Quick start guide for molecular circadian clock model


Gonze-Goldbeter model[1]

For example, let us consider the circadian oscillator model, which has been proposed by Goldbeter and the co-workers, with an external periodic force in Neurospora. The external periodic force corresponds to a light-dark cycle. Then, the circadian oscillator can be entrained to the external periodic force under a certain condition. However, when the period of the external force and the amplitude of one are changed, the Neimark-Sacker bifurcation occurs. Let us calculate the Neimark-Sacker bifurcation set in the index.php?plugin=attach&refer=GGmodel&openfile=eq_img01.gif-plane. The circadian oscillator model is described by:

index.php?plugin=attach&refer=GGmodel&openfile=GG_eq1.png

where index.php?plugin=attach&refer=GGmodel&openfile=eq_img02.gif represents as follows:

index.php?plugin=attach&refer=GGmodel&openfile=GG_eq2.png

where, the period index.php?plugin=attach&refer=GGmodel&openfile=eq_img03.gif of the external force is replaced by index.php?plugin=attach&refer=GGmodel&openfile=GG_eq3.png, see Ref.[1], [2], and [3] for detail.

In the following, we make the following programs, and use these:

  1. PP
  2. FIX
  3. BF


Configurations

  • First, please refer to the Installation section to install BunKi, then go to the Preferences.
  • Complement. 1
    • If the Symbolic Math Toolbox (MATLAB additional package) is not available, please consult the following sample for a demo. It is a file where we gathered all the analysis tools required for the analysis of the GG model, and the configuration files in a compressed project directory. Please extract its content in any location.
    • Attention:starting BunKi from the MATLAB command in the expanded directory, the SE program starts. If choosing "system create" there, an install without the "Symbolic Math Toolbox" will return an error. If the "Symbolic Math Toolbox" is not available, please skip the Creation of the Programs and directly continue to PP.
  • Complement. 2
    • In the extracted directory of the BunKi program, there is a sample directory.
      /BUNKI/Sample/Gonze-Goldbeter
      The various files placed down this directory are used in the following explanations. It is possible to save some time by using such files instead of filling up each configuration field manually. The files prepared are as follows:
      • fileGG.sed : file containing the model equations: configuration file for use with SE
      • filePd_point.ppd : The example of a stable periodic solution : configuration file for use with PP
      • filePd_point.fixd : For the detection of the Neimark-Sacker bifurcation : configuration file for use with FIX
      • filePd_point_NS.bfd : Setup file for the calculation of the Neimark-Sacker bifurcation of a periodic solution : configuration file for use with BF


Creation of the Programs

The following steps are done in a sub-directory. Any name can be used for the sub-directory. First, the following is executed in the MATLAB command window:

>> BUNKI

After the execution, a window named Select BUNKI Project Directory will open.

index.php?plugin=attach&refer=GGmodel&openfile=project.png

Here the Project to be analyzed is created or selected. At this stage, no Project is generated. Therefore, input a Project name for identification in the Name field and click the OK button. In this case, since the circadian oscillator model is proposed by Goldbeter and is analyzed in detailed by Gonze, GG is used as the project name.*1

index.php?plugin=attach&refer=GGmodel&openfile=project2.png

When creating a new Project, you are asked whether it is okay to create a directory (or folder). Please select Yes. SE (system editor) will start.

index.php?plugin=attach&refer=GGmodel&openfile=startSE.png

We now create the programs for analysis.*2

  • Description of system
    • First, we select what type of system is going to be analyzed. The GG model has three variables and is an Non-autonomous system. Therefore, the Dimension is set to 3,and Nonautonomous is checked as the system type.

      index.php?plugin=attach&refer=GGmodel&openfile=startSE2.png

    • The equations for the model equation is input into the Equations field. Please follow the following rules.
      • State variables : Each state variable should be written as x[i], i = 1,2,...,n . In other words, equations are input using the substitutions M->x[1]index.php?plugin=attach&refer=GGmodel&openfile=fc.png->x[2]index.php?plugin=attach&refer=GGmodel&openfile=fn.png->x[3].
      • Parameters : Parameters are placed between two $ symbols to identify that the variable is a parameter. For example, items placed between two $ symbols, such as $para$,$Vm$,$alpha$,etc, are considered parameters. Parameter names can be chosen freely.
      • Sinusoidal force : When handling non-autonomous equations, external forces that can be applied are limited to sinusoidal forces with single frequency. In this case, in the equations, the force is described as cos(t) or sin(t). The angular frequency or the frequency is normalized with a parameter index.php?plugin=attach&refer=GGmodel&openfile=eq_img04.gif.*3.

        index.php?plugin=attach&refer=GGmodel&openfile=startSE3.png

  • Syntax check & program creation
    • The above figure shows the screenshot after the input of the system equation. Click the check button to check whether there are no inconsistencies in the equations. The syntax checking will start.

      index.php?plugin=attach&refer=GGmodel&openfile=startSE4.png

    • After the syntax check, if there are no errors, then programs are created. The bottom-right check button controls which programs are created. To obtain a bifurcation curve, check all of PPFIXBF. Please click Create button to start program creations.

      index.php?plugin=attach&refer=GGmodel&openfile=startSE5.png

      (a): The status bar will be displayed when the system creation starts.

      index.php?plugin=attach&refer=GGmodel&openfile=startSE6.png

      (b): After system creation, a message is shown to indicate that the programs were created without problem in the log window.



PP

Here, we import the initial values of parameters and state variables to draw a phase portrait and to capture a stable periodic solution.

First, start the PP (phase portrait) program from SE. Select Program -> PP.

index.php?plugin=attach&refer=GGmodel&openfile=pp.png

The drawing window and the control panel will start. In the first run, the drawing window is the (x[1],x[2]) plane.*4 Here, there is no need to change the drawing window.

index.php?plugin=attach&refer=GGmodel&openfile=pp2.png

Next, we deal with the control panel. First, we input the values of Initial states and Parameters as the initial condition.*5. The initial points to start the simulations are input (x[1],x[2]) in the Initial states field. Here, let us start from the origin (0,0) !

Please do not forget to input the parameter values before drawing. When first ran, 1 is entered for all parameters. Parameters are configured as follows.

parametervalue
KI1
Kd0.13
Km0.5
k10.5
k20.6
ks0.5
vd1.4
vm0.505
vsmax3
vsmin1.6
omega0.261


Next, click the Start button to start simulations. Can you see that the orbit of a solution starting from the origin converges to a periodic solution ? The blue line and red point indicate the orbit of the periodic solution and the points of the stroboscopic map every time of index.php?plugin=attach&refer=GGmodel&openfile=eq_img05.gif, i.e., the point of the Poincare map.

index.php?plugin=attach&refer=GGmodel&openfile=pp3.png

Parameters can be changed in real time. Let us change the parameter omega. The parameter Omega is related to the period of the sinusoidal force. Currently, the period of this external force is set to about 24 [h](T = 2π/omega) . Since the default step size to change the parameter is 0.1, which is too large, first we make changes here.

Select menu: Setting -> Parameter.

index.php?plugin=attach&refer=GGmodel&openfile=pp4.png

Then, the configuration window will open.*6 Change the step size for omega displayed at the bottom from 0.1 to 0.001. It is OK after the change is completed.

index.php?plugin=attach&refer=GGmodel&openfile=pp5.png

Now, what happens when the parameter omega is increased ? By increasing the parameter omega, the stable periodic solution changes to the unstable one. For instance, the following figure shows an attractor observed in the system when omega = 0.317.

index.php?plugin=attach&refer=GGmodel&openfile=pp7.png

We make this figure more easier to understand. Uncheck the check box Orbit in Plot field in main panel. The orbit of the solution will not be displayed, and only a point sequence is displayed in the phase portrait. As time advances, the sequence of points settle on a closed curve. This is called the invariant closed curve (ICC). The emergence of the ICC is involved in the generation of the Neimark-Sacker bifurcation.*7 In this case, by increasing the value of omega, the stable periodic solution changes to unstable one, and then the Neimark-Sacker bifurcation (supercritical NS bifurcation, which a stable ICC can be observed.) occurs.~

index.php?plugin=attach&refer=GGmodel&openfile=pp8.png

Here, we return to the original settings by reducing omega. Then, it can be achived to click the button to decrease the parameter or to directly change the value of the parameter to its origibal value. You can see that the orbit of the solution converges to a stable periodic solution as expected.

index.php?plugin=attach&refer=GGmodel&openfile=pp9.png

Next, we check the check box states flow in the Plot field. In this way, we can check the current values of the state variables in the list box during numerical integration. The values of the state variables sampled once per 2π/omega-period is output in the list box. In this example, we can see to converge to a stable fixed point of the Poincare map since the same number repeatedly appears in the list box.

index.php?plugin=attach&refer=GGmodel&openfile=pp10.png

Here, by clicking the Clear button, the orbit during the transient response shown in the drawing window can be erased, and the steady state can be observed. Uncheck the check box Orbit, and stop drawing the orbit of the solution. Then only one point corresponding to the stable fixed point is shown on the drawing window since the system converges to a stable fixed point.

index.php?plugin=attach&refer=GGmodel&openfile=pp11.png

Next, let us investigate bifurcations of this periodic solution (the fixed point of the Poincare map). The eigenvalues of the variational equation for the fixed point are evaluated when the parameters are changed. For this purpose, the FIX tool can be used. We pass the initial values to use FIX from PP. The important thing is that the values of the state variables and parameters at the steady state must be passed to FIX.

To confirm whether the system reached a steady state, either check the check box states flow switch and confirm that the values of the state variables no longer change as described above, or click the Status button a few times and confirm that the information displayed in the list box does not change.

index.php?plugin=attach&refer=GGmodel&openfile=pp12.png

After confirming that there are no more changes, select Export current status from Tools in the menu bar. With this procedure, the values of state variables, the parameter values, and other configuration values at the steady state are preserved and can be passed to the next program (see section on FIX).

index.php?plugin=attach&refer=GGmodel&openfile=pp13.png

(a): Select export.

index.php?plugin=attach&refer=GGmodel&openfile=pp14.png

(b): The export of configuration values finished.

FIX

Here, the objective is to track the accurate location of a periodic solution when the parameters are changed and also to monitor the eigenvalues during tracking.

Users must adjust the configurations to run FIX. For simplicity, here let us run FIX as is. Normally, the FIX program is run after the values saved by the PP program (using Export Current status) is imported as the initial values for Newton’s method.*8

First, let us start up the FIX program from the PP control panel. Select Program -> FIX.

index.php?plugin=attach&refer=GGmodel&openfile=fix.png

When initially starting up the FIX, default values are entered in each field.

index.php?plugin=attach&refer=GGmodel&openfile=fix2.png

Information on the fixed point for a periodic solution obtained with PP is imported as the initial configuration value. Select Tools ->Import initial point.

index.php?plugin=attach&refer=GGmodel&openfile=fix3.png

After the selection, the information obtained with PP such as the parameter values and the coordinates of the fixed point for the periodic solution are reflected.

index.php?plugin=attach&refer=GGmodel&openfile=fix4.png

By using PP, we saw that the dynamics of the system changes by increasing the parameter omega. Therefore, select the parameter to be changed as omega and configure the step size to increase the parameter.

index.php?plugin=attach&refer=GGmodel&openfile=fix5.png

Click the Start button to start calculations. A new window will open when the calculations start. This window is to monitor the location of the eigenvalues on a Gauss plane. The Neimark-Sacker bifurcation of a periodic solution occurs when the absolute value of the complex conjugate eigenvalues becomes "1", that is when passing through the unit circle. We can observe that as the parameter changes, the location of the eigenvalues change, and the pair of eigenvalues on the window approaches "unit circle".

Parameters are continuously changed, and when a bifurcation point is detected, the program stops.*9 In this example, the Neimark-Sacker bifurcation is detected because the absolute value of the complex conjugate eigenvalues became |1|.

index.php?plugin=attach&refer=GGmodel&openfile=fix6.png

The values of the parameters and the state variables when the program stopped can be used as the initial point for BF, the program to track bifurcation curves. The information when the bifurcation point is detected are exported for BF. When Export BF point is selected here, the status, parameter values, and other necessary configuration values are preserved. The preserved values are passed to the next program (see section on BF).

index.php?plugin=attach&refer=GGmodel&openfile=fix7.png

(a): Select the Export BF point.

index.php?plugin=attach&refer=GGmodel&openfile=fix8.png

(b): The exported information is displayed in the list box.

BF

The BF tool is used to obtain a bifurcation curve by changing system parameters.

The user must adjust the configuration to run BF. For simplicity, here let us run BF as is. Normally, the BF program is run after the values of the state variables and parameters at the bifurcation point detected by the FIX program are imported as the initial values for Newton's method.*10

First, let us run the BF program from the FIX main panel. Select Program -> BF.

index.php?plugin=attach&refer=GGmodel&openfile=bf.png

When starting up BF at the first time, default values are entered in each field.

index.php?plugin=attach&refer=GGmodel&openfile=bf2.png

Various information on the bifurcation point obtained with FIX is imported as the initial value. Select Tools ->Import BF point from main panel.

index.php?plugin=attach&refer=GGmodel&openfile=bf3.png

After the selection, various information such as the type of bifurcation, the period of the periodic solution, and the angle of eigenvalues are input automatically into each field. To tracke the bifurcation set on the (omega,vsmax) plane, check the radio button omega as the X parameter and vsmax as the Y parameter. Furthermore, configure items such as step size to change the parameters and the value to stop the BF when reaching end parameter. To save the calculated results as a data file, input the output file name in Output.

index.php?plugin=attach&refer=GGmodel&openfile=bf4.png

After these configurations, click Start button to start tracking of the bifurcation curve. Then, the Eigenvalues window and Bifurcation diagram window will open as new windows. The Eigenvalues window is used to visualize the change in the root locus of the eigenvalues during calculation of the bifurcation curve. Furthermore, to monitor the change of the root locus, we can visually understand occurrence of a bifurcation point with co-dimension 2. The Bifurcation diagram window is a window to monitor how the Neimark-Sacker bifurcation is being tracked in real time.

index.php?plugin=attach&refer=GGmodel&openfile=bf5.png

(a):The root locus of eigenvalues and the status of the list box when BF runs.

index.php?plugin=attach&refer=GGmodel&openfile=bf6.png

(b):Bifurcation diagram in the (omega,vsmax)-plane. The calculation result can be monitored in real time.


References


  1. J.C. Leloup ,D. Gonze, A. Goldbeter, Limit cycle models for circadian rhythms based on transcriptional regulation in Drosophila and Neurospora, J. Biol. Rhythms. 14, pp.443-448, 1999.
  2. D. Gonze and A. Goldbeter, Entrainment versus chaos in a model for a circadian oscillator driven by light-dark cycles, J. Stat. Phys. 101, pp.649-663, 2000.
  3. G. Kurosawa and A. Goldbeter, Amplitude of circadian oscillations entrained by 24-h light-dark cycles, J. Theor. Biol., 242, pp.478-488, 2006.




*1 Project name can be changed freely. The user should choose a name that is easy to identify.
*2 There is a configuration file for SE as fileGG.sed in the sample directory. To import this file, it's able to apply several important information to each field in SE. You can create the programs to only click the Create button. Thus, you can skip various configurations for SE. Please refer to see How to import the configuration file for SE.
*3 The parameters are automatically added during tool generation.
*4 This configuration can be changed from the setting menu of the Main panel.
*5 There is a configuration file of PP as filePd_point.ppd in the sample directory. This file is the configuration file saved the values of state variables and parameters to simulate the behavior related to a periodic solution. To import this file, it's able to simulate the system behavior to only click the Start button. Please refer to see How to import a configuration files for PP. In addition, there is another sample of the configuration file of PP as filechoas.ppd in the sample directory. The configuration files differ to the above example. Please enjoy simulations to import their interesting configuration files.
*6 For detail, see How to change parameters using key input.
*7 The Neimark-Sacker bifurcation is also called the Hopf bifurcation for the fixed point.
*8 Another method is to import a file describing the values of the parameters and the state variables. There is a file named filePd_point.fixd under the sample directory where configurations for FIX are saved. This file is a FIX settings file where parameters and initial conditions of the periodic solution described below are saved. Numerical calculations can start after just clicking the Start Button after loading this file. For directions on how to load the file, see How to import configuration files for FIX.
*9 Details of this operation can be controlled by using the option of the detection from Settings. For detail, see manual or User's Guides:About FIX.
*10 Another method is to import a file describing parameter and state variable values. There is a file named filePd_point_NS.bfd under the sample directory where configurations for BF are saved. This file is a BF settings file where parameters and initial conditions to track the Neimark-Sacker bifurcation of the fixed point for a periodic solution described below are saved. The bifurcation curve can be tracked after just clicking the start button after loading this file. For directions on how to load the file, see How to import a configuration files for BF.

Attach file: fileeq_img05.gif 185 download [Information] fileeq_img04.gif 194 download [Information] fileeq_img03.gif 226 download [Information] fileeq_img02.gif 182 download [Information] fileeq_img01.gif 200 download [Information]

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Last-modified: 2009-07-23 (Thu) 20:13:30 (3229d)