Bridging the gap between stochastic and deterministic regimes in the kinetic simulations of the biochemical reaction networks

Analysis and design of antithetic proportional-integral-derivative biocontrol-systems with species dilution

The nonlinearity and non-separability of the antithetic PID (aPID) controller have provided greater flexibility in the design of biochemical reaction networks (BCRNs), resulting in significant impacts on biocontrol-systems. Nevertheless, the dilution of control species is disregarded in designs of aPID controllers, which would lead to the failure of inhibition mechanism in the controller and loss of robust perfect adaptation (RPA)-the biological counterpart of robust steady-state tracking. Here, the impact of dilution processes on the structure of aPID is investigated in this study. It is discovered that the proportional and low-pass filters are altered when the dilution processes is present in control species, which increases the coupling between the controller parameters. Moreover, additional integrations for the reference signal and control output generated by control species dilution further leads to the loss of RPA. Subsequently, a novel aPID controller represented by BCRNs, termed quasi-aPID, has been designed to eliminate the detrimental effects of the dilution processes. In an effort to ameliorate the interdependencies among controller parameters, a degradation inhibition mechanism is employed within this controller. Furthermore, this work establishes the limiting relationship between the controller's reaction rates in order to guarantee RPA, while abstaining from the introduction of supplementary species and biochemical reactions. By using the quasi-aPID controller in both the Escherichia coli gene expression model and the whole-body cholesterol metabolism model, its effectiveness is confirmed. Simulation results demonstrate that, the quasi-aPID exhibits a smaller absolute steady-state error in both models and guarantees the RPA property.

Conversion of fat to cellular fuel-Fatty acids β-oxidation model

β-oxidation of fatty acids plays a significant role in the energy metabolism of the cell. This paper presents a β-oxidation model of fatty acids based on queueing theory. It uses Michaelis-Menten enzyme kinetics, and literature data on metabolites' concentration and enzymatic constants. A genetic algorithm was used to optimize the parameters for the pathway reactions. The model enables real-time tracking of changes in the concentrations of metabolites with different carbon chain lengths. Another application of the presented model is to predict the changes caused by system disturbance, such as altered enzyme activity or abnormal fatty acid concentration. The model has been validated against experimental data. There are diseases that change the metabolism of fatty acids and the presented model can be used to understand the cause of these changes, analyze metabolites abnormalities, and determine the initial target of treatment.

Nets-within-nets for modeling emergent patterns in ontogenetic processes

Ontogenesis is the development of an organism from its earliest stage to maturity, including homeostasis maintenance throughout adulthood despite environmental perturbations. Almost all cells of a multicellular organism share the same genomic information. Nevertheless, phenotypic diversity and complex supra-cellular architectures emerge at every level, starting from tissues and organs. This is possible thanks to a robust and dynamic interplay of regulative mechanisms. To study ontogenesis, it is necessary to consider different levels of regulation, both genetic and epigenetic. Each cell undergoes a specific path across a landscape of possible regulative states affecting both its structure and its functions during development. This paper proposes using the Nets-Within-Nets formalism, which combines Petri Nets' simplicity with the capability to represent and simulate the interplay between different layers of regulation connected by non-trivial and context-dependent hierarchical relations. In particular, this work introduces a modeling strategy based on Nets-Within-Nets that can model several critical processes involved in ontogenesis. Moreover, it presents a case study focusing on the first phase of Vulval Precursor Cells specification in C.Elegans. The case study shows that the proposed model can simulate the emergent morphogenetic pattern corresponding to the observed developmental outcome of that phase, in both the physiological case and different mutations. The model presented in the results section is available online at https://github.com/sysbio-polito/NWN_CElegans_VPC_model/.

Cell-to-Cell Heterogeneity in Trypanosomes

Recent developments in single-cell and single-molecule techniques have revealed surprising levels of heterogeneity among isogenic cells. These advances have transformed the study of cell-to-cell heterogeneity into a major area of biomedical research, revealing that it can confer essential advantages, such as priming populations of unicellular organisms for future environmental stresses. Protozoan parasites, such as trypanosomes, face multiple and often hostile environments, and to survive, they undergo multiple changes, including changes in morphology, gene expression, and metabolism. But why does only a subset of proliferative cells differentiate to the next life cycle stage? Why do only some bloodstream parasites undergo antigenic switching while others stably express one variant surface glycoprotein? And why do some parasites invade an organ while others remain in the bloodstream? Building on extensive research performed in bacteria, here we suggest that biological noise can contribute to the fitness of eukaryotic pathogens and discuss the importance of cell-to-cell heterogeneity in trypanosome infections. Expected final online publication date for the Annual Review of Microbiology, Volume 75 is October 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Analytic solutions for stochastic hybrid models of gene regulatory networks

Discrete-state stochastic models are a popular approach to describe the inherent stochasticity of gene expression in single cells. The analysis of such models is hindered by the fact that the underlying discrete state space is extremely large. Therefore hybrid models, in which protein counts are replaced by average protein concentrations, have become a popular alternative. The evolution of the corresponding probability density functions is given by a coupled system of hyperbolic PDEs. This system has Markovian nature but its hyperbolic structure makes it difficult to apply standard functional analytical methods. We are able to prove convergence towards the stationary solution and determine such equilibrium explicitly by combining abstract methods from the theory of positive operators and elementary ideas from potential analysis.

Computation of Single-Cell Metabolite Distributions Using Mixture Models

Metabolic heterogeneity is widely recognized as the next challenge in our understanding of non-genetic variation. A growing body of evidence suggests that metabolic heterogeneity may result from the inherent stochasticity of intracellular events. However, metabolism has been traditionally viewed as a purely deterministic process, on the basis that highly abundant metabolites tend to filter out stochastic phenomena. Here we bridge this gap with a general method for prediction of metabolite distributions across single cells. By exploiting the separation of time scales between enzyme expression and enzyme kinetics, our method produces estimates for metabolite distributions without the lengthy stochastic simulations that would be typically required for large metabolic models. The metabolite distributions take the form of Gaussian mixture models that are directly computable from single-cell expression data and standard deterministic models for metabolic pathways. The proposed mixture models provide a systematic method to predict the impact of biochemical parameters on metabolite distributions. Our method lays the groundwork for identifying the molecular processes that shape metabolic heterogeneity and its functional implications in disease.

Stochastic Simulation of Cellular Metabolism

Increased technological methods have enabled the investigation of biology at nanoscale levels. Such systems require the use of computational methods to comprehend the complex interactions that occur. The dynamics of metabolic systems have been traditionally described utilizing differential equations without fully capturing the heterogeneity of biological systems. Stochastic modeling approaches have recently emerged with the capacity to incorporate the statistical properties of such systems. However, the processing of stochastic algorithms is a computationally intensive task with intrinsic limitations. Alternatively, the queueing theory approach, historically used in the evaluation of telecommunication networks, can significantly reduce the computational power required to generate simulated results while simultaneously reducing the expansion of errors. We present here the application of queueing theory to simulate stochastic metabolic networks with high efficiency. With the use of glycolysis as a well understood biological model, we demonstrate the power of the proposed modeling methods discussed herein. Furthermore, we describe the simulation and pharmacological inhibition of glycolysis to provide an example of modeling capabilities.

Finite state projection for approximating the stationary solution to the chemical master equation using reaction rate equations

When modeling a physical system using a Markov chain, it is often instructive to compute its probability distribution at statistical equilibrium, thereby gaining insight into the stationary, or long-term, behavior of the system. Computing such a distribution directly is problematic when the state space of the system is large. Here, we look at the case of a chemical reaction system that models the dynamics of cellular processes, where it has become popular to constrain the computational burden by using a finite state projection, which aims only to capture the most likely states of the system, rather than every possible state. We propose an efficient method to further narrow these states to those that remain highly probable in the long run, after the transient behavior of the system has dissipated. Our strategy is to quickly estimate the local maxima of the stationary distribution using the reaction rate formulation, which is of considerably smaller size than the full-blown chemical master equation, and from there develop adaptive schemes to profile the distribution around the maxima. We include numerical tests that show the efficiency of our approach.

Computational modelling folate metabolism and DNA methylation: implications for understanding health and ageing

Dietary folates have a key role to play in health, as deficiencies in the intake of these B vitamins have been implicated in a wide variety of clinical conditions. The reason for this is folates function as single carbon donors in the synthesis of methionine and nucleotides. Moreover, folates have a vital role to play in the epigenetics of mammalian cells by supplying methyl groups for DNA methylation reactions. Intriguingly, a growing body of experimental evidence suggests that DNA methylation status could be a central modulator of the ageing process. This has important health implications because the methylation status of the human genome could be used to infer age-related disease risk. Thus, it is imperative we further our understanding of the processes which underpin DNA methylation and how these intersect with folate metabolism and ageing. The biochemical and molecular mechanisms, which underpin these processes, are complex. However, computational modelling offers an ideal framework for handling this complexity. A number of computational models have been assembled over the years, but to date, no model has represented the full scope of the interaction between the folate cycle and the reactions, which governs the DNA methylation cycle. In this review, we will discuss several of the models, which have been developed to represent these systems. In addition, we will present a rationale for developing a combined model of folate metabolism and the DNA methylation cycle.

Stochastic Simulation of Pattern Formation in Growing Tissue: A Multilevel Approach

We take up the challenge of designing realistic computational models of large interacting cell populations. The goal is essentially to bring Gillespie's celebrated stochastic methodology to the level of an interacting population of cells. Specifically, we are interested in how the gold standard of single-cell computational modeling, here taken to be spatial stochastic reaction-diffusion models, may be efficiently coupled with a similar approach at the cell population level. Concretely, we target a recently proposed set of pathways for pattern formation involving Notch-Delta signaling mechanisms. These involve cell-to-cell communication as mediated both via direct membrane contact sites and via cellular protrusions. We explain how to simulate the process in growing tissue using a multilevel approach and we discuss implications for future development of the associated computational methods.

Kinetic models of hematopoietic differentiation

As cell and molecular biology is becoming increasingly quantitative, there is an upsurge of interest in mechanistic modeling at different levels of resolution. Such models mostly concern kinetics and include gene and protein interactions as well as cell population dynamics. The final goal of these models is to provide experimental predictions, which is now taking on. However, even without matured predictions, kinetic models serve the purpose of compressing a plurality of experimental results into something that can empower the data interpretation, and importantly, suggesting new experiments by turning "knobs" in silico. Once formulated, kinetic models can be executed in terms of molecular rate equations for concentrations or by stochastic simulations when only a limited number of copies are involved. Developmental processes, in particular those of stem and progenitor cell commitments, are not only topical but also particularly suitable for kinetic modeling due to the finite number of key genes involved in cellular decisions. Stem and progenitor cell commitment processes have been subject to intense experimental studies over the last decade with some emphasis on embryonic and hematopoietic stem cells. Gene and protein interactions governing these processes can be modeled by binary Boolean rules or by continuous-valued models with interactions set by binding strengths. Conceptual insights along with tested predictions have emerged from such kinetic models. Here we review kinetic modeling efforts applied to stem cell developmental systems with focus on hematopoiesis. We highlight the future challenges including multi-scale models integrating cell dynamical and transcriptional models. This article is categorized under: Models of Systems Properties and Processes > Mechanistic Models Developmental Biology > Stem Cell Biology and Regeneration.

Network Analyses in Plant Pathogens

Even in the age of big data in Biology, studying the connections between the biological processes and the molecular mechanisms behind them is a challenging task. Systems biology arose as a transversal discipline between biology, chemistry, computer science, mathematics, and physics to facilitate the elucidation of such connections. A scenario, where the application of systems biology constitutes a very powerful tool, is the study of interactions between hosts and pathogens using network approaches. Interactions between pathogenic bacteria and their hosts, both in agricultural and human health contexts are of great interest to researchers worldwide. Large amounts of data have been generated in the last few years within this area of research. However, studies have been relatively limited to simple interactions. This has left great amounts of data that remain to be utilized. Here, we review the main techniques in network analysis and their complementary experimental assays used to investigate bacterial-plant interactions. Other host-pathogen interactions are presented in those cases where few or no examples of plant pathogens exist. Furthermore, we present key results that have been obtained with these techniques and how these can help in the design of new strategies to control bacterial pathogens. The review comprises metabolic simulation, protein-protein interactions, regulatory control of gene expression, host-pathogen modeling, and genome evolution in bacteria. The aim of this review is to offer scientists working on plant-pathogen interactions basic concepts around network biology, as well as an array of techniques that will be useful for a better and more complete interpretation of their data.

MUFINS: multi-formalism interaction network simulator

Systems Biology has established numerous approaches for mechanistic modeling of molecular networks in the cell and a legacy of models. The current frontier is the integration of models expressed in different formalisms to address the multi-scale biological system organization challenge. We present MUFINS (MUlti-Formalism Interaction Network Simulator) software, implementing a unique set of approaches for multi-formalism simulation of interaction networks. We extend the constraint-based modeling (CBM) framework by incorporation of linear inhibition constraints, enabling for the first time linear modeling of networks simultaneously describing gene regulation, signaling and whole-cell metabolism at steady state. We present a use case where a logical hypergraph model of a regulatory network is expressed by linear constraints and integrated with a Genome-Scale Metabolic Network (GSMN) of mouse macrophage. We experimentally validate predictions, demonstrating application of our software in an iterative cycle of hypothesis generation, validation and model refinement. MUFINS incorporates an extended version of our Quasi-Steady State Petri Net approach to integrate dynamic models with CBM, which we demonstrate through a dynamic model of cortisol signaling integrated with the human Recon2 GSMN and a model of nutrient dynamics in physiological compartments. Finally, we implement a number of methods for deriving metabolic states from ~omics data, including our new variant of the iMAT congruency approach. We compare our approach with iMAT through the analysis of 262 individual tumor transcriptomes, recovering features of metabolic reprogramming in cancer. The software provides graphics user interface with network visualization, which facilitates use by researchers who are not experienced in coding and mathematical modeling environments.

Mathematical Modeling and Dynamic Simulation of Metabolic Reaction Systems Using Metabolome Time Series Data

The high-throughput acquisition of metabolome data is greatly anticipated for the complete understanding of cellular metabolism in living organisms. A variety of analytical technologies have been developed to acquire large-scale metabolic profiles under different biological or environmental conditions. Time series data are useful for predicting the most likely metabolic pathways because they provide important information regarding the accumulation of metabolites, which implies causal relationships in the metabolic reaction network. Considerable effort has been undertaken to utilize these data for constructing a mathematical model merging system properties and quantitatively characterizing a whole metabolic system in toto. However, there are technical difficulties between benchmarking the provision and utilization of data. Although, hundreds of metabolites can be measured, which provide information on the metabolic reaction system, simultaneous measurement of thousands of metabolites is still challenging. In addition, it is nontrivial to logically predict the dynamic behaviors of unmeasurable metabolite concentrations without sufficient information on the metabolic reaction network. Yet, consolidating the advantages of advancements in both metabolomics and mathematical modeling remain to be accomplished. This review outlines the conceptual basis of and recent advances in technologies in both the research fields. It also highlights the potential for constructing a large-scale mathematical model by estimating model parameters from time series metabolome data in order to comprehensively understand metabolism at the systems level.

Synthetic biology and regulatory networks: where metabolic systems biology meets control engineering

Metabolic pathways can be engineered to maximize the synthesis of various products of interest. With the advent of computational systems biology, this endeavour is usually carried out through in silico theoretical studies with the aim to guide and complement further in vitro and in vivo experimental efforts. Clearly, what counts is the result in vivo, not only in terms of maximal productivity but also robustness against environmental perturbations. Engineering an organism towards an increased production flux, however, often compromises that robustness. In this contribution, we review and investigate how various analytical approaches used in metabolic engineering and synthetic biology are related to concepts developed by systems and control engineering. While trade-offs between production optimality and cellular robustness have already been studied diagnostically and statically, the dynamics also matter. Integration of the dynamic design aspects of control engineering with the more diagnostic aspects of metabolic, hierarchical control and regulation analysis is leading to the new, conceptual and operational framework required for the design of robust and productive dynamic pathways.

Discrete-continuous reaction-diffusion model with mobile point-like sources and sinks

In many applications in soft and biological physics, there are multiple time and length scales involved but often with a distinct separation between them. For instance, in enzyme kinetics, enzymes are relatively large, move slowly and their copy numbers are typically small, while the metabolites (being transformed by these enzymes) are often present in abundance, are small in size and diffuse fast. It seems thus natural to apply different techniques to different time and length levels and couple them. Here we explore this possibility by constructing a stochastic-deterministic discrete-continuous reaction-diffusion model with mobile sources and sinks. Such an approach allows in particular to separate different sources of stochasticity. We demonstrate its application by modelling enzyme-catalysed reactions with freely diffusing enzymes and a heterogeneous source of metabolites. Our calculations suggest that using a higher amount of less active enzymes, as compared to fewer more active enzymes, reduces the metabolite pool size and correspondingly the lag time, giving rise to a faster response to external stimuli. The methodology presented can be extended to more complex systems and offers exciting possibilities for studying problems where spatial heterogeneities, stochasticity or discreteness play a role.

Noise propagation in synthetic gene circuits for metabolic control

Dynamic control of enzyme expression can be an effective strategy to engineer robust metabolic pathways. It allows a synthetic pathway to self-regulate in response to changes in bioreactor conditions or the metabolic state of the host. The implementation of this regulatory strategy requires gene circuits that couple metabolic signals with the genetic machinery, which is known to be noisy and one of the main sources of cell-to-cell variability. One of the unexplored design aspects of these circuits is the propagation of biochemical noise between enzyme expression and pathway activity. In this article, we quantify the impact of a synthetic feedback circuit on the noise in a metabolic product in order to propose design criteria to reduce cell-to-cell variability. We consider a stochastic model of a catalytic reaction under negative feedback from the product to enzyme expression. On the basis of stochastic simulations and analysis, we show that, depending on the repression strength and promoter strength, transcriptional repression of enzyme expression can amplify or attenuate the noise in the number of product molecules. We obtain analytic estimates for the metabolic noise as a function of the model parameters and show that noise amplification/attenuation is a structural property of the model. We derive an analytic condition on the parameters that lead to attenuation of metabolic noise, suggesting that a higher promoter sensitivity enlarges the parameter design space. In the theoretical case of a switch-like promoter, our analysis reveals that the ability of the circuit to attenuate noise is subject to a trade-off between the repression strength and promoter strength.

Direct solution of the Chemical Master Equation using quantized tensor trains

The Chemical Master Equation (CME) is a cornerstone of stochastic analysis and simulation of models of biochemical reaction networks. Yet direct solutions of the CME have remained elusive. Although several approaches overcome the infinite dimensional nature of the CME through projections or other means, a common feature of proposed approaches is their susceptibility to the curse of dimensionality, i.e. the exponential growth in memory and computational requirements in the number of problem dimensions. We present a novel approach that has the potential to “lift” this curse of dimensionality. The approach is based on the use of the recently proposed Quantized Tensor Train (QTT) formatted numerical linear algebra for the low parametric, numerical representation of tensors. The QTT decomposition admits both, algorithms for basic tensor arithmetics with complexity scaling linearly in the dimension (number of species) and sub-linearly in the mode size (maximum copy number), and a numerical tensor rounding procedure which is stable and quasi-optimal. We show how the CME can be represented in QTT format, then use the exponentially-converging -discontinuous Galerkin discretization in time to reduce the CME evolution problem to a set of QTT-structured linear equations to be solved at each time step using an algorithm based on Density Matrix Renormalization Group (DMRG) methods from quantum chemistry. Our method automatically adapts the “basis” of the solution at every time step guaranteeing that it is large enough to capture the dynamics of interest but no larger than necessary, as this would increase the computational complexity. Our approach is demonstrated by applying it to three different examples from systems biology: independent birth-death process, an example of enzymatic futile cycle, and a stochastic switch model. The numerical results on these examples demonstrate that the proposed QTT method achieves dramatic speedups and several orders of magnitude storage savings over direct approaches.

Stochastic models of chemical networks are necessary to quantitatively describe random fluctuations and other probabilistic phenomena within living cells. The Chemical Master Equation (CME) describes the time evolution of molecular abundance probabilities in these models, and is a basis for many stochastic simulation and analysis methods. Yet the CME is difficult to solve directly except for very simple structures. Indeed current approaches are susceptible to the curse of dimensionality, that is, the exponential growth of memory and computational requirements in the number of problem dimensions. In this paper, we propose a novel approach that has the potential to overcome the curse of dimensionality. It is based on the use of the recently proposed Quantized Tensor Train (QTT) formatted numerical linear algebra for numerical representation of tensors, using algorithms for basic tensor arithmetics with complexity scaling linearly in the number of reacting species considered, and sub-linearly in the maximum allowed copy number per species. We present this approach and demonstrate its effectiveness by applying it to three problems from systems biology. Numerical experiments are reported which show that several orders of magnitude memory savings is typically afforded by the new approach presented here.

QSSPN: dynamic simulation of molecular interaction networks describing gene regulation, signalling and whole-cell metabolism in human cells

MOTIVATION

Dynamic simulation of genome-scale molecular interaction networks will enable the mechanistic prediction of genotype-phenotype relationships. Despite advances in quantitative biology, full parameterization of whole-cell models is not yet possible. Simulation methods capable of using available qualitative data are required to develop dynamic whole-cell models through an iterative process of modelling and experimental validation.

RESULTS

We formulate quasi-steady state Petri nets (QSSPN), a novel method integrating Petri nets and constraint-based analysis to predict the feasibility of qualitative dynamic behaviours in qualitative models of gene regulation, signalling and whole-cell metabolism. We present the first dynamic simulations including regulatory mechanisms and a genome-scale metabolic network in human cell, using bile acid homeostasis in human hepatocytes as a case study. QSSPN simulations reproduce experimentally determined qualitative dynamic behaviours and permit mechanistic analysis of genotype-phenotype relationships.

AVAILABILITY AND IMPLEMENTATION

The model and simulation software implemented in C++ are available in supplementary material and at http://sysbio3.fhms.surrey.ac.uk/qsspn/.

Stochastic chemical kinetics : A review of the modelling and simulation approaches

A review of the physical principles that are the ground of the stochastic formulation of chemical kinetics is presented along with a survey of the algorithms currently used to simulate it. This review covers the main literature of the last decade and focuses on the mathematical models describing the characteristics and the behavior of systems of chemical reactions at the nano- and micro-scale. Advantages and limitations of the models are also discussed in the light of the more and more frequent use of these models and algorithms in modeling and simulating biochemical and even biological processes.

Stochastic analysis of complex reaction networks using binomial moment equations

The stochastic analysis of complex reaction networks is a difficult problem because the number of microscopic states in such systems increases exponentially with the number of reactive species. Direct integration of the master equation is thus infeasible and is most often replaced by Monte Carlo simulations. While Monte Carlo simulations are a highly effective tool, equation-based formulations are more amenable to analytical treatment and may provide deeper insight into the dynamics of the network. Here, we present a highly efficient equation-based method for the analysis of stochastic reaction networks. The method is based on the recently introduced binomial moment equations [Barzel and Biham, Phys. Rev. Lett. 106, 150602 (2011)]. The binomial moments are linear combinations of the ordinary moments of the probability distribution function of the population sizes of the interacting species. They capture the essential combinatorics of the reaction processes reflecting their stoichiometric structure. This leads to a simple and transparent form of the equations, and allows a highly efficient and surprisingly simple truncation scheme. Unlike ordinary moment equations, in which the inclusion of high order moments is prohibitively complicated, the binomial moment equations can be easily constructed up to any desired order. The result is a set of equations that enables the stochastic analysis of complex reaction networks under a broad range of conditions. The number of equations is dramatically reduced from the exponential proliferation of the master equation to a polynomial (and often quadratic) dependence on the number of reactive species in the binomial moment equations. The aim of this paper is twofold: to present a complete derivation of the binomial moment equations; to demonstrate the applicability of the moment equations for a representative set of example networks, in which stochastic effects play an important role.

NONLINEAR DYNAMICS OF CELL CYCLES WITH STOCHASTIC MATHEMATICAL MODELS

Cell cycles are fundamental components of all living organisms and their systematic studies extend our knowledge about the interconnection between regulatory, metabolic, and signaling networks, and therefore open new opportunities for our ultimate efficient control of cellular processes for disease treatments, as well as for a wide variety of biomedical and biotechnological applications. In the study of cell cycles, nonlinear phenomena play a paramount role, in particular in those cases where the cellular dynamics is in the focus of attention. Quantification of this dynamics is a challenging task due to a wide range of parameters that require estimations and the presence of many stochastic effects. Based on the originally deterministic model, in this paper we develop a hierarchy of models that allow us to describe the nonlinear dynamics accounting for special events of cell cycles. First, we develop a model that takes into account fluctuations of relative concentrations of proteins during special events of cell cycles. Such fluctuations are induced by varying rates of relative concentrations of proteins and/or by relative concentrations of proteins themselves. As such fluctuations may be responsible for qualitative changes in the cell, we develop a new model that accounts for the effect of cellular dynamics on the cell cycle. Finally, we analyze numerically nonlinear effects in the cell cycle by constructing phase portraits based on the newly developed model and carry out a parametric sensitivity analysis in order to identify parameters for an efficient cell cycle control. The results of computational experiments demonstrate that the metabolic events in gene regulatory networks can qualitatively influence the dynamics of the cell cycle.

A method for efficient calculation of diffusion and reactions of lipophilic compounds in complex cell geometry

A general description of effects of toxic compounds in mammalian cells is facing several problems. Firstly, most toxic compounds are hydrophobic and partition phenomena strongly influence their behaviour. Secondly, cells display considerable heterogeneity regarding the presence, activity and distribution of enzymes participating in the metabolism of foreign compounds i.e. bioactivation/biotransformation. Thirdly, cellular architecture varies greatly. Taken together, complexity at several levels has to be addressed to arrive at efficient in silico modelling based on physicochemical properties, metabolic preferences and cell characteristics. In order to understand the cellular behaviour of toxic foreign compounds we have developed a mathematical model that addresses these issues. In order to make the system numerically treatable, methods motivated by homogenization techniques have been applied. These tools reduce the complexity of mathematical models of cell dynamics considerably thus allowing to solve efficiently the partial differential equations in the model numerically on a personal computer. Compared to a compartment model with well-stirred compartments, our model affords a more realistic representation. Numerical results concerning metabolism and chemical solvolysis of a polycyclic aromatic hydrocarbon carcinogen show good agreement with results from measurements in V79 cell culture. The model can easily be extended and refined to include more reactants, and/or more complex reaction chains, enzyme distribution etc, and is therefore suitable for modelling cellular metabolism involving membrane partitioning also at higher levels of complexity.

Stochastic simulations of the tetracycline operon

Background

The tetracycline operon is a self-regulated system. It is found naturally in bacteria where it confers resistance to antibiotic tetracycline. Because of the performance of the molecular elements of the tetracycline operon, these elements are widely used as parts of synthetic gene networks where the protein production can be efficiently turned on and off in response to the presence or the absence of tetracycline. In this paper, we investigate the dynamics of the tetracycline operon. To this end, we develop a mathematical model guided by experimental findings. Our model consists of biochemical reactions that capture the biomolecular interactions of this intriguing system. Having in mind that small biological systems are subjects to stochasticity, we use a stochastic algorithm to simulate the tetracycline operon behavior. A sensitivity analysis of two critical parameters embodied this system is also performed providing a useful understanding of the function of this system.

Results

Simulations generate a timeline of biomolecular events that confer resistance to bacteria against tetracycline. We monitor the amounts of intracellular TetR2 and TetA proteins, the two important regulatory and resistance molecules, as a function of intrecellular tetracycline. We find that lack of one of the promoters of the tetracycline operon has no influence on the total behavior of this system inferring that this promoter is not essential for Escherichia coli. Sensitivity analysis with respect to the binding strength of tetracycline to repressor and of repressor to operators suggests that these two parameters play a predominant role in the behavior of the system. The results of the simulations agree well with experimental observations such as tight repression, fast gene expression, induction with tetracycline, and small intracellular TetR2 amounts.

Conclusions

Computer simulations of the tetracycline operon afford augmented insight into the interplay between its molecular components. They provide useful explanations of how the components and their interactions have evolved to best serve bacteria carrying this operon. Therefore, simulations may assist in designing novel gene network architectures consisting of tetracycline operon components.

Quantifying stochastic effects in biochemical reaction networks using partitioned leaping

"Leaping" methods show great promise for significantly accelerating stochastic simulations of complex biochemical reaction networks. However, few practical applications of leaping have appeared in the literature to date. Here, we address this issue using the "partitioned leaping algorithm" (PLA) [L. A. Harris and P. Clancy, J. Chem. Phys. 125, 144107 (2006)], a recently introduced multiscale leaping approach. We use the PLA to investigate stochastic effects in two model biochemical reaction networks. The networks that we consider are simple enough so as to be accessible to our intuition but sufficiently complex so as to be generally representative of real biological systems. We demonstrate how the PLA allows us to quantify subtle effects of stochasticity in these systems that would be difficult to ascertain otherwise as well as not-so-subtle behaviors that would strain commonly used "exact" stochastic methods. We also illustrate bottlenecks that can hinder the approach and exemplify and discuss possible strategies for overcoming them. Overall, our aim is to aid and motivate future applications of leaping by providing stark illustrations of the benefits of the method while at the same time elucidating obstacles that are often encountered in practice.

Hybrid stochastic simulations of intracellular reaction-diffusion systems

With the observation that stochasticity is important in biological systems, chemical kinetics have begun to receive wider interest. While the use of Monte Carlo discrete event simulations most accurately capture the variability of molecular species, they become computationally costly for complex reaction-diffusion systems with large populations of molecules. On the other hand, continuous time models are computationally efficient but they fail to capture any variability in the molecular species. In this study a hybrid stochastic approach is introduced for simulating reaction-diffusion systems. We developed an adaptive partitioning strategy in which processes with high frequency are simulated with deterministic rate-based equations, and those with low frequency using the exact stochastic algorithm of Gillespie. Therefore the stochastic behavior of cellular pathways is preserved while being able to apply it to large populations of molecules. We describe our method and demonstrate its accuracy and efficiency compared with the Gillespie algorithm for two different systems. First, a model of intracellular viral kinetics with two steady states and second, a compartmental model of the postsynaptic spine head for studying the dynamics of Ca+2 and NMDA receptors.

Stochastic modelling for quantitative description of heterogeneous biological systems

Two related developments are currently changing traditional approaches to computational systems biology modelling. First, stochastic models are being used increasingly in preference to deterministic models to describe biochemical network dynamics at the single-cell level. Second, sophisticated statistical methods and algorithms are being used to fit both deterministic and stochastic models to time course and other experimental data. Both frameworks are needed to adequately describe observed noise, variability and heterogeneity of biological systems over a range of scales of biological organization.

Regulation by transcription factors in bacteria: beyond description

Transcription is an essential step in gene expression and its understanding has been one of the major interests in molecular and cellular biology. By precisely tuning gene expression, transcriptional regulation determines the molecular machinery for developmental plasticity, homeostasis and adaptation. In this review, we transmit the main ideas or concepts behind regulation by transcription factors and give just enough examples to sustain these main ideas, thus avoiding a classical ennumeration of facts. We review recent concepts and developments: cis elements and trans regulatory factors, chromosome organization and structure, transcriptional regulatory networks (TRNs) and transcriptomics. We also summarize new important discoveries that will probably affect the direction of research in gene regulation: epigenetics and stochasticity in transcriptional regulation, synthetic circuits and plasticity and evolution of TRNs. Many of the new discoveries in gene regulation are not extensively tested with wetlab approaches. Consequently, we review this broad area in Inference of TRNs and Dynamical Models of TRNs. Finally, we have stepped backwards to trace the origins of these modern concepts, synthesizing their history in a timeline schema.

Biochemical simulations: stochastic, approximate stochastic and hybrid approaches

Computer simulations have become an invaluable tool to study the sometimes counterintuitive temporal dynamics of (bio-)chemical systems. In particular, stochastic simulation methods have attracted increasing interest recently. In contrast to the well-known deterministic approach based on ordinary differential equations, they can capture effects that occur due to the underlying discreteness of the systems and random fluctuations in molecular numbers. Numerous stochastic, approximate stochastic and hybrid simulation methods have been proposed in the literature. In this article, they are systematically reviewed in order to guide the researcher and help her find the appropriate method for a specific problem.

Kinetic modeling of biological systems

The dynamics of how the constituent components of a natural system interact defines the spatio-temporal response of the system to stimuli. Modeling the kinetics of the processes that represent a biophysical system has long been pursued with the aim of improving our understanding of the studied system. Due to the unique properties of biological systems, in addition to the usual difficulties faced in modeling the dynamics of physical or chemical systems, biological simulations encounter difficulties that result from intrinsic multi-scale and stochastic nature of the biological processes.This chapter discusses the implications for simulation of models involving interacting species with very low copy numbers, which often occur in biological systems and give rise to significant relative fluctuations. The conditions necessitating the use of stochastic kinetic simulation methods and the mathematical foundations of the stochastic simulation algorithms are presented. How the well-organized structural hierarchies often seen in biological systems can lead to multi-scale problems and the possible ways to address the encountered computational difficulties are discussed. We present the details of the existing kinetic simulation methods and discuss their strengths and shortcomings. A list of the publicly available kinetic simulation tools and our reflections for future prospects are also provided.

Elimination of fast variables in chemical Langevin equations

Internal and external fluctuations are ubiquitous in cellular signaling processes. Because biochemical reactions often evolve on disparate time scales, mathematical perturbation techniques can be invoked to reduce the complexity of stochastic models. Previous work in this area has focused on direct treatment of the master equation. However, eliminating fast variables in the chemical Langevin equation is also an important problem. We show how to solve this problem by utilizing a partial equilibrium assumption. Our technique is applied to a simple birth-death-dimerization process and a more involved gene regulation network, demonstrating great computational efficiency. Excellent agreement is found with results computed from exact stochastic simulations. We compare our approach with existing reduction schemes and discuss avenues for future improvement.

Validity of rate equation results for reaction rates in reaction networks with fluctuations

Systems of reacting particles, which are well mixed or distributed homogeneously in space, are commonly modeled by deterministic rate equations. These equations, which are based on the mean-field approximation, are valid for macroscopic systems. However, they neglect fluctuations in the particle populations. As a result, under conditions of strong fluctuations, the reaction rates obtained from the rate equations are highly inaccurate. To account for the fluctuations, stochastic methods are required. However, these methods are computationally intensive and may become infeasible for complex reaction networks. Therefore, it is useful to identify the conditions under which the rate equations provide accurate results. Naively, one expects strong fluctuations when the average population sizes of some of the reactants are of order one or lower. Here we present a systematic approach, for testing the validity of the rate equations, in which we define characteristic scales in terms of the rate constants of the network. We show that the rate equations fail to accurately reproduce the reaction rates when the system size is reduced below these scales. Surprisingly, the rate equations are found to be applicable in a wider range than expected. Their validity depends not only on the population sizes of the reactive species but also on the kinetic properties of the reaction network.

FERN - a Java framework for stochastic simulation and evaluation of reaction networks

Background

Stochastic simulation can be used to illustrate the development of biological systems over time and the stochastic nature of these processes. Currently available programs for stochastic simulation, however, are limited in that they either a) do not provide the most efficient simulation algorithms and are difficult to extend, b) cannot be easily integrated into other applications or c) do not allow to monitor and intervene during the simulation process in an easy and intuitive way. Thus, in order to use stochastic simulation in innovative high-level modeling and analysis approaches more flexible tools are necessary.

Results

In this article, we present FERN (Framework for Evaluation of Reaction Networks), a Java framework for the efficient simulation of chemical reaction networks. FERN is subdivided into three layers for network representation, simulation and visualization of the simulation results each of which can be easily extended. It provides efficient and accurate state-of-the-art stochastic simulation algorithms for well-mixed chemical systems and a powerful observer system, which makes it possible to track and control the simulation progress on every level. To illustrate how FERN can be easily integrated into other systems biology applications, plugins to Cytoscape and CellDesigner are included. These plugins make it possible to run simulations and to observe the simulation progress in a reaction network in real-time from within the Cytoscape or CellDesigner environment.

Conclusion

FERN addresses shortcomings of currently available stochastic simulation programs in several ways. First, it provides a broad range of efficient and accurate algorithms both for exact and approximate stochastic simulation and a simple interface for extending to new algorithms. FERN's implementations are considerably faster than the C implementations of gillespie2 or the Java implementations of ISBJava. Second, it can be used in a straightforward way both as a stand-alone program and within new systems biology applications. Finally, complex scenarios requiring intervention during the simulation progress can be modelled easily with FERN.

The SBML discrete stochastic models test suite

MOTIVATION

Stochastic simulation is a very important tool for mathematical modelling. However, it is difficult to check the correctness of a stochastic simulator, since any two realizations from a single model will typically be different.

RESULTS

We have developed a test suite of stochastic models that have been solved either analytically or using numerical methods. This allows the accuracy of stochastic simulators to be tested against known results. The test suite is already being used by a number of stochastic simulator developers.

AVAILABILITY

The latest version of the test suite can be obtained from http://www.calibayes.ncl.ac.uk/Resources/dsmts/ and is licensed under GNU Lesser General Public License.

Models for synthetic biology

Synthetic biological engineering is emerging from biology as a distinct discipline based on quantification. The technologies propelling synthetic biology are not new, nor is the concept of designing novel biological molecules. What is new is the emphasis on system behavior. The objective is the design and construction of new biological devices and systems to deliver useful applications. Numerous synthetic gene circuits have been created in the past decade, including bistable switches, oscillators, and logic gates, and possible applications abound, including biofuels, detectors for biochemical and chemical weapons, disease diagnosis, and gene therapies. More than fifty years after the discovery of the molecular structure of DNA, molecular biology is mature enough for real quantification that is useful for biological engineering applications, similar to the revolution in modeling in chemistry in the 1950s. With the excitement that synthetic biology is generating, the engineering and biological science communities appear remarkably willing to cross disciplinary boundaries toward a common goal.

Deterministic and stochastic models of NFkappaB pathway

In the article, we discuss the state of art and perspectives in deterministic and stochastic models of NFkappaB regulatory module. The NFkappaB is a transcription factor controlling various immune responses including inflammation and apoptosis. It is tightly regulated by at least two negative feedback loops involving IkappaBalpha and A20. This mode of regulation results in nucleus-to-cytoplasm oscillations in NFkappaB localization, which induce subsequent waves of NFkappaB responsive genes. Single cell experiments carried by several groups provided comprehensive evidence that stochastic effects play an important role in NFkappaB regulation. From modeling point of view, living cells might be considered noisy or stochastic biochemical reactors. In eukaryotic cells, in which the number of protein or mRNA molecules is relatively large, stochastic effects primarily originate in regulation of gene activity. Transcriptional activity of a gene can be initiated by trans-activator molecules binding to the specific regulatory site(s) in the target gene. The stochastic event of gene activation is amplified by transcription and translation, since it results in a burst of mRNA molecules, and each copy of mRNA then serves as a template for numerous protein molecules. Another potential source of variability can be receptors activation. At low-dose stimulation, important in cell-to-cell signaling, the number of active receptors can be low enough to introduce substantial noise to downstream signaling. Stochastic modeling confirms the large variability in cell responses and shows that no cell behaves like an "average" cell. This high cell-to-cell variability can be one of the weapons of the immune defense. Such non-deterministic defense may be harder to overcome by relatively simple programs coded in viruses and other pathogens.

The interplay between discrete noise and nonlinear chemical kinetics in a signal amplification cascade

We used various analytical and numerical techniques to elucidate signal propagation in a small enzymatic cascade which is subjected to external and internal noises. The nonlinear character of catalytic reactions, which underlie protein signal transduction cascades, renders stochastic signaling dynamics in cytosol biochemical networks distinct from the usual description of stochastic dynamics in gene regulatory networks. For a simple two-step enzymatic cascade which underlies many important protein signaling pathways, we demonstrated that the commonly used techniques such as the linear noise approximation and the Langevin equation become inadequate when the number of proteins becomes too low. Consequently, we developed a new analytical approximation, based on mixing the generating function and distribution function approaches, to the solution of the master equation that describes nonlinear chemical signaling kinetics for this important class of biochemical reactions. Our techniques work in a much wider range of protein number fluctuations than the methods used previously. We found that under certain conditions the burst phase noise may be injected into the downstream signaling network dynamics, resulting possibly in unusually large macroscopic fluctuations. In addition to computing first and second moments, which is the goal of commonly used analytical techniques, our new approach provides the full time-dependent probability distributions of the colored non-Gaussian processes in a nonlinear signal transduction cascade.

On the origins of approximations for stochastic chemical kinetics

This paper considers the derivation of approximations for stochastic chemical kinetics governed by the discrete master equation. Here, the concepts of (1) partitioning on the basis of fast and slow reactions as opposed to fast and slow species and (2) conditional probability densities are used to derive approximate, partitioned master equations, which are Markovian in nature, from the original master equation. Under different conditions dictated by relaxation time arguments, such approximations give rise to both the equilibrium and hybrid (deterministic or Langevin equations coupled with discrete stochastic simulation) approximations previously reported. In addition, the derivation points out several weaknesses in previous justifications of both the hybrid and equilibrium systems and demonstrates the connection between the original and approximate master equations. Two simple examples illustrate situations in which these two approximate methods are applicable and demonstrate the two methods' efficiencies.

Developing Itô stochastic differential equation models for neuronal signal transduction pathways

Mathematical modeling and simulation of dynamic biochemical systems are receiving considerable attention due to the increasing availability of experimental knowledge of complex intracellular functions. In addition to deterministic approaches, several stochastic approaches have been developed for simulating the time-series behavior of biochemical systems. The problem with stochastic approaches, however, is the larger computational time compared to deterministic approaches. It is therefore necessary to study alternative ways to incorporate stochasticity and to seek approaches that reduce the computational time needed for simulations, yet preserve the characteristic behavior of the system in question. In this work, we develop a computational framework based on the Itô stochastic differential equations for neuronal signal transduction networks. There are several different ways to incorporate stochasticity into deterministic differential equation models and to obtain Itô stochastic differential equations. Two of the developed models are found most suitable for stochastic modeling of neuronal signal transduction. The best models give stable responses which means that the variances of the responses with time are not increasing and negative concentrations are avoided. We also make a comparative analysis of different kinds of stochastic approaches, that is the Itô stochastic differential equations, the chemical Langevin equation, and the Gillespie stochastic simulation algorithm. Different kinds of stochastic approaches can be used to produce similar responses for the neuronal protein kinase C signal transduction pathway. The fine details of the responses vary slightly, depending on the approach and the parameter values. However, when simulating great numbers of chemical species, the Gillespie algorithm is computationally several orders of magnitude slower than the Itô stochastic differential equations and the chemical Langevin equation. Furthermore, the chemical Langevin equation produces negative concentrations. The Itô stochastic differential equations developed in this work are shown to overcome the problem of obtaining negative values.