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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_fL(R^T-C-2C₂ ), unbinding rate k_rC, dimerization rate k_dimC² and undimerization rate 2k_undimC₂, 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 C₂ 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 k₀ represents the basal transcription rate of Z_m, k₁ represents the transcription rate activated by C₂, k₃ represents the translational rate of Z_p, and k₂ and k₄ 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.

Module 4
In this module, the tumbling activity of E coli is characterized by the so-called “bias”, which is defined as the ratio of the time of directed movement and the total time. It is experimentally measured that the bias is a Hill function dependent on the concentration of phosphorylated CheY (Cluzel, Surette et al. 2000).



Using the steady-state value of Yp in the previous two figures, that is 4.08 µM or 14.25µM, the final bias is 6.6% or .01%.

References
Cluzel, P., M. Surette, et al. (2000). "An ultrasensitive bacterial motor revealed by monitoring signaling proteins in single cells." Science 287(5458): 1652-5.
Forsten-Williams, K., C. C. Chua, et al. (2005). "The kinetics of FGF-2 binding to heparan sulfate proteoglycans and MAP kinase signaling." J Theor Biol 233(4): 483-99.
Spiro, P. A., J. S. Parkinson, et al. (1997). "A model of excitation and adaptation in bacterial chemotaxis." Proc Natl Acad Sci U S A 94(14): 7263-8.