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Allostericenzymes

by Nicolas Descostes (Submitted: 11/26/2008)

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WHAT IS IT?

‘The enzymes are important and essential components of biological systems, their function being to catalyse the chemical reactions that are essential to life. Without the efficient aid of the enzymes these chemical processes would occur at greatly diminished rates, or not at all’ (by Keith J. Laidler and Peter S. Bunting, 1973).

Basic mechanism of non allosteric enzymes:

          E+S <-> ES -> P
with E = enzyme
S = substrate
-> binding of the enzyme with the substrate
ES = enzyme/substrate complex
p = product

This model is simulating action of ALLOSTERIC enzymes.

Allosteric enzymes bind small and physiologically important molecules by non covalent binding. These small regulatory molecules are called effectors.
The binding of these effectors will lead to a change in the catalytic function of the enzyme and in its structural conformation. This will modify the affinity of the enzyme for the substrate; it will have a higher or lower affinity for the substrate.

There are different types of effectors:
The first one is the heterotropic effectors: There are two kinds of heterotropic effectors; activators and inhibitors. Activators will bind the enzyme and increase the affinity for the substrate whereas the inhibitors, still by changing the enzyme conformation, will decrease the affinity for the substrate.
The second kind of effector is said homotrophic; the substrate itself induces change in affinity and conformation of the enzyme: we talk about negative or positive cooperativity with the substrate.

The first part of the model is simulating the kinetics of allosteric enzymes with homotrophic effectors.
Mechanism:
The first line is describing the action of allosteric enzymes by underlying how it has been represented in the model. The second line is closer from reality because of the conformation change induced by the binding of substrates.

          E+S1 <-> ES1 + S2 <-> ES1S2 -> P
E+S1 <-> C1 + S2 <-> C2 -> P
With E = enzyme
S1 = substrate1
S2 = substrate2
ES1 = enzyme-substrate1 complex or C1
ES1S2 = enzyme-substrate1-substrate2 complex or C2.
P = product

Each reaction:

    <-> 

is regulated by an affinity constant K, which enables to represent the change of affinity. Each combination induces also a change of conformation as it has been said before.

The second part of the model is simulating the action of effectors (inhibitors/activators) on the model (see cooperativity area).

WARNING

The first part of the model is a theoretical one. It does not correspond to any "validated" models (Monod-Wyman-Changeux, Koshland-Némethy-Filmer, Association-dissociation model). The aim of this model is to make the user understand the main principles of allosteric enzymes kinetics by looking at different parameters. This 'non conventional' representation has been chosen because of the complexity of the models named above, but it represents most of ideas of these ones.

The model must be think in two parts wich are indepant. The first part enables to understand how an allosteric enzyme is working, the second enables to study the effect on effectors (activators/inhibitors) on the cooperativity.

HOW TO USE IT

The interface contains three main parts: THE SCREEN (in the center), THE RATE (on the left) AND THE AFFINITY (on the right) AREA.
As it has been said before, the first one enables to setup, run and display the model. The second one enables to follow the evolution of the system, and the third one enables to display the variation of cooperativity between enzymes and substrates.

________________________________The Screen Area________________________________:
---------------> Setup Button:
Enables to initialize the system by taking into account the values chosen on each slide bars.
---------------> Go Button and the Screen:
The Go button enables to run the model. substrate1 turtles (in green) are going to combine with free enzymes (in red), a new element will be created: the substrate1-enzyme complex (in pink). Substrate2 (in yellow) will combine a free substrate1-enzyme complex, the substrate1-enzyme-substrate2 complex will be created (in black). Then, substrate1-enzyme-substrate2 complex will be transformed into product (in blue).
---------------> Clear-every-plot:
Enables to clear the rate plot and the cooperativity plot. These plots can also be cleared separately with the buttons situated under each plot screen.

________________________________The Rate Area________________________________:
---------------> enzyme-substrate1 affinity slide:
Enables to chose the constant of the first reaction seen in the first part of this manual.

     E + S1 <-> ES1
---------------> enzyme-substrate2 affinity slide:
Enables to chose the constant of the second reaction seen in the first part of this manual.
     ES1 + S2 <-> ES1S2
---------------> product-formation-constant slide:
Enables to chose the constant of the third reaction seen in the first part of this manual.
    ES1S2 -> P
---------------> substrate1_units slide:
Enables to fix the initial rate of substrate1.
---------------> substrate2_units slide:
Enables to fix the initial rate of substrate2.
---------------> enzyme_units slide:
Enables to fix the initial rate of enzyme.
---------------> Monitors:
Enable to follow the rate of each element of the system.
---------------> Frame number:
It ranges from 1 to 3. This enables to plot several times the rate curves without deleting the previous values. Hence, the user would be able to compare the rate variations when the paramaters are changed.
WARNING: It is not possible to plot two (or more) times (by resetup) using the same frame.
---------------> The Rate Plot:
Enables to visualize the different rate values using a graph. It is possible to clear it at any time using the "clear-rate-plot" button.

________________________________The Affinity Area________________________________
---------------> The Cooperativity plot:
Enables to visualize the variation of cooperativity between enzymes and substrates. It is plot thanks to the Hill Equation.
---------------> pen-number slide:
Has the same role than the 'frame-number' slide. It enables to compare the variation of cooperativity directly on the same screen and, a same pen could not be used several times without clearing the plot.
---------------> K value slide:
Regarding the Hill Equation

    ((V * (a ^ h)) / ((K.O.5 ^ h) + (a ^ h)))
K is the value of the substrate concentration 'a' at which v = 0.5*V ie the velocity v is the half of the maximum velocity V.
---------------> Cooperativity slide:
Here Cooperativity is in fact the 'h' element in the equation. Cooperativity can be well estimated by the number of subunits the enzyme has. The slide enables to choose a positive or negative (between 0 and 1) cooperativity.

THINGS TO NOTICE/ THINGS TO TRY

->Enzymes/substrates, complex/substrate2 can combine only if they are close in space.

-> Notice the change of shape of each elements on the screen.

-> Notice S1-E-complex turtles and S1-E-S2-complex turtles are not moving on the screen to explicit transformations

-> Look at the different rates of each element at the beginning and at the end of the run. Try to pause the model at different moments to understand how an allosteric enzyme is working, by looking at the rate of each element.

-> Look at the rate plot. Link each curve with its rate monitor.

->Change the affinity slides and product formation values and look at the effect on the plot (Do not forget to not use the same frame two times). What is the relation between affinity constant and power of binding? Does a high affinity constant mean a strong binding?

-> Look at the values of the slides in the 'Rate Area', minimum and maximum. What does it means on a biological range?

->Change the initial rate of substrate1, substrate2 and enzyme. What are the effects? How the enzyme/substrate1/substrate2 concentration determines the speed of product formation? Look at it on the plot.

-> Start with only one substrate1, one substrate2 and one enzyme. Slow down the speed of the model and look at the mechanism of allostery.

-> An enzyme is made free each time a product is produced.

-> The cooperativity area is independant. The Rate and the Screen Area are linked.

-> The cooperativity study is a K-system. All curves are heading towards the same Vmax which has been set to 10. A V-system contains effectors that are changing the Vmax, that system has not been chosen because the modification of Vmax corresponds to a change of scale; hence, the information we would be able to obtain on the plot is not interesting.

-> Try to change the K value and cooperativity value. What are the difference between positive and negative cooperativity? What does a cooperativity of 1 mean? How does the plot look like with a such cooperativity? Could we use a cooperativity of 0?

EXTENDING THE MODEL

The main improvement we can do with this model is to adapt it to an existing theoretical one: the Monod, Wyman and Changeux model; the Koshland, Némethy and Filmer model; or the Association-dissociation model.

--------------> What could we do to create a Monod, Wyman and Changeux model?

********* What is this model?
The Monod, Wyman and Changeux model is often called the 'allosteric model' but the term of 'symmetry model' is more accurate because 'it emphasizes the principal difference between it and alternative models' (by Athel Cornish-Bowden, 1995).
The symmetry model is based on the following postulates:
1) Each subunit can be:
- In the R conformation: when the protein is relaxed it can bind the substrate.
- In the T conformation: when the protein is tensed it cannot bind the substrate.
2) All the subunits have to be in the same conformation.
3) The two states of the protein are in equilibrium.
4) A ligand can bind to a subunit in either conformation but the dissociation constants are different

Without any substrate or effectors, the enzyme tends to be in the conformation T. We can say that there is something like 99% of T-shaped enzymes at time 0. There are two ways of thinking the model of Monod-Wyman-Changeux: with the substrate and with the effectors. The most important thing is to think in mass attraction law and equilibrium:
The binding of the substrate will put the equilibrium on the high affinity side (R shape) because of the mass attraction law; thereby this model is often used to explain a positive cooperativity. We can reproach this model to not explain the negative cooperativity.
On the other side, an activator will bind and put the enzyme in its high affinity conformation and an inhibitor will obviously put it in the T shape (low affinity).

********* What do we have to change in the current model?
First and foremost, we can model the R and T state by using the S1-E complex. As we have seen, the enzyme is in T shape at time 0 and the binding of substrate lead to the R shape. So the element 'enzyme' becomes the T-shaped enzyme, and the 'S1-E complex' becomes the R-shaped enzyme. The S1-E-S2 complex element has to be deleted.
A product can be formed only if all subunits are associated with a substrate.

Two possibilities:
- A simple one: An improvement would be to still use the homotrophic effectors (substrate) and thus, the only part of programming would be to switch from a monomeric enzyme to a multimeric enzyme. You just have to integrate a structure in each function recording the number of subunits bound. When all of them are bound you can release a product and an enzyme.
- A complex one:
Effectors can be added to the system, both type (activators/inhibitors), hence, the binding of an inhibitor will lead to the T shape and the binding of an activator will lead to the R shape. This have to be integrated in the program.

NETLOGO FEATURES

Note the use of the 'frame-number system' and the 'pen-number system' makes the two functions 'update-rate-plot' and 'update-cooperativity-plot' not very efficient in the way of programming. A smarter solution could maybe be found to enable to plot several curves on the same screen.

Note the use of links and destruction of turtles with the 'die' procedure. The destruction/creation of turtles might consume more memory but the code is more efficient then with the use of hidden turtles.

RELATED MODELS

-> Enzyme Kinetics
-> Simple Kinetics 1
-> Simple Kinetics 2
-> Simple kinetics 3

CREDITS AND REFERENCES

• Keith J. Laidler and Peter S. Bunting, the chemical kinetics of enzyme action, clarendon press, Oxford, 1973, P1.

• Robert K.Murray, Daryl K. Granner, Peter A. Mayes and Victor W. Rodwell, Harper’s Biochemistry, International edition, 24 edition, Prentice Hall International, p66

• Keitaro Hiromi, Kinetics of fast enzyme reactions theory and practice, A Halsted press book, Tokyo, 1979, p ix, p 3

• Athel Cornish-Bowden, fundamentals of enzyme kinetics, Portland Press Ltd, London, 1995, p 2, p 223, p228, p234

• B. I. Kurganov, Allosteric Enzymes, A Wiley-Interscience Publication, 1982, p4, p128

http://web.indstate.edu/thcme/mwking/enzyme-kinetics.html#allosteric

http://www-biol.paisley.ac.uk/kinetics/Chapter_5/contents_chap5.html

• Netlogo user manual, p1, p 34

• Brian Harvey, computer science logo style volume 1: intermediate programming, The MIT Press, Cambridge, Massachusetts, London, England, 1985

http://el.media.mit.edu/Logo-foundation/logo/index.html

COPYRIGHTS

• ------------> This model has been created within the framework of the projet of Honors Year in Biology with Information Technologies at the University of the West of Scotland. It has been headed by Doctor Peter Birch. I want to thank him for his advices and his time spent to explain me the mechanisms of allostery.

• ------------> To refer to this model, please use: Descostes, N. (2008). NetLogo Allosteric Enzymes Kinetics model. University of the West of Scotland, Scotland.

• ------------> I will be very pleased if you could email-me if you are using this model for academic purposes. contact: nicolas_descostes@hotmail.com

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