High-throughput genome sequencing and transcriptome analysis have provided researchers with a

High-throughput genome sequencing and transcriptome analysis have provided researchers with a quantitative basis for detailed modeling of gene expression using a wide variety of mathematical models. that provides a dynamical description of gene regulatory systems using detailed DNA-based information as well as spatial and temporal transcription factor concentration data. We also developed a semi-implicit numerical algorithm for solving the model equations and demonstrate here the efficiency of this algorithm through stability and convergence analyses. To test the model we used it together with the semi-implicit algorithm to simulate a gene regulatory circuit that drives development in the dorsal-ventral axis of the blastoderm-stage embryo involving three genes. For model validation we have done both mathematical and statistical comparisons between the experimental data and the model’s simulated data. Where operon and protein in [34]. The difficulty of applying such models to eukaryotic systems is the increased complexity of = all successful states is the contribution from state is mogroside IIIe the sum of contributions of all possible states of the enhancer. The continuing state of the enhancer with nothing bound is defined to have contribution 1. Each state consists of a product of protein concentrations and binding affinities (i.e. a product of terms of the form [with and bound cooperatively is given by = [is the number of binding sites bound by a TF in state is the binding affinity of binding site is the cooperativity parameter representing the product of all cooperative interactions between the TFs bound in state parameter is always in the interval [0 1 representing the proportion of time that the state is successful. So in general ≤ 1 is the quenching parameter representing the product of mogroside IIIe all repressive interactions mogroside IIIe between the activators and repressors bound in state and 0 ≤ ≤ 1 for all = {2 4 5 6 7 8 and term is defined in Table 2.1. Table 2.1 State contributions. Note that the above example (and any other following the assumptions given in section 2.1) results in an equation of the form represents the gene of interest is a vector of concentration values and Φand Ψare polynomials with coefficients depending on the parameter values. The mogroside IIIe binding be included by these parameter values affinities ∈ ∈ ?is a diagonal matrix containing diffusion constants and Δis the one-dimensional (1-D) Laplacian operator. The 1-D simplifying assumption here is based on biological observations that gradients of TFs in the embryo tend to run in a 1-D axis either dorsal-ventral or anterior-posterior. Note that this is a PDE due to mogroside IIIe its dependence on both time and space. This equation is a starting point for a realistic model of gene regulation during development. It can incorporate mRNA and protein concentrations their spatial diffusion and decay over time and the intricate relationships between these concentrations can. In the setting of the blastoderm-stage embryo cell membranes have not yet separated the nuclei from each other so proteins and mRNAs are able to diffuse in this syncytial environment. mRNA production the process of transcription is Rabbit Polyclonal to 5-HT-6. heavily regulated by protein (TF) binding and protein production the process of translation is dependent on mRNA production. It is therefore essential to employ an equation that allows for diffusion decay and an easily modified synthesis term that can incorporate natural feedforward and feedback loops. We discretize (3 first.1) in space to change the PDE system to an ODE system. Unlike the standard practice in numerical analysis where the continuous spatial variable is divided into mesh points in the domain we fix our discretization points as the positions of the nuclei. This treatment is based on the assumption that diffusion takes place between nuclei and that over the time period we are modeling the positions of the nuclei are fixed. This leads to the following ODE: ∈ ?represents the specific mRNA or protein that the equation corresponds to and and are the corresponding diffusion and decay rates. We will assume that we are modeling a mogroside IIIe given set of mRNAs = {= {∈ represents a specific type of mRNA and for all ∈ represents a specific type of protein. represents the nucleus (spatial) location. The most important term in (3.2) is the synthesis term (: ?→ ?. The function represents the vector of these functions for the mRNA or protein : ?→ ?for ≤ ∈ and models translation when ∈ ∈ (at each nucleus and takes the form is the scaling constant associated with gene and Φand Ψare polynomials with coefficients depending on the parameter values. These parameter.