Abstract
In this paper, the delay differential equations of Gene expression models with mechanisms of signal-dependent transcription regulation are solved and studied in two cases: When there is (i) competition and (ii) without competition(non-competition) for Deoxy ribo Nucleic Acid (DNA) regulatory binding sites in a cell. Also, we studied the effect of both increasing the inhibitor or decreasing the abundance of the activator (inhibition mechanism), and decreasing the inhibitor or increasing the abundance of the activator (activator mechanism) on the steady-state of the solutions. A new analytical approximation approach derived from Taylor series expansion is used for solving the delay differential equations of gene expression models. From the analytical approximate solutions of gene expression models that are resulting from using the proposed method, we found that the behavior of the solution in the activation mechanisms whether in the competitive or non-competitive model is more stable than the abundance of the activator increases, while the inhibition mechanisms are less stable. We also noticed that the convergence of these solutions is achieved with a few iterations. Keywords: Taylors’ technique , activator, inhibitor , Gen, mRNA, protein, convergence analysis. DOI: 10.7176/MTM/12-1-01 Publication date: January 31 st 2022
Highlights
Gene expression is a process by which genetic information is used to obtain genetic products or a type of Ribonucleic Acid (RNA)
Wang et al [2] introduced a genetic transcriptional regulatory model, subject to associated noise and the role of time delay in gene switching with random resonance, took two cases of time delay: the first is the linear delay that occurs during the degradation process and the second is the non-linear delay that occurs during the synthesis process
Sharma et al [4] took the gene expression model of strain E. coli (TJK16) which consists of a system of differential equations independent of the delay time that occurs during the process of transcription and translation, the solution was found using the fuzzy method of gene expression and noticed that when the translation rate and transcription rate values change, the steady-state of the model solutions are satisfied
Summary
To model the signal-induced suppression, we note in the competitive model, the activation level can be reduced by the signal ( ) =. Induced activation in the competitive model the level of the activator can be increased by the signal ( ) =. 3-Taylors’ technique and application: The series method can develop analytical methods in finding an exact and /or approximate analytical solution to many linear and non-linear differential equations[ 14,15,16]. Apply the above steps to find analytical approximate solution of equations activator bound gene, repressor bound gene, mRNA, Protein (2.1-2.9). The following equation we found from the relationship (2.5), which represents the rate of change in the free gene with the initial condition (0) = (0) because we need it to find solutions to the equations(2.1-2.4):-. To find the activator binding concentration, take equation (2.1) with the initial condition (2.11)
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