Stochastic Kinetic Models Research Articles

A thermal–kinetic subgrid model for supernova feedback in simulations of galaxy formation

ABSTRACT We present a subgrid model for supernova feedback designed for cosmological simulations of galaxy formation that may include a cold interstellar medium (ISM). The model uses thermal and kinetic channels of energy injection, which are built upon the stochastic kinetic and thermal models for stellar feedback used in the owls and eagle simulations, respectively. In the thermal channel, the energy is distributed statistically isotropically and injected stochastically in large amounts per event, which minimizes spurious radiative energy losses. In the kinetic channel, we inject the energy in small portions by kicking gas particles in pairs in opposite directions. The implementation of kinetic feedback is designed to conserve energy, linear and angular momentum, and is statistically isotropic. To test the model, we run simulations of isolated Milky Way-mass and dwarf galaxies, in which the gas is allowed to cool down to 10 K. Using the thermal and kinetic channels together, we obtain smooth star formation histories and powerful galactic winds with realistic mass loading factors. Furthermore, the model produces spatially resolved star formation rates (SFRs) and velocity dispersions that are in agreement with observations. We vary the numerical resolution by several orders of magnitude and find excellent convergence of the global SFRs and wind mass loading. We show that large thermal energy injections generate a hot phase of the ISM and modulate the star formation by ejecting gas from the disc, while the low-energy kicks increase the turbulent velocity dispersion in the neutral ISM, which in turn helps suppress star formation.

Accelerating inference for stochastic kinetic models

Stochastic kinetic models (SKMs) are increasingly used to account for the inherent stochasticity exhibited by interacting populations of species in areas such as epidemiology, population ecology and systems biology. Species numbers are modelled using a continuous-time stochastic process, and, depending on the application area of interest, this will typically take the form of a Markov jump process or an Itô diffusion process. Widespread use of these models is typically precluded by their computational complexity. In particular, performing exact fully Bayesian inference in either modelling framework is challenging due to the intractability of the observed data likelihood, necessitating the use of computationally intensive techniques such as particle Markov chain Monte Carlo (particle MCMC). It is proposed to increase the computational and statistical efficiency of this approach by leveraging the tractability of an inexpensive surrogate derived directly from either the jump or diffusion process. The surrogate is used in three ways: in the design of a gradient-based parameter proposal, to construct an appropriate bridge and in the first stage of a delayed-acceptance step. The resulting approach, which exactly targets the posterior of interest, offers substantial gains in efficiency over a standard particle MCMC implementation.

Inference for stochastic kinetic models from multiple data sources for joint estimation of infection dynamics from aggregate reports and virological data

Before the current pandemic, influenza and respiratory syncytial virus (RSV) were the leading etiological agents of seasonal acute respiratory infections (ARI) around the world. In this setting, medical doctors typically based the diagnosis of ARI on patients’ symptoms alone and did not routinely conduct virological tests necessary to identify individual viruses, limiting the ability to study the interaction between multiple pathogens and to make public health recommendations. We consider a stochastic kinetic model (SKM) for two interacting ARI pathogens circulating in a large population and an empirically-motivated background process for infections with other pathogens causing similar symptoms. An extended marginal sampling approach, based on the linear noise approximation to the SKM, integrates multiple data sources and additional model components. We infer the parameters defining the pathogens’ dynamics and interaction within a Bayesian model and explore the posterior trajectories of infections for each illness based on aggregate infection reports from six epidemic seasons collected by the state health department and a subset of virological tests from a sentinel program at a general hospital in San Luis Potosí, México. We interpret the results and make recommendations for future data collection strategies.

Parameter inference for stochastic biochemical models from perturbation experiments parallelised at the single cell level.

Understanding and characterising biochemical processes inside single cells requires experimental platforms that allow one to perturb and observe the dynamics of such processes as well as computational methods to build and parameterise models from the collected data. Recent progress with experimental platforms and optogenetics has made it possible to expose each cell in an experiment to an individualised input and automatically record cellular responses over days with fine time resolution. However, methods to infer parameters of stochastic kinetic models from single-cell longitudinal data have generally been developed under the assumption that experimental data is sparse and that responses of cells to at most a few different input perturbations can be observed. Here, we investigate and compare different approaches for calculating parameter likelihoods of single-cell longitudinal data based on approximations of the chemical master equation (CME) with a particular focus on coupling the linear noise approximation (LNA) or moment closure methods to a Kalman filter. We show that, as long as cells are measured sufficiently frequently, coupling the LNA to a Kalman filter allows one to accurately approximate likelihoods and to infer model parameters from data even in cases where the LNA provides poor approximations of the CME. Furthermore, the computational cost of filtering-based iterative likelihood evaluation scales advantageously in the number of measurement times and different input perturbations and is thus ideally suited for data obtained from modern experimental platforms. To demonstrate the practical usefulness of these results, we perform an experiment in which single cells, equipped with an optogenetic gene expression system, are exposed to various different light-input sequences and measured at several hundred time points and use parameter inference based on iterative likelihood evaluation to parameterise a stochastic model of the system.

Power of stochastic kinetic models: From biological signaling and antibiotic activities to T cell activation and cancer initiation dynamics

AbstractAll chemical processes exhibit two main universal features. They are stochastic because chemical reactions might happen only after random successful collisions of reacting species, and they are dynamic because the amount of reactants and products changes with time. Since biological processes rely heavily on specific chemical reactions, the stochasticity and dynamics are also crucial features for all living systems. To understand the molecular mechanisms of chemical and biological processes, it is important to develop and apply theoretical methods that fully incorporate the randomness and dynamic nature of these systems. In recent years, there have been significant advances in formulating and exploring such theoretical methods. As an illustration of such developments, in this review the recent applications of stochastic kinetic models for various biological processes are discussed. Specifically, we focus on applying these theoretical approaches to investigate the biological signaling, clearance of bacteria under antibiotics, T cells activation in the immune system, and cancer initiation dynamics. The main advantage of the presented stochastic kinetic models is that they generally can be solved analytically, allowing to clarify the underlying microscopic picture, as well as to explain the existing experimental observations and to make new testable predictions. This theoretical approach becomes a powerful tool in uncovering the molecular mechanisms of complex natural phenomena.This article is categorized under: Molecular and Statistical Mechanics > Free Energy Methods Structure and Mechanism > Computational Biochemistry and Biophysics Theoretical and Physical Chemistry > Statistical Mechanics

Using single-cell models to predict the functionality of synthetic circuits at the population scale

SignificanceAt the single-cell level, biochemical processes are inherently stochastic. For many natural systems, the resulting cell-to-cell variability is exploited by microbial populations. In synthetic biology, however, the interplay of cell-to-cell variability and population processes such as selection or growth often leads to circuits not functioning as predicted by simple models. Here we show how multiscale stochastic kinetic models that simultaneously track single-cell and population processes can be obtained based on an augmentation of the chemical master equation. These models enable us to quantitatively predict complex population dynamics of a yeast optogenetic differentiation system from a specification of the circuit's components and to demonstrate how cell-to-cell variability can be exploited to purposefully create unintuitive circuit functionality.

Effective Finite-Difference Techniques for Estimating Sensitivities for Stochastic Biochemical Systems

Cellular environments are in essence stochastic, owing to the random character of the biochemical reaction events in a single cell. Stochastic fluctuations may substantially contribute to the dynamics of systems with small copy numbers of some biochemical species. Then, stochastic models are indispensable for properly portraying the behaviour of the system. Sensitivity analysis is one of the central tools for studying stochastic models of cellular dynamics. Here, we propose some finite-difference strategies for estimating parametric sensitivities of higher-order moments of the system state for stochastic discrete biochemical kinetic models. To reduce the variance of the sensitivity estimator, we employ various coupling techniques. The advantages of the proposed methods are illustrated in several models of biochemical systems of practical relevance.

Implementation of simplified stochastic microdosimetric kinetic models into PHITS for application to radiation treatment planning

Purpose The stochastic microdosimetric kinetic (SMK) model is one of the most sophisticated and precise models used in the estimation of the relative biological effectiveness of carbon-ion radiotherapy (CRT) and boron neutron capture therapy (BNCT). However, because of its complicated and time-consuming calculation procedures, it is nearly impractical to directly incorporate this model into a radiation treatment-planning system. Materials and methods Through the introduction of Taylor expansion (TE) or fast Fourier transform (FFT), we developed two simplified SMK models and implemented them into the Particle and Heavy Ion Transport code System (PHITS). To verify the implementation, we calculated the photon isoeffective doses in a cylindrical phantom placed in the radiation fields of passive CRT and accelerator-based BNCT. Results and discussion Our calculation suggested that both TE-based and FFT-based SMK models can reproduce the data obtained from the original SMK model very well for absorbed doses approximately below 5 Gy, whereas the TE-based SMK model overestimates the original data at higher doses. In terms of computational efficiency, the TE-based SMK model is much faster than the FFT-based SMK model. Conclusion This study enables the instantaneous calculation of the photo isoeffective dose for CRT and BNCT, considering their cellular-scale dose heterogeneities. Treatment-planning systems that use the improved PHITS as a dose-calculation engine are under development.

Study of Interfacial Charge Transfer from an Electron Rich Organic Molecule to CdTe Quantum Dot by using Stern-Volmer and Stochastic Kinetic Models.

Photoinduced electron transfer (PET) from N-methylaniline (NMA) to a photoexcited CdTe quantum dot (QD*) is studied in toluene. The PET mechanism at low to moderate quencher (NMA) concentrations (<0.08 M) remains mostly collisional with some contributions from QD-NMA complex formation. However, at high quencher concentrations (>0.10 M), QDs form larger numbers of static complexes with NMA molecules leading to a steep positive deviation in the steady-state Stern-Volmer curves. An isothermal titration calorimetry (ITC) study confirms the formation of QD-NMA complexes (K∼150 M-1 ) at high quencher concentrations. Fitting our experimental data using a stochastic kinetic model indicates that the number of NMA molecules attached per QD at highest NMA concentration (∼0.16 M) used in this study decreases from ∼0.76 to ∼0.47 with reducing the QD size from ∼5.2 nm to ∼3.2 nm. However, the PET rate increases with decreasing QD size, which is commensurate with the observation that the chemical driving force (ΔG) increases with decreasing the QD particle size. We have analyzed the PET kinetics mainly by using Stern-Volmer fittings. However, in some cases Tachiya's stochastic kinetic model is used for stoichiometric analysis, which seems to be useful only at high quencher concentrations. The measured PET rate coefficients in all the cases are found to be at least an order of magnitude lower when compared to the diffusion-controlled rate of the reaction medium.

Correlated pseudo-marginal schemes for time-discretised stochastic kinetic models

The challenging problem of conducting fully Bayesian inference for the reaction rate constants governing stochastic kinetic models (SKMs) is considered. Given the challenges underlying this problem, the Markov jump process representation is routinely replaced by an approximation based on a suitable time-discretisation of the system of interest. Improving the accuracy of these schemes amounts to using an ever finer discretisation level, which in the context of the inference problem, requires integrating over the uncertainty in the process at a predetermined number of intermediate times between observations. Pseudo-marginal Metropolis–Hastings schemes are increasingly used, since for a given discretisation level, the observed data likelihood can be unbiasedly estimated using a particle filter. When observations are particularly informative, an auxiliary particle filter can be implemented, by propagating particles conditional on the next observation. Recent work in state-space settings has shown how the pseudo-marginal approach can be made much more efficient by correlating the underlying pseudo-random numbers used to form the estimate of likelihood at the current and proposed values of the unknown parameters. This approach is extended to the time-discretised SKM framework by correlating the innovations that drive the auxiliary particle filter. The resulting approach is found to offer substantial gains in efficiency over a standard implementation.

Bayesian Computation Methods for Inference in Stochastic Kinetic Models

In this paper we investigate Monte Carlo methods for the approximation of the posterior probability distributions in stochastic kinetic models (SKMs). SKMs are multivariate Markov jump processes that model the interactions among species in biological systems according to a set of usually unknown parameters. The tracking of the species populations together with the estimation of the interaction parameters is a Bayesian inference problem for which Markov chain Monte Carlo (MCMC) methods have been a typical computational tool. Specifically, the particle MCMC (pMCMC) method has been shown to be effective, while computationally demanding method applicable to this problem. Recently, it has been shown that an alternative approach to Bayesian computation, namely, the class of adaptive importance samplers, may be more efficient than classical MCMC‐like schemes, at least for certain applications. For example, the nonlinear population Monte Carlo (NPMC) algorithm has yielded promising results with a low dimensional SKM (the classical predator‐prey model). In this paper we explore the application of both pMCMC and NPMC to analyze complex autoregulatory feedback networks modelled by SKMs. We demonstrate numerically how the populations of the relevant species in the network can be tracked and their interaction rates estimated, even in scenarios with partial observations. NPMC schemes attain an appealing trade‐off between accuracy and computational cost that can make them advantageous in many practical applications.

A generalized electron energy probability function for inductively coupled plasmas under conditions of nonlocal electron kinetics

A generalized equation for the electron energy probability function (EEPF) of inductively coupled Ar plasmas is proposed under conditions of nonlocal electron kinetics and diffusive cooling. The proposed equation describes the local EEPF in a discharge and the independent variable is the kinetic energy of electrons. The EEPF consists of a bulk and a depleted tail part and incorporates the effect of the plasma potential, Vp, and pressure. Due to diffusive cooling, the break point of the EEPF is eVp. The pressure alters the shape of the bulk and the slope of the tail part. The parameters of the proposed EEPF are extracted by fitting to measure EEPFs (at one point in the reactor) at different pressures. By coupling the proposed EEPF with a hybrid plasma model, measurements in the gaseous electronics conference reference reactor concerning (a) the electron density and temperature and the plasma potential, either spatially resolved or at different pressure (10–50 mTorr) and power, and (b) the ion current density of the electrode, are well reproduced. The effect of the choice of the EEPF on the results is investigated by a comparison to an EEPF coming from the Boltzmann equation (local electron kinetics approach) and to a Maxwellian EEPF. The accuracy of the results and the fact that the proposed EEPF is predefined renders its use a reliable alternative with a low computational cost compared to stochastic electron kinetic models at low pressure conditions, which can be extended to other gases and/or different electron heating mechanisms.

Boltzmann-type models with uncertain binary interactions

In this paper we study binary interaction schemes with uncertain parameters for a general class of Boltzmann-type equations with applications in classical gas and aggregation dynamics. We consider deterministic (i.e., a priori averaged) and stochastic kinetic models, corresponding to different ways of understanding the role of uncertainty in the system dynamics, and compare some thermodynamic quantities of interest, such as the mean and the energy, which characterise the asymptotic trends. Furthermore, via suitable scaling techniques we derive the corresponding deterministic and stochastic Fokker-Planck equations in order to gain more detailed insights into the respective asymptotic distributions. We also provide numerical evidences of the trends estimated theoretically by resorting to recently introduced structure preserving uncertainty quantification methods.

Efficient hbox {SMC}^2 schemes for stochastic kinetic models

Fitting stochastic kinetic models represented by Markov jump processes within the Bayesian paradigm is complicated by the intractability of the observed-data likelihood. There has therefore been considerable attention given to the design of pseudo-marginal Markov chain Monte Carlo algorithms for such models. However, these methods are typically computationally intensive, often require careful tuning and must be restarted from scratch upon receipt of new observations. Sequential Monte Carlo (SMC) methods on the other hand aim to efficiently reuse posterior samples at each time point. Despite their appeal, applying SMC schemes in scenarios with both dynamic states and static parameters is made difficult by the problem of particle degeneracy. A principled approach for overcoming this problem is to move each parameter particle through a Metropolis-Hastings kernel that leaves the target invariant. This rejuvenation step is key to a recently proposed hbox {SMC}^2 algorithm, which can be seen as the pseudo-marginal analogue of an idealised scheme known as iterated batch importance sampling. Computing the parameter weights in hbox {SMC}^2 requires running a particle filter over dynamic states to unbiasedly estimate the intractable observed-data likelihood up to the current time point. In this paper, we propose to use an auxiliary particle filter inside the hbox {SMC}^2 scheme. Our method uses two recently proposed constructs for sampling conditioned jump processes, and we find that the resulting inference schemes typically require fewer state particles than when using a simple bootstrap filter. Using two applications, we compare the performance of the proposed approach with various competing methods, including two global MCMC schemes.

Hard-sphere kinetic models for inert and reactive mixtures

I consider stochastic variants of a simple reacting sphere (SRS) kinetic model (Xystris and Dahler 1978 J. Chem. Phys. 68 387–401, Qin and Dahler 1995 J. Chem. Phys. 103 725–50, Dahler and Qin 2003 J. Chem. Phys. 118 8396–404) for dense reacting mixtures. In contrast to the line-of-center models of chemical reactive models, in the SRS kinetic model, the microscopic reversibility (detailed balance) can be easily shown to be satisfied, and thus all mathematical aspects of the model can be fully justified. In the SRS model, the molecules behave as if they were single mass points with two internal states. Collisions may alter the internal states of the molecules, and this occurs when the kinetic energy associated with the reactive motion exceeds the activation energy. Reactive and non-reactive collision events are considered to be hard sphere-like. I consider a four component mixture A, B, A*, B*, in which the chemical reactions are of the type , with A* and B* being distinct species from A and B.This work extends the joined works with George Stell to the kinetic models of dense inert and reactive mixtures. The idea of introducing smearing-type effect in the collisional process results in a new class of stochastic kinetic models for both inert and reactive mixtures. In this paper the important new mathematical properties of such systems of kinetic equations are proven. The new results for stochastic revised Enskog system for inert mixtures are also provided.

Multivariate moment closure techniques for stochastic kinetic models

Stochastic effects dominate many chemical and biochemical processes. Their analysis, however, can be computationally prohibitively expensive and a range of approximation schemes have been proposed to lighten the computational burden. These, notably the increasingly popular linear noise approximation and the more general moment expansion methods, perform well for many dynamical regimes, especially linear systems. At higher levels of nonlinearity, it comes to an interplay between the nonlinearities and the stochastic dynamics, which is much harder to capture correctly by such approximations to the true stochastic processes. Moment-closure approaches promise to address this problem by capturing higher-order terms of the temporally evolving probability distribution. Here, we develop a set of multivariate moment-closures that allows us to describe the stochastic dynamics of nonlinear systems. Multivariate closure captures the way that correlations between different molecular species, induced by the reaction dynamics, interact with stochastic effects. We use multivariate Gaussian, gamma, and lognormal closure and illustrate their use in the context of two models that have proved challenging to the previous attempts at approximating stochastic dynamics: oscillations in p53 and Hes1. In addition, we consider a larger system, Erk-mediated mitogen-activated protein kinases signalling, where conventional stochastic simulation approaches incur unacceptably high computational costs.

Stochastic kinetics of the circular gene hypothesis: Feedback effects and protein fluctuations

Stochastic kinetic models of genetic expression are able to describe protein fluctuations. A comparative study of the canonical and a feedback model is given here by using stochastic simulation methods. The feedback model is a skeleton model implementation of the circular gene hypothesis, which suggests the interaction between the synthesis and degradation of mRNA. Qualitative and quantitative changes in the shape and in the numerical characteristics of the stationary distributions suggest that more combined experimental and theoretical studies should be done to uncover the details of the kinetic mechanisms of gene expressions.

Likelihood free inference for Markov processes: a comparison.

Approaches to Bayesian inference for problems with intractable likelihoods have become increasingly important in recent years. Approximate Bayesian computation (ABC) and "likelihood free" Markov chain Monte Carlo techniques are popular methods for tackling inference in these scenarios but such techniques are computationally expensive. In this paper we compare the two approaches to inference, with a particular focus on parameter inference for stochastic kinetic models, widely used in systems biology. Discrete time transition kernels for models of this type are intractable for all but the most trivial systems yet forward simulation is usually straightforward. We discuss the relative merits and drawbacks of each approach whilst considering the computational cost implications and efficiency of these techniques. In order to explore the properties of each approach we examine a range of observation regimes using two example models. We use a Lotka-Volterra predator-prey model to explore the impact of full or partial species observations using various time course observations under the assumption of known and unknown measurement error. Further investigation into the impact of observation error is then made using a Schlögl system, a test case which exhibits bi-modal state stability in some regions of parameter space.

Bayesian inference for Markov jump processes with informative observations.

In this paper we consider the problem of parameter inference for Markov jump process (MJP) representations of stochastic kinetic models. Since transition probabilities are intractable for most processes of interest yet forward simulation is straightforward, Bayesian inference typically proceeds through computationally intensive methods such as (particle) MCMC. Such methods ostensibly require the ability to simulate trajectories from the conditioned jump process. When observations are highly informative, use of the forward simulator is likely to be inefficient and may even preclude an exact (simulation based) analysis. We therefore propose three methods for improving the efficiency of simulating conditioned jump processes. A conditioned hazard is derived based on an approximation to the jump process, and used to generate end-point conditioned trajectories for use inside an importance sampling algorithm. We also adapt a recently proposed sequential Monte Carlo scheme to our problem. Essentially, trajectories are reweighted at a set of intermediate time points, with more weight assigned to trajectories that are consistent with the next observation. We consider two implementations of this approach, based on two continuous approximations of the MJP. We compare these constructs for a simple tractable jump process before using them to perform inference for a Lotka-Volterra system. The best performing construct is used to infer the parameters governing a simple model of motility regulation in Bacillus subtilis.

Bayesian inference for hybrid discrete-continuous stochastic kinetic models

We consider the problem of efficiently performing simulation and inference for stochastic kinetic models. Whilst it is possible to work directly with the resulting Markov jump process (MJP), computational cost can be prohibitive for networks of realistic size and complexity. In this paper, we consider an inference scheme based on a novel hybrid simulator that classifies reactions as either ‘fast’ or ‘slow’ with fast reactions evolving as a continuous Markov process whilst the remaining slow reaction occurrences are modelled through a MJP with time-dependent hazards. A linear noise approximation (LNA) of fast reaction dynamics is employed and slow reaction events are captured by exploiting the ability to solve the stochastic differential equation driving the LNA. This simulation procedure is used as a proposal mechanism inside a particle MCMC scheme, thus allowing Bayesian inference for the model parameters. We apply the scheme to a simple application and compare the output with an existing hybrid approach and also a scheme for performing inference for the underlying discrete stochastic model.