Mathematical Modelling

Introduction

After mating, the fused yeast cell gains both the sensor and receiver system. Then the yeast cell is capable of detection AHL in the environment and report them.

There are three stages in the detection of AHL from bacteria. First, AHL in the environment diffuses across the cell membrane of the yeast. Second, AHL binds to modified LuxR receptor and forms a complex, which enters the nucleus and bind to the LuxR promoter. Upon binding, the AHL-LuxR complex activates the expression of the transcription factor tTA¹. tTA enters the nucleus and binds to TetO operator, activating the reporter gene ADE2. Expression of ADE2 changes the color of the yeast from red to white. An overview of the biochemical process is shown in Figure 1. The figure is drawn with CellDesigner² 4.3.

Figure 1. Overview of the biochemical process

Assumptions

AHL is secreted by bacteria and diffused across the cell membrane of the yeast. It is assumed that the diffusion process reaches equilibrium within a short time so the concentration of AHL inside and outside the yeast cell membrane is the same.

After AHL binds to modified LuxR protein to form an AHL-LuxR complex, the complex must be transported into the cell nucleus. The nuclear localization sequence on the LuxR protein is recognized by importin and then imported into the cell nucleus. To model the cell more accurately, the rate of transportation must be considered. However, without sufficient experiment data, it is difficult to estimate the kinetic parameters. In a simplified model, the concentrations of transcription factor inside and outside cell nucleus are assumed to be equal.

Three steps are required to activate expression of a protein: transcription factor binding, transcription and translation. If transportation of proteins and mRNAs are considered, there will be more steps. To simplify the model, we assume that the concentrations of transcription factors and mRNAs inside and outside the cell nucleus are equal. Transcription and translation can be modeled as a single process as they are tightly coupled.

Activation of transcription is modeled as a stochastic process. A promoter is either bound or unbound by one transcription factor molecule at a moment. Binding of transcription factor increases transcription rate of the target gene. The probability of transcription factor binding is determined by the concentration of transcription factor, gene copy number and binding affinity (or disassociation rate).

Model

The biochemical process is modeled as ordinary differential equations. The variables and equations are list as follows.

Species

• AHL (concentration remains constant)
• LuxR – LuxR in cytoplasm
• LuxRC – LuxR-AHL complex (dimer)
• tTA
• ADE2

Kinetic parameters

Equations

LuxR protein is synthesize at a constant rate k1. AHL binds to LuxR to form a complex. Then AHL-LuxR complex dimerizes to form a transcription factor³.

Activation of tTA expression is modeled using Hill function. Hill functions is commonly used to model the interactions between transcription factors and promoters4. The transcription factor cooperativity is 1 (single binding site). k5 is the expression rate of tTA if the promoter is fully activated.

Activation of ADE2 expression is also modeled in Hill function.

Initial Conditions

Parameter Estimation

There are many parameters in the ODE equations. While some of the parameters can be found or adapted from literature, yet other parameters were estimated from experiment data. The parameters are estimated by fitting the model to experiment data. The model is a set of ODE equations depending on time t with unknown parameters: where x are concentration of the species and θ are the parameters.

The distance between the experiment observations and model predictions are expressed as a cost function. A simple cost function is the Euclid distance. The objective is to search for parameters that minimize the cost function: where x_exp is the experiment data.

Sensitivity Analysis

Among all species considered in the model, initial AHL concentration is the main factor that determines the output of the system. The main output of the system is the color of the yeast which is correlated with the concentration of ADE2. The relationship between the concentration of ADE2 and the initial concentration of AHL will be analyzed.

References

1. Gossen, M. & Bujard, H. Tight control of gene expression in mammalian cells by tetracycline-responsive promoters. Proc. Natl. Acad. Sci. 89, 5547–5551 (1992).
2. Funahashi, A., Morohashi, M., Kitano, H. & Tanimura, N. CellDesigner: a process diagram editor for gene-regulatory and biochemical networks. BIOSILICO 1, 159–162 (2003).
3. Basu, S., Gerchman, Y., Collins, C. H., Arnold, F. H. & Weiss, R. A synthetic multicellular system for programmed pattern formation. Nature 434, 1130–1134 (2005).
4. Goutelle, S. et al. The Hill equation: a review of its capabilities in pharmacological modelling. Fundam. Clin. Pharmacol. 22, 633–648 (2008).