http://2009.igem.org/wiki/index.php?title=Team:McGill/Modeling&feed=atom&action=historyTeam:McGill/Modeling - Revision history2024-03-29T07:38:44ZRevision history for this page on the wikiMediaWiki 1.16.5http://2009.igem.org/wiki/index.php?title=Team:McGill/Modeling&diff=164331&oldid=prevFnaqib at 01:54, 22 October 20092009-10-22T01:54:09Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The purpose of the following mathematical modeling is to gain insight into activation-inhibition signaling. We begin by describing the mathematical model employed to simulate our biological system. Then we examine the dynamics of one activation site and one inhibitory site, what we call an oscillator, and the effect of distance. We also examine the role two parameters, the Hill coefficient and diffusion rate, have on the oscillatory behavior. This leads to the next examination of two oscillators and the effect of varying the distance between the oscillators but holding the distance between the sites within an oscillator constant. Here we find several unexpected dynamics that we further explore. Finally we look into two dimensional modeling. However because of the size of the .avi files we will not discuss this modeling here but save it for our presentation at the Jamboree.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The purpose of the following mathematical modeling is to gain insight into activation-inhibition signaling. We begin by describing the mathematical model employed to simulate our biological system. Then we examine the dynamics of one activation site and one inhibitory site, what we call an oscillator, and the effect of distance. We also examine the role two parameters, the Hill coefficient and diffusion rate, have on the oscillatory behavior. This leads to the next examination of two oscillators and the effect of varying the distance between the oscillators but holding the distance between the sites within an oscillator constant. Here we find several unexpected dynamics that we further explore. Finally we look into two dimensional modeling. However because of the size of the .avi files we will not discuss this modeling here but save it for our presentation at the Jamboree.</div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">The following represents a brief overview of all of our modeling exercises. We unfortunately did not make use of this wiki throughout the summer and so many of our results have been left out for lack of time to upload. However, all of our results will be available on our poster and oral presentations at the Jamboree. </ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Introduction'''==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Introduction'''==</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image: McGill09Phaseplot_17(V2).jpg|frame|center|Figure 17 - Activation molecule at the activation site of oscillator 1 vs the activation molecule at the activation site of oscillator 2 for distance of 17 - This was compiled from a simulation lasting 0.156 time units. Notice the colours correspond to numerical simulation time not real time.]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image: McGill09Phaseplot_17(V2).jpg|frame|center|Figure 17 - Activation molecule at the activation site of oscillator 1 vs the activation molecule at the activation site of oscillator 2 for distance of 17 - This was compiled from a simulation lasting 0.156 time units. Notice the colours correspond to numerical simulation time not real time.]]</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Comparing figure 11 to figure 12 <del class="diffchange diffchange-inline">let's </del>us believe that the oscillations in figure 11 are periodic while those from figure 12 are quasiperiodic. This is further illustrated in figure 13 where we can see that on each cycle there is a small shift in the oscillations, which is characteristic of quaisperiodic curves. In order to further investigate these dynamics we look at the Poincare maps of each curve.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Comparing figure 11 to figure 12 <ins class="diffchange diffchange-inline">lets </ins>us believe that the oscillations in figure 11 are periodic while those from figure 12 are quasiperiodic. This is further illustrated in figure 13 where we can see that on each cycle there is a small shift in the oscillations, which is characteristic of quaisperiodic curves. In order to further investigate these dynamics we look at the Poincare maps of each curve.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image: McGill90Poincare_15.jpg|frame|center|Figure 18 - Poincare map for separation distance of 15 intervals - Since the number of dots do not increase with more iterations we are convinced that this is a periodic oscillation.]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image: McGill90Poincare_15.jpg|frame|center|Figure 18 - Poincare map for separation distance of 15 intervals - Since the number of dots do not increase with more iterations we are convinced that this is a periodic oscillation.]]</div></td></tr>
</table>Fnaqibhttp://2009.igem.org/wiki/index.php?title=Team:McGill/Modeling&diff=163856&oldid=prevFnaqib: /* Two Oscillators */2009-10-22T01:38:10Z<p><span class="autocomment">Two Oscillators</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image: McGill09Amplitude_activation_activationsite_oscillator2_vs_distance.png|frame|center|Figure 10 - Amplitude of Oscillations of Activation Molecule at Activation Site of Oscillator 2 vs Separation Distance - Notice that between the distances 14-18 there appears to be more than 2 amplitude values per distance.]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image: McGill09Amplitude_activation_activationsite_oscillator2_vs_distance.png|frame|center|Figure 10 - Amplitude of Oscillations of Activation Molecule at Activation Site of Oscillator 2 vs Separation Distance - Notice that between the distances 14-18 there appears to be more than 2 amplitude values per distance.]]</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>We can now begin to understand why the frequency calculating method failed. This method assumes simple periodic motion (similar to a <del class="diffchange diffchange-inline">sin </del>or cosine wave), however the curves for which a frequency could not be calculated have more than 2 values where their derivative equals zero. This hints that they are not simple periodic curves. The following figure illustrates the concentration of the activation molecule at the activation site of oscillator 2 for the first distance that results in an oscillation whose frequency cannot be computed.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>We can now begin to understand why the frequency calculating method failed. This method assumes simple periodic motion (similar to a <ins class="diffchange diffchange-inline">sine </ins>or cosine wave), however the curves for which a frequency could not be calculated have more than 2 values where their derivative equals zero. This hints that they are not simple periodic curves. The following figure illustrates the concentration of the activation molecule at the activation site of oscillator 2 for the first distance that results in an oscillation whose frequency cannot be computed.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image: McGill09Activation_activationsite_oscillator2_13.jpg|frame|center|Figure 11 - Concentration of activation molecule at the activation site of oscillator 2 for a separation distance of 13 - There appears to be a constant "resetting" of the oscillations. As if a threshold is constantly being crossed that induces a reset of the oscillatory behaviour.]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image: McGill09Activation_activationsite_oscillator2_13.jpg|frame|center|Figure 11 - Concentration of activation molecule at the activation site of oscillator 2 for a separation distance of 13 - There appears to be a constant "resetting" of the oscillations. As if a threshold is constantly being crossed that induces a reset of the oscillatory behaviour.]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>It is immediately obvious why the frequency measuring numerical method failed, the oscillations are far from simple. Before investigating this curve further, let's take a look at the other separation distances that resulted in complex oscillations.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>It is immediately obvious why the frequency measuring numerical method failed, the oscillations are far from simple. Before investigating this curve further, let's take a look at the other separation distances that resulted in complex oscillations.</div></td></tr>
</table>Fnaqibhttp://2009.igem.org/wiki/index.php?title=Team:McGill/Modeling&diff=163781&oldid=prevFnaqib at 01:34, 22 October 20092009-10-22T01:34:12Z<p></p>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">The purpose of the following mathematical modeling is to gain insight into activation-inhibition signaling. We begin by describing the mathematical model employed to simulate our biological system. Then we examine the dynamics of one activation site and one inhibitory site, what we call an oscillator, and the effect of distance. We also examine the role two parameters, the Hill coefficient and diffusion rate, have on the oscillatory behavior. This leads to the next examination of two oscillators and the effect of varying the distance between the oscillators but holding the distance between the sites within an oscillator constant. Here we find several unexpected dynamics that we further explore. Finally we look into two dimensional modeling. However because of the size of the .avi files we will not discuss this modeling here but save it for our presentation at the Jamboree.</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Introduction'''==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Introduction'''==</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The distance between the two oscillators was varied while the distance between two sites within an oscillator was held fixed at 5 intervals. This value was chosen for demonstration purposes, however the dynamics to be described have been observed at various separation distances.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The distance between the two oscillators was varied while the distance between two sites within an oscillator was held fixed at 5 intervals. This value was chosen for demonstration purposes, however the dynamics to be described have been observed at various separation distances.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>We first <del class="diffchange diffchange-inline">look </del>at the change in frequency as the oscillators are moved apart.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>We first <ins class="diffchange diffchange-inline">looked </ins>at the change in frequency as the oscillators are moved apart.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image: McGill09Frequency_vs_Distance_for_Osc2_Geometry1.png|frame|center|Figure 9 - Frequency of Activation Molecule at Activation Site of Oscillator 2 vs Separation Distance – There are eight separation distances that resulted in a frequency of -1: 14,15,16,17,474,475,476, and 477. This value is used as a marker to denote a curve who's frequency did not converge to a single value. As will be seen later, these distances resulted in complex dynamics.]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image: McGill09Frequency_vs_Distance_for_Osc2_Geometry1.png|frame|center|Figure 9 - Frequency of Activation Molecule at Activation Site of Oscillator 2 vs Separation Distance – There are eight separation distances that resulted in a frequency of -1: 14,15,16,17,474,475,476, and 477. This value is used as a marker to denote a curve who's frequency did not converge to a single value. As will be seen later, these distances resulted in complex dynamics.]]</div></td></tr>
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</table>Fnaqibhttp://2009.igem.org/wiki/index.php?title=Team:McGill/Modeling&diff=163397&oldid=prevFnaqib: /* One Oscillator */2009-10-22T01:24:09Z<p><span class="autocomment">One Oscillator</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Notice the range of distances that have two values plotted correspond to the range of distances that have a nonzero frequency in figure 4.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Notice the range of distances that have two values plotted correspond to the range of distances that have a nonzero frequency in figure 4.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">A </del>concern <del class="diffchange diffchange-inline">we have </del>is that it is unlikely the parameters <del class="diffchange diffchange-inline">we have chosen </del>are representative of the synthetic genetic network we are constructing <del class="diffchange diffchange-inline">in parallel</del>. The major parameter that we currently have no control over <del class="diffchange diffchange-inline">whatsoever </del>is the Hill coefficient n. We believe that if the Hill function is steep enough then the circuit will act as an on/off switch and oscillations are possible. Increasing n will only increase the steepness of the Hill function so we are safe in assuming that n > 8 will still lead to oscillations. However what is the smallest n that would still allow oscillatory behaviour? We calculated frequency for the range of separation distances with the following Hill coefficients n=2,3 and 4.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">One </ins>concern is that it is unlikely the <ins class="diffchange diffchange-inline">chosen </ins>parameters are representative of the synthetic genetic network <ins class="diffchange diffchange-inline">that </ins>we are constructing. The major parameter that we currently have no control over is the Hill coefficient n. We believe that if the Hill function is steep enough then the circuit will act as an on/off switch and oscillations are possible. Increasing n will only increase the steepness of the Hill function so we are safe in assuming that n > 8 will still lead to oscillations. However what is the smallest n that would still allow oscillatory behaviour? We calculated frequency for the range of separation distances with the following Hill coefficients n=2,3 and 4.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image: Mcgill09OneOscFreq_vs_Distance_n4.png|frame|center|Figure 6 - Frequency of Activation Molecule at Activation Site vs Separation Distance (n=4) - Oscillations still present for a similar range of separation distances as n=8 but with lower frequencies.]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image: Mcgill09OneOscFreq_vs_Distance_n4.png|frame|center|Figure 6 - Frequency of Activation Molecule at Activation Site vs Separation Distance (n=4) - Oscillations still present for a similar range of separation distances as n=8 but with lower frequencies.]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image: Mcgill09OneOscFreq_vs_Distance_n3.png|frame|center|Figure 7 - Frequency of Activation Molecule at Activation Site vs Separation Distance (n=3) - No oscillations observed.]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image: Mcgill09OneOscFreq_vs_Distance_n3.png|frame|center|Figure 7 - Frequency of Activation Molecule at Activation Site vs Separation Distance (n=3) - No oscillations observed.]]</div></td></tr>
</table>Fnaqibhttp://2009.igem.org/wiki/index.php?title=Team:McGill/Modeling&diff=163335&oldid=prevFnaqib: /* Two Dimensional Modeling */2009-10-22T01:22:20Z<p><span class="autocomment">Two Dimensional Modeling</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Two Dimensional Modeling'''==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Two Dimensional Modeling'''==</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">We were able to further observe oscillatory behaviour in two dimensions, unfortunately the .avi files we have are too large to upload onto the wiki. This is all we have right now to describe our efforts into two dimensional modeling. We'll hopefully be describing them in detail during our presentation at the Jamboree.</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Appendix A - Parameters'''==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Appendix A - Parameters'''==</div></td></tr>
</table>Fnaqibhttp://2009.igem.org/wiki/index.php?title=Team:McGill/Modeling&diff=162797&oldid=prevFnaqib at 01:05, 22 October 20092009-10-22T01:05:44Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>It is currently not well understood why a range of separation distances past the quasiperiodic curves remain 180 degrees out of phase.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>It is currently not well understood why a range of separation distances past the quasiperiodic curves remain 180 degrees out of phase.</div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">=='''Two Dimensional Modeling'''==</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Appendix A - Parameters'''==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Appendix A - Parameters'''==</div></td></tr>
</table>Fnaqibhttp://2009.igem.org/wiki/index.php?title=Team:McGill/Modeling&diff=162542&oldid=prevFnaqib: /* One Oscillator */2009-10-22T00:58:59Z<p><span class="autocomment">One Oscillator</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In order to observe oscillations we require a Hill coefficient of at least n=4.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In order to observe oscillations we require a Hill coefficient of at least n=4.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">We also looked into the dependence on the diffusion rate, D, and the presence of oscillations. As can be imagined, if the diffusion rate is too slow or too fast oscillations will not occur. This was confirmed with preliminary simulations where we found a particular separation distance that loses oscillatory behaviour if the diffusion rate is outside a critical range. We endeavoured to form a bifurcation diagram showing the ranges of separation distance and diffusion rate where oscillations occur, however this task is computationally intensive and we have not completed it as of this writing. We hope to have this result in time for the Jamboree.</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Two Oscillators'''==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Two Oscillators'''==</div></td></tr>
</table>Fnaqibhttp://2009.igem.org/wiki/index.php?title=Team:McGill/Modeling&diff=162292&oldid=prevFnaqib: /* Two Oscillators */2009-10-22T00:51:58Z<p><span class="autocomment">Two Oscillators</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Two Oscillators'''==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Two Oscillators'''==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>We next looked at a system consisting of two oscillators, where each consists of an activation and inhibitory site.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>We next looked at a system consisting of two oscillators, where each consists of an activation and inhibitory site.</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Image: Mcgill09TwoOscillatorsGeometry.png|frame|center|Figure <del class="diffchange diffchange-inline">5 </del>– Two Oscillators – Each oscillator consists of one activation and one inhibitory site. There are two arrangements of the system: BA AB, termed the symmetrical system, and BA BA, the unsymmetrical system. In this document we only discuss the symmetrical system. Oscillator 2 travels around the ring.]]</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Image: Mcgill09TwoOscillatorsGeometry.png|frame|center|Figure <ins class="diffchange diffchange-inline">8 </ins>– Two Oscillators – Each oscillator consists of one activation and one inhibitory site. There are two arrangements of the system: BA AB, termed the symmetrical system, and BA BA, the unsymmetrical system. In this document we only discuss the symmetrical system. Oscillator 2 travels around the ring.]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The distance between the two oscillators was varied while the distance between two sites within an oscillator was held fixed at 5 intervals. This value was chosen for demonstration purposes, however the dynamics to be described have been observed at various separation distances.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The distance between the two oscillators was varied while the distance between two sites within an oscillator was held fixed at 5 intervals. This value was chosen for demonstration purposes, however the dynamics to be described have been observed at various separation distances.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>We first look at the change in frequency as the oscillators are moved apart.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>We first look at the change in frequency as the oscillators are moved apart.</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Image: McGill09Frequency_vs_Distance_for_Osc2_Geometry1.png|frame|center|Figure <del class="diffchange diffchange-inline">6 </del>- Frequency of Activation Molecule at Activation Site of Oscillator 2 vs Separation Distance – There are eight separation distances that resulted in a frequency of -1: 14,15,16,17,474,475,476, and 477. This value is used as a marker to denote a curve who's frequency did not converge to a single value. As will be seen later, these distances resulted in complex dynamics.]]</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Image: McGill09Frequency_vs_Distance_for_Osc2_Geometry1.png|frame|center|Figure <ins class="diffchange diffchange-inline">9 </ins>- Frequency of Activation Molecule at Activation Site of Oscillator 2 vs Separation Distance – There are eight separation distances that resulted in a frequency of -1: 14,15,16,17,474,475,476, and 477. This value is used as a marker to denote a curve who's frequency did not converge to a single value. As will be seen later, these distances resulted in complex dynamics.]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The first striking feature of this graph is the appearance of separation distances that have negative frequencies. The value of -1 is assigned to a separation distance that results in a complex oscillation whose frequency cannot be resolved using a simple method, which assumes simple periodic motion. Before we look into these curves, let's look over the entire graph. When the two oscillators are very close together, their frequency is always less than an isolated oscillator (look at figure 4 for two sites with a separation distance of 5 intervals). Interestingly, between the separation distances of 1 and 10 there appears to be a local minimum frequency. Meaning as the oscillators are moved apart, their frequency initially decreases and then begins increasing. This trend of increasing frequency continues until the numerical method calculating frequency breaks down. To get a better idea of why the frequency calculating method is not converging let's look at the amplitude of oscillations vs separation distance:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The first striking feature of this graph is the appearance of separation distances that have negative frequencies. The value of -1 is assigned to a separation distance that results in a complex oscillation whose frequency cannot be resolved using a simple method, which assumes simple periodic motion. Before we look into these curves, let's look over the entire graph. When the two oscillators are very close together, their frequency is always less than an isolated oscillator (look at figure 4 for two sites with a separation distance of 5 intervals). Interestingly, between the separation distances of 1 and 10 there appears to be a local minimum frequency. Meaning as the oscillators are moved apart, their frequency initially decreases and then begins increasing. This trend of increasing frequency continues until the numerical method calculating frequency breaks down. To get a better idea of why the frequency calculating method is not converging let's look at the amplitude of oscillations vs separation distance:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Image: McGill09Amplitude_activation_activationsite_oscillator2_vs_distance.png|frame|center|Figure <del class="diffchange diffchange-inline">7 </del>- Amplitude of Oscillations of Activation Molecule at Activation Site of Oscillator 2 vs Separation Distance - Notice that between the distances 14-18 there appears to be more than 2 amplitude values per distance.]]</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Image: McGill09Amplitude_activation_activationsite_oscillator2_vs_distance.png|frame|center|Figure <ins class="diffchange diffchange-inline">10 </ins>- Amplitude of Oscillations of Activation Molecule at Activation Site of Oscillator 2 vs Separation Distance - Notice that between the distances 14-18 there appears to be more than 2 amplitude values per distance.]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>We can now begin to understand why the frequency calculating method failed. This method assumes simple periodic motion (similar to a sin or cosine wave), however the curves for which a frequency could not be calculated have more than 2 values where their derivative equals zero. This hints that they are not simple periodic curves. The following figure illustrates the concentration of the activation molecule at the activation site of oscillator 2 for the first distance that results in an oscillation whose frequency cannot be computed.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>We can now begin to understand why the frequency calculating method failed. This method assumes simple periodic motion (similar to a sin or cosine wave), however the curves for which a frequency could not be calculated have more than 2 values where their derivative equals zero. This hints that they are not simple periodic curves. The following figure illustrates the concentration of the activation molecule at the activation site of oscillator 2 for the first distance that results in an oscillation whose frequency cannot be computed.</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Image: McGill09Activation_activationsite_oscillator2_13.jpg|frame|center|Figure <del class="diffchange diffchange-inline">7 </del>- Concentration of activation molecule at the activation site of oscillator 2 for a separation distance of 13 - There appears to be a constant "resetting" of the oscillations. As if a threshold is constantly being crossed that induces a reset of the oscillatory behaviour.]]</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Image: McGill09Activation_activationsite_oscillator2_13.jpg|frame|center|Figure <ins class="diffchange diffchange-inline">11 </ins>- Concentration of activation molecule at the activation site of oscillator 2 for a separation distance of 13 - There appears to be a constant "resetting" of the oscillations. As if a threshold is constantly being crossed that induces a reset of the oscillatory behaviour.]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>It is immediately obvious why the frequency measuring numerical method failed, the oscillations are far from simple. Before investigating this curve further, let's take a look at the other separation distances that resulted in complex oscillations.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>It is immediately obvious why the frequency measuring numerical method failed, the oscillations are far from simple. Before investigating this curve further, let's take a look at the other separation distances that resulted in complex oscillations.</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Image: McGill09Activation_activationsite_oscillator2_14.jpg|frame|center|Figure <del class="diffchange diffchange-inline">8 </del>- Concentration of activation molecule at the activation site of oscillator 2 for a separation distance of 14]]</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Image: McGill09Activation_activationsite_oscillator2_14.jpg|frame|center|Figure <ins class="diffchange diffchange-inline">12 </ins>- Concentration of activation molecule at the activation site of oscillator 2 for a separation distance of 14]]</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Image: McGill09Activation_activationsite_oscillator2_15.jpg|frame|center|Figure <del class="diffchange diffchange-inline">9 </del>- Concentration of activation molecule at the activation site of oscillator 2 for a separation distance of 15]]</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Image: McGill09Activation_activationsite_oscillator2_15.jpg|frame|center|Figure <ins class="diffchange diffchange-inline">13 </ins>- Concentration of activation molecule at the activation site of oscillator 2 for a separation distance of 15]]</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Image: McGill09Activation_activationsite_oscillator2_16.jpg|frame|center|Figure <del class="diffchange diffchange-inline">10 </del>- Concentration of activation molecule at the activation site of oscillator 2 for a separation distance of 16]]</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Image: McGill09Activation_activationsite_oscillator2_16.jpg|frame|center|Figure <ins class="diffchange diffchange-inline">14 </ins>- Concentration of activation molecule at the activation site of oscillator 2 for a separation distance of 16]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The separation distances 474,475,476, and 477 resulted in similar dynamics and are not shown for brevity. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The separation distances 474,475,476, and 477 resulted in similar dynamics and are not shown for brevity. </div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In order to further classify these dynamics as either aperiodic, quasiperiodic, or chaotic we must look further than the time evolution graphs. The phase plots begin to describe what type of oscillations are present. We plot the concentration of the activation molecule at the activation site of oscillator 1 vs the activation molecule at the activation site of oscillator 2.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In order to further classify these dynamics as either aperiodic, quasiperiodic, or chaotic we must look further than the time evolution graphs. The phase plots begin to describe what type of oscillations are present. We plot the concentration of the activation molecule at the activation site of oscillator 1 vs the activation molecule at the activation site of oscillator 2.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Image: McGill09Phaseplot_15.jpg|frame|center|Figure <del class="diffchange diffchange-inline">11 </del>- Activation molecule at the activation site of oscillator 1 vs the activation molecule at the activation site of oscillator 2 for distance of 15 - This was compiled from a simulation lasting 100 time units.]]</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Image: McGill09Phaseplot_15.jpg|frame|center|Figure <ins class="diffchange diffchange-inline">15 </ins>- Activation molecule at the activation site of oscillator 1 vs the activation molecule at the activation site of oscillator 2 for distance of 15 - This was compiled from a simulation lasting 100 time units.]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Image: McGill09Phaseplot_17.jpg|frame|center|Figure <del class="diffchange diffchange-inline">12 </del>- Activation molecule at the activation site of oscillator 1 vs the activation molecule at the activation site of oscillator 2 for distance of 17 - This was compiled from a simulation lasting 100 time units.]]</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Image: McGill09Phaseplot_17.jpg|frame|center|Figure <ins class="diffchange diffchange-inline">16 </ins>- Activation molecule at the activation site of oscillator 1 vs the activation molecule at the activation site of oscillator 2 for distance of 17 - This was compiled from a simulation lasting 100 time units.]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Image: McGill09Phaseplot_17(V2).jpg|frame|center|Figure <del class="diffchange diffchange-inline">13 </del>- Activation molecule at the activation site of oscillator 1 vs the activation molecule at the activation site of oscillator 2 for distance of 17 - This was compiled from a simulation lasting 0.156 time units. Notice the colours correspond to numerical simulation time not real time.]]</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Image: McGill09Phaseplot_17(V2).jpg|frame|center|Figure <ins class="diffchange diffchange-inline">17 </ins>- Activation molecule at the activation site of oscillator 1 vs the activation molecule at the activation site of oscillator 2 for distance of 17 - This was compiled from a simulation lasting 0.156 time units. Notice the colours correspond to numerical simulation time not real time.]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Comparing figure 11 to figure 12 let's us believe that the oscillations in figure 11 are periodic while those from figure 12 are quasiperiodic. This is further illustrated in figure 13 where we can see that on each cycle there is a small shift in the oscillations, which is characteristic of quaisperiodic curves. In order to further investigate these dynamics we look at the Poincare maps of each curve.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Comparing figure 11 to figure 12 let's us believe that the oscillations in figure 11 are periodic while those from figure 12 are quasiperiodic. This is further illustrated in figure 13 where we can see that on each cycle there is a small shift in the oscillations, which is characteristic of quaisperiodic curves. In order to further investigate these dynamics we look at the Poincare maps of each curve.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Image: McGill90Poincare_15.jpg|frame|center|Figure <del class="diffchange diffchange-inline">14 </del>- Poincare map for separation distance of 15 intervals - Since the number of dots do not increase with more iterations we are convinced that this is a periodic oscillation.]]</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Image: McGill90Poincare_15.jpg|frame|center|Figure <ins class="diffchange diffchange-inline">18 </ins>- Poincare map for separation distance of 15 intervals - Since the number of dots do not increase with more iterations we are convinced that this is a periodic oscillation.]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Image: McGill90Poincare_17.jpg|frame|center|Figure <del class="diffchange diffchange-inline">15 </del>- Poincare map for separation distance of 17 intervals - The number of dots increase with each iteration, meaning this is truly a quasiperiodic motion.]]</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Image: McGill90Poincare_17.jpg|frame|center|Figure <ins class="diffchange diffchange-inline">19 </ins>- Poincare map for separation distance of 17 intervals - The number of dots increase with each iteration, meaning this is truly a quasiperiodic motion.]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Since the Poincare map of the oscillations present at a separation distance of 17 intervals leads to a filled in closed shape we are fairly confident that we have found quasiperiodic motion. Although the curve seen at a separation distance of 15 also looks quasiperiodic, the Poincare map does not form a filled in closed shape we can conclude that this separation distance leads to complex oscillations but not quasiperiodic.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Since the Poincare map of the oscillations present at a separation distance of 17 intervals leads to a filled in closed shape we are fairly confident that we have found quasiperiodic motion. Although the curve seen at a separation distance of 15 also looks quasiperiodic, the Poincare map does not form a filled in closed shape we can conclude that this separation distance leads to complex oscillations but not quasiperiodic.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The final analysis that we will perform on these simulations is to look at the phase lag between the two oscillators. We again move the second oscillator around the ring while measuring the phase lag between the activation molecule at the activation site of the first oscillator and the activation molecule at the activation site of the second oscillator.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The final analysis that we will perform on these simulations is to look at the phase lag between the two oscillators. We again move the second oscillator around the ring while measuring the phase lag between the activation molecule at the activation site of the first oscillator and the activation molecule at the activation site of the second oscillator.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Image: McGill09PhaselagG1.png|frame|center|Figure <del class="diffchange diffchange-inline">16 </del>- Phase lag between the activation molecule at the activation site of the first oscillator and the activation molecule at the activation site of the second oscillator - The phase lag is calculated by finding the lag in real time and then dividing by the period of oscillations. Negative phase lag values correspond to those separation distances where a frequency could not be calculated.]]</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Image: McGill09PhaselagG1.png|frame|center|Figure <ins class="diffchange diffchange-inline">20 </ins>- Phase lag between the activation molecule at the activation site of the first oscillator and the activation molecule at the activation site of the second oscillator - The phase lag is calculated by finding the lag in real time and then dividing by the period of oscillations. Negative phase lag values correspond to those separation distances where a frequency could not be calculated.]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>A striking result of the phase lag calculation is that for some range of distances past the quasiperiodic forming distances, all the oscillations are exactly 180 degrees out of phase! A phase plot gives us more insight into these dynamics.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>A striking result of the phase lag calculation is that for some range of distances past the quasiperiodic forming distances, all the oscillations are exactly 180 degrees out of phase! A phase plot gives us more insight into these dynamics.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Image: McGill09Phaseplot_20.png|frame|center|Figure <del class="diffchange diffchange-inline">17 </del>- Activation molecule at the activation site of oscillator 1 vs the activation molecule at the activation site of oscillator 2 for distance of 20 - The perfect reflection across the line y=x means the two curves are always 180 degrees out of phase.]]</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Image: McGill09Phaseplot_20.png|frame|center|Figure <ins class="diffchange diffchange-inline">21 </ins>- Activation molecule at the activation site of oscillator 1 vs the activation molecule at the activation site of oscillator 2 for distance of 20 - The perfect reflection across the line y=x means the two curves are always 180 degrees out of phase.]]</div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">It is currently not well understood why a range of separation distances past the quasiperiodic curves remain 180 degrees out of phase.</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Appendix A - Parameters'''==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Appendix A - Parameters'''==</div></td></tr>
</table>Fnaqibhttp://2009.igem.org/wiki/index.php?title=Team:McGill/Modeling&diff=161933&oldid=prevFnaqib: /* Introduction */2009-10-22T00:38:49Z<p><span class="autocomment">Introduction</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Two types of signaling molecules exist: activating and inhibiting. Each molecule is synthesized by a unique strain of cells and affects the synthesis rate of the other strain.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Two types of signaling molecules exist: activating and inhibiting. Each molecule is synthesized by a unique strain of cells and affects the synthesis rate of the other strain.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Image: Mcgill09Projectfig1.png|frame|center|Figure 1 – Activation-inhibition intercellular signaling – Activating molecule (A) <del class="diffchange diffchange-inline">synthesized </del>and diffuses to increase synthesis of inhibiting molecule (B) in secondary strain. Inhibiting molecule also diffuses back to initial cell and decreases synthesis of activating molecule.]]</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Image: Mcgill09Projectfig1.png|frame|center|Figure 1 – Activation-inhibition intercellular signaling – Activating molecule (A) <ins class="diffchange diffchange-inline">synthesizes </ins>and diffuses to increase synthesis of inhibiting molecule (B) in secondary strain. Inhibiting molecule also diffuses back to initial cell and decreases synthesis of activating molecule.]]</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>This is modeled using the following system of PDEs:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>This is modeled using the following system of PDEs:</div></td></tr>
</table>Fnaqibhttp://2009.igem.org/wiki/index.php?title=Team:McGill/Modeling&diff=160355&oldid=prevFnaqib: /* Two Oscillators */2009-10-21T23:50:14Z<p><span class="autocomment">Two Oscillators</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image: McGill09PhaselagG1.png|frame|center|Figure 16 - Phase lag between the activation molecule at the activation site of the first oscillator and the activation molecule at the activation site of the second oscillator - The phase lag is calculated by finding the lag in real time and then dividing by the period of oscillations. Negative phase lag values correspond to those separation distances where a frequency could not be calculated.]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image: McGill09PhaselagG1.png|frame|center|Figure 16 - Phase lag between the activation molecule at the activation site of the first oscillator and the activation molecule at the activation site of the second oscillator - The phase lag is calculated by finding the lag in real time and then dividing by the period of oscillations. Negative phase lag values correspond to those separation distances where a frequency could not be calculated.]]</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">A striking result of the phase lag calculation is that for some range of distances past the quasiperiodic forming distances, all the oscillations are exactly 180 degrees out of phase! A phase plot gives us more insight into these dynamics.</ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">[[Image: McGill09Phaseplot_20.png|frame|center|Figure 17 - Activation molecule at the activation site of oscillator 1 vs the activation molecule at the activation site of oscillator 2 for distance of 20 - The perfect reflection across the line y=x means the two curves are always 180 degrees out of phase.]]</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Appendix A - Parameters'''==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=='''Appendix A - Parameters'''==</div></td></tr>
</table>Fnaqib