Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art

Stochasticity is a key characteristic of intracellular processes such as gene regulation and chemical signalling. Therefore, characterising stochastic effects in biochemical systems is essential to understand the complex dynamics of living things. Mathematical idealisations of biochemically reacting systems must be able to capture stochastic phenomena. While robust theory exists to describe such stochastic models, the computational challenges in exploring these models can be a significant burden in practice since realistic models are analytically intractable. Determining the expected behaviour and variability of a stochastic biochemical reaction network requires many probabilistic simulations of its evolution. Using a biochemical reaction network model to assist in the interpretation of time course data from a biological experiment is an even greater challenge due to the intractability of the likelihood function for determining observation probabilities. These computational challenges have been subjects of active research for over four decades. In this review, we present an accessible discussion of the major historical developments and state-of-the-art computational techniques relevant to simulation and inference problems for stochastic biochemical reaction network models. Detailed algorithms for particularly important methods are described and complemented with MATLAB implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community

An Interpretation of the Dual Problem of the THREE-like Approaches

Spectral estimation can be preformed using the so called THREE-like approach. Such method leads to a convex optimization problem whose solution is characterized through its dual problem. In this paper, we show that the dual problem can be seen as a new parametric spectral estimation problem. This interpretation implies that the THREE-like solution is optimal in terms of closeness to the correlogram over a certain parametric class of spectral densities, enriching in this way its meaningfulness

Sensitivity Analyses of Optimized Attitude Estimators Using Sensor Fusion Solutions for Low-Cost MEMS Configurations

Since the 1990’s, there has been increased focus on creating navigation systems for small unmanned systems, particularly small unmanned aerial systems (SUAS). Due to size, weight, and cost restrictions, compared to larger more costly manned systems, navigation systems for SUAS have evolved to be quite different from the proven systems of the past. Today, there are many solutions for the problem of navigation for SUAS. The problem has now become choosing the most fitting navigation solution for a given application/mission. This is particularly true for evaluating solutions that are fundamentally different. This research analyses the performance and sensitivity of four sensor fusion solutions for attitude estimation under multiple simulated flight conditions. There are three different hardware configurations between the four estimators. For this reason, each estimator is tuned to be experimentally optimal, as to provide a fair comparison between different estimators. With each estimator tuned to its highest performance, the estimators are compared based on their sensitivity to tuning error, sensor bias, and estimator initialization error. Finally the estimators\u27 accuracy performances are directly compared. This thesis also provides methods to tune different configuration estimators to their individual best performances. These methods show that choosing tuning parameters based on sensor noise covariance, as is typically done in research, does not produce optimal performance for all estimator formulations. After comparing multiple sensitivity and performance properties of the estimators, observations are provided regarding the efficacy of the analyses, including the applicability of the metrics used to determine performance. Some metrics where shown to be misleading for particular estimators or analyses. Ultimately, guidance is given for choosing performance metrics capable of comparing different solutions

A unified framework for solving a general class of conditional and robust set-membership estimation problems

In this paper we present a unified framework for solving a general class of problems arising in the context of set-membership estimation/identification theory. More precisely, the paper aims at providing an original approach for the computation of optimal conditional and robust projection estimates in a nonlinear estimation setting where the operator relating the data and the parameter to be estimated is assumed to be a generic multivariate polynomial function and the uncertainties affecting the data are assumed to belong to semialgebraic sets. By noticing that the computation of both the conditional and the robust projection optimal estimators requires the solution to min-max optimization problems that share the same structure, we propose a unified two-stage approach based on semidefinite-relaxation techniques for solving such estimation problems. The key idea of the proposed procedure is to recognize that the optimal functional of the inner optimization problems can be approximated to any desired precision by a multivariate polynomial function by suitably exploiting recently proposed results in the field of parametric optimization. Two simulation examples are reported to show the effectiveness of the proposed approach.Comment: Accpeted for publication in the IEEE Transactions on Automatic Control (2014

Coarse Projective kMC Integration: Forward/Reverse Initial and Boundary Value Problems

In "equation-free" multiscale computation a dynamic model is given at a fine, microscopic level; yet we believe that its coarse-grained, macroscopic dynamics can be described by closed equations involving only coarse variables. These variables are typically various low-order moments of the distributions evolved through the microscopic model. We consider the problem of integrating these unavailable equations by acting directly on kinetic Monte Carlo microscopic simulators, thus circumventing their derivation in closed form. In particular, we use projective multi-step integration to solve the coarse initial value problem forward in time as well as backward in time (under certain conditions). Macroscopic trajectories are thus traced back to unstable, source-type, and even sometimes saddle-like stationary points, even though the microscopic simulator only evolves forward in time. We also demonstrate the use of such projective integrators in a shooting boundary value problem formulation for the computation of "coarse limit cycles" of the macroscopic behavior, and the approximation of their stability through estimates of the leading "coarse Floquet multipliers".Comment: Submitted to Journal of Computational Physic

Aircraft adaptive learning control

The optimal control theory of stochastic linear systems is discussed in terms of the advantages of distributed-control systems, and the control of randomly-sampled systems. An optimal solution to longitudinal control is derived and applied to the F-8 DFBW aircraft. A randomly-sampled linear process model with additive process and noise is developed

Local error estimates for adaptive simulation of the Reaction-Diffusion Master Equation via operator splitting

The efficiency of exact simulation methods for the reaction-diffusion master equation (RDME) is severely limited by the large number of diffusion events if the mesh is fine or if diffusion constants are large. Furthermore, inherent properties of exact kinetic-Monte Carlo simulation methods limit the efficiency of parallel implementations. Several approximate and hybrid methods have appeared that enable more efficient simulation of the RDME. A common feature to most of them is that they rely on splitting the system into its reaction and diffusion parts and updating them sequentially over a discrete timestep. This use of operator splitting enables more efficient simulation but it comes at the price of a temporal discretization error that depends on the size of the timestep. So far, existing methods have not attempted to estimate or control this error in a systematic manner. This makes the solvers hard to use for practitioners since they must guess an appropriate timestep. It also makes the solvers potentially less efficient than if the timesteps are adapted to control the error. Here, we derive estimates of the local error and propose a strategy to adaptively select the timestep when the RDME is simulated via a first order operator splitting. While the strategy is general and applicable to a wide range of approximate and hybrid methods, we exemplify it here by extending a previously published approximate method, the Diffusive Finite-State Projection (DFSP) method, to incorporate temporal adaptivity