Allosteric Control Explained Using ATCase

Introduction

Aspartate carbamoyltransferase (ATCase) is an important component of the initial, committed step of pyrimidine synthesis, and pyrimidine in turn is used as a derivative for three out of the five DNA and RNA bases. ATCase does not follow simple Michaelis–Menten kinetics, but rather is an allosteric enzyme, meaning it behaves differently based on the concentration of substances, in this case adenosine triphosphate (ATP), uridine triphosphate (UTP) and the final product: cytidine triphosphate (CTP). The model in this example is a simplified version of the ATCase catalyzed reaction, with the purpose of explaining allosteric enzyme kinetics.

Explore the available scenarios by evaluating each cell in order. Evaluation is done by selecting a cell and pressing Shift+Enter.

In order to get the full experience of this example, you need the following:

Wolfram System Modeler; a free trial download is available at www.wolfram.com/system-modeler

The model that came in the same bundle as this notebook

For more information about Wolfram System Modeler, visit www.wolfram.com/system-modeler.

Initialize

Define the model name:

mmodel="AllostericControl";

Load the model:

Import[FileNameJoin[{NotebookDirectory[],"AllostericControl.mo"}],"MO"];

Model

SystemModel[mmodel]

The enzyme shifts between tense and relaxed forms depending on how much product (CTP) and other modifiers (ATP and UTP) are present. ATCase is therefore represented as both ATCase_R and ATCase_T. ES_complex stands for enzyme-substrate complex. 

When in its relaxed state, the enzyme ATCase increases the reaction rate to a larger degree than when in its tense state (which has a rate that is negligible). This is because the active site is hidden by other parts of the enzyme when it is tense. 

UTP increases the rate of reformation of ATCase_T. Together with CTP, this pyrimidine base can decrease the reaction rate almost completely.

ATP decreases the rate of reformation of ATCase_T. This means that the more ATP you have, the more favorable the relaxed state is for the enzyme, and the reaction rate is increased.

CTP increases the rate of reformation of ATCase_T. This means that as more product (CTP) is produced, the tense form for the enzyme becomes more favorable, slowing the reaction down. An increased amount of CTP also decreases the transformation of CTP synthesis in the final step. This step is more complicated in reality, though, as several intermediaries have been skipped in favor of simplicity.

In order to see all of this happen in real time, you can simulate the model and create a plot in which you can manipulate different initial conditions and see how the different substances are affected as time passes.

Analysis

In order to properly analyze the model, a GUI for manipulating some key initial conditions can be used.

CTP=1;
Asp=1;
UTP=1;
ATCaseT=1;
ATP=1;
plotVars={"Aspartate.c","Carbamoyl_Aspartate.c","CTP.c","Phosphate.c","Carbamoyl_Phosphate.c","complex.c","ATCase_T.c","ATCase_R.c","ATP.c","UTP.c"};
Column[{Manipulate[
SystemModelPlot[Quiet@SystemModelSimulate[mmodel,{0,20},<|"InitialValues"->
{
"CTP.c"->10^CTP,
"Aspartate.c"->10^Asp,
"ATCase_T.c"-> 10^ATCaseT,
"ATP.c"->10^ATP,
"UTP.c"->10^UTP
}|>,
Method->{"CVODES"}],
states,PlotRange->Full,ImageSize->{360,300},Filling->Axis,PlotLegends-> Placed[Automatic,Bottom]],
"States:",{{states,{"Aspartate.c","Carbamoyl_Aspartate.c","CTP.c"},""},ListPicker[#,plotVars,ImageSize->{Automatic,220}]&},
"Initial Conditions:",
{{Asp,1,""},Row[{Pane["Asp",ImageSize->50],Slider[#,{-3,2,1},ContinuousAction->False],ToString[N[10^Asp]]<>" mol/l"}]&},
{{ATP,0,""},Row[{Pane["ATP",ImageSize->50],Slider[#,{-3,2,1},ContinuousAction->False],ToString[N[10^ATP]]<>" mol/l"}]&},
{{CTP,0,""},Row[{Pane["CTP",ImageSize->50],Slider[#,{-3,2,1},ContinuousAction->False],ToString[N[10^CTP]]<>" mol/l"}]&},
{{UTP,0,""},Row[{Pane["UTP",ImageSize->50],Slider[#,{-3,2,1},ContinuousAction->False],ToString[N[10^UTP]]<>" mol/l"}]&},
{{ATCaseT,0,""},Row[{Pane["ATCaseT",ImageSize->50],Slider[#,{-3,2,1},ContinuousAction->False],ToString[N[10^ATCaseT]]<>" mol/l"}]&},ControlPlacement->Right,FrameMargins->Small,SynchronousInitialization->False,SynchronousUpdating->False]}]

Several states can be viewed at a time, and can be selected for simultaneous viewing by left-clicking them while holding down the Ctrl key on Windows and ⌘ on Mac. The states displayed are in the concentration unit of mol/l.

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