Stochastic approaches for modelling in vivo reactions

Quantifying Parameter Interdependence in Stochastic Discrete Models of Biochemical Systems

Stochastic modeling of biochemical processes at the cellular level has been the subject of intense research in recent years. The Chemical Master Equation is a broadly utilized stochastic discrete model of such processes. Numerous important biochemical systems consist of many species subject to many reactions. As a result, their mathematical models depend on many parameters. In applications, some of the model parameters may be unknown, so their values need to be estimated from the experimental data. However, the problem of parameter value inference can be quite challenging, especially in the stochastic setting. To estimate accurately the values of a subset of parameters, the system should be sensitive with respect to variations in each of these parameters and they should not be correlated. In this paper, we propose a technique for detecting collinearity among models' parameters and we apply this method for selecting subsets of parameters that can be estimated from the available data. The analysis relies on finite-difference sensitivity estimations and the singular value decomposition of the sensitivity matrix. We illustrated the advantages of the proposed method by successfully testing it on several models of biochemical systems of practical interest.

Predicting DNA kinetics with a truncated continuous-time Markov chain method

Predicting the kinetics of reactions involving nucleic acid strands is a fundamental task in biology and biotechnology. Reaction kinetics can be modeled as an elementary step continuous-time Markov chain, where states correspond to secondary structures and transitions correspond to base pair formation and breakage. Since the number of states in the Markov chain could be large, rates are determined by estimating the mean first passage time from sampled trajectories. As a result, the cost of kinetic predictions becomes prohibitively expensive for rare events with extremely long trajectories. Also problematic are scenarios where multiple predictions are needed for the same reaction, e.g., under different environmental conditions, or when calibrating model parameters, because a new set of trajectories is needed multiple times. We propose a new method, called pathway elaboration, to handle these scenarios. Pathway elaboration builds a truncated continuous-time Markov chain through both biased and unbiased sampling. The resulting Markov chain has moderate state space size, so matrix methods can efficiently compute reaction rates, even for rare events. Also the transition rates of the truncated Markov chain can easily be adapted when model or environmental parameters are perturbed, making model calibration feasible. We illustrate the utility of pathway elaboration on toehold-mediated strand displacement reactions, show that it well-approximates trajectory-based predictions of unbiased elementary step models on a wide range of reaction types for which such predictions are feasible, and demonstrate that it performs better than alternative truncation-based approaches that are applicable for mean first passage time estimation. Finally, in a small study, we use pathway elaboration to optimize the Metropolis kinetic model of Multistrand, an elementary step simulator, showing that the optimized parameters greatly improve reaction rate predictions. Our framework and dataset are available at https://github.com/DNA-and-Natural-Algorithms-Group/PathwayElaboration.

Analytical and Numerical Treatment of Continuous Ageing in the Voter Model

The conventional voter model is modified so that an agent's switching rate depends on the 'age' of the agent-that is, the time since the agent last switched opinion. In contrast to previous work, age is continuous in the present model. We show how the resulting individual-based system with non-Markovian dynamics and concentration-dependent rates can be handled both computationally and analytically. The thinning algorithm of Lewis and Shedler can be modified in order to provide an efficient simulation method. Analytically, we demonstrate how the asymptotic approach to an absorbing state (consensus) can be deduced. We discuss three special cases of the age-dependent switching rate: one in which the concentration of voters can be approximated by a fractional differential equation, another for which the approach to consensus is exponential in time, and a third case in which the system reaches a frozen state instead of consensus. Finally, we include the effects of a spontaneous change of opinion, i.e., we study a noisy voter model with continuous ageing. We demonstrate that this can give rise to a continuous transition between coexistence and consensus phases. We also show how the stationary probability distribution can be approximated, despite the fact that the system cannot be described by a conventional master equation.

Narrow capture problem: An encounter-based approach to partially reactive targets

A general topic of current interest is the analysis of diffusion problems in singularly perturbed domains with small interior targets or traps (the narrow capture problem). One major application is to intracellular diffusion, where the targets typically represent some form of reactive biochemical substrate. Most studies of the narrow capture problem treat the target boundaries as totally absorbing (Dirichlet), that is, the chemical reaction occurs immediately on first encounter between particle and target surface. In this paper, we analyze the three-dimensional narrow capture problem in the more realistic case of partially reactive target boundaries. We begin by considering classical Robin boundary conditions. Matching inner and outer solutions of the single-particle probability density, we derive an asymptotic expansion of the Laplace transformed flux into each reactive surface in powers of ε, where ερ is a given target size. In turn, the fluxes determine the splitting probabilities for target absorption. We then extend our analysis to more general types of reactive targets by combining matched asymptotic analysis with an encounter-based formulation of diffusion-mediated surface reactions. That is, we derive an asymptotic expansion of the joint probability density for particle position and the so-called boundary local time, which characterizes the amount of time that a Brownian particle spends in the neighborhood of a point on a totally reflecting boundary. The effects of surface reactions are then incorporated via an appropriate stopping condition for the boundary local time. Robin boundary conditions are recovered in the special case of an exponential law for the stopping local times. Finally, we illustrate the theory by exploring how the leading-order contributions to the splitting probabilities depend on the choice of surface reactions. In particular, we show that there is an effective renormalization of the target radius of the form ρ→ρ-Ψ[over ̃](1/ρ), where Ψ[over ̃] is the Laplace transform of the stopping local time distribution.

Introduction of the Effective Photon Concentration Variable for Studying the Mechanism of Crystal Violet Photodegradation

In this work, we have proposed a new approach to study the mechanism of crystal violet (CV) photodegradation on TiO₂ surface using kinetic Monte Carlo simulation. The TiO₂ surface was considered as a set of reactive centers, which is essential in dye photodegradation. A new variable "the effective photon concentration" (I_eff ) is defined. A detailed chemical understanding of the photocatalytic reaction is provided. This approach provides a simple and effective method to find the optimal conditions of the studied system. This goal was achieved by investigating the effects of some operational parameters, including initial concentration of CV, pH, loading TiO₂ , light intensity and volume, on the degradation percent, and also, on the effective photon concentration. The perfect agreements between the experimental and simulated data at different conditions confirmed the proposed approach for describing the CV photodestruction. Also, the simulation results indicated that: (1) a significant fraction of the scattered UV irradiation into the reaction vessel does not lead to charge carrier generation; (2) the generation and recombination of charge carriers have crucial roles in the photodegradation. This is the first time that a method based on the reactive centers is employed to investigate the dye degradation by a photocatalyst.

Identifiability analysis for stochastic differential equation models in systems biology

Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.

The quasi-steady-state approximations revisited: Timescales, small parameters, singularities, and normal forms in enzyme kinetics

In this work, we revisit the scaling analysis and commonly accepted conditions for the validity of the standard, reverse and total quasi-steady-state approximations through the lens of dimensional Tikhonov-Fenichel parameters and their respective critical manifolds. By combining Tikhonov-Fenichel parameters with scaling analysis and energy methods, we derive improved upper bounds on the approximation error for the standard, reverse and total quasi-steady-state approximations. Furthermore, previous analyses suggest that the reverse quasi-steady-state approximation is only valid when initial enzyme concentrations greatly exceed initial substrate concentrations. However, our results indicate that this approximation can be valid when initial enzyme and substrate concentrations are of equal magnitude. Using energy methods, we find that the condition for the validity of the reverse quasi-steady-state approximation is far less restrictive than was previously assumed, and we derive a new "small" parameter that determines the validity of this approximation. In doing so, we extend the established domain of validity for the reverse quasi-steady-state approximation. Consequently, this opens up the possibility of utilizing the reverse quasi-steady-state approximation to model enzyme catalyzed reactions and estimate kinetic parameters in enzymatic assays at much lower enzyme to substrate ratios than was previously thought. Moreover, we show for the first time that the critical manifold of the reverse quasi-steady-state approximation contains a singular point where normal hyperbolicity is lost. Associated with this singularity is a transcritical bifurcation, and the corresponding normal form of this bifurcation is recovered through scaling analysis.

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

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

Intrinsic noise, Delta-Notch signalling and delayed reactions promote sustained, coherent, synchronized oscillations in the presomitic mesoderm

Using a stochastic individual-based modelling approach, we examine the role that Delta-Notch signalling plays in the regulation of a robust and reliable somite segmentation clock. We find that not only can Delta-Notch signalling synchronize noisy cycles of gene expression in adjacent cells in the presomitic mesoderm (as is known), but it can also amplify and increase the coherence of these cycles. We examine some of the shortcomings of deterministic approaches to modelling these cycles and demonstrate how intrinsic noise can play an active role in promoting sustained oscillations, giving rise to noise-induced quasi-cycles. Finally, we explore how translational/transcriptional delays can result in the cycles in neighbouring cells oscillating in anti-phase and we study how this effect relates to the propagation of noise-induced stochastic waves.

The Effects of Statistical Multiplicity of Infection on Virus Quantification and Infectivity Assays

Many biological assays are employed in virology to quantify parameters of interest. Two such classes of assays, virus quantification assays (VQAs) and infectivity assays (IAs), aim to estimate the number of viruses present in a solution and the ability of a viral strain to successfully infect a host cell, respectively. VQAs operate at extremely dilute concentrations, and results can be subject to stochastic variability in virus-cell interactions. At the other extreme, high viral-particle concentrations are used in IAs, resulting in large numbers of viruses infecting each cell, enough for measurable change in total transcription activity. Furthermore, host cells can be infected at any concentration regime by multiple particles, resulting in a statistical multiplicity of infection and yielding potentially significant variability in the assay signal and parameter estimates. We develop probabilistic models for statistical multiplicity of infection at low and high viral-particle-concentration limits and apply them to the plaque (VQA), endpoint dilution (VQA), and luciferase reporter (IA) assays. A web-based tool implementing our models and analysis is also developed and presented. We test our proposed new methods for inferring experimental parameters from data using numerical simulations and show improvement on existing procedures in all limits.

Simulating Stochastic Populations. Direct Averaging Methods

A method of directly computing the average behavior of stochastic populations is established, which obviates the time-consuming process of generating detailed sample paths. The method relies on suitably discretized time intervals in which nonlinearities are quasi-linearized to produce random variables with known expectations and variances. The pair of equations is directly solved to obtain the average behavior of the system at the end of a time interval based on its knowledge at the beginning of the interval. The sample path requirement for this process is considerably lower than that for the process over the entire simulation period. The efficiency of the method is demonstrated on the transfer of antibiotics resistance between two bacterial species which is a problem of mounting concern in fighting disease.

Gene Regulatory Networks: A Primer in Biological Processes and Statistical Modelling

Modelling gene regulatory networks requires not only a thorough understanding of the biological system depicted, but also the ability to accurately represent this system from a mathematical perspective. Throughout this chapter, we aim to familiarize the reader with the biological processes and molecular factors at play in the process of gene expression regulation. We first describe the different interactions controlling each step of the expression process, from transcription to mRNA and protein decay. In the second section, we provide statistical tools to accurately represent this biological complexity in the form of mathematical models. Among other considerations, we discuss the topological properties of biological networks, the application of deterministic and stochastic frameworks, and the quantitative modelling of regulation. We particularly focus on the use of such models for the simulation of expression data that can serve as a benchmark for the testing of network inference algorithms.

In Silico Prediction of Large-Scale Microbial Production Performance: Constraints for Getting Proper Data-Driven Models

Industrial bioreactors range from 10.000 to 700.000 L and characteristically show different zones of substrate availabilities, dissolved gas concentrations and pH values reflecting physical, technical and economic constraints of scale-up. Microbial producers are fluctuating inside the bioreactors thereby experiencing frequently changing micro-environmental conditions. The external stimuli induce responses on microbial metabolism and on transcriptional regulation programs. Both may deteriorate the expected microbial production performance in large scale compared to expectations deduced from ideal, well-mixed lab-scale conditions. Accordingly, predictive tools are needed to quantify large-scale impacts considering bioreactor heterogeneities. The review shows that the time is right to combine simulations of microbial kinetics with calculations of large-scale environmental conditions to predict the bioreactor performance. Accordingly, basic experimental procedures and computational tools are presented to derive proper microbial models and hydrodynamic conditions, and to link both for bioreactor modeling. Particular emphasis is laid on the identification of gene regulatory networks as the implementation of such models will surely gain momentum in future studies.

Investigation of Biotransport in a Tumor With Uncertain Material Properties Using a Nonintrusive Spectral Uncertainty Quantification Method

In this study, statistical models are developed for modeling uncertain heterogeneous permeability and porosity in tumors, and the resulting uncertainties in pressure and velocity fields during an intratumoral injection are quantified using a nonintrusive spectral uncertainty quantification (UQ) method. Specifically, the uncertain permeability is modeled as a log-Gaussian random field, represented using a truncated Karhunen-Lòeve (KL) expansion, and the uncertain porosity is modeled as a log-normal random variable. The efficacy of the developed statistical models is validated by simulating the concentration fields with permeability and porosity of different uncertainty levels. The irregularity in the concentration field bears reasonable visual agreement with that in MicroCT images from experiments. The pressure and velocity fields are represented using polynomial chaos (PC) expansions to enable efficient computation of their statistical properties. The coefficients in the PC expansion are computed using a nonintrusive spectral projection method with the Smolyak sparse quadrature. The developed UQ approach is then used to quantify the uncertainties in the random pressure and velocity fields. A global sensitivity analysis is also performed to assess the contribution of individual KL modes of the log-permeability field to the total variance of the pressure field. It is demonstrated that the developed UQ approach can effectively quantify the flow uncertainties induced by uncertain material properties of the tumor.

Quantitative computational models of molecular self-assembly in systems biology

Molecular self-assembly is the dominant form of chemical reaction in living systems, yet efforts at systems biology modeling are only beginning to appreciate the need for and challenges to accurate quantitative modeling of self-assembly. Self-assembly reactions are essential to nearly every important process in cell and molecular biology and handling them is thus a necessary step in building comprehensive models of complex cellular systems. They present exceptional challenges, however, to standard methods for simulating complex systems. While the general systems biology world is just beginning to deal with these challenges, there is an extensive literature dealing with them for more specialized self-assembly modeling. This review will examine the challenges of self-assembly modeling, nascent efforts to deal with these challenges in the systems modeling community, and some of the solutions offered in prior work on self-assembly specifically. The review concludes with some consideration of the likely role of self-assembly in the future of complex biological system models more generally.

A straightforward method to compute average stochastic oscillations from data samples

BACKGROUND

Many biological systems exhibit sustained stochastic oscillations in their steady state. Assessing these oscillations is usually a challenging task due to the potential variability of the amplitude and frequency of the oscillations over time. As a result of this variability, when several stochastic replications are averaged, the oscillations are flattened and can be overlooked. This can easily lead to the erroneous conclusion that the system reaches a constant steady state.

RESULTS

This paper proposes a straightforward method to detect and asses stochastic oscillations. The basis of the method is in the use of polar coordinates for systems with two species, and cylindrical coordinates for systems with more than two species. By slightly modifying these coordinate systems, it is possible to compute the total angular distance run by the system and the average Euclidean distance to a reference point. This allows us to compute confidence intervals, both for the average angular speed and for the distance to a reference point, from a set of replications.

CONCLUSIONS

The use of polar (or cylindrical) coordinates provides a new perspective of the system dynamics. The mean trajectory that can be obtained by averaging the usual cartesian coordinates of the samples informs about the trajectory of the center of mass of the replications. In contrast to such a mean cartesian trajectory, the mean polar trajectory can be used to compute the average circular motion of those replications, and therefore, can yield evidence about sustained steady state oscillations. Both, the coordinate transformation and the computation of confidence intervals, can be carried out efficiently. This results in an efficient method to evaluate stochastic oscillations.

Stochastic modelling of PTEN regulation in brain tumors: A model for glioblastoma multiforme

This work is the outcome of the partnership between the mathematical group of Department DISBEF and the biochemical group of Department DISB of the University of Urbino "Carlo Bo" in order to better understand some crucial aspects of brain cancer oncogenesis. Throughout our collaboration we discovered that biochemists are mainly attracted to the instantaneous behaviour of the whole cell, while mathematicians are mostly interested in the evolution along time of small and different parts of it. This collaboration has thus been very challenging. Starting from [23,24,25], we introduce a competitive stochastic model for post-transcriptional regulation of PTEN, including interactions with the miRNA and concurrent genes. Our model also covers protein formation and the backward mechanism going from the protein back to the miRNA. The numerical simulations show that the model reproduces the expected dynamics of normal glial cells. Moreover, the introduction of translational and transcriptional delays offers some interesting insights for the PTEN low expression as observed in brain tumor cells.

The effects of intrinsic noise on the behaviour of bistable cell regulatory systems under quasi-steady state conditions

We analyse the effect of intrinsic fluctuations on the properties of bistable stochastic systems with time scale separation operating under quasi-steady state conditions. We first formulate a stochastic generalisation of the quasi-steady state approximation based on the semi-classical approximation of the partial differential equation for the generating function associated with the chemical master equation. Such approximation proceeds by optimising an action functional whose associated set of Euler-Lagrange (Hamilton) equations provides the most likely fluctuation path. We show that, under appropriate conditions granting time scale separation, the Hamiltonian can be re-scaled so that the set of Hamilton equations splits up into slow and fast variables, whereby the quasi-steady state approximation can be applied. We analyse two particular examples of systems whose mean-field limit has been shown to exhibit bi-stability: an enzyme-catalysed system of two mutually inhibitory proteins and a gene regulatory circuit with self-activation. Our theory establishes that the number of molecules of the conserved species is order parameters whose variation regulates bistable behaviour in the associated systems beyond the predictions of the mean-field theory. This prediction is fully confirmed by direct numerical simulations using the stochastic simulation algorithm. This result allows us to propose strategies whereby, by varying the number of molecules of the three conserved chemical species, cell properties associated to bistable behaviour (phenotype, cell-cycle status, etc.) can be controlled.

Control of Stochastic and Induced Switching in Biophysical Networks

Noise caused by fluctuations at the molecular level is a fundamental part of intracellular processes. While the response of biological systems to noise has been studied extensively, there has been limited understanding of how to exploit it to induce a desired cell state. Here we present a scalable, quantitative method based on the Freidlin-Wentzell action to predict and control noise-induced switching between different states in genetic networks that, conveniently, can also control transitions between stable states in the absence of noise. We apply this methodology to models of cell differentiation and show how predicted manipulations of tunable factors can induce lineage changes, and further utilize it to identify new candidate strategies for cancer therapy in a cell death pathway model. This framework offers a systems approach to identifying the key factors for rationally manipulating biophysical dynamics, and should also find use in controlling other classes of noisy complex networks.

Emergent properties arising from the assembly of amphiphiles. Artificial vesicle membranes as reaction promoters and regulators

This article deals with artificial vesicles and their membranes as reaction promoters and regulators. Among the various molecular assemblies which can form in an aqueous medium from amphiphilic molecules, vesicle systems are unique. Vesicles compartmentalize the aqueous solution in which they exist, independent on whether the vesicles are biological vesicles (existing in living systems) or whether they are artificial vesicles (formed in vitro from natural or synthetic amphiphiles). After the formation of artificial vesicles, their aqueous interior (the endovesicular volume) may become - or may be made - chemically different from the external medium (the exovesicular solution), depending on how the vesicles are prepared. The existence of differences between endo- and exovesicular composition is one of the features on the basis of which biological vesicles contribute to the complex functioning of living organisms. Furthermore, artificial vesicles can be formed from mixtures of amphiphiles in such a way that the vesicle membranes become molecularly, compositionally and organizationally highly complex, similarly to the lipidic matrix of biological membranes. All the various properties of artificial vesicles as membranous compartment systems emerge from molecular assembly as these properties are not present in the individual molecules the system is composed of. One particular emergent property of vesicle membranes is their possible functioning as promoters and regulators of chemical reactions caused by the localization of reaction components, and possibly catalysts, within or on the surface of the membranes. This specific feature is reviewed and highlighted with a few selected examples which range from the promotion of decarboxylation reactions, the selective binding of DNA or RNA to suitable vesicle membranes, and the reactivation of fragmented enzymes to the regulation of the enzymatic synthesis of polymers. Such type of emergent properties of vesicle membranes may have been important for the prebiological evolution of protocells, the hypothetical compartment systems preceding the first cells in those chemical and physico-chemical processes that led to the origin of life.

Stochastic quasi-steady state approximations for asymptotic solutions of the chemical master equation

In this paper, we propose two methods to carry out the quasi-steady state approximation in stochastic models of enzyme catalytic regulation, based on WKB asymptotics of the chemical master equation or of the corresponding partial differential equation for the generating function. The first of the methods we propose involves the development of multiscale generalisation of a WKB approximation of the solution of the master equation, where the separation of time scales is made explicit which allows us to apply the quasi-steady state approximation in a straightforward manner. To the lowest order, the multi-scale WKB method provides a quasi-steady state, Gaussian approximation of the probability distribution. The second method is based on the Hamilton-Jacobi representation of the stochastic process where, as predicted by large deviation theory, the solution of the partial differential equation for the corresponding characteristic function is given in terms of an effective action functional. The optimal transition paths between two states are then given by those paths that maximise the effective action. Such paths are the solutions of the Hamilton equations for the Hamiltonian associated to the effective action functional. The quasi-steady state approximation is applied to the Hamilton equations thus providing an approximation to the optimal transition paths and the transition time between two states. Using this approximation we predict that, unlike the mean-field quasi-steady approximation result, the rate of enzyme catalysis depends explicitly on the initial number of enzyme molecules. The accuracy and validity of our approximated results as well as that of our predictions regarding the behaviour of the stochastic enzyme catalytic models are verified by direct simulation of the stochastic model using Gillespie stochastic simulation algorithm.

Methodologies for the modeling and simulation of biochemical networks, illustrated for signal transduction pathways: a primer

Biochemical networks depict the chemical interactions that take place among elements of living cells. They aim to elucidate how cellular behavior and functional properties of the cell emerge from the relationships between its components, i.e. molecules. Biochemical networks are largely characterized by dynamic behavior, and exhibit high degrees of complexity. Hence, the interest in such networks is growing and they have been the target of several recent modeling efforts. Signal transduction pathways (STPs) constitute a class of biochemical networks that receive, process, and respond to stimuli from the environment, as well as stimuli that are internal to the organism. An STP consists of a chain of intracellular signaling processes that ultimately result in generating different cellular responses. This primer presents the methodologies used for the modeling and simulation of biochemical networks, illustrated for STPs. These methodologies range from qualitative to quantitative, and include structural as well as dynamic analysis techniques. We describe the different methodologies, outline their underlying assumptions, and provide an assessment of their advantages and disadvantages. Moreover, publicly and/or commercially available implementations of these methodologies are listed as appropriate. In particular, this primer aims to provide a clear introduction and comprehensive coverage of biochemical modeling and simulation methodologies for the non-expert, with specific focus on relevant literature of STPs.

Simulation of anaerobic digestion processes using stochastic algorithm

BACKGROUND

The Anaerobic Digestion (AD) processes involve numerous complex biological and chemical reactions occurring simultaneously. Appropriate and efficient models are to be developed for simulation of anaerobic digestion systems. Although several models have been developed, mostly they suffer from lack of knowledge on constants, complexity and weak generalization. The basis of the deterministic approach for modelling the physico and bio-chemical reactions occurring in the AD system is the law of mass action, which gives the simple relationship between the reaction rates and the species concentrations. The assumptions made in the deterministic models are not hold true for the reactions involving chemical species of low concentration. The stochastic behaviour of the physicochemical processes can be modeled at mesoscopic level by application of the stochastic algorithms.

METHOD

In this paper a stochastic algorithm (Gillespie Tau Leap Method) developed in MATLAB was applied to predict the concentration of glucose, acids and methane formation at different time intervals. By this the performance of the digester system can be controlled. The processes given by ADM1 (Anaerobic Digestion Model 1) were taken for verification of the model.

RESULTS

The proposed model was verified by comparing the results of Gillespie's algorithms with the deterministic solution for conversion of glucose into methane through degraders. At higher value of 'τ' (timestep), the computational time required for reaching the steady state is more since the number of chosen reactions is less. When the simulation time step is reduced, the results are similar to ODE solver.

CONCLUSION

It was concluded that the stochastic algorithm is a suitable approach for the simulation of complex anaerobic digestion processes. The accuracy of the results depends on the optimum selection of tau value.

A stochastic chemical dynamic approach to correlate autoimmunity and optimal vitamin-D range

Motivated by several recent experimental observations that vitamin-D could interact with antigen presenting cells (APCs) and T-lymphocyte cells (T-cells) to promote and to regulate different stages of immune response, we developed a coarse grained but general kinetic model in an attempt to capture the role of vitamin-D in immunomodulatory responses. Our kinetic model, developed using the ideas of chemical network theory, leads to a system of nine coupled equations that we solve both by direct and by stochastic (Gillespie) methods. Both the analyses consistently provide detail information on the dependence of immune response to the variation of critical rate parameters. We find that although vitamin-D plays a negligible role in the initial immune response, it exerts a profound influence in the long term, especially in helping the system to achieve a new, stable steady state. The study explores the role of vitamin-D in preserving an observed bistability in the phase diagram (spanned by system parameters) of immune regulation, thus allowing the response to tolerate a wide range of pathogenic stimulation which could help in resisting autoimmune diseases. We also study how vitamin-D affects the time dependent population of dendritic cells that connect between innate and adaptive immune responses. Variations in dose dependent response of anti-inflammatory and pro-inflammatory T-cell populations to vitamin-D correlate well with recent experimental results. Our kinetic model allows for an estimation of the range of optimum level of vitamin-D required for smooth functioning of the immune system and for control of both hyper-regulation and inflammation. Most importantly, the present study reveals that an overdose or toxic level of vitamin-D or any steroid analogue could give rise to too large a tolerant response, leading to an inefficacy in adaptive immune function.

A computational framework for analyzing stochasticity in gene expression

Stochastic fluctuations in gene expression give rise to distributions of protein levels across cell populations. Despite a mounting number of theoretical models explaining stochasticity in protein expression, we lack a robust, efficient, assumption-free approach for inferring the molecular mechanisms that underlie the shape of protein distributions. Here we propose a method for inferring sets of biochemical rate constants that govern chromatin modification, transcription, translation, and RNA and protein degradation from stochasticity in protein expression. We asked whether the rates of these underlying processes can be estimated accurately from protein expression distributions, in the absence of any limiting assumptions. To do this, we (1) derived analytical solutions for the first four moments of the protein distribution, (2) found that these four moments completely capture the shape of protein distributions, and (3) developed an efficient algorithm for inferring gene expression rate constants from the moments of protein distributions. Using this algorithm we find that most protein distributions are consistent with a large number of different biochemical rate constant sets. Despite this degeneracy, the solution space of rate constants almost always informs on underlying mechanism. For example, we distinguish between regimes where transcriptional bursting occurs from regimes reflecting constitutive transcript production. Our method agrees with the current standard approach, and in the restrictive regime where the standard method operates, also identifies rate constants not previously obtainable. Even without making any assumptions we obtain estimates of individual biochemical rate constants, or meaningful ratios of rate constants, in 91% of tested cases. In some cases our method identified all of the underlying rate constants. The framework developed here will be a powerful tool for deducing the contributions of particular molecular mechanisms to specific patterns of gene expression.

Proteins, the molecular machines encoded by our genes, serve essential roles in every living cell. Investigators were therefore surprised to find widely variable levels of a particular protein within populations of genetically identical cells. This variation in protein level, called stochasticity, arises from the chemical nature of the processes that underlie protein production. The “central dogma” of biology dictates that the DNA encoding a particular gene transmits information via RNA to molecular factories called ribosomes in order to create proteins. Each step in transcription and translation introduces some variation, or stochasticity, into the production of the protein. In the current work, we tackled how one might learn more about the machinery responsible for creating proteins by the character of the stochasticity in the central dogma process. We find that many different mechanisms can explain any given stochastic protein signature. Even though there were many explanations for any particular pattern of stochasticity, the set of explanations still inform on how a given gene creates its protein. Our mathematical and computational framework will permit others to better understand how the machinery that expresses genes works. This, in turn, will enable investigators to better predict how a given mutation is likely to affect gene expression.

A Bayesian hierarchical model for maximizing the vascular adhesion of nanoparticles

The complex vascular dynamics and wall deposition of systemically injected nanoparticles is regulated by their geometrical properties (size, shape) and biophysical parameters (ligand-receptor bond type and surface density, local shear rates). Although sophisticated computational models have been developed to capture the vascular behavior of nanoparticles, it is increasingly recognized that purely deterministic approaches, where the governing parameters are known a priori and conclusively describe behaviors based on physical characteristics, may be too restrictive to accurately reflect natural processes. Here, a novel computational framework is proposed by coupling the physics dictating the vascular adhesion of nanoparticles with a stochastic model. In particular, two governing parameters (i.e. the ligand-receptor bond length and the ligand surface density on the nanoparticle) are treated as two stochastic quantities, whose values are not fixed a priori but would rather range in defined intervals with a certain probability. This approach is used to predict the deposition of spherical nanoparticles with different radii, ranging from 750 to 6,000 nm, in a parallel plate flow chamber under different flow conditions, with a shear rate ranging from 50 to 90 sec^-1. It is demonstrated that the resulting stochastic model can predict the experimental data more accurately than the original deterministic model. This approach allows one to increase the predictive power of mathematical models of any natural process by accounting for the experimental and intrinsic biological uncertainties.

Transition model for ricin-aptamer interactions with multiple pathways and energy barriers

We develop a transition model to interpret single-molecule ricin-aptamer interactions with multiple unbinding pathways and energy barriers measured by atomic force microscopy dynamic force spectroscopy. Molecular simulations establish the relationship between binding conformations and the corresponding unbinding pathways. Each unbinding pathway follows a Bell-Evans multiple-barrier model. Markov-type transition matrices are developed to analyze the redistribution of unbinding events among the pathways under different loading rates. Our study provides detailed information about complex behaviors in ricin-aptamer unbinding events.

Training an asymmetric signal perceptron through reinforcement in an artificial chemistry

State-of-the-art biochemical systems for medical applications and chemical computing are application-specific and cannot be reprogrammed or trained once fabricated. The implementation of adaptive biochemical systems that would offer flexibility through programmability and autonomous adaptation faces major challenges because of the large number of required chemical species as well as the timing-sensitive feedback loops required for learning. In this paper, we begin addressing these challenges with a novel chemical perceptron that can solve all 14 linearly separable logic functions. The system performs asymmetric chemical arithmetic, learns through reinforcement and supports both Michaelis-Menten as well as mass-action kinetics. To enable cascading of the chemical perceptrons, we introduce thresholds that amplify the outputs. The simplicity of our model makes an actual wet implementation, in particular by DNA-strand displacement, possible.

FOUNDATIONS FOR MODELING THE DYNAMICS OF GENE REGULATORY NETWORKS: A MULTILEVEL-PERSPECTIVE REVIEW

A promising alternative for unraveling the principles under which the dynamic interactions among genes lead to cellular phenotypes relies on mathematical and computational models at different levels of abstraction, from the molecular level of protein-DNA interactions to the system level of functional relationships among genes. This review article presents, under a bottom–up perspective, a hierarchy of approaches to modeling gene regulatory network dynamics, from microscopic descriptions at the single-molecule level in the spatial context of an individual cell to macroscopic models providing phenomenological descriptions at the population-average level. The reviewed modeling approaches include Molecular Dynamics, Particle-Based Brownian Dynamics, the Master Equation approach, Ordinary Differential Equations, and the Boolean logic abstraction. Each of these frameworks is motivated by a particular biological context and the nature of the insight being pursued. The setting of gene network dynamic models from such frameworks involves assumptions and mathematical artifacts often ignored by the non-specialist. This article aims at providing an entry point for biologists new to the field and computer scientists not acquainted with some recent biophysically-inspired models of gene regulation. The connections promoting intuition between different abstraction levels and the role that approximations play in the modeling process are highlighted throughout the paper.

Real-time in silico experiments on gene regulatory networks and surgery simulation on handheld devices

Abstract

Simulation of all phenomena taking place in a surgical procedure is a formidable task that involves, when possible, the use of supercomputing facilities over long time periods. However, decision taking in the operating room needs for fast methods that provide an accurate response in real time. To this end, Model Order Reduction (MOR) techniques have emerged recently in the field of Computational Surgery to help alleviate this burden. In this paper, we review the basics of classical MOR and explain how a technique recently developed by the authors and coined as Proper Generalized Decomposition could make real-time feedback available with the use of simple devices like smartphones or tablets. Examples are given on the performance of the technique for problems at different scales of the surgical procedure, form gene regulatory networks to macroscopic soft tissue deformation and cutting.

Parameter discovery in stochastic biological models using simulated annealing and statistical model checking

Stochastic models are increasingly used to study the behaviour of biochemical systems. While the structure of such models is often readily available from first principles, unknown quantitative features of the model are incorporated into the model as parameters. Algorithmic discovery of parameter values from experimentally observed facts remains a challenge for the computational systems biology community. We present a new parameter discovery algorithm that uses simulated annealing, sequential hypothesis testing, and statistical model checking to learn the parameters in a stochastic model. We apply our technique to a model of glucose and insulin metabolism used for in-silico validation of artificial pancreata and demonstrate its effectiveness by developing parallel CUDA-based implementation for parameter synthesis in this model.

Mosaic physiology from developmental noise: within-organism physiological diversity as an alternative to phenotypic plasticity and phenotypic flexibility

A key problem in organismal biology is to explain the origins of functional diversity. In the context of organismal biology, functional diversity describes the set of phenotypes, across scales of biological organization and through time, that a single genotype, or genome, or organism, can produce. Functional diversity encompasses many phenomena: differences in cell types within organisms; physiological and morphological differences among tissues and organs; differences in performance; morphological shifts in external phenotype; and changes in behavior. How can single genomes produce so many different phenotypes? Modern biology proposes two general mechanisms. The first is developmental programs, by which single cells and their single genomes diversify, via relatively deterministic processes, into the sets of cell types, tissues and organs that we see in most multicellular organisms. The second general mechanism is phenotypic modification stemming from interactions between organisms and their environments – modifications known either as phenotypic plasticity or as phenotypic flexibility, depending on the time scale of the response and the degree of reversibility. These two diversity-generating mechanisms are related because phenotypic modifications may sometimes arise as a consequence of environments influencing developmental programs. Here, I propose that functional diversity also arises via a third fundamental mechanism: stochastic developmental events giving rise to mosaics of physiological diversity within individual organisms. In biological systems, stochasticity stems from the inherently random actions of small numbers of molecules interacting with one another. Although stochastic effects occur in many biological contexts, available evidence suggests that they can be especially important in gene networks, specifically as a consequence of low transcript numbers in individual cells. I briefly review known mechanisms by which organisms control such stochasticity, and how they may use it to create adaptive functional diversity. I then fold this idea into modern thinking on phenotypic plasticity and flexibility, proposing that multicellular organisms exhibit ‘mosaic physiology’. Mosaic physiology refers to sets of diversified phenotypes, within individual organisms, that carry out related functions at the same time, but that are distributed in space. Mosaic physiology arises from stochasticity-driven differentiation of cells, early during cell diversification, which is then amplified by cell division and growth into macroscopic phenotypic modules (cells, tissues, organs) making up the physiological systems of later life stages. Mosaic physiology provides a set of standing, diversified phenotypes, within single organisms, that raise the likelihood of the organism coping well with novel environmental challenges. These diversified phenotypes can be distinct, akin to polyphenisms at the organismal level; or they can be continuously distributed, creating a kind of standing, simultaneously expressed reaction norm of physiological capacities.

Mathematical modeling of biochemical systems with PottersWheel

The program PottersWheel has been developed to provide an intuitive and yet powerful framework for data-based modeling of dynamical systems like biochemical reaction networks. Its key functionality is multi-experiment fitting, where several experimental data sets from different laboratory conditions are fitted simultaneously in order to improve the estimation of unknown model parameters, to check the validity of a given model, and to discriminate competing model hypotheses. New experiments can be designed interactively. Models are either created text-based or using a visual model designer. Dynamically generated and compiled C files provide fast simulation and fitting procedures. Each function can either be accessed using a graphical user interface or via command line, allowing for batch processing within custom Matlab scripts. PottersWheel is designed as a Matlab toolbox, comprises 250,000 lines of Matlab and C code, and is freely available for academic usage at www.potterswheel.de .

Converting differential-equation models of biological systems to membrane computing

This paper presents a method to convert the deterministic, continuous representation of a biological system by ordinary differential equations into a non-deterministic, discrete membrane computation. The dynamics of the membrane computation is governed by rewrite rules operating at certain rates. That has the advantage of applying accurately to small systems, and to expressing rates of change that are determined locally, by region, but not necessary globally. Such spatial information augments the standard differentiable approach to provide a more realistic model. A biological case study of the ligand-receptor network of protein TGF-β is used to validate the effectiveness of the conversion method. It demonstrates the sense in which the behaviours and properties of the system are better preserved in the membrane computing model, suggesting that the proposed conversion method may prove useful for biological systems in particular.

Comparison of models for IP3 receptor kinetics using stochastic simulations

Inositol 1,4,5-trisphosphate receptor (IP₃R) is a ubiquitous intracellular calcium (Ca²⁺) channel which has a major role in controlling Ca²⁺ levels in neurons. A variety of computational models have been developed to describe the kinetic function of IP₃R under different conditions. In the field of computational neuroscience, it is of great interest to apply the existing models of IP₃R when modeling local Ca²⁺ transients in dendrites or overall Ca²⁺ dynamics in large neuronal models. The goal of this study was to evaluate existing IP₃R models, based on electrophysiological data. This was done in order to be able to suggest suitable models for neuronal modeling. Altogether four models (Othmer and Tang, 1993; Dawson etal., 2003; Fraiman and Dawson, 2004; Doi etal., 2005) were selected for a more detailed comparison. The selection was based on the computational efficiency of the models and the type of experimental data that was used in developing the model. The kinetics of all four models were simulated by stochastic means, using the simulation software STEPS, which implements the Gillespie stochastic simulation algorithm. The results show major differences in the statistical properties of model functionality. Of the four compared models, the one by Fraiman and Dawson (2004) proved most satisfactory in producing the specific features of experimental findings reported in literature. To our knowledge, the present study is the first detailed evaluation of IP₃R models using stochastic simulation methods, thus providing an important setting for constructing a new, realistic model of IP₃R channel kinetics for compartmental modeling of neuronal functions. We conclude that the kinetics of IP₃R with different concentrations of Ca²⁺ and IP₃ should be more carefully addressed when new models for IP₃R are developed.

Logic-based models in systems biology: a predictive and parameter-free network analysis method

Highly complex molecular networks, which play fundamental roles in almost all cellular processes, are known to be dysregulated in a number of diseases, most notably in cancer. As a consequence, there is a critical need to develop practical methodologies for constructing and analysing molecular networks at a systems level. Mathematical models built with continuous differential equations are an ideal methodology because they can provide a detailed picture of a network's dynamics. To be predictive, however, differential equation models require that numerous parameters be known a priori and this information is almost never available. An alternative dynamical approach is the use of discrete logic-based models that can provide a good approximation of the qualitative behaviour of a biochemical system without the burden of a large parameter space. Despite their advantages, there remains significant resistance to the use of logic-based models in biology. Here, we address some common concerns and provide a brief tutorial on the use of logic-based models, which we motivate with biological examples.

Online learning in a chemical perceptron

Autonomous learning implemented purely by means of a synthetic chemical system has not been previously realized. Learning promotes reusability and minimizes the system design to simple input-output specification. In this article we introduce a chemical perceptron, the first full-featured implementation of a perceptron in an artificial (simulated) chemistry. A perceptron is the simplest system capable of learning, inspired by the functioning of a biological neuron. Our artificial chemistry is deterministic and discrete-time, and follows Michaelis-Menten kinetics. We present two models, the weight-loop perceptron and the weight-race perceptron, which represent two possible strategies for a chemical implementation of linear integration and threshold. Both chemical perceptrons can successfully identify all 14 linearly separable two-input logic functions and maintain high robustness against rate-constant perturbations. We suggest that DNA strand displacement could, in principle, provide an implementation substrate for our model, allowing the chemical perceptron to perform reusable, programmable, and adaptable wet biochemical computing.

Modeling cellular compartmentation in one-carbon metabolism

Folate-mediated one-carbon metabolism (FOCM) is associated with risk for numerous pathological states including birth defects, cancers, and chronic diseases. Although the enzymes that constitute the biological pathways have been well described and their interdependency through the shared use of folate cofactors appreciated, the biological mechanisms underlying disease etiologies remain elusive. The FOCM network is highly sensitive to nutritional status of several B-vitamins and numerous penetrant gene variants that alter network outputs, but current computational approaches do not fully capture the dynamics and stochastic noise of the system. Combining the stochastic approach with a rule-based representation will help model the intrinsic noise displayed by FOCM, address the limited flexibility of standard simulation methods for coarse-graining the FOCM-associated biochemical processes, and manage the combinatorial complexity emerging from reactions within FOCM that would otherwise be intractable.

Discrete-state stochastic models of calcium-regulated calcium influx and subspace dynamics are not well-approximated by ODEs that neglect concentration fluctuations

Cardiac myocyte calcium signaling is often modeled using deterministic ordinary differential equations (ODEs) and mass-action kinetics. However, spatially restricted "domains" associated with calcium influx are small enough (e.g., 10(-17) liters) that local signaling may involve 1-100 calcium ions. Is it appropriate to model the dynamics of subspace calcium using deterministic ODEs or, alternatively, do we require stochastic descriptions that account for the fundamentally discrete nature of these local calcium signals? To address this question, we constructed a minimal Markov model of a calcium-regulated calcium channel and associated subspace. We compared the expected value of fluctuating subspace calcium concentration (a result that accounts for the small subspace volume) with the corresponding deterministic model (an approximation that assumes large system size). When subspace calcium did not regulate calcium influx, the deterministic and stochastic descriptions agreed. However, when calcium binding altered channel activity in the model, the continuous deterministic description often deviated significantly from the discrete stochastic model, unless the subspace volume is unrealistically large and/or the kinetics of the calcium binding are sufficiently fast. This principle was also demonstrated using a physiologically realistic model of calmodulin regulation of L-type calcium channels introduced by Yue and coworkers.

Accelerating the Gillespie Exact Stochastic Simulation Algorithm using hybrid parallel execution on graphics processing units

The Gillespie Stochastic Simulation Algorithm (GSSA) and its variants are cornerstone techniques to simulate reaction kinetics in situations where the concentration of the reactant is too low to allow deterministic techniques such as differential equations. The inherent limitations of the GSSA include the time required for executing a single run and the need for multiple runs for parameter sweep exercises due to the stochastic nature of the simulation. Even very efficient variants of GSSA are prohibitively expensive to compute and perform parameter sweeps. Here we present a novel variant of the exact GSSA that is amenable to acceleration by using graphics processing units (GPUs). We parallelize the execution of a single realization across threads in a warp (fine-grained parallelism). A warp is a collection of threads that are executed synchronously on a single multi-processor. Warps executing in parallel on different multi-processors (coarse-grained parallelism) simultaneously generate multiple trajectories. Novel data-structures and algorithms reduce memory traffic, which is the bottleneck in computing the GSSA. Our benchmarks show an 8×−120× performance gain over various state-of-the-art serial algorithms when simulating different types of models.

Mathematical simulation of membrane protein clustering for efficient signal transduction

Initiation and propagation of cell signaling depend on productive interactions among signaling proteins at the plasma membrane. These diffusion-limited interactions can be influenced by features of the membrane that introduce barriers, such as cytoskeletal corrals, or microdomains that transiently confine both transmembrane receptors and membrane-tethered peripheral proteins. Membrane topographical features can lead to clustering of receptors and other membrane components, even under very dynamic conditions. This review considers the experimental and mathematical evidence that protein clustering impacts cell signaling in complex ways. Simulation approaches used to consider these stochastic processes are discussed.

A computational approach to increase time scales in Brownian dynamics-based reaction-diffusion modeling

Particle-based Brownian dynamics simulations offer the opportunity to not only simulate diffusion of particles but also the reactions between them. They therefore provide an opportunity to integrate varied biological data into spatially explicit models of biological processes, such as signal transduction or mitosis. However, particle based reaction-diffusion methods often are hampered by the relatively small time step needed for accurate description of the reaction-diffusion framework. Such small time steps often prevent simulation times that are relevant for biological processes. It is therefore of great importance to develop reaction-diffusion methods that tolerate larger time steps while maintaining relatively high accuracy. Here, we provide an algorithm, which detects potential particle collisions prior to a BD-based particle displacement and at the same time rigorously obeys the detailed balance rule of equilibrium reactions. We can show that for reaction-diffusion processes of particles mimicking proteins, the method can increase the typical BD time step by an order of magnitude while maintaining similar accuracy in the reaction diffusion modelling.

Towards uncertainty quantification and inference in the stochastic SIR epidemic model

In this paper we address the problem of estimating the parameters of Markov jump processes modeling epidemics and introduce a novel method to conduct inference when data consists on partial observations in one of the state variables. We take the classical stochastic SIR model as a case study. Using the inverse-size expansion of van Kampen we obtain approximations for the first and second moments of the state variables. These approximate moments are in turn matched to the moments of an inputed Generic Discrete distribution aimed at generating an approximate likelihood that is valid both for low count or high count data. We conduct a full Bayesian inference using informative priors. Estimations and predictions are obtained both in a synthetic data scenario and in two Dengue fever case studies.

A paradox of state-dependent diffusion and how to resolve it

Consider a particle diffusing in a confined volume which is divided into two equal regions. In one region, the diffusion coefficient is twice the value of the diffusion coefficient in the other region. Will the particle spend equal proportions of time in the two regions in the long term? Statistical mechanics would suggest yes, since the number of accessible states in each region is presumably the same. However, another line of reasoning suggests that the particle should spend less time in the region with faster diffusion, since it will exit that region more quickly. We demonstrate with a simple microscopic model system that both predictions are consistent with the information given. Thus, specifying the diffusion rate as a function of position is not enough to characterize the behaviour of a system, even assuming the absence of external forces. We propose an alternative framework for modelling diffusive dynamics in which both the diffusion rate and equilibrium probability density for the position of the particle are specified by the modeller. We introduce a numerical method for simulating dynamics in our framework that samples from the equilibrium probability density exactly and is suitable for discontinuous diffusion coefficients.