A stochastic simulation algorithm for biochemical reactions with delays

Fine-Grained Parallel and Distributed Spatial Stochastic Simulation of Biological Reactions

Numerical simulations of seeded batch cooling crystallization were performed to investigate the effect of stochastic nucleation on crystal quality parameters. In a typical deterministic mathematical model for crystallization, which contains population and mass balance equations, primary and secondary nucleation are regarded as Poisson processes in order to derive a stochastic model. In this study, these stochastic model equations were repeatedly solved to achieve a stochastic simulation, and statistical analysis of the results revealed differences in product quality when the simulations were run under certain conditions. In particular, the statistics, such as the mean and the coefficient of variation, of the product crystal size distribution were found to fluctuate as a result of stochastic primary and secondary nucleation. Further, stochastic primary nucleation was also found to be the source of the differences between the statistics obtained using the deterministic and stochastic simulations when the seed-loading ratio was very low. This difference was attributed to the growth of crystals when the total number of crystals was less than one, as well as the accompanying secondary nucleation, in the standard deterministic simulation. Thus, a novel deterministic model in which secondary nucleation does not occur until the total number of crystals reaches one was used, and the results and statistics were found to agree with those obtained using the stochastic numerical simulation. In addition, a stochastic model ignoring stochastic secondary nucleation omitted to predict the significant statistical fluctuations under certain conditions. Finally, the statistical fluctuations were predicted for several crystallizer scales, and crystallizer scale-up was found to reduce the fluctuations caused by stochastic primary and secondary nucleation.

Unraveling pine wilt disease: Comparative study of stochastic and deterministic model using spectral method

Event Abstract Back to Event Stochastic modeling of neuronal signaling Jukka Intosalmi1*, Tiina Manninen1, Keijo Ruohonen1 and Marja-Leena Linne1 1 Tampere University of Technology, Finland The time evolution of biochemical systems in neurons is traditionally modeled using deterministic ordinary differential equations (ODEs). Chemical reactions, however, are random in nature, and the deterministic approach is valid only for a restricted class of systems. Stochastic models take random fluctuations into account and are thus more realistic. Biochemical reactions can be modeled stochastically using numerous different methods. An ideal model would have the following three important properties. First, the model should be as realistic as possible, second, the mathematical method should be easily implementable as a computer algorithm, and third, the algorithm should be computationally effective. Naturally, all these three conditions cannot be fulfilled at the same time. Some realistic modeling approaches can be derived directly from chemical kinetics without making any approximations. Such approaches are called exact. A good example of an exact modeling approach is the stochastic simulation algorithm (SSA) developed by Gillespie. The SSA is applicable when the molecular populations in the system are small, but it becomes computationally inefficient when the numbers of molecules increase. In order to construct stochastic models that can be effectively simulated, new mathematical approaches have to be explored. As an approximate method also stochastic differential equations (SDEs) have been considered a promising way to model biochemical reactions stochastically. The SDE approach is attractive especially if we consider a system for which the SSA is computationally inefficient and the traditional deterministic ODE approach cannot be used as a good approximation. SDE models treat the chemical populations as real numbers and the construction of the model is based on the law of mass action, similarly as the construction of the traditional deterministic modeling approach. The SDE model, however, takes the random fluctuations into account by describing the time evolution of the system with the stochastic Itô process instead of the deterministic set of ODEs. The Itô process is basically a set of coupled stochastic differential equations. Although SDE modeling usually leads to equations that cannot be solved analytically, the solutions can be approximated numerically using different numerical integration methods. Thus, SDEs provide a mathematically rigorous way to construct models that can be simulated effectively. In order to investigate how well the SDE approach models biochemical systems, the results have to be compared with experimental data or with the simulation results from some exact simulation procedure. In this study, we use the methodology of spectral analysis to compare the simulation results of the SDE model and the SSA. As case studies, we consider e.g. calcium binding and protein kinase C (PKC) signal transduction pathway. We investigate the nature of noise in different modeling approaches and try to interpret the meaning of the noise in real biological systems. The main goal of our study is to find the most essential components and parameters in the SDE model and to adjust them so that the model is capable of giving similar results as the SSA. The simulations are carried out using the distributed computing resources (GRID) provided by Techila Technologies Ltd. Conference: Neuroinformatics 2008, Stockholm, Sweden, 7 Sep - 9 Sep, 2008. Presentation Type: Poster Presentation Topic: Computational Neuroscience Citation: Intosalmi J, Manninen T, Ruohonen K and Linne M (2008). Stochastic modeling of neuronal signaling. Front. Neuroinform. Conference Abstract: Neuroinformatics 2008. doi: 10.3389/conf.neuro.11.2008.01.023 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 28 Jul 2008; Published Online: 28 Jul 2008. * Correspondence: Jukka Intosalmi, Tampere University of Technology, Tampere, Finland, jukka.intosalmi@tut.fi Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Scholar Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne PubMed Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Stochastic Simulation Algorithm (SSA) is crucial for modeling biochemical reactions and gene regulatory networks. Traditional SSA is characterized by Markovian property and cannot naturally model systems with time delays. Several algorithms have already been designed to handle delayed reactions, yet few easy-to-use implementations exist. To address these challenges, we have developed DelaySSA, an R package that implements currently available algorithms for SSA with or without delays. Meanwhile, we also provided Matlab and Python versions to support wider applications. We demonstrated its accuracy and validity by simulating two classical models: the Bursty model and Refractory model. We then tested its capability to simulate the RNA Velocity model, where it successfully reproduced both the up- and down-regulation stages in the phase portrait. Finally, we extended its application to simulate a gene regulatory network of lung cancer adeno-to-squamous transition (AST) and qualitatively analyzed its bistability behavior by approximating the Waddington's landscape. Modeling the therapeutic intervention of a SOX2 degrader as a delayed degradation reaction, AST is effectively blocked and reprogrammed back to the adenocarcinoma state, providing a useful clue for targeting drug-resistant AST in the future. Taken together, DelaySSA is a powerful and easy-to-use software suite, facilitating accurate modeling of various kinds of biological systems and broadening the scope of stochastic simulations in systems biology.

For an engineered genetic oscillator, deterministic analysis indicates sustained oscillations and stochastic simulations show irregular or absent oscillations. Since the major difference is in the modeling of the promoters, we have performed a detailed analysis of a generic repressible promoter system. Deterministic, stochastic, thermodynamic, and hybrid models were developed for the promoter. The average behavior of the stochastic model converged to the thermodynamic model. The deterministic model is a special case of the thermodynamic model. The hybrid model could lock into the off state. Adding an unrelated background reaction allowed it to exit that state. Increasing the background rate allowed the hybrid model to converge to thermodynamic and stochastic model. Adding a background reaction to the stochastic oscillator simulation noticeably improved its performance.

This work presents a deterministic and a stochastic model for the simulation of industrial-size deionized water and water for injection (DI/WFI) systems. The objective of the simulations is to determine if additional DI/WFI demand from future production processes can be supported by an existing DI/WFI system. The models utilize discrete event simulation to compute the demand profile from the distribution system; they also use a continuous simulation to calculate the variation of the water level in the storage tank. Whereas the deterministic model ignores uncertainties, the stochastic model allows for both volume and schedule uncertainties. The Monte Carlo method is applied to solve the stochastic method. This paper compares the deterministic and stochastic models and shows that the deterministic model may be suitable for most applications and that the stochastic model should only be used if found necessary by the deterministic simulation. The models are programmed within Excel 2003 and are available for download as open public domain software (1), allowing for public modifications and improvements of the model. The proposed models may also be utilized to determine size or analyze the performance of other utilities, such as heat transfer media, drinking water, etc. Water for injection (WFI) and other pharmaceutical water distribution systems are notoriously difficult to analyze analytically due to the highly dynamic variable demand that is drawn from these systems. Discrete event simulation may provide an answer where the typical engineering approach of utilizing a diversity factor fails. This paper develops an Excel based deterministic and stochastic model for a WFI system with the latter allowing for the modeling of offtake volume and schedule uncertainty. The paper also compares the deterministic and stochastic models and shows that the deterministic model may be suitable for most applications while the stochastic model should only be used if found necessary. The models are available for download as open public domain software allowing for modifications and improvements of the model.

We look at the behavior of biological oscillators, specifically analyzing a genetic circuit that has oscillatory behavior. Implementing the system in deterministic and stochastic models, we compare the two models using various tests and analyze the effect of stochastic noise on these oscillations. We first investigate the effect of volume and find that at small system volumes, stochastic events cause the system to exhibit more sporadic oscillations and a longer period of oscillation. Next, we develop classification systems for discerning the boundary in the parameter space where the model begins to show oscillations. This is more challenging in the stochastic model, due to persistent fluctuations. We compare different methods of analysis of our deterministic and stochastic models and visualize the conditions which lead to oscillation in each model. We find that the parameter range of oscillation is larger in the stochastic model than in the deterministic model. Finally, we find an increased rate of mRNA production can create greater noise in a system and amplify the difference between a stochastic and deterministic system.

This study compares the newly developed stochastic agent-based model (ABM) and deterministic system within the context of COVID-19 transmission dynamics, using the Susceptible-Exposed-Infectious-Removed-Perception (SEIRP) compartments. The primary aim is to analyse the similarities and differences between these modelling approaches, providing insights into the emergent behaviours of the epidemiological systems. We also investigate how social phenomena like public perception affect the two epidemiological models’ outcomes, focusing on scenarios with varying awareness levels and proportions of severe cases. Our results show that while deterministic model outputs align well with the ABM for large populations, discrepancies emerge for small populations in the ABM due to the impacts of stochastic extinction and discreteness of individuals. In scenarios with high proportions of severe cases and for large population sizes, the deterministic model exhibits oscillatory behaviour. In this situation, the averaged ABM densities initially capture the fluctuation dynamics when a substantial number of realisations is used in simulation; however, this stochastic system exhibits diminishing fluctuations across different realisations, contributing to a consistent average akin to an endemic steady state over a longer period. Interestingly, as the number of realisations is reduced, the agreement between stochastic and deterministic systems in depicting recurrent outbreaks is evident (i.e., realisations-dependent dynamical behaviour). In the next case study, the joint effect of recovery, latency period, disease severity and public perception is explored, highlighting how different factors can combine to influence the systems’ outcomes. Notably, our case study finds that even when the deterministic model demonstrates the persistence of sustained oscillations, the ABM can depict an extinction state; this stochastic ABM observation is caused by the realisations in this epidemiological system fluctuating to a very low population density and being excluded over a long run. Overall, our findings suggest the importance of considering both deterministic and stochastic models in infectious disease modelling, highlighting the need for comprehensive analyses to guide evidence-based decision-making in public health and epidemiology.

Thermodynamic modeling and uncertainty quantification of CO2-loaded aqueous MEA solutions

Gillespie's stochastic simulation algorithm (SSA) has been a conventional method for stochastic modeling and simulation of biochemical systems. However, as a population-based algorithm it faces the challenge of combinatorial complexity in many biochemical models where species may present with multiple states. To solve this problem, the rules-based modeling was proposed by Hlavacek's group and the particle-based Network-Free Algorithm (NFA) was used for its simulation. In this paper, we first improve the NFA to efficiently integrate the population-based and particle-based features. Then we propose a population-based SSA scheme for the stochastic simulation of rule-based models. Complexity analysis is presented for the proposed methods. Numerical experiments on two rule-based models demonstrate the power of the proposed methods.

Recent research in both the experimental and mathematical communities has focused on biochemical interaction systems that satisfy an "absolute concentration robustness" (ACR) property. The ACR property was first discovered experimentally when, in a number of different systems, the concentrations of key system components at equilibrium were observed to be robust to the total concentration levels of the system. Followup mathematical work focused on deterministic models of biochemical systems and demonstrated how chemical reaction network theory can be utilized to explain this robustness. Later mathematical work focused on the behavior of this same class of reaction networks, though under the assumption that the dynamics were stochastic. Under the stochastic assumption, it was proven that the system will undergo an extinction event with a probability of one so long as the system is conservative, showing starkly different long-time behavior than in the deterministic setting. Here we consider a general class of stochastic models that intersects with the class of ACR systems studied previously. We consider a specific system scaling over compact time intervals and prove that in a limit of this scaling the distribution of the abundances of the ACR species converges to a certain product-form Poisson distribution whose mean is the ACR value of the deterministic model. This result is in agreement with recent conjectures pertaining to the behavior of ACR networks endowed with stochastic kinetics, and helps to resolve the conflicting theoretical results pertaining to deterministic and stochastic models in this setting.

Comparison of deterministic and stochastic SIS and SIR models in discrete time

Deterministic models have played an important role in biology over the past few centuries, and continue to play. However, there is a need to introduce a random factor due to the nature of biological processes, which led to the emergence of stochastic models. To bridge the gap between deterministic and stochastic models, and in appreciation of the huge efforts made in deterministic models, which have played a fundamental role in understanding, predicting, and controlling the transmission dynamics of infectious diseases, this study aims to clarify a method for creating stochastic model for one of most vital and recent deterministic, that is Covid-19 model with two doses, which showed the effect of receiving both doses in containing the disease. To achieve this goal, the deterministic model was studied extensively before converting to corresponding stochastic. The deterministic and stochastic systems were solved numerically and represented using MATAB tools. Keywords: Covid-19 deterministic model, stochastic model, next generation matrix method (NGMM), The diffusion matrix G.

Introduction: This study aims to provide a mathematical description of the production planning model (method). There will occasionally be restrictions on the raw material availability for production activities due to this element of uncertainty. Objectives: The researcher will outline and analyze the various perspectives, arguments, and opinions that exist in the literature, and look for relationships between them Methods: This research was compiled using qualitative research methods and descriptive and analytical journal literature review approaches. Most of the data used are secondary data. Result: Deterministic and stochastic mathematical models are the two broad categories into which production planning mathematical models fall. In deterministic models, the data is assumed to be confident at this point. In the meantime, the stochastic model uses data that reflects the "best prediction" of unknown values. These deterministic methods finally address average or worst-case value issues even though many human judgment and quantitative models have been developed to anticipate variables with this uncertainty. Even though they have been studied extensively in the literature, deterministic production planning models are generally not helpful when uncertainty is considered. There are two types of manufacturing-related uncertainties: system uncertainty and manufacturing environment uncertainty. The second category consists of supply and demand uncertainties and other uncertainties outside the purview of production process management. Conclusion: Even though they have been studied extensively in the literature, deterministic production planning models are generally not helpful when uncertainty is considered. There are two types of manufacturing-related uncertainties: system uncertainty and manufacturing environment uncertainty. Uncertainties like supply and demand issues outside the purview of industrial process management are included in the second category.