Steady-state expression of self-regulated genes

MOTIVATION: Regulatory gene networks contain generic modules such as feedback loops that are essential for the regulation of many biological functions. The study of the stochastic mechanisms of gene regulation is instrumental for the understanding of how cells maintain their expression at levels commensurate with their biological role, as well as to engineer gene expression switches of appropriate behavior. The lack of precise knowledge on the steady-state distribution of gene expression requires the use of Gillespie algorithms and Monte-Carlo approximations. Methodology: In this study, we provide new exact formulas and efficient numerical algorithms for computing/modeling the steady-state of a class of self-regulated genes, and we use it to model/compute the stochastic expression of a gene of interest in an engineered network introduced in mammalian cells. The behavior of the genetic network is then analyzed experimentally in living cells. RESULTS: Stochastic models often reveal counter-intuitive experimental behaviors, and we find that this genetic architecture displays a unimodal behavior in mammalian cells, which was unexpected given its known bimodal response in unicellular organisms. We provide a molecular rationale for this behavior, and we implement it in the mathematical picture to explain the experimental results obtained from this network.     

Time-continuous branching walk models of unstable gene amplification

We consider a stochastic mechanism of the loss of resistance of cancer cells to cytotoxic agents, in terms of unstable gene amplification. Two models being different versions of a time-continuous branching random walk are presented. Both models assume strong dependence in replication and segregation of the extrachromosomal elements. The mathematical part of the paper includes the expression for the expected number of cells with a given number of gene copies in terms of modified Bessel functions. This adds to the collection of rare explicit solutions to branching process models. Original asymptotic expansions are also demonstrated. Fitting the model to experimental data yields estimates of the probabilities of gene amplification and deamplification. The thesis of the paper is that purely stochastic mechanisms may explain the dynamics of reversible drug resistance of cancer cells. Various stochastic approaches and their limitations are discussed.     

Hill coefficient-based stochastic switch-like signal directly governs damage-recovery dynamics in freshwater fish in response to pulse copper

Growing evidence demonstrates that fluctuating metal stressors can have profound impact on the ecophysiological responses in aquatic species. However, how environmental stochasticity affects the complex damage-recovery dynamics in organisms remains difficult to predict. The objective of this paper was to investigate the stochastic behavior in the damage-recovery dynamics in tilapia in response to pulse waterborne copper (Cu). We developed a mathematical framework that allows discrimination between damage and recovery processes in tilapia exposed to designed pulse Cu scenarios. We built deterministic nonlinear models for the damage-recovery dynamics that produce response surfaces describing killing/recovery rate–Cu-pulse interval interactions. Here we showed that the stochastic switching behavior arose from competition among killing, recovery rates, and Cu pulse frequency. This competition resulted in an ultrasensitivity appeared in whole body, gills, muscle, liver, and kidney with Hill coefficients of ≥7, 4, 7, 5, and 5, respectively, at Cu 3 mg L⁻¹, dilution rate 0.05 h⁻¹, and pulse interval 72 h, indicating that a stochastic switch-like response was generated. We argue that the role of gill-associated Hill coefficient as a direct signal of the stochastic switch-like response in the damage-recovery dynamics in response to pulse metal stressor can serve as a sensitive indicator for risk detection in fluctuating environments. Our approach constitutes a general method to identify the stochastic switch-like response for aquatic species exposed to fluctuating metal stressors, which may help to predict and, eventually, expand our understanding of the damage-recovery dynamics. Finally, we implicate that Hill coefficient-based switch-like signal and its damage with hazard response can be linked in an information theoretic framework to handle environmental stochasticity.     

Comparing Stochastic Differential Equations and Agent-Based Modelling and Simulation for Early-Stage Cancer

There is great potential to be explored regarding the use of agent-based modelling and simulation as an alternative paradigm to investigate early-stage cancer interactions with the immune system. It does not suffer from some limitations of ordinary differential equation models, such as the lack of stochasticity, representation of individual behaviours rather than aggregates and individual memory. In this paper we investigate the potential contribution of agent-based modelling and simulation when contrasted with stochastic versions of ODE models using early-stage cancer examples. We seek answers to the following questions: (1) Does this new stochastic formulation produce similar results to the agent-based version? (2) Can these methods be used interchangeably? (3) Do agent-based models outcomes reveal any benefit when compared to the Gillespie results? To answer these research questions we investigate three well-established mathematical models describing interactions between tumour cells and immune elements. These case studies were re-conceptualised under an agent-based perspective and also converted to the Gillespie algorithm formulation. Our interest in this work, therefore, is to establish a methodological discussion regarding the usability of different simulation approaches, rather than provide further biological insights into the investigated case studies. Our results show that it is possible to obtain equivalent models that implement the same mechanisms; however, the incapacity of the Gillespie algorithm to retain individual memory of past events affects the similarity of some results. Furthermore, the emergent behaviour of ABMS produces extra patters of behaviour in the system, which was not obtained by the Gillespie algorithm.     

Modeling stochastic noise in gene regulatory systems

The Master equation is considered the gold standard for modeling the stochastic mechanisms of gene regulation in molecular detail, but it is too complex to solve exactly in most cases, so approximation and simulation methods are essential. However, there is still a lack of consensus about the best way to carry these out. To help clarify the situation, we review Master equation models of gene regulation, theoretical approximations based on an expansion method due to N.G. van Kampen and R. Kubo, and simulation algorithms due to D.T. Gillespie and P. Langevin. Expansion of the Master equation shows that for systems with a single stable steady-state, the stochastic model reduces to a deterministic model in a first-order approximation. Additional theory, also due to van Kampen, describes the asymptotic behavior of multistable systems. To support and illustrate the theory and provide further insight into the complex behavior of multistable systems, we perform a detailed simulation study comparing the various approximation and simulation methods applied to synthetic gene regulatory systems with various qualitative characteristics. The simulation studies show that for large stochastic systems with a single steady-state, deterministic models are quite accurate, since the probability distribution of the solution has a single peak tracking the deterministic trajectory whose variance is inversely proportional to the system size. In multistable stochastic systems, large fluctuations can cause individual trajectories to escape from the domain of attraction of one steady-state and be attracted to another, so the system eventually reaches a multimodal probability distribution in which all stable steady-states are represented proportional to their relative stability. However, since the escape time scales exponentially with system size, this process can take a very long time in large systems.     

A Multi-stage Representation of Cell Proliferation as a Markov Process

The stochastic simulation algorithm commonly known as Gillespie’s algorithm (originally derived for modelling well-mixed systems of chemical reactions) is now used ubiquitously in the modelling of biological processes in which stochastic effects play an important role. In well-mixed scenarios at the sub-cellular level it is often reasonable to assume that times between successive reaction/interaction events are exponentially distributed and can be appropriately modelled as a Markov process and hence simulated by the Gillespie algorithm. However, Gillespie’s algorithm is routinely applied to model biological systems for which it was never intended. In particular, processes in which cell proliferation is important (e.g. embryonic development, cancer formation) should not be simulated naively using the Gillespie algorithm since the history-dependent nature of the cell cycle breaks the Markov process. The variance in experimentally measured cell cycle times is far less than in an exponential cell cycle time distribution with the same mean.Here we suggest a method of modelling the cell cycle that restores the memoryless property to the system and is therefore consistent with simulation via the Gillespie algorithm. By breaking the cell cycle into a number of independent exponentially distributed stages, we can restore the Markov property at the same time as more accurately approximating the appropriate cell cycle time distributions. The consequences of our revised mathematical model are explored analytically as far as possible. We demonstrate the importance of employing the correct cell cycle time distribution by recapitulating the results from two models incorporating cellular proliferation (one spatial and one non-spatial) and demonstrating that changing the cell cycle time distribution makes quantitative and qualitative differences to the outcome of the models. Our adaptation will allow modellers and experimentalists alike to appropriately represent cellular proliferation—vital to the accurate modelling of many biological processes—whilst still being able to take advantage of the power and efficiency of the popular Gillespie algorithm.     

Probabilistic polynomial dynamical systems for reverse engineering of gene regulatory networks

Elucidating the structure and/or dynamics of gene regulatory networks from experimental data is a major goal of systems biology. Stochastic models have the potential to absorb noise, account for un-certainty, and help avoid data overfitting. Within the frame work of probabilistic polynomial dynamical systems, we present an algorithm for the reverse engineering of any gene regulatory network as a discrete, probabilistic polynomial dynamical system. The resulting stochastic model is assembled from all minimal models in the model space and the probability assignment is based on partitioning the model space according to the likeliness with which a minimal model explains the observed data. We used this method to identify stochastic models for two published synthetic network models. In both cases, the generated model retains the key features of the original model and compares favorably to the resulting models from other algorithms.     

A multi-approach and multi-scale platform to model CD4+ T cells responding to infections

Immune responses rely on a complex adaptive system in which the body and infections interact at multiple scales and in different compartments. We developed a modular model of CD4+ T cells, which uses four modeling approaches to integrate processes at three spatial scales in different tissues. In each cell, signal transduction and gene regulation are described by a logical model, metabolism by constraint-based models. Cell population dynamics are described by an agent-based model and systemic cytokine concentrations by ordinary differential equations. A Monte Carlo simulation algorithm allows information to flow efficiently between the four modules by separating the time scales. Such modularity improves computational performance and versatility and facilitates data integration. We validated our technology by reproducing known experimental results, including differentiation patterns of CD4+ T cells triggered by different combinations of cytokines, metabolic regulation by IL2 in these cells, and their response to influenza infection. In doing so, we added multi-scale insights to single-scale studies and demonstrated its predictive power by discovering switch-like and oscillatory behaviors of CD4+ T cells that arise from nonlinear dynamics interwoven across three scales. We identified the inflamed lymph node’s ability to retain naive CD4+ T cells as a key mechanism in generating these emergent behaviors. We envision our model and the generic framework encompassing it to serve as a tool for understanding cellular and molecular immunological problems through the lens of systems immunology.     

Modeling stochasticity and variability in gene regulatory networks

Modeling stochasticity in gene regulatory networks is an important and complex problem in molecular systems biology. To elucidate intrinsic noise, several modeling strategies such as the Gillespie algorithm have been used successfully. This article contributes an approach as an alternative to these classical settings. Within the discrete paradigm, where genes, proteins, and other molecular components of gene regulatory networks are modeled as discrete variables and are assigned as logical rules describing their regulation through interactions with other components. Stochasticity is modeled at the biological function level under the assumption that even if the expression levels of the input nodes of an update rule guarantee activation or degradation there is a probability that the process will not occur due to stochastic effects. This approach allows a finer analysis of discrete models and provides a natural setup for cell population simulations to study cell-to-cell variability. We applied our methods to two of the most studied regulatory networks, the outcome of lambda phage infection of bacteria and the p53-mdm2 complex.     

Cellular growth and division in the Gillespie algorithm

Recent experimental studies elucidating the importance of noise in gene regulation have ignited widespread interest in Gillespie's stochastic simulation technique for biochemical networks. We formulate modifications to the Gillespie algorithm which are necessary to correctly simulate chemical reactions with time-dependent reaction rates. We concentrate on time dependence of kinetic rates arising from the periodic process of growth and division of the cellular volume, and demonstrate that a careful re-derivation of the Gillespie algorithm is important when all stochastically simulated reactions have rates slower or comparable to the cellular growth rate. For an unregulated single-gene system, we illustrate our findings using recently proposed hybrid simulation techniques, and systematically compare our algorithm with analytic results obtained from the chemical master equation.     

Stochastic simulations on a model of circadian rhythm generation

Biological phenomena are often modeled by differential equations, where states of a model system are described by continuous real values. When we consider concentrations of molecules as dynamical variables for a set of biochemical reactions, we implicitly assume that numbers of the molecules are large enough so that their changes can be regarded as continuous and they are described deterministically. However, for a system with small numbers of molecules, changes in their numbers are apparently discrete and molecular noises become significant. In such cases, models with deterministic differential equations may be inappropriate, and the reactions must be described by stochastic equations. In this study, we focus a clock gene expression for a circadian rhythm generation, which is known as a system involving small numbers of molecules. Thus it is appropriate for the system to be modeled by stochastic equations and analyzed by methodologies of stochastic simulations. The interlocked feedback model proposed by Ueda et al. as a set of deterministic ordinary differential equations provides a basis of our analyses. We apply two stochastic simulation methods, namely Gillespie's direct method and the stochastic differential equation method also by Gillespie, to the interlocked feedback model. To this end, we first reformulated the original differential equations back to elementary chemical reactions. With those reactions, we simulate and analyze the dynamics of the model using two methods in order to compare them with the dynamics obtained from the original deterministic model and to characterize dynamics how they depend on the simulation methodologies.     

A model of sequential branching in hierarchical cell fate determination

Multipotent stem or progenitor cells undergo a sequential series of binary fate decisions, which ultimately generate the diversity of differentiated cells. Efforts to understand cell fate control have focused on simple gene regulatory circuits that predict the presence of multiple stable states, bifurcations and switch-like transitions. However, existing gene network models do not explain more complex properties of cell fate dynamics such as the hierarchical branching of developmental paths. Here, we construct a generic minimal model of the genetic regulatory network controlling cell fate determination, which exhibits five elementary characteristics of cell differentiation: stability, directionality, branching, exclusivity, and promiscuous expression. We argue that a modular architecture comprising repeated network elements reproduces these features of differentiation by sequentially repressing selected modules and hence restricting the dynamics to lower dimensional subspaces of the high-dimensional state space. We implement our model both with ordinary differential equations (ODEs), to explore the role of bifurcations in producing the one-way character of differentiation, and with stochastic differential equations (SDEs), to demonstrate the effect of noise on the system. We further argue that binary cell fate decisions are prevalent in cell differentiation due to general features of the underlying dynamical system. This minimal model makes testable predictions about the structural basis for directional, discrete and diversifying cell phenotype development and thus can guide the evaluation of real gene regulatory networks that govern differentiation.     

Modelling of translation of human protein disulfide isomerase in Escherichia coli-A case study of gene optimisation

Recombinant human protein disulfide isomerase (PDI) was expressed in vivo in Escherichia coli using a non-optimised gene sequence and an optimised sequence with four 5' codons substituted by synonymous codons that take less time to translate. The optimisation resulted in a 2-fold increase of total PDI concentration and by successive optimisation with expression at low temperature in a 10-fold increase of the amount of soluble PDI in comparison with the original wild-type construct. The improvement can be due to a faster clearing of the ribosome binding site on the mRNA, elevating the translation initiation rate and resulting in higher ribosome loading and better ribosome protection of the PDI mRNA against endonucleolytic cleavage by RNase. This hypothesis was supported by a novel computer simulation model of E. coli translational ribosome traffic based upon the stochastic Gillespie algorithm. The study indicates the applicability of such models in optimisation of recombinant protein sequences.     

Deterministic Versus Stochastic Models for Circadian Rhythms

Circadian rhythms which occur with a period close to 24 h in nearly all living organisms originate from the negative autoregulation of gene expression.Deterministic models based on genetic regulatory processes account for theoccurrence of circadian rhythms in constant environmental conditions (e.g.constant darkness), for entrainment of these rhythms by light-dark cycles, and for their phase-shifting by light pulses. At low numbers of protein and mRNA molecules, it becomes necessary to resort to stochastic simulations to assess the influence of molecular noise on circadian oscillations. We address the effect of molecular noise by considering two stochastic versions of a core model for circadian rhythms. The deterministic version of this core modelwas previously proposed for circadian oscillations of the PER protein in Drosophila and of the FRQ protein in Neurospora. In the first, non-developed version of the stochastic model, we introduce molecular noise without decomposing the deterministic mechanism into detailed reaction steps while in the second, developed version we carry out such a detailed decomposition. Numerical simulations of the two stochastic versions of the model are performed by means of the Gillespie method. We compare the predictions of the deterministic approach with those of the two stochastic models, with respect both to sustained oscillations of the limit cycle type and to the influence of the proximity of a bifurcation point beyond which the system evolves to a stable steady state. The results indicate that robust circadian oscillations can occur even when the numbers of mRNA and nuclear protein involved in the oscillatory mechanism are reduced to a few tens orhundreds, respectively. The non-developed and developed versions of the stochastic model yield largely similar results and provide good agreement with the predictions of the deterministic model for circadian rhythms.