Model and simulation
The signaling network from the input of external ligand signal to the output of the tumbling state of a E coli cell can be quantitatively described by a modular model. The model is formulated based on the law of mass action and Michaelis-Menten mechanism and contains four relatively independent modules that are explained in detail below.
Module 1: Activation of ToxR receptor
In this module, the ligand signal activates the ToxR receptor into a dimerized complex formed with the ligand, which is the active state of the receptor. The biochemical reaction can be illustrated as:
where L, R, C and C2 represent the concentration of external ligand, free receptor, recptor-ligand complex, and the dimerized receptor-ligand complex. The first reaction represents the binding/unbinding between the free receptor and the ligand. The second reaction represents the dimerization /undimerization conversions between C and C2. The conservation of the total concentration of the receptor can be written as:
where R^T is the total receptor concentration. The governing differential equations of this module are:
The equations contain binding rate k_f L(R^T-C-2C_2 ), unbinding rate k_r C, dimerization rate k_dim C^2 and undimerization rate 2k_undim C_2, where k_f, k_r, k_dim and k_undim represent the rate constants for binding, unbinding, dimerization and undimerization reactions. The values of the parameters are obtained from (Forsten-Williams, Chua et al. 2005). Typical simulation results in response to L=1 µM are:
where the dissociation constant for receptor-ligand binding used in the right figure is 100 times than that in the left figure.
Module 2. Transcription/translation of CheZ
In this module, the dimerized complex C_2 activates the transcription of the cheZ mRNA. The reactions are the standard transcription and translation reactions illustrated as:
Where Zm and Zp represent the concentration of the mRNA and the protein product of CheZ. The formulation of the reactions follow the standard equations for transcription and translation.
Where k_0 represents the basal transcription rate of Z_m, k_1 represents the transcription rate activated by C_2, k_3 represents the translational rate of Z_p, and k_2 and k_4 represent the degradation rates of Z_m and Z_p. Typical simulation results are:
where the degradation rate for Zp in the right figure is 10 times than that in the left figure.
Module 3.
In this module, the CheY protein is dephosphorylated by the CheZ protein and the reaction can be illustrated as:
The governing differential equation follows the format given in (Spiro, Parkinson et al. 1997)
Where Y^Tis the total concentration of the CheY protein, and k_p and k_d are the phosphorylation and the dephosphorylation rate constants of CheY. The parameter values are obtained from (Spiro, Parkinson et al. 1997). Typical simulation results are:
where again the degradation rate for Zp in the right figure is 10 times than that in the left figure.
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