Delay Stochastic Simulation Algorithm Research Articles

Applications of the Delay Stochastic Simulation Algorithm (DSSA) in Mathematical Epidemiology

The calculation of the probability of a minor outbreak is crucial in analyzing a stochastic epidemic model. For stochastic epidemic models with fixed delays, the linear chain trick is applied to transform the delayed models into a family of ODE models with increasing shape parameters. We then prove that the well-established results on the probability of a minor outbreak for continuous-time Markov chain (CTMC) epidemic models also hold for the stochastic epidemic models with fixed delays. All theoretical results are verified by numerical simulations implemented by the delay stochastic simulation algorithm (DSSA) in Python. It is shown that DSSA is able to generate exact realizations for underlying delayed models in the context of mathematical epidemiology, and therefore, provides insights into the effect of delays during the outbreak phases of epidemics.

DelaySSAToolkit.jl: stochastic simulation of reaction systems with time delays in Julia.

DelaySSAToolkit.jl is a Julia package for modelling reaction systems with non-Markovian dynamics, specifically those with time delays. These delays implicitly capture multiple intermediate reaction steps and hence serve as an effective model reduction technique for complex systems in biology, chemistry, ecology and genetics. The package implements a variety of exact formulations of the delay stochastic simulation algorithm. The source code and documentation of DelaySSAToolkit.jl are available at https://github.com/palmtree2013/DelaySSAToolkit.jl.

The distributed delay rearranges the bimodal distribution at protein level

BackgroundIn bimodal distribution, we refer ON to the mode with a higher protein level and OFF to a lower protein level. The traditional wisdom of manipulating the population in each mode is done by adding an inducer or inhibitor to change the protein number. MethodsBy delay stochastic simulation algorithm, we applied transcriptional or translational delay to the gene expression system of toggle switches. We also incorporated distributed delay in the model. From a deterministic aspect, reaction delay does NOT alter the steady-state value. Thus, our approach is different from the conventional method. Significant FindingsThe transcriptional delay rearranged the bimodal distribution at the protein level and the influence of the translational delay is even higher. The increment of delay time pushes more cells to have the protein in the ON mode. Remarkably, the distributed delay of translation guides 98% of cells to have protein in the ON mode. We then displayed the time-course trajectories and discovered the cause leading to the rearrangement of bimodal distribution. We also exclude the possibility that rearrangement of bimodal distribution simply comes from the trivial reason, the change of the input distribution, and concluded that the change of dynamic stochastic processes is crucial.

Interplay of cellular states: Role of delay as control mechanism

Delay is ubiquitous, no matter how small or big it is. Experimental evidences show the existence and importance of time delayed reactions specially in biological systems. We studied the interplay of delay and noise in some biological models, analytically and computationally. The analytically solved P(x,t) of gene regulation process shows universal class of Poisson process at large initial population and large x. Fano factor which measures noise in the system is found to be delay dependent. Analytical results give the possibility of noise-induced control of the system dynamics driven by delay. From analytical results, time delay creates possible conditions to the system to trigger various mechanisms such as correlated activity, coherence, etc. This indicates the interplay of delay and noise in regulating and controlling the system. To see this, we simulated some biological systems (gene regulation, circadian rhythm, repressilator, brusselator) using delay stochastic simulation algorithm. Simulation results show various mechanisms such as delay-induced onset of oscillating states (which could be active states where the system can establish coherence among the system variables), delay-induced switching-off of the oscillating states (which may correspond to inactive state or system failure), delay-induced coherent bistable states (where the system can stay longer to make decisions about the fate of the system). From simulation studies, we thus observed that delay and noise could play significant roles in regulating and controlling system dynamics, as also pointed out by analytical results.

Stochastic fluctuations as a driving force to dissipative non-equilibrium states

Most natural complex systems exhibit fluctuations-driven processes, which work at far from equilibrium states, and are generally dissipative processes, for instance living cells. We studied this phenomenon within the stochastic framework by taking a set of nonequilibrium, bimolecular, autocatalytic reactions, originally proposed by Nicolis (1972). We also extended this model to incorporate the concept of time delay. Firstly, for both non-delay and delay cases, we calculated the exact non-stationary probability distribution solutions of the corresponding Master equations, which are found to deviate from the Maxwell–Boltzmann distribution. The analytically calculated probability distribution P of an autocatalyst X in the chemically reacting model system is found to follow some universal class of probability distributions at different situations. At the thermodynamic limit with a large population, P obeys Normal distribution. Again, we showed that one of the causes of this peculiar behaviour is the fluctuations in the reacting system. The analytical result of the Fano factor F in the non-delay case predicted a noise-enhanced process for our dynamical stochastic system which could probably drive the system far from equilibrium. For the delay case, the analytically calculated F was found to depend on the time delay function, which predicts that time delay could play an important role in regulating the system dynamics. These analytical predictions were then verified using numerical experiments with the stochastic simulation algorithm (SSA) and delay stochastic simulation algorithm (DSSA). Indeed, numerical results from SSA and DSSA confirmed noise-enhanced processes which are far from equilibrium and dissipative in nature.

A Practicable Method of Tuning the Noise Intensity at Protein Level.

The expression of genes is inevitably subject to intracellular noise. Noise, for some regulatory networks, is constructive but detrimental to many others. The intensity of the noise is a determinant factor and the method of tuning it is of great value. In this study, we illustrated that the transcriptional delay in an incoherent feedforward loop (FFL) grants the target protein modulation the intensity of noise. Remarkably, for a wide range, the coefficient of variation (COV) of the target protein appeared to be about linear to the time span of the transcriptional delay. Without a noise-buffering method, the COV of the target protein is 0.455. While applying incoherent FFL, the COV reduced to 0.236. Then, it changed from 0.236 to 0.630 as the transcriptional delay raised from 0 to 1000 seconds. If we further increased the delay out of the linear range, the COV finally reached 0.779. In addition, we incorporated the distribution of the transcriptional delay in the delay stochastic simulation algorithm. This distribution is based on the experimental observation in the literature. The outcome suggested that the distributed delay slightly improved the ability of tuning noise. In conclusion, we demonstrated a noise-tuning method that altered only the intensity of noise without changing the deterministic steady-state behaviors. It is ready to be applied to various systems in the field of synthetic biology.

Transcriptional processes: Models and inference

Many biochemical events involve multistep reactions. One of the most important biological processes that involve multistep reaction is the transcriptional process. Models for multistep reaction necessarily need multiple states and it is a challenge to compute model parameters that best agree with experimental data. Therefore, the aim of this work is to design a multistep promoter model which accurately characterizes transcriptional bursting and is consistent with observed data. To address this issue, we develop a model for promoters with several OFF states and a single ON state using Erlang distribution. To explore the combined effects of model and data, we combine Monte Carlo extension of Expectation Maximization (MCEM) and delay Stochastic Simulation Algorithm (DSSA) and call the resultant algorithm as delay Bursty MCEM. We apply this algorithm to time-series data of endogenous mouse glutaminase promoter to validate the model assumptions and infer the kinetic parameters. Our results show that with multiple OFF states, we are able to infer and produce a model which is more consistent with experimental data. Our results also show that delay Bursty MCEM inference is more efficient.

Delay chemical master equation: direct and closed-form solutions

The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived.

Multi-scale approach for simulating time-delay biochemical reaction systems.

This study presents a multi-scale approach for simulating time-delay biochemical reaction systems when there are wide ranges of molecular numbers. The authors construct a new efficient approach based on partitioning into slow and fast subsets in conjunction with predictor-corrector methods. This multi-scale approach is shown to be much more efficient than existing methods such as the delay stochastic simulation algorithm and the modified next reaction method. Numerical testing on several important problems in systems biology confirms the accuracy and computational efficiency of this approach.

Reduction of chemical reaction networks through delay distributions

Accurate modelling and simulation of dynamic cellular events require two main ingredients: an adequate description of key chemical reactions and simulation of such chemical events in reasonable time spans. Quite logically, posing the right model is a crucial step for any endeavour in Computational Biology. However, more often than not, it is the associated computational costs which actually limit our capabilities of representing complex cellular behaviour. In this paper, we propose a methodology aimed at representing chains of chemical reactions by much simpler, reduced models. The abridgement is achieved by generation of model-specific delay distribution functions, consecutively fed to a delay stochastic simulation algorithm. We show how such delay distributions can be analytically described whenever the system is solely composed of consecutive first-order reactions, with or without additional "backward" bypass reactions, yielding an exact reduction. For models including other types of monomolecular reactions (constitutive synthesis, degradation, or "forward" bypass reactions), we discuss why one must adopt a numerical approach for its accurate stochastic representation, and propose two alternatives for this. In these cases, the accuracy depends on the respective numerical sample size. Our model reduction methodology yields significantly lower computational costs while retaining accuracy. Quite naturally, computational costs increase alongside network size and separation of time scales. Thus, we expect our model reduction methodologies to significantly decrease computational costs in these instances. We anticipate the use of delays in model reduction will greatly alleviate some of the current restrictions in simulating large sets of chemical reactions, largely applicable in pharmaceutical and biological research.

Improved delay-leaping simulation algorithm for biochemical reaction systems with delays

In biochemical reaction systems dominated by delays, the simulation speed of the stochastic simulation algorithm depends on the size of the wait queue. As a result, it is important to control the size of the wait queue to improve the efficiency of the simulation. An improved accelerated delay stochastic simulation algorithm for biochemical reaction systems with delays, termed the improved delay-leaping algorithm, is proposed in this paper. The update method for the wait queue is effective in reducing the size of the queue as well as shortening the storage and access time, thereby accelerating the simulation speed. Numerical simulation on two examples indicates that this method not only obtains a more significant efficiency compared with the existing methods, but also can be widely applied in biochemical reaction systems with delays.

Bio-PEPAd: A non-Markovian extension of Bio-PEPA

Delays in biological systems may be used to model events for which the underlying dynamics cannot be precisely observed, or to provide abstraction of some behavior of the system resulting in more compact models. In this paper, we enrich the stochastic process algebra Bio-PEPA, with the possibility of assigning delays to actions, yielding a new non-Markovian stochastic process algebra: Bio-PEPAd. This is a conservative extension meaning that the original syntax of Bio-PEPA is retained and the delay specification which can now be associated with actions may be added to existing Bio-PEPA models. The semantics of the firing of the actions with delays is the delay-as-duration approach, earlier presented in papers on the stochastic simulation of biological systems with delays. This semantics of the algebra is given in the Starting–Terminating style, meaning that the state and the completion of an action are observed as two separate events, as required by delays. We formally define the encoding of Bio-PEPAd systems in Generalized Semi-Markov Processes (GSMPs), as input for a Delay Stochastic Simulation Algorithm (DSSA) and as sets of Delay Differential Equations (DDEs), the deterministic framework for modeling of biological systems with delays. Finally, we prove theorems stating the relation between Bio-PEPA and Bio-PEPAd models. We end the paper with an example model of biological systems with delays to illustrate the approach.

A partial-propensity formulation of the stochastic simulation algorithm for chemical reaction networks with delays

Several real-world systems, such as gene expression networks in biological cells, contain coupled chemical reactions with a time delay between reaction initiation and completion. The non-Markovian kinetics of such reaction networks can be exactly simulated using the delay stochastic simulation algorithm (dSSA). The computational cost of dSSA scales with the total number of reactions in the network. We reduce this cost to scale at most with the smaller number of species by using the concept of partial reaction propensities. The resulting delay partial-propensity direct method (dPDM) is an exact dSSA formulation for well-stirred systems of coupled chemical reactions with delays. We detail dPDM and present a theoretical analysis of its computational cost. Furthermore, we demonstrate the implications of the theoretical cost analysis in two prototypical benchmark applications. The dPDM formulation is shown to be particularly efficient for strongly coupled reaction networks, where the number of reactions is much larger than the number of species.

Probability distributed time delays: integrating spatial effects into temporal models

BackgroundIn order to provide insights into the complex biochemical processes inside a cell, modelling approaches must find a balance between achieving an adequate representation of the physical phenomena and keeping the associated computational cost within reasonable limits. This issue is particularly stressed when spatial inhomogeneities have a significant effect on system's behaviour. In such cases, a spatially-resolved stochastic method can better portray the biological reality, but the corresponding computer simulations can in turn be prohibitively expensive.ResultsWe present a method that incorporates spatial information by means of tailored, probability distributed time-delays. These distributions can be directly obtained by single in silico or a suitable set of in vitro experiments and are subsequently fed into a delay stochastic simulation algorithm (DSSA), achieving a good compromise between computational costs and a much more accurate representation of spatial processes such as molecular diffusion and translocation between cell compartments. Additionally, we present a novel alternative approach based on delay differential equations (DDE) that can be used in scenarios of high molecular concentrations and low noise propagation.ConclusionsOur proposed methodologies accurately capture and incorporate certain spatial processes into temporal stochastic and deterministic simulations, increasing their accuracy at low computational costs. This is of particular importance given that time spans of cellular processes are generally larger (possibly by several orders of magnitude) than those achievable by current spatially-resolved stochastic simulators. Hence, our methodology allows users to explore cellular scenarios under the effects of diffusion and stochasticity in time spans that were, until now, simply unfeasible. Our methodologies are supported by theoretical considerations on the different modelling regimes, i.e. spatial vs. delay-temporal, as indicated by the corresponding Master Equations and presented elsewhere.

Generalized binomial τ-leap method for biochemical kinetics incorporating both delay and intrinsic noise

The delay stochastic simulation algorithm (DSSA) by Barrio et al. [Plos Comput. Biol. 2, 117(E) (2006)] was developed to simulate delayed processes in cell biology in the presence of intrinsic noise, that is, when there are small-to-moderate numbers of certain key molecules present in a chemical reaction system. These delayed processes can faithfully represent complex interactions and mechanisms that imply a number of spatiotemporal processes often not explicitly modeled such as transcription and translation, basic in the modeling of cell signaling pathways. However, for systems with widely varying reaction rate constants or large numbers of molecules, the simulation time steps of both the stochastic simulation algorithm (SSA) and the DSSA can become very small causing considerable computational overheads. In order to overcome the limit of small step sizes, various tau-leap strategies have been suggested for improving computational performance of the SSA. In this paper, we present a binomial tau-DSSA method that extends the tau-leap idea to the delay setting and avoids drawing insufficient numbers of reactions, a common shortcoming of existing binomial tau-leap methods that becomes evident when dealing with complex chemical interactions. The resulting inaccuracies are most evident in the delayed case, even when considering reaction products as potential reactants within the same time step in which they are produced. Moreover, we extend the framework to account for multicellular systems with different degrees of intercellular communication. We apply these ideas to two important genetic regulatory models, namely, the hes1 gene, implicated as a molecular clock, and a Her1/Her 7 model for coupled oscillating cells.

Modeling Intrinsic Noise and Delays in Chemical Kinetics of Coupled Autoregulated Oscillating Cells

Delays are an important feature in temporal models of genetic regulation due to slow biochemical processes, such as transcription and translation. In this paper, we show how to model intrinsic noise effects in a delayed setting by either using a delay stochastic simulation algorithm (DSSA) or, for larger and more complex systems, a generalized Binomial τ-leap method (Bτ-DSSA). As a particular application, we apply these ideas to modeling somite segmentation in zebra fish across a number of cells in which two linked oscillatory genes (her1 and her7) are synchronized via Notch signaling between the cells.

Stochastic delay differential equations for genetic regulatory networks

Time delay is an important aspect in the modelling of genetic regulation due to slow biochemical reactions such as gene transcription and translation, and protein diffusion between the cytosol and nucleus. In this paper we introduce a general mathematical formalism via stochastic delay differential equations for describing time delays in genetic regulatory networks. Based on recent developments with the delay stochastic simulation algorithm, the delay chemical master equation and the delay reaction rate equation are developed for describing biological reactions with time delay, which leads to stochastic delay differential equations derived from the Langevin approach. Two simple genetic regulatory networks are used to study the impact of intrinsic noise on the system dynamics where there are delays.