Chemical master versus chemical langevin for first-order reaction networks

Markov jump processes are widely used to model interacting species in circumstances where discreteness and stochasticity are relevant. Such models have been particularly successful in computational cell biology, and in this case, the interactions are typically rst-order. The Chemical Langevin Equation is a stochastic dierential equation that can be regarded as an approximation to the underlying jump process. In particular, the Chemical Langevin Equation allows simulations to be performed more eectively. In this work, we obtain expressions for the rst and second moments of the Chemical Langevin Equation for a generic rst-order reaction network. Moreover, we show that these moments exactly match those of the under-lying jump process. Hence, in terms of means, variances and correlations, the Chemical Langevin Equation is an excellent proxy for the Chemical Master Equation. Our work assumes that a unique solution exists for the Chemical Langevin Equation. We also show that the moment matching re- sult extends to the case where a gene regulation model of Raser and O'Shea (Science, 2004) is replaced by a hybrid model that mixes elements of the Master and Langevin equations. We nish with numerical experiments on a dimerization model that involves second order reactions, showing that the two regimes continue to give similar results

Synthetic biology of genetic circuits

The combination of positive and negative feedback loops has been shown to increase the robustness of oscillations. Such breakthrough has enabled to understand the importance of that dual control in a few biological systems. The reason is that most biological systems are non-linear. One of the obstacles that must be overcome when dealing with non-linear systems, even if they are simple, is that the use of intuition to predict its behavior is no longer valid. The comprehension of the behavior of the system can only be achieved mathematical modeling and computational simulations This bachelor thesis aims to develop a mathematical model of the relaxoscillator, a gene regulatory network in which two genes with identical promoters are regulated by the same activator and repressor. At the same time, the binding of those depends on the concentrations of two inducers: arabinose and IPTG, which correspond to the control parameters of the system. The obtained model, derived from the chemical reactions, was simulated under different inducer concentrations in an attempt to comprehend the long term behavior of the system. The results show that varying these inducer concentrations allows to tune the period and the amplitude of the oscillations observed in the system. In order to analyze changes in the long term behavior of the system it will be required to include a third control parameter, the transcription repressor rate, so that the system displays different dynamic behaviors. The analysis of the simulations indicates the presence of a supercritical Hopf bifurcation for a given value of the transcription repressor rate that would explain the transition between damped oscillations and persistent oscillations. Nevertheless, due to the theoretical nature of the project, experimental studies as well as two-parameter bifurcation analysis should be performed in order to confirm such hypothesis and gain understanding of the behavior of the system as a function of inducers concentrations.Ingeniería Biomédic

Switching and diffusion models for gene regulation networks

We analyze a hierarchy of three regimes for modeling gene regulation. The most complete model is a continuous time, discrete state space, Markov jump process. An intermediate 'switch plus diffusion' model takes the form of a stochastic differential equation driven by an independent continuous time Markov switch. In the third 'switch plus ODE' model the switch remains but the diffusion is removed. The latter two models allow for multi-scale simulation where, for the sake of computational efficiency, system components are treated differently according to their abundance. The 'switch plus ODE' regime was proposed by Paszek (Modeling stochasticity in gene regulation: characterization in the terms of the underlying distribution function, Bulletin of Mathematical Biology, 2007), who analyzed the steady state behavior, showing that the mean was preserved but the variance only approximated that of the full model. Here, we show that the tools of stochastic calculus can be used to analyze first and second moments for all time. A technical issue to be addressed is that the state space for the discrete-valued switch is infinite. We show that the new 'switch plus diffusion' regime preserves the biologically relevant measures of mean and variance, whereas the 'switch plus ODE' model uniformly underestimates the variance in the protein level. We also show that, for biologically relevant parameters, the transient behaviour can differ significantly from the steady state, justifying our time-dependent analysis. Extra computational results are also given for a protein dimerization model that is beyond the scope of the current analysis