Team:KULeuven/Model/Memory

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Pictogram memory.png

Contents

Memory

Position in the system

Describing the system

Memory BioBrick.jpg

This system must activate the cell death system after one light pulse. As long as there is no light, there is no P2ogr, no CIIP22 and a lot of antimRNA_LuxI. The antimRNA_LuxI blocks the cell death system. When light is turned on OmpF increases. This causes P2ogr and CIIP22 to increase and antimRNA_LuxI to decrease. This activates the system. When light is turned off, the P2ogr concentration is large enough to maintain itself. This way antimRNA doesn't increase. The system stays activated.

ODE's

Parameters

Parameter values (Memory)
Name Value Comments Reference
Degradation Rates
dP2ogr 0.002265 s-1
dRNA_P2ogr 0.002265 s-1 [http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=124983&blobtype=pdf link]
dP22CII 0.002311 s-1 This value is to low. The correct value is used in the final model. [http://www.pnas.org/content/88/12/5217.abstract?sid=27806463-33af-4a78-b964-f15872433865 link]
dRNA_P22CII 0,0022651 s-1 [http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=124983&blobtype=pdf link]
dantimRNA_luxI 0.0045303 s-1 [http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=124983&blobtype=pdf link]
Transcription Rates
kP2ogr 0.0125 s-1 estimate
kP22CII 0.0125 s-1 estimate
kAntimRNA_LuxI 0.0094 s-1 estimate
Dissociation Constants
KP2ogr 4.2156 Used in two reactions for activator control at the transcription of P2ogr mRNA and CIIP22 mRNA [http://www.jbc.org/cgi/content/abstract/258/17/10536?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&fulltext=P22+c2+repressor&andorexactfulltext=and&searchid=1&FIRSTINDEX=0&sortspec=relevance&volume=258&resourcetype=HWCIT link]
KR0053_P22CII 0.1099 [http://www.jbc.org/cgi/content/abstract/258/17/10536?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&andorexactfulltext=and&searchid=1&FIRSTINDEX=0&sortspec=relevance&firstpage=10536&resourcetype=HWCIT link]
Hill Cooperativity
n 2 Used for all reactions throughout the memory submodel using Hill kinetics

Models

CellDesigner (SBML file)

Memory
Figure: CellDesigner system representation

Matlab

Problem

The OmpF promoter is not ideal. When there is no light the transcription rate is still 0.00005. This means that P2ogr will slowly build up, activating the system. In this case the memory is in 0-state when the stationary state isn't reached yet. The 1-state is the stationary state. So the system automatically ends up in state 1 after some time (300s). This can be seen in the figure below.

mem_no_act
Figure: Celldesigner simulation of the memory system. CIIP22(purple), P2ogr(green), AntimRNA(pink).

The system can only stay in 0-state for 300 sec. This makes it completely useless.

Alternative

In the previous system the 0-state isn't actively maintained. It's just 'not stationary state'. So we need to search mathematical system that has 2 stationary states. A possible solution is given below.

Describing the system

alt
Figure: Part representation of alternative system

When this system starts Rep build up because Rep represses the Act promoter better then Act represses the Rep promoter. The Rep concentration stays high and the Act stays low. This is the 0-state. When there is light, the OmpF promoter is activated and the Act concentration is increased. This represses Rep promoter. The Rep concentration decreases and the Act promoter is activated. The Act concentration keeps increasing. When the light pulse ends the Act concentration is high enough to repress the Rep promoter, Act concentration stays high and Rep concentration stays low. This is the 1-state.

Using typical values this system can be tested in CellDesigner.

alt_CD
Figure: Celldesigner system representation

Parameters

These typical parameters are:

Parameter values (alternative Memory)
Name Value
Degradation Rates
dact 0.0011552 s-1
dRNA_act 0.0023105 s-1
drep 0.0011552 s-1
dRNA_rep 0.0023105 s-1
Transcription Rates
kact 0.006 s-1
krep 0.0125 s-1
Translation rate
kact 0,166666
krep 0,166666
Dissociation Constants
Kact-rep 4.2156
Krep-act 4.2156
Hill Cooperativity
n 2

Results

CellDesigner gives the following simulation when OmpF transcription rate changes from 0.0001 to 0.01 at t=6000 sec for 2000 sec.

alt_CDplot
Figure: Celldesigner simulation of the alterative system. Act(grey), Rep(yellow)

The OmpF peak causes the Act concentration to rise and the Rep to decrease. The high Act concentration keeps the Rep concentration low. This causes the Act concentration to stay high.

New model

Describing the system

New Mem symbols.PNG

This is the same system as the alternative. The role of act is taken up by cI434 and rep by cIIP22.The output of this system is given in by RNA production that repressed the LuxI production.

Models

Matlab (SBML file)

New Memory

ODE's

Parameters

Parameter values (New Memory)
Name Value Comments Reference
Degradation Rates
dcIIP22 2.8822E-4 s-1 this part has a LVA-tag, so we estimate a 40 min half-life
dmRNA_cIIP22 0.00462 s-1 [http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=124983&blobtype=pdf link]
dcI434 2.8822E-4 s-1 this part has a LVA-tag, so we estimate a 40 min half-life
dmRNA_cI434 0.00462 s-1 [http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=124983&blobtype=pdf link]
dantimRNA_luxI 0.00462 s-1 [http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=124983&blobtype=pdf link]
cII434 regulated promotor
ktranscr 0.125 s-1 estimate
Km 0.8708 [http://jb.asm.org/cgi/content/abstract/187/16/5624?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&searchid=1&FIRSTINDEX=0&volume=187&firstpage=5624&resourcetype=HWCIT link]
cIIP22 regulated promotor
ktranscr 0.125 s-1 estimate
Km 0.1099 [http://www.jbc.org/cgi/content/abstract/258/17/10536?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&andorexactfulltext=and&searchid=1&FIRSTINDEX=0&sortspec=relevance&firstpage=10536&resourcetype=HWCIT link]
translation rates
ktransl_cI434 0.038888 s-1 eff. 0.07
ktransl_cIIP22 0,555555 s-1 eff. 1
Hill Cooperativity
n 2 Used for all reactions throughout the memory submodel using Hill kinetics

Results

Simulation

CellDesigner gives the following simulation when OmpF transcription rate changes from 0.0001 to 0.0125 at t=20000 sec for 1000 sec (17min).

New Memory

There is a clear change. The cIIP22 amount drops and the cI434 increases. The high cI434 amount will repress the RNA production. These RNA amount are small and can’t be see on the graph. It might be necessary to increase the RNA production.

Mathematical Analysis (Maple file)

The existence of the two stable states of the memory (cI434 high and cIIP22 low / cI434 low and cIIP22 high) can be mathematically proven.

First we define the equilibrium points of the following differential equation system:

Differential.png

The equilibrium points are defined as the points for which all the derivatives are zero. Solving this non-linear system results in finding the roots of the equation

Equation.png

If [OmpF] equals 0.0001, this equation has three real zeros ([CI434 = 2.551596228], [CI434 = 40.65751021], [CI434 = 320.7902620]) and two conjugated imaginary zeros ([CI434 = -21.30179624+33.02466798*I], [CI434 = -21.30179624-33.02466798*I]) as can be seen in the following figure:

Equation.jpg

The real roots of the system for all the variables are:

Real Roots
Zero 1 Zero 2 Zero 3
[CI434] 2.551596228 40.65751021 320.7902620
[mRNACI434] 0.02165211656 0.3450080152 2.722134509
[CIIP22] 67.86681025 0.2982960778 0.004793831084
[mRNACIIP22] 0.2822500555 0.001240578188 0.00001993697780

The stability of the three real zeros is defined by the eigenvalues of the Jacobian of the differential equation system. This Jacobian equals

Jacobian.png

The eigenvalues for Zero 1 are all negative (-0.2312299809e-2, -0.4575424931e-2, -0.4663289165e-2, -0.3294760947e-3). This is also the case for Zero 3 (-0.2322149252e-2, -0.4504274069e-2, -0.4727420465e-2, -0.3266462137e-3). These two zero's represent the two stable states of the memory system. Zero 2 has a positive eigenvalue and is therefore unstable.

The following figure shows some trajectories in the phase plane ([CIIP22],[CI434]): there is a clear boundary between the two stable equilibrium points (the green dots) that goes through the unstable equilibrium point (the red dot). This boundary divides the two dimensional phase plane in two separate basins of attraction.

Memory PhasePlot.png