 81-82).The mass fluctuation kinetics (MFK) model is a set of coupled first-order differential equations describing the temporal evolution of means, variances and covariances of species concentrations in systems of chemical reactions. It generalizes classical mass action kinetics (MAK) in which fluctuations around the mean are ignored. This thesis begins with the motivating background theory for the development of MFK. The model equations follow from the time-evolution of the molecule number moment generating function obtained from the chemical master equation (CME). A closed-form expression for the MFK Jacobian matrix that describes small deviations from equilibrium is derived. An MFK software toolbox prototype, developed in MATLAB (and available at http://www.mit.edu/~azunre/MFK), applies this Jacobian in the context of single substrate enzyme kinetics to exploring the local dynamics of MFK equilibria. MFK means and covariances are observed to be locally decoupled at the equilibrium in the large volume thermodynamic limit, providing an alternative explanation for why MAK is an accurate approximation for system behavior there. Increasing discreteness of system behavior with decreasing system volume, a characteristic that the MAK model cannot capture, is captured by the MFK model via the growth of its variance. This ability is limited to a threshold beyond which MFK ceases to be a useful approximation for system behavior. Systematic extensions to higher order moments to correct for this are suggested.by Paul Azunre.S.M

Mass fluctuation kinetics: analysis and computation of equilibria and local dynamics

The mass fluctuation kinetics (MFK) model is a set of coupled ordinary differential equations approximating the time evolution of means and covariances of species concentrations in chemical reaction networks. It generalises classical mass action kinetics (MAK), in which fluctuations around the mean are ignored. MFK may be used to approximate stochasticity in system trajectories when stochastic simulation methods are prohibitively expensive computationally. This study presents a set of tools to aid in the analysis of systems within the MFK framework. A closed-form expression for the MFK Jacobian matrix is derived. This expression facilitates the computation of MFK equilibria and the characterisation of the dynamics of small deviations from the equilibria (i.e. local dynamics). Software developed in MATLAB to analyse systems within the MFK framework is also presented. The authors outline a homotopy continuation method that employs the Jacobian for bifurcation analysis, that is, to generate a locus of steady-state Jacobian eigenvalues corresponding to changing a chosen MFK parameter such as system volume or a rate constant. This method is applied to study the effect of small-volume stochasticity on local dynamics at equilibria in a pair of example systems, namely the formation and dissociation of an enzyme-substrate complex and a genetic oscillator. For both systems, this study reveals volume regimes where MFK provides a quantitatively and/or qualitatively correct description of system behaviour, and regimes where the MFK approximation is inaccurate. Moreover, our analysis provides evidence that decreasing volume from the MAK regime (infinite volume) has a destabilising effect on system dynamics

Probabilistic Model Checking for Continuous-Time Markov Chains via Sequential Bayesian Inference

Probabilistic model checking for systems with large or unbounded state space is a challenging computational problem in formal modelling and its applications. Numerical algorithms require an explicit representation of the state space, while statistical approaches require a large number of samples to estimate the desired properties with high confidence. Here, we show how model checking of time-bounded path properties can be recast exactly as a Bayesian inference problem. In this novel formulation the problem can be efficiently approximated using techniques from machine learning. Our approach is inspired by a recent result in statistical physics which derived closed form differential equations for the first-passage time distribution of stochastic processes. We show on a number of non-trivial case studies that our method achieves both high accuracy and significant computational gains compared to statistical model checking

A parallel branch-and-bound algorithm for thin-film optical systems, with application to realizing a broadband omnidirectional antireflection coating for silicon solar cells

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2014.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 124-129).For the class of nondispersive, nonabsorbing, multilayer thin-film optical systems, this thesis work develops a parallel branch-and-bound computational system on Amazon's EC2 platform, using the Taylor model mathematical/computational system due to Berz and Makino to construct tight rigorous bounds on the merit function on subsets of the search space (as required by a branch-and-bound algorithm). This represents the first, to the best of our knowledge, deterministic global optimization algorithm for this important class of problems, i.e., the first algorithm that can guarantee that a global solution to an optimization problem in this class has been found. For the particular problem of reducing reflection using multilayer systems, it is shown that a gradient index constraint on the solution can be exploited to significantly reduce the search space and thereby make the algorithm more practical. This optimization system is then used to design a broadband omnidirectional antireflection coating for silicon solar energy. The design is experimentally validated using RF sputtering, and shows performance that is competitive with existing solutions based on impractical sophisticated nano-deposition techniques, as well as the more practical but also more narrowly applicable solutions based on texturing. This makes it arguably the best practical solution to this important problem to date. In addition, this thesis develops a mathematical theory for cheaply (in the computational sense) and tightly bounding solutions to parametric weakly-coupled semilinear parabolic (reaction-diffusion) partial differential equation systems, as motivated by the design of tandem organic solar cell structures (which are governed by the drift-diffusion-Poisson system of equations). This represents the first theoretical foundation, to the best of our knowledge, to enable guaranteed global optimization of this important class of problems, which includes, but is broader, than many semiconductor design problems. A serial branch-and-bound algorithm implementation illustrates the applicability of the bounds on a pair of simple examples.by Paul Azunre.Ph. D

Guaranteed global optimization of thin-film optical systems

© 2019 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft. A parallel deterministic global optimization algorithm for thin-film multilayer optical coatings is developed. This algorithm enables locating a global solution to an optimization problem in this class to within a user-specified tolerance. More specifically, the algorithm is a parallel branch-And-bound method with applicable bounds on the merit function computed using Taylor models. This study is the first one, to the best of our knowledge, to attempt guaranteed global optimization of this important class of problems, thereby providing an overview and an assessment of the current state of such techniques in this domain. As a proof of concept on a small scale, the method is illustrated numerically and experimentally in the context of antireflection coatings for silicon solar cells-we design and fabricate a three-layer dielectric stack on silicon that exhibits an average reflectance of (2.53 0.10)%, weighted over a broad range of incident angles and the solar spectrum. The practicality of our approach is assessed by comparing its computational cost relative to traditional stochastic global optimization techniques which provide no guarantees on their solutions. While our method is observed to be significantly more computationally expensive, we demonstrate via our proof of concept that it is already feasible to optimize sufficiently simple practical problems at a reasonable cost, given the current accessibility of cloud computing resources. Ongoing advances in distributed computing are likely to bring more design problems within the reach of deterministic global optimization methods, yielding rigorous guaranteed solutions in the presence of practical manufacturing constraints