Team:iHKU/modeling

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background-image:url(http://2008.igem.org/wiki/images/e/ef/Leftext.jpg); float:left; } {	width:745px; float:left; } {	width:705px; float:left; } {	width:40px; height:700px; float:right; background-image:url(http://2008.igem.org/wiki/images/7/7e/Rightside.jpg); } {	width: 40px; height: 746px; background-image:url(http://2008.igem.org/wiki/images/7/7e/Rightside.jpg); } {	width:40px; background-image:url(http://2008.igem.org/wiki/images/6/6b/Rightext.jpg); } {	width:705px; height:65px; background-image:url(http://2008.igem.org/wiki/images/4/4e/Up2.jpg); float:left; } {	width:705px; height:59px; background-image:url(http://2008.igem.org/wiki/images/5/5e/Header.jpg); float:left; } {	width:705px; background-image:url(http://2008.igem.org/wiki/images/3/38/Contentbg.jpg); float: left; } {	width:705px; height:59px; background-image:url(http://2008.igem.org/wiki/images/e/e9/Footer.jpg); float:left; } .style3 {font-size: 12px} .style7 {font-size: 18px}   Fig. 2 The movie of the wild type pattern obtained by model in 2D(left) and 3D(right)  (Click here to see the movie obtained by experiments                            )                            [Back to Top]  Density Dependent Motility  Fig. 3 Designed genetic circuit In experiments, we designed a circuit that the cell motility was repressed by cell density                                                                      ρ. When the cell density    ρ was high, the  diffusion coefficient     D became small. Therefore, the fisher’s equation as equation (2.1) was not valid in this  case any more. In order to be simple, we firstly considered  the one dimension problem again. We assumed the cell density at point x was     ρ (x) at time t. In a very short time τ, there would be two groups of cell at x moving into its nearby points x - δ , x + δ, due to the random walk. And the amount of cell in each group were proportional to the product of D ( ρ ( x )) and ρ ( x ). Therefore, (3.1) In the limit τ --&gt;0 and δ --&gt;0, we obtain (3.2)  And the function Dρ ( ρ ) was a decrease  function. For a simplest case, we considered a Heaviside Function(Fig.4). There was a threshold of cell density above which the cell can not swim forward but always tumble in the same place. Considering five points with the different cell denisty, among which there were group cell swimming to the nearby sites, we assumed that only the cell density in the middle of which was larger than the threshold. Then the cell in the middle would not go out of it. At the same time, the cell nearby would incessantly come into this point. As a result, the cell density here would increase, while the cell density nearby would continually decrease until zero.  The numerical simulations gave us the results shown in Fig.5. The cell density showed a periodical-narrow-peak structure. These peaks were what we wanted, as they produced some regions of low cell density, though the whole pattern was not quite similar with that of experiments. And the exact pattern would come out when we took account of the other parts in the whole genetic circuit(See Full Model</a>). <img src="/wiki/images/d/dc/Modelnew2.png" width="561" height="272" /> Fig. 5 Periodical-narrow-peak pattern in 2D(left); the cell density distribution along the radius(right) [Back to Top]</a> </a> Full Model of Density Dependent Motility <img src="/wiki/images/0/01/Modelling_pic10.png" width="386" height="242" /> Fig. 6 The entire designed genetic circuit Actually, in our genetic circuit, we transformed a plasmid which can secrete AHL to environment (Fig.6). When the AHL density h of environment was high, the AHL came into the cell. Then AHL combining with LuxR repressed the expression level of cheZ which controled the motility of E.coli. So it was necessary to take account of the AHL effect. First, the AHL was synthesized by the E.coli cell at the rate λ. And the degradation rate β of AHL whose half life is normally about 15 to 30 minutes. Considering the diffusion of AHL, we obtained <img src="/wiki/images/e/ee/HkumodelEq5.gif" width="500" height="69">(4.1) where Dh is the diffusion coefficient of AHL, which is about 0.001mm2/min[6]. And the cell diffusion coefficient of E.coli is determined by the density of AHL. Therefore <img src="/wiki/images/6/60/ModelEq6.gif" width="500" height="69">(4.2) where Dρ ( h ) is a decreasing function of h. Furthermore, the nutrient consumption influences the growth rate of E.coli. Therefore <img src="/wiki/images/0/00/HkumodelEq7.gif" width="500" height="69"> (4.3) where n is the nutrient concentration.
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<img src="http://igem.org/wiki/images/1/1f/Fig7.gif" width="164" height="140" /> </a>Fig. 7 The growth rate relates to nutrient concentration For the nutrient, we made an assumption that the amount of nutrient consumed were proportional to the amount of cell increasing. Taking account of the nutrient diffusion, we had <img src="/wiki/images/5/58/Modelling_pic12.JPG" width="500" height="58" /> (4.4) where k was the ratio that nutrient converts to cell mass, D n is nutrient diffusion coefficient which is about that of small molecule[7</a>]. We used several possible forms of function Dρ ( h )(see Results Section2</a>). The results showed that if the cell diffusion coefficient decreases fast near the AHL  density threshold, the pattern came out as a multiple-ring one(Results Section2 (a),(e)</a>). On the contrary, there was only one-ring pattern. Therefore, in order to to make this curve Dρ ( h ) decreasing sharply near the threshold, it followed a prediction that we can get a multiple-ring pattern by making an auto-activate genetic circuit which the combination of LuxR and AHL can activate the expression level  of itself. <img src="/wiki/images/f/f4/Modelling_pic13.png" width="329" height="221" /> Fig.8 The designed genetic circuit for predicted pattern Furthermore, we did some two-spot patterns which initially have two points of E.coli cell, so as to compare with the experiments culture (Results Section3</a>). [Back to Top]</a> </a> Modeling Result s                            In the simulation, we tried different values of undetermined parameters which were possible to change in experiments, to see how they influenced the pattern. Generally, we can vary the parameters of cell growth and cell motility which, in experiments, are easy to  change. So we list the results of varying several parameters in the cell growth part and cell motility  part. Cell Growth</a> <ul> <li> Maximum growth rate γ 0</a> </li> <li class="STYLE23"> Initial nutrient concentration n </a> i </li> <li class="STYLE23"> κ</a> </li> <li> k</a> </li> </ul> Cell movement</a> Multiple initial Spots</a> </a> 1. Cell Growth In the cell growth term, we considered the growth rate was related to nutrient concentration. When the nutrient was consumed by the cell, the grow rate of the cell would decrease  (Fig. 7</a>). There was a maximum growth rate γ 0, when the nutrient was rich. The parameter κ described the sharp of this curve(Fig. 7</a>). The larger κ gave us more smooth increasing curve. And we made an assumption that one unit of nutrient would change to cell number with a ratio k. Also in  the initial condition, the nutrient concentration n i can influence  the patterns. Therefore, the effects of all the four parameters were studied in our model. </a><img src="/wiki/images/b/b7/Modelnew3_v2.PNG" width="541" height="441" /> Above reuslts indicated that, the faster cell growth (larger γ 0 and  smaller doubling time) formed a smaller and a little more unclear ring pattern. And the inner ring diameter would become a litter more large when the growth rate increased. </a><img src="/wiki/images/6/6d/Modelnew4.png" width="535" height="441" /> From above figures, it was found that the initial nutrient concentration ni had  little influence to the pattern, especially when it was large. The only effect we observed was that smaller initial nutrient concentration ni  made the ring slightly wider and more clear. </a><img src="/wiki/images/6/6c/Modelnew5.png" width="543" height="444" /> Here, the value of κ was related to the  initial nutrient concentration ni. When initial nutrient concentration ni was fixed, a smaller κ parameter gave us the similar results with that of faster cell  growth. </a><img src="/wiki/images/5/51/Modelnew6.png" width="545" height="447" /> When we increased the value of k, the  patterns did not change a lot, but only became a little more clearly. In conclusion, the simulation results showed that the parameters of maximum growth rate γ o and κ had a more important role on the ring-like pattern. And the other two parameters k and ni did not influence the  pattern much. <p class="style7 style7 style7">2. Cell Motility Dρ ( h ) <a name="Drho" id="Drho"></a> Forms of Dρ ( h ) As Dρ ( h ) was a decreasing function of h, the possible formed of it can be                            The  above movies showed that if the cell diffusion coefficient decreases fast near the AHL  density threshold, the pattern came out as a multiple-ring one(<a href="http://2008.igem.org/Team:iHKU/modeling/movie#a">Results Section2 (a),(e)</a>). On the contrary, there was only one-ring pattern<a href="http://2008.igem.org/Team:iHKU/modeling/movie#b">(Results Section2 (b),(c),(d))</a>. <a name="result3" id="result3"></a>3. Mulitple initial spots With the initial conditions of two or more spots, we obtained the below funny patterns which are amazing similar with that of experiments<a href="/Team:iHKU/result#fun">(results)</a>. <img src="/wiki/images/8/85/Modelnew12.png" width="384" height="508" /> <p class="STYLE28"> </a>Reference: <ul> [1] P.K.Pathria, Statistical Mechanics (Pergamon Press, Headington Hill Hall, Oxford; 1972)</li> [2] Howard C.Berg, Random Walks in Biology (Princeton University Press, Princeton, Hew Jersey; 1993)</li> [3] Nikhil Mittal, et al, Proc Natl Acad Sci 100, 13259 (2003)</li> [4] R. A. Fisher, Ann. Eugenics 7, 353 (1937)</li> [5] A.H.Bokhari, et al., Nonlinear Analysis (In Press)</li> [6]Basu S, et al., Nature 434, 1130(2005)</li> [7]J.W. Costerton, Naomi Balaban, Control of Biofilm Infections by Signal Manipulation (Springer,2008)</li> </ul> <a href="#top">[Back to Top]</a> <td width="10%">