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{{Tsinghua:Common-Style}}
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Modeling
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<div class="normal">
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<h1>Mathematical Modelling</h1>
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<h2>Introduction</h2>
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<p>
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After mating, the fused yeast cell gains both the sensor and receiver system.
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Then the yeast cell is capable of detection AHL in the environment and report them.
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</p>
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<p>
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There are three stages in the detection of AHL from bacteria.
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First, AHL in the environment diffuses across the cell membrane of the yeast.
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Second, AHL binds to modified LuxR receptor and forms a complex,
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which enters the nucleus and bind to the LuxR promoter.
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Upon binding, the AHL-LuxR complex activates the expression of the transcription factor tTA<sup>1</sup>.
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tTA enters the nucleus and binds to TetO operator, activating the reporter gene ADE2.
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Expression of ADE2 changes the color of the yeast from red to white.
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An overview of the biochemical process is shown in Figure 1. The figure is drawn with CellDesigner<sup>2</sup> 4.3.
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</p>
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<div class="figure">
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<img class="center" src="NotFound"/>
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<p class="legend">
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Figure 1. Overview of the biochemical process
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</p>
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</div>
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<h2>Assumptions</h2>
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<p>
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AHL is secreted by bacteria and diffused across the cell membrane of the yeast.
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It is assumed that the diffusion process reaches equilibrium within a
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short time so the concentration of AHL inside and outside the yeast cell membrane is the same.
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</p>
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<p>
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After AHL binds to modified LuxR protein to form an AHL-LuxR complex,
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the complex must be transported into the cell nucleus.
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The nuclear localization sequence on the LuxR protein is recognized by importin and then imported into the cell nucleus.
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To model the cell more accurately, the rate of transportation must be considered. However,
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without sufficient experiment data, it is difficult to estimate the kinetic parameters.
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In a simplified model, the concentrations of transcription factor inside and outside cell nucleus are assumed to be equal.
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</p>
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<p>
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Three steps are required to activate expression of a protein:
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transcription factor binding, transcription and translation.
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If transportation of proteins and mRNAs are considered,
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there will be more steps. To simplify the model,
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we assume that the concentrations of transcription factors and mRNAs inside and outside the cell nucleus are equal.
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Transcription and translation can be modeled as a single process as they are tightly coupled.
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</p>
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<p>
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Activation of transcription is modeled as a stochastic process.
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A promoter is either bound or unbound by one transcription factor molecule at a moment.
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Binding of transcription factor increases transcription rate of the target gene.
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The probability of transcription factor binding is determined by the concentration of transcription factor,
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gene copy number and binding affinity (or disassociation rate).
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</p>
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<h2>Model</h2>
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<p>
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The biochemical process is modeled as ordinary differential equations. The variables and equations are list as follows.
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</p>
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<h3>Species</h3>
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<ul>
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<li>
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AHL (concentration remains constant)
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</li>
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<li>
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LuxR – LuxR in cytoplasm
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</li>
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<li>
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LuxRC – LuxR-AHL complex (dimer)
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</li>
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<li>
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tTA
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</li>
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<li>
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ADE2
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</li>
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</ul>
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<h3>Kinetic parameters</h3>
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<table border="1" class="center">
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<tr>
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<th>Name</th><th>Value</th><th>Comment</th>
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</tr>
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<tr>
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<td>k1</td><td></td><td>basal expression rate under constitutive promoter</td>
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</tr>
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<tr>
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<td>k2</td><td></td><td>dimerization rate of AHL and LuxR</td>
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</tr>
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<tr>
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<td>k3</td><td></td><td>degradation rate of LuxR</td>
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</tr>
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<tr>
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<td>k4</td><td></td><td>degradation rate of LuxRC</td>
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</tr>
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<tr>
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<td>k5</td><td></td><td>expression rate of tTA</td>
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</tr>
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<tr>
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<td>k6</td><td></td><td>activation coefficient of LuxRC</td>
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</tr>
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<tr>
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<td>k7</td><td></td><td>degradation rate of tTA</td>
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</tr>
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<tr>
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<td>k8</td><td></td><td>basal expression rate of tTA</td>
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</tr>
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<tr>
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<td>k9</td><td></td><td>expression rate of ADE2</td>
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</tr>
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<tr>
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<td>k10</td><td></td><td>activation coefficient of tTA</td>
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</tr>
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<tr>
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<td>k11</td><td></td><td>degradation rate of ADE2</td>
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</tr>
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<tr>
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<td>k12</td><td></td><td>basal expression rate of ADE2</td>
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</tr>
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</table>
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<h3>Equations</h3>
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<p>
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LuxR protein is synthesize at a constant rate k1.
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AHL binds to LuxR to form a complex.
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Then AHL-LuxR complex dimerizes to form a transcription factor<sup>3</sup>.
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</p>
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<img class="center" src="NotFound"/>
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<img class="center" src="NotFound"/>
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<p>
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Activation of tTA expression is modeled using Hill function.
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Hill functions is commonly used to model the interactions between transcription factors and promoters4.
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The transcription factor cooperativity is 1 (single binding site).
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k5 is the expression rate of tTA if the promoter is fully activated.
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</p>
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<img class="center" src="NotFound"/>
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<p>
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Activation of ADE2 expression is also modeled in Hill function.
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</p>
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<img class="center" src="NotFound"/>
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<h3>Initial Conditions</h3>
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<table border="1" class="center">
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<tr>
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<th>Species</th><th>Concentration</th>
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</tr>
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<tr>
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<td>AHL</td><td></td>
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</tr>
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<tr>
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<td>LuxR</td><td></td>
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</tr>
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<tr>
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<td>LuxRC</td><td></td>
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</tr>
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<tr>
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<td>tTA</td><td></td>
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</tr>
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<tr>
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<td>tTAC</td><td></td>
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</tr>
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<tr>
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<td>ADE2</td><td></td>
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</tr>
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</table>
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<h2>Parameter Estimation</h2>
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<p>
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There are many parameters in the ODE equations.
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While some of the parameters can be found or adapted from literature,
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yet other parameters were estimated from experiment data.
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The parameters are estimated by fitting the model to experiment data.
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The model is a set of ODE equations depending on time t with unknown parameters:
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<img class="center" src="NotFound"/>
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where <span class="math"><b>x</b></span> are concentration of the species and 
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<span class="math"><b>θ</b></span> are the parameters.
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</p>
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<p>
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The distance between the experiment observations and model predictions are expressed as a cost function.
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A simple cost function is the Euclid distance.
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The objective is to search for parameters that minimize the cost function:
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<img class="center" src="NotFound"/>
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where <span class="math">x<sub>exp</sub></span> is the experiment data.
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</p>
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<h2>Sensitivity Analysis</h2>
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<p>
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Among all species considered in the model,
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initial AHL concentration is the main factor that determines the output of the system.
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The main output of the system is the color of the yeast which is correlated with the concentration of ADE2.
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The relationship between the concentration of ADE2 and the initial concentration of AHL will be analyzed.
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</p>
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<h2>References</h2>
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<ol>
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<li>Gossen, M. &amp; Bujard, H. Tight control of gene expression in mammalian cells by tetracycline-responsive promoters.
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<i>Proc. Natl. Acad. Sci.</i> <b>89</b>, 5547–5551 (1992).</li>
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<li>Funahashi, A., Morohashi, M., Kitano, H. &amp; Tanimura, N.
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CellDesigner: a process diagram editor for gene-regulatory and biochemical networks.
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<i>BIOSILICO</i> <b>1</b>, 159–162 (2003).</li>
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<li>Basu, S., Gerchman, Y., Collins, C. H., Arnold, F. H. &amp; Weiss, R.
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A synthetic multicellular system for programmed pattern formation. <i>Nature</i> <b>434</b>, 1130–1134 (2005).</li>
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<li>Goutelle, S. et al.
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The Hill equation: a review of its capabilities in pharmacological modelling.
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<i>Fundam. Clin. Pharmacol.</i> <b>22</b>, 633–648 (2008).</li>
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</ol>
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</div>
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</div>
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</div>
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</body></html>
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Latest revision as of 13:03, 25 September 2013

Modeling

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