Stochastic Biological Systems Research Articles

Adaptive Fuzzy Tracking Control and Its Application in Stochastic Biological Systems

Adaptive Fuzzy Tracking Control and Its Application in Stochastic Biological Systems

Interval neutrosophic stochastic dynamical systems driven by Brownian motion

Stochastic and neutrosophic theory are effective tools for modeling real-world phenomena and natural dynamical systems, where inputs are often affected by stochastic noises and outputs often contain both randomness and indeterminacy. In this work, we present a new type of stochastic differential equations (SDE) driven by an one-dimensional Brownian motion that can be considered as an efficient tool to describe the uncertain behavior of dynamical systems operating in interval neutrosophic environments with stochastic noises. After introducing some basic foundations on neutrosophic arithmetic, neutrosophic calculus and neutrosophic stochastic process, we define the new form of interval neutrosophic stochastic differential equations taking values in neutrosophic environment. Under some suitable conditions, the unique existence result of stochastic solution is obtained based on the use of Picard successive approximation. We also introduce an efficient numerical algorithm, namely Euler–Maruyama method, to solve the numerical solution of proposed problem and further demonstrate the effectiveness of the numerical method by solving some examples in stochastic biological systems such as stochastic logistic growth model, stochastic Lotka–Volterra predator–prey model, and stochastic SARS model, respectively.

An Approximate Method of System Entropy in Discrete-Time Nonlinear Biological Networks

This study discusses the calculation of entropy of discrete-time stochastic biological systems. First, measurement methods of the system entropy of discrete-time linear stochastic networks are introduced. The system entropy is found to be characterized by system matrices of the discrete-time biological systems. Secondly, the system entropy of nonlinear discrete-time stochastic biological systems is discussed and is calculated based on a global linearization method. The approximation of the values of system entropy of nonlinear stochastic systems needs to solve an optimization problem that is constrained by a kind of linear matrix inequality (LMI). Finally, a practical biochemical system is provided to verify the effectiveness of the proposed calculation method.

Robust finite automata in stochastic chemical reaction networks.

Finite-state automata (FSA) are simple computational devices that can nevertheless illustrate interesting behaviours. We propose that FSA can be employed as control circuits for engineered stochastic biological and biomolecular systems. We present an implementation of FSA using counts of chemical species in the range of hundreds to thousands, which is relevant for the counts of many key molecules such as mRNAs in prokaryotic cells. The challenge here is to ensure a robust representation of the current state in the face of stochastic noise. We achieve this by using a multistable approximate majority algorithm to stabilize and store the current state of the system. Arbitrary finite state machines can thus be compiled into robust stochastic chemical automata. We present two variants: one that consumes its input signals to initiate state transitions and one that does not. We characterize the state change dynamics of these systems and demonstrate their application to solve the four-bit binary square root problem. Our work lays the foundation for the use of chemical automata as control circuits in bioengineered systems and biorobotics.

Stochastic reaction and diffusion systems in molecular communications: Recent results and open problems

Chemical reactions and diffusion are two basic mechanisms governing the dynamics of molecules in a fluid. As such, they play a critical role in molecular communication for channel modeling, design of detection rules, implementation of molecular circuits for computation, and modeling interactions with external biochemical systems. For finite numbers of information-carrying molecules, stochastic models naturally arise with the simplest example given by the Wiener process, often known as Brownian motion. Nevertheless, the Wiener process fails to be accurate when external forces, friction, and chemical reactions are present. Recently, there have been several contributions that tailor molecular communication systems to these more challenging channel conditions. In this paper, we first overview a general family of stochastic models of reaction and diffusion systems, including both Langevin diffusion and the reaction-diffusion master equation. These models form a basis for the use of these models as molecular communication channels, from which modulation and detection schemes can be developed. We survey recent results on the design of these schemes, with a focus on a recently developed approach which is robust to a wide range of channel models, known as equilibrium signaling. We then turn to the implementation of these detection schemes and related parameter estimation problems via stochastic molecular circuits, based on stochastic chemical reaction networks. Finally, interactions between molecular communication systems and stochastic biological systems as well as open problems are discussed. Our overarching goal is to highlight how the consideration of general stochastic models of reaction and diffusion can be utilized in order to widen the application of molecular communications both within engineered systems, and also as motivation for advances in the mathematical characterization of these models.

Importance sampling for a robust and efficient multilevel Monte Carlo estimator for stochastic reaction networks

The multilevel Monte Carlo (MLMC) method for continuous-time Markov chains, first introduced by Anderson and Higham (SIAM Multiscal Model. Simul. 10(1), 2012), is a highly efficient simulation technique that can be used to estimate various statistical quantities for stochastic reaction networks (SRNs), in particular for stochastic biological systems. Unfortunately, the robustness and performance of the multilevel method can be affected by the high kurtosis, a phenomenon observed at the deep levels of MLMC, which leads to inaccurate estimates of the sample variance. In this work, we address cases where the high-kurtosis phenomenon is due to \textit{catastrophic coupling (characteristic of pure jump processes where coupled consecutive paths are identical in most of the simulations, while differences only appear in a tiny proportion) and introduce a pathwise-dependent importance sampling (IS) technique that improves the robustness and efficiency of the multilevel method. Our theoretical results, along with the conducted numerical experiments, demonstrate that our proposed method significantly reduces the kurtosis of the deep levels of MLMC, and also improves the strong convergence rate from $\beta=1$ for the standard case (without IS), to $\beta=1+\delta$, where $0<\delta<1$ is a user-selected parameter in our IS algorithm. Due to the complexity theorem of MLMC, and given a pre-selected tolerance, $\text{TOL}$, this results in an improvement of the complexity from $\mathcal{O}\left(\text{TOL}^{-2} \log(\text{TOL})^2\right)$ in the standard case to $\mathcal{O}\left(\text{TOL}^{-2}\right)$, which is the optimal complexity of the MLMC estimator. We achieve all these improvements with a negligible additional cost since our IS algorithm is only applied a few times across each simulated path.

In-Silico Proportional-Integral Moment Control of Stochastic Gene Expression

In-silico control of complex stochastic biological systems is a promising area of cybergenetics, which provides enabling theoretical and practical tools for the real-time control of living cells with digital computers. We demonstrate here that simple proportional-integral control laws can be used to control both the mean and the variance of the expression product of a simple gene expression network. Extensions to more general unimolecular networks, networks with input delay, and to a gene expression network involving protein dimerization are also provided. Notably, no moment closure method is used for dealing with the network with dimerization. The obtained results are illustrated by simulations.

Optimal Design of Single-Cell Experiments within Temporally Fluctuating Environments.

Modern biological experiments are becoming increasingly complex, and designing these experiments to yield the greatest possible quantitative insight is an open challenge. Increasingly, computational models of complex stochastic biological systems are being used to understand and predict biological behaviors or to infer biological parameters. Such quantitative analyses can also help to improve experiment designs for particular goals, such as to learn more about specific model mechanisms or to reduce prediction errors in certain situations. A classic approach to experiment design is to use the Fisher information matrix (FIM), which quantifies the expected information a particular experiment will reveal about model parameters. The Finite State Projection based FIM (FSP-FIM) was recently developed to compute the FIM for discrete stochastic gene regulatory systems, whose complex response distributions do not satisfy standard assumptions of Gaussian variations. In this work, we develop the FSP-FIM analysis for a stochastic model of stress response genes in S. cerevisae under time-varying MAPK induction. We verify this FSP-FIM analysis and use it to optimize the number of cells that should be quantified at particular times to learn as much as possible about the model parameters. We then extend the FSP-FIM approach to explore how different measurement times or genetic modifications help to minimize uncertainty in the sensing of extracellular environments, and we experimentally validate the FSP-FIM to rank single-cell experiments for their abilities to minimize estimation uncertainty of NaCl concentrations during yeast osmotic shock. This work demonstrates the potential of quantitative models to not only make sense of modern biological data sets, but to close the loop between quantitative modeling and experimental data collection.

Permanence of Stochastic Biological Systems

Area of research related to prey-predator systems is an important topic. The concept of permanence is an important issues related to biological systems. In general permanence is considered as a combination of persistence and boundedness. Following this, this paper reviews few existing definitions of stochastic permanence. Following the existing definition, in this paper a new definition of permanence for stochastic biological systems has been proposed, which modifies the existing ones. The permanence of a general Lotka-Volterra stochastic two species cooperation, competition and predation systems is proved by new definition.

Note on the permanence of stochastic population models

Abstract The concept of permanence of any system is an important technical issue. This concept is very significant to all kind of systems, e.g., social, medical, biological, population, mechanical, or electrical. It is desirable by scientists and investigators that any system under consideration must be long time survival. For example, if we consider any ecosystem, it is always pre-requisite that this system is permanent. In general language, permanence is just the persistent and bounded system in a particular surface time frame. But the meaning may vary with the type of systems. For example, deterministic and stochastic biological systems have different concepts of permanence in an abstract mathematical platform. The reason is simple: it is due to the mathematical nature of parameters, methods of derivations of the model, biological assumptions, details of the study, etc. In this short note, we consider the stochastic models for their permanence. To address stochastic permanence of biological systems, many different approaches have been proposed in the literature. In this note, we propose a new definition of permanence for stochastic population models (SPM). The proposed definition is applied to the well-known Lotka–Volterra two species stochastic population model. The note is closed with the open ended discussion on the topic.

On the System Entropy and Energy Dissipativity of Stochastic Systems and Their Application in Biological Systems

This study discusses the relationship between the entropy and the dissipativity of stochastic systems under the background of biological systems. First, measurement methods of the system entropy and energy dissipativity of linear stochastic biological systems are introduced. We found that the system entropy is negatively proportional to the energy dissipativity in logarithmic scale. Some opposite effects between system entropy and energy dissipativity are also discussed and compared based on their measured values to get insight into the understanding of the system mechanisms and the system characteristics. We found that the intrinsic random fluctuation and the enhancement of the system robust stability both can increase the system entropy but decrease the system dissipativity. The system entropy and the energy dissipativity of nonlinear stochastic biological systems are also discussed and compared based on a global linearization method. Computation methods are also provided. Finally, two numerical examples are demonstrated to verify theoretical prediction.

Extending the Multi-level Method for the Simulation of Stochastic Biological Systems.

The multi-level method for discrete-state systems, first introduced by Anderson and Higham (SIAM Multiscale Model Simul 10(1):146-179, 2012), is a highly efficient simulation technique that can be used to elucidate statistical characteristics of biochemical reaction networks. A single point estimator is produced in a cost-effective manner by combining a number of estimators of differing accuracy in a telescoping sum, and, as such, the method has the potential to revolutionise the field of stochastic simulation. In this paper, we present several refinements of the multi-level method which render it easier to understand and implement, and also more efficient. Given the substantial and complex nature of the multi-level method, the first part of this work reviews existing literature, with the aim of providing a practical guide to the use of the multi-level method. The second part provides the means for a deft implementation of the technique and concludes with a discussion of a number of open problems.

An adaptive multi-level simulation algorithm for stochastic biological systems.

Discrete-state, continuous-time Markov models are widely used in the modeling of biochemical reaction networks. Their complexity often precludes analytic solution, and we rely on stochastic simulation algorithms (SSA) to estimate system statistics. The Gillespie algorithm is exact, but computationally costly as it simulates every single reaction. As such, approximate stochastic simulation algorithms such as the tau-leap algorithm are often used. Potentially computationally more efficient, the system statistics generated suffer from significant bias unless tau is relatively small, in which case the computational time can be comparable to that of the Gillespie algorithm. The multi-level method [Anderson and Higham, "Multi-level Monte Carlo for continuous time Markov chains, with applications in biochemical kinetics," SIAM Multiscale Model. Simul. 10(1), 146-179 (2012)] tackles this problem. A base estimator is computed using many (cheap) sample paths at low accuracy. The bias inherent in this estimator is then reduced using a number of corrections. Each correction term is estimated using a collection of paired sample paths where one path of each pair is generated at a higher accuracy compared to the other (and so more expensive). By sharing random variables between these paired paths, the variance of each correction estimator can be reduced. This renders the multi-level method very efficient as only a relatively small number of paired paths are required to calculate each correction term. In the original multi-level method, each sample path is simulated using the tau-leap algorithm with a fixed value of τ. This approach can result in poor performance when the reaction activity of a system changes substantially over the timescale of interest. By introducing a novel adaptive time-stepping approach where τ is chosen according to the stochastic behaviour of each sample path, we extend the applicability of the multi-level method to such cases. We demonstrate the efficiency of our method using a number of examples.

Automatic validation of computational models using pseudo-3D spatio-temporal model checking.

BackgroundComputational models play an increasingly important role in systems biology for generating predictions and in synthetic biology as executable prototypes/designs. For real life (clinical) applications there is a need to scale up and build more complex spatio-temporal multiscale models; these could enable investigating how changes at small scales reflect at large scales and viceversa. Results generated by computational models can be applied to real life applications only if the models have been validated first. Traditional in silico model checking techniques only capture how non-dimensional properties (e.g. concentrations) evolve over time and are suitable for small scale systems (e.g. metabolic pathways). The validation of larger scale systems (e.g. multicellular populations) additionally requires capturing how spatial patterns and their properties change over time, which are not considered by traditional non-spatial approaches.ResultsWe developed and implemented a methodology for the automatic validation of computational models with respect to both their spatial and temporal properties. Stochastic biological systems are represented by abstract models which assume a linear structure of time and a pseudo-3D representation of space (2D space plus a density measure). Time series data generated by such models is provided as input to parameterised image processing modules which automatically detect and analyse spatial patterns (e.g. cell) and clusters of such patterns (e.g. cellular population). For capturing how spatial and numeric properties change over time the Probabilistic Bounded Linear Spatial Temporal Logic is introduced. Given a collection of time series data and a formal spatio-temporal specification the model checker Mudi (http://mudi.modelchecking.org) determines probabilistically if the formal specification holds for the computational model or not. Mudi is an approximate probabilistic model checking platform which enables users to choose between frequentist and Bayesian, estimate and statistical hypothesis testing based validation approaches. We illustrate the expressivity and efficiency of our approach based on two biological case studies namely phase variation patterning in bacterial colony growth and the chemotactic aggregation of cells.ConclusionsThe formal methodology implemented in Mudi enables the validation of computational models against spatio-temporal logic properties and is a precursor to the development and validation of more complex multidimensional and multiscale models.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-014-0124-0) contains supplementary material, which is available to authorized users.

Efficient parametric inference for stochastic biological systems with measured variability.

Stochastic systems in biology often exhibit substantial variability within and between cells. This variability, as well as having dramatic functional consequences, provides information about the underlying details of the system's behavior. It is often desirable to infer properties of the parameters governing such systems given experimental observations of the mean and variance of observed quantities. In some circumstances, analytic forms for the likelihood of these observations allow very efficient inference: we present these forms and demonstrate their usage. When likelihood functions are unavailable or difficult to calculate, we show that an implementation of approximate Bayesian computation (ABC) is a powerful tool for parametric inference in these systems. However, the calculations required to apply ABC to these systems can also be computationally expensive, relying on repeated stochastic simulations. We propose an ABC approach that cheaply eliminates unimportant regions of parameter space, by addressing computationally simple mean behavior before explicitly simulating the more computationally demanding variance behavior. We show that this approach leads to a substantial increase in speed when applied to synthetic and experimental datasets.

Numerical simulations and modeling for stochastic biological systems with jumps

This paper gives a numerical method to simulate sample paths for stochastic differential equations (SDEs) driven by Poisson random measures. It provides us a new approach to simulate systems with jumps from a different angle. The driving Poisson random measures are assumed to be generated by stationary Poisson point processes instead of Lévy processes. Methods provided in this paper can be used to simulate SDEs with Lévy noise approximately. The simulation is divided into two parts: the part of jumping integration is based on definition without approximation while the continuous part is based on some classical approaches. Biological explanations for stochastic integrations with jumps are motivated by several numerical simulations. How to model biological systems with jumps is showed in this paper. Moreover, method of choosing integrands and stationary Poisson point processes in jumping integrations for biological models are obtained. In addition, results are illustrated through some examples and numerical simulations. For some examples, earthquake is chose as a jumping source which causes jumps on the size of biological population.

External noise control in inherently stochastic biological systems

Biological systems are often subject to external noise from signal stimuli and environmental perturbations, as well as noises in the intracellular signal transduction pathway. Can different stochastic fluctuations interact to give rise to new emerging behaviors? How can a system reduce noise effects while still being capable of detecting changes in the input signal? Here, we study analytically and computationally the role of nonlinear feedback systems in controlling external noise with the presence of large internal noise. In addition to noise attenuation, we analyze derivatives of Fano factor to study systems' capability of differentiating signal inputs. We find effects of internal noise and external noise may be separated in one slow positive feedback loop system; in particular, the slow loop can decrease external noise and increase robustness of signaling with respect to fluctuations in rate constants, while maintaining the signal output specific to the input. For two feedback loops, we demonstrate that the influence of external noise mainly depends on how the fast loop responds to fluctuations in the input and the slow loop plays a limited role in determining the signal precision. Furthermore, in a dual loop system of one positive feedback and one negative feedback, a slower positive feedback always leads to better noise attenuation; in contrast, a slower negative feedback may not be more beneficial. Our results reveal interesting stochastic effects for systems containing both extrinsic and intrinsic noises, suggesting novel noise filtering strategies in inherently stochastic systems.

Existence and exponential stability of almost periodic solution for stochastic cellular neural networks with delay

The paper considers the problems of existence of quadratic mean almost periodic and global exponential stability for stochastic cellular neural networks with delays. By employing the Holder’s inequality and fixed points principle, we present some new criteria ensuring existence and uniqueness of a quadratic mean almost periodic and global exponential stability. These criteria are important in signal processing and the design of networks. Moreover, these criteria are also applied in others stochastic biological neural systems.

On a Nonlinear Master Equation and the Haken-Kelso-Bunz Model

A nonlinear master equation (NLME) is proposed basedon general information measures.Classical and cut-off solutions of the NLME are considered.In the former case, the NLME exhibits uniquely defined stationary distributions. In the latter case, there are multiple stationary distributions.In particular, for classical solutions, it is shown that transient solutions converge to stationary distributions that maximize information measures (H-theorem). Cut-off distributions arestudied numerically for the Haken-Kelso-Bunz model. The Haken-Kelso-Bunz modelis known to describe multistable human motor control systems. It is shownthat a stochastic Haken-Kelso-Bunz model based on a NLME can exhibit multiplestationary cut-off distributions.In doing so, we illustrate that multistability in stochastic biological systems can beestablished by means of cut-off distributions.