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Biochemistry
Biostatistics and Bioinformatics
Simulating Enzyme Kinetics
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Biochemistry
Biostatistics and Bioinformatics Simulating Enzyme Kinetics
Description of Module Subject Name Biochemistry
Paper Name 13 Biostatistics and Bioinformatics Module Name/Title 15 Simulating Enzyme Kinetics
Dr. Vijaya Khader Dr. MC Varadaraj
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Biochemistry
Biostatistics and Bioinformatics Simulating Enzyme Kinetics
1. Objectives: In this module, the students will understand 1. Kinetic data of enzyme catalysed reactions
2. Downloading kinetic data from reaction kinetics database
3. Simulating irreversible enzyme kinetics using downloaded enzyme kinetic data with Complex Pathway Simulation (COPASI) software, and
4. Simulating reversible enzyme kinetics using COPASI
Enzyme Kinetics Data Brief Description
Reaction kinetics Databases Simulating Irreversible Enzyme Kinetics
Simulating Reversible Enzyme Kinetics Summary
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2. Brief Description
Dear students, Biochemistry is the study of chemical reactions, catalysed by enzymes produced by living cells of bacteria, plants and animals. Enzymes are biological catalysts and they enhance the rates of the reactions i. e. they speed up the chemical reactions. The enzyme catalysed reactions can run in both forward and backward direction. Therefore, the enzyme catalysed reactions are technically reversible and achieve equilibrium levels of product and substrate concentrations after some time of the start of the reaction. However, if the Gibbs energy of reaction is highly exothermic, then the reaction virtually behaves as irreversible, because it cannot be reversed without input of external energy.
Each of the enzyme catalysed reaction has associated data values for rate constants and concentrations of species such as enzyme, substrate, product, activators, inhibitors. The experimentally determined Kinetic values of enzyme catalysed reactions are published in literature and are manually curated to develop enzyme reaction kinetics databases. Two significant Enzyme kinetics databases are SABIO-RK and BRENDA.
These data can be downloaded from reaction kinetics database and can be used for simulating enzyme kinetics, following an adequate kinetic rate law model.
A model of an enzyme catalysed reaction is in the form a mathematical equation which can be used for simulating the enzyme catalysed reactions, in silico i.e. using a simulation software. The model may be used to predict time course of a reaction, as well as to derive steady state/equilibrium concentrations of substrates and products. The model is actually built from limited experimental data. It encompasses only the necessary kinetic parameters. This reduces complexity for understating the experimental data.
Once an enzyme kinetic model and associated data values for kinetic parameters are available, the same may be used for simulating enzyme kinetics in silico. The most popular simulation software is Complex Pathway Simulation i.e. COPASI. When the enzyme kinetic models and associated data values for kinetic parameters each enzyme in a biochemical pathway are available, the same may be used for constructing biochemical pathway in silico, without conducting further experiments.
Therefore, in the present module the learning objectives includes understanding Kinetic data of simple enzyme catalysed reaction and Enzyme kinetics databases for Downloading kinetic data for Simulating irreversible enzyme kinetics using downloaded enzyme kinetic data with Complex Pathway Simulation (i.e.
COPASI) software, and also Simulating reversible enzyme kinetics using COPASI
Back to Concept Map
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Biochemistry
Biostatistics and Bioinformatics Simulating Enzyme Kinetics
2.1. Enzyme Kinetics Data
Each of the enzyme catalysed reaction has associated kinetics parameters such as velocity called Rate of the reaction. It is defined as the amount of product formed or substrate consumed per unit time. The factors affecting Rates of Enzyme catalyzed reactions include Concentrations and conditions. More the enzyme concentration more is the rate of reaction. In addition, the concentration of substrates has a profound effect on rate or velocity of the reaction. The products produced also affect the rate of the ongoing reaction through product inhibition and finally achieve fixed level of substrate and product concentrations at equilibrium. Further, certain ligands such as activators enhance and certain ligands such as inhibitors reduce the rate of a reaction. The conditions such as pH, Ionic Strength and Temperature also affect the rate of an enzyme catalysed reaction.
The simplest enzyme reaction in the cells is the one in which only a single reactant is present on each side of reaction i.e. single substrate and single product. An example is provided by the reaction catalysed by glucose-6-phosphate isomerase, an enzyme involved in glycolysis, in which glucose-6-phosphate and fructose-6-phosphate are interconverted.
The Gibbs free energy change for this reaction is 0.6 kcal/mole, which is very low, showing that the reaction catalysed by glucose-6-phosphate isomerase is reversible, i.e. it can also run in the reverse direction to produce glucose-6-phosphate from fructose-6-phosphate, as shown next:
Therefore, either glucose-6-phosphate or fructose-6-phosphate can act as substrate, depending upon the levels of the metabolites and metabolic needs of cell.
Cleland devised a shorthand notation for writing down kinetic mechanisms of enzyme catalysed reactions.
The letters used for substrates are A, B, C and D, in the order, they are added to the active site of the enzyme. The letters Products are P, Q, R and S, in the order, they leave the active site of the enzyme.
Stable enzyme forms are lettered E, F and G, in the order, they occur during chemical reaction mechanism.
The number of reactants in the reaction are designated by the terms Uni for one, Bi for two, Ter for three
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and Quad for four. For two substrate entering active site of the enzyme and producing two products, the reaction is named as Bi-Bi reaction.
If the substrate and products enter and leave in orderred manner then it is named as Orderred Bi-Bi. When any of substrate and any of product enter or leave in random order, then it is call Random Bi-Bi reaction in Cleland kinetics nomenclature. When first substarte enters and then first product leaves, followed by entry of second substrate to altered enzyme F and then exit of second product, then it is named as Ping Pong Mechanism. The symbols used in Cleland nomenclature for writing down kinetic mechanism are different from symbols used in writing reaction mechanism for developing enzyme rate law equations.
The reversible reaction sequence for glucose-6-phosphate isomerase can be represented using symbols, E for enzyme, S for substrate glucose-6-phosphate, P for product, fructose-6-phosphate and ES for enzyme- substrate complex, using reversible arrow for the reaction, as shown next:
2.1.1. Uni Uni irreversible reaction kinetics
This reaction involving a single substrate and a single product is called Uni Uni reaction. When the breakdown of ES complex to E and P is fast enough as compared to its back breakdown to E and S, then, the reaction sequence is represented as irreversible reaction, as shown next:
The velocity, v, or the rate of overall reaction for the formation of one product from one substrate for an irreversible reaction is given by Michaelis-Menten (irreversible) equation
where, [S] is molar concentration of substrate, for the reaction shown above. Km is Michaelis-Menten constant for the overall reaction, given by
k-1
kcat
k1
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Where k1 is the rate constant for formation of ES complex and k-1 & Kcat are the rate constants for degradation of ES complex. Numerical Value of Km is of interest for several reasons
1. Km is equal to [S] concentration when Vmax of the reaction is ½ Vmax. Therefore, it helps to adjust [S]
concentration to determine Vmax
2. It is an indicator of apparent affinity of a substrate for an enzyme. A low numerical value of Km means high affinity and a high numerical value of Km means low affinity of the enzyme for its substrate.
3. It establishes an approximate level of a substrate within cell. high affinity means very low concentration of the substrate will be present.
4. It indicates the way the enzyme activity is regulated by activators and inhibitors under physiological conditions, in vivo
5. It provides a means for comparing enzymes for medical importance of the same reaction such as lactate dehydrogenase reaction from heart, muscle and liver for instance.
Vmax for the overall reaction is given by
Vmax = [E]t * kc at
Where [E]t is the total molar concentration of enzyme and Kcat is the rate constant for irreversible formation of product. We will use this model to simulate time-course of Uni Uni irreversible reaction.
2.1.2. Uni Uni reversible reaction kinetics
The actual reaction sequence for completely reversible reaction sequence begins with diffusion of substrate to the active site of the enzyme for formation of enzyme -substrate complex, ES, as shown next:
For example glucose -6-phosphate binds with “glucose-6-phosphate isomerase” to form ES complex. This complex then passes through a transition state to form product, bound to the enzyme, EP complex. The product i.e. fructose -6-phosphate, then diffuses away from the enzyme. For simplicity, let us represent ES complex, transition state complex and EP complex with single symbol, i.e. ES.
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This reduces sequence for a Uni Uni kinetic mechanism, to four reactions:
1. Binding of S with E to form ES complex
2. Decomposition of ES complex to release product P 3. Binding of P with E to form ES complex
4. Decomposition of ES complex to release substrate
The rate constants for each of the four reactions, can also be shown by following a hypothetical scheme, labelling with the “reaction number” and symbol k1 for forward reaction and symbol k2 for reverse reaction.
The (reaction _1).k1 is the rate constant (k1 in some schemes) in the first reaction for the formation of ES complex from E and S. The (reaction _2).k1 is the rate constant (k2 in some schemes) in the second reaction for the formation of product P from Enzyme-substrate complex. The (reaction _2).k2 is the rate constant (k-2 in some schemes) in the third reaction for the backward reaction, i.e. binding of product with Enzyme to form enzyme substrate complex, ES. The (reaction _1).k2 is the rate constant (k-1 in some schemes) in the fourth reaction, i.e. dissociation of Enzyme substrate complex, ES, to form free enzyme E and substrate S. The rate constants in alternative scheme are shown next.
The rate of overall reaction for the formation of one product from one substrate for a reversible reaction is given by reversible Michaelis-Menten equation. Let us expand the rate law equation, as shown next:
k-1
k1
k-2
k2
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where, v is the rate or velocity of the reaction, Vf is the maximum rate of the forward reaction for formation of product (represented as Vmaxf in biochemical literature), Vr is the maximum rate of the reaction in the reverse direction for the formation of substrate (represented as Vmaxr in biochemical literature), Kms is the Michaelis-Menten rate constant (represented as Km for substrate in biochemical literature), Kmp is the Michaelis-Menten rate constant Km for the product. Substrate is the molar concentrations of substrate and product is the molar concentrations of product. Vf and Vr is the maximum velocity which can be achieved at a given enzyme concentration, usually achieved at very large substrate and product concentrations, respectively.
The Vf i.e. Vmaxf for the forward reaction is given by the product of total enzyme concentration and the rate constant for product formation:
The Vr i.e. Vmaxr for the reverse reaction is given by the product of total enzyme concentration and the rate constant for substrate formation:
The Kms i.e. Km for the substrate, and is given by the ratio between sum of rate constants for decomposition of ES [(reaction_1).k2 + (reaction_2).k1 )] and rate constant for formation of ES from substrate:
The Kmp i.e. Km for the product, and is given by the ratio between sum of rate constants for decomposition ES [(reaction_1).k2 + (reaction_2).k1 )] and rate constant for formation of ES from product:
The ratio of concentrations of product to substrate at equilibrium can be calculated using enzyme kinetic parameters with haldane equation, shown next
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Keq = (Vf * Kmp) / (Vr * Kms)
Therefore we have two models for a uni-uni reaction. One for the enzymes following irreversible kinetics and other for enzymes following reversible kinetics. We can use Enzyme catalysed reaction kinetic data values to verify which of these models is actually followed by the enzyme under study.
Back to Concept Map 2.2. Enzyme Kinetics Databases
Enzyme catalysed reaction kinetic data i.e. Km Vmax, Kcat etc. are stored in online databases and can be downloaded in a platform independent language called SBML (Systems Biology Markup Language). SBML is recognized international format and enable scientists to import data for Enzyme catalysed reactions using several simulation software. This helps in sharing of enzyme kinetic models for modifications, simulation and analysis. The SBML is not required to be understood by the users of enzyme catalysed reactions or users of Biochemical pathways models. With passage of time, there have been various levels with different versions for SBML. As in December 2015, available Level 3 version 1 of SBML, supported marking various features of a biochemical process. These include the following.
1. Compartment: whether extracellular, membranous, cytoplasmic etc.
2. Species: whether substrate, product, modifier etc.
3. Parameter: whether Vmax , Km, equilibrium constants etc.
4. Unit Definition: for time, concentration and volume.
5. Rule: to be followed during simulation
6. Reaction: whether reversible or irreversible and following a particular reaction rate law such as simple Michaelis-Menten, allosteric activation or competitive inhibition etc.
7. Event: to be initiated such as addition of additional substrate during time course of pathway, export of a product after achieving a specified concentration level etc., with the support for including Events with priority.
8. Function Definition: for the individual reaction rate laws to be followed during pathway simulation
9. Initial Assignment: based on fixed level or a mathematical assignment 10. Constraint: to be followed during simulation.
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The information about the biochemical reaction kinetic parameters such as Vmax, Km, Inhibition constants and other kinetic parameters is available in published research. The enzyme reaction kinetic databases houses biochemical reaction kinetic parameters and SABIO-RK is a database, compiled and annotated by biological experts. Another is BRENDA. In SABIO-RK database, the information about biochemical reactions, their kinetic rate equations or kinetic rate laws alongwith parameters values and experimental conditions, is compiled from published research papers. It also includes data directly submitted from laboratory experiments. It includes information about biological sources, i.e. source organisms and is not specific for a single organism. The kinetic rate laws such as Michaelis-Menten equation under experimental conditions are also stored. The data can be downloaded with its annotations in SBML and can be used for import in modelling software. We will learn BRENDA in the next module on simulating biochemical pathways. The SABIO-RK can be can be searched using keywords including name the enzyme combined with Boolean operators for other fields such as Enzyme Commission number. The information about the ECNumber is available in ‘ENZYME’ database at ExPASy server.
To gather the ECNumber, visit ENZYME Database at http://enzyme.expasy.org/ and search for “glucose- 6-phosphate isomerase”, as shown next:
Click “Search” button and results are displayed as shown next:
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This shows that glucose-6-phosphate isomeras has enzyme number is 5.3.1.9 but in addition it has several alternative names.
The SABIO-RK can be accessed via web-based user interfaces at http://sabio.villa-bosch.de/ or http://sabio.h-its.org/. Visit SABIO-RK and Click Search menu to open search dialog box. In the search text box, start typing ‘glucose-6-phosphate isomerase’. The available entries will be suggested. Choose the one in which you are interested. Then select “AND” Boolean operator and enter ECNumber 5.3.1.9 as shown next: Press enter,
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select “AND” operator again and open drop down list to choose “Organism” field to search. The following display will appear.
Enter “Homo Sapiens” in the text box and click “Add and Search” button. This display the search results page showing 24 entires.
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Now to use filter options on the right top of the page, click on the “Rate equation” check box. This will update the search results page to display 14 entries.
Now using filter options further, to exclude mutant enzyme just uncheck the “Mutant” check box. This will update the search results page to display 6 entries for only wildtype enzymes. Click on “Right Arrow”
button of first entry in the “Kinetic data” column i.e. first column, to display the kinetic data.
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Click on “Right Arrow” button of first entry in the “Kinetic data” column to display the kinetic data. The first section displays the general information for the organism, tissue EC class etc. and substrates, products, modifiers.
Scroll down to find information about enzyme protein data, kinetic law, parameter values, experimental conditions and reference. The last section has information of the reference literature. The data shows Km 445 and Vmax 400.
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In the record for Entry ID:21894, check the checkbox in the last column, as shown next:
A hyperlink for “entries to export will appear. Click hyperlink “Entries to Export” button to download the enzyme reaction kinetic data.
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In the pursuing page, click ‘Write SBML’ button.
Save model in SBML on your disk.
As on 03st May, 2016, saving models in pdf format (adobe Acrobat Reader) for SBML level 3 version 1 is not implemented. Therefore, one can save the model on disk as SBML Level 3, Version 1. But for other levels,
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user can save the model in pdf format also. In addition, Click on PubMed hyperlink in the Reference to download literature of this enzyme.
The PubMed page will present the abstract, as shown next.
Full text link is also provided alongwith abstract. Click on “Full text link” to read the full text paper, shown next.
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Click on “Download PDF” to be opened with Adobe Acrobat Reader, as shown next.
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Back to Concept Map
2.3. Simulating Irreversible Enzyme Kinetics
Enzyme catalysed reactions saved in SBML can be simulated using COmplex PAthway Simulation software, i.e. COPASI package. Visit http://copasi.org/Download/ to download COPASI.
and install COPASI in your computer.
Now run COPASI and From the file menu, select ”Import SBML” and open
the file downloaded and saved for glucose-6-phosphate Isomerase enzyme.
This will display several warnings as shown next. These warnings are displayed to convey that COPASI requires certain data values and parameter units, which are not provided by the SBML file imported.
However just ignore these warnings at this stage and click “OK” button.
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The COPASI window will open and has four GUI i.e. graphic user interface elements.
On the top is “menu bar“, below menu bar is a “tool bar“, and below tool bar is “Navigation Tree” nodes in the left panel and “Working Space” in the right panel. The opening working space panel displays the general information regarding units and literature for publication of the model. Open all nodes in
“Navigation Tree” by double click on each, i.e. open Model, then Biochemical, then Species, then Reactions. Double click “Model” node in “navigation tree” panel and set quantity unit to Mole.
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Now click on REAC_0 and in working space panel, uncheck Reversible checkbox and select Henri-Michaelis Menten (irreversible) in the Rate Law drop down list, as shown next:
This will change the Km and Vmax to 0.1 by default. Therefore, set Km to 445 and V to 400 to display working panel as shown next:
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Click the “Commit” button on left bottom corner of working panel. Now click the “Edit Rate Law” button on right top corner of the working panel corner. This will display the Michaelis-Menten equation
Double click “Parameter Overview” node in “navigation tree” panel to display the values for initial species values as well as kinetic parameters. Now, Set D-Fructose 6-phosphate initial concentration to 0 micromole. Now click “Commit” button at the bottom of working space panel, as shown.
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Expand Output specification node in “navigation tree” panel and select Plots(0) by double click followed by double clicking New Plot in working space panel.
This will open new plot_1. Click on “New Curve” Button
This will open a dialog box. Select Model Time in left panel and [Fructose 6-phosphate]/(t) and [Glucose 6- phosphate]/(t) in the right panel under Species node of the right panel and click “OK” button
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This will insert a new plot for display of the transient concentrations of selected species on y-axis with model time on x-axis of the plot_1.
Now open the Tasks node in navigation tree panel and click Time Course. Set the Time Course parameter values in the working space.
Now click Run Button at the left bottom corner of working panel. This will display the time course for irreversible conversion of glucose-6-phosphate to fructose-6-phosphate, as shown next.
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This shows that within 10 seconds, all of the Glucose-6-phosphate is converted to Fructose-6-phosphate under Henri-Michaelis Menten (irreversible) kinetics with Km 445 micromolar and Vmax 400.
Back to Concept Map
2.4. Simulating Reversible Enzyme Kinetics
Run COPASI afresh and expand “Model”, “Biochemical” and “Reactions” Nodes.
Double click “New Reaction” in the Working Panel and a new reaction will be added. The reaction will be automatically named as reaction_1. Enter the isomerisation reaction between "D-Glucose 6-phosphate"
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between "D-Fructose 6-phosphate" by typing characters exactly, as "D-Glucose 6-phosphate" -> "D- Fructose 6-phosphate". User can copy the above reaction and paste in the “Reaction” text box and press
“Enter” Key.
The Reversible check box “ ” is presented as unchecked. This will be clear when we look at the rules for entering a reaction, which are as follows:
1. The name of any species i.e. substrate, product, modifier (enzyme, activator, inhibitor) or any ligand, is to be enclosed in “Double Quotes”. However, if the name of any species does not contain any “space character”, i.e. name of any species is a single word without gap, then one can enter the name without enclosing in “Double Quotes”.
2. Join multiple substrates or products by typing “+” sign and then typing the name of next substrate or product.
3. The Substartes and Products on each side of the reaction are joined by reaction type sign. If the reaction is “irreversible”, then type “->”, i.e type hyphen (“-“) followed by greater than (”>”) sign, without any space in between these two, but a space before and after to separate from substrates and products.
On the other hand, if the reaction is “reversible”, then type “equal to” sign (“=”) with a space in between “equal to sign and substrtaes and a space between “equal to sign and products. This will convey to COPASI that the reaction is reversible and the checkbox for “Reversible” will be checked automatically. However, in the present example, check the checkbox for reaction to be reversible, manually, as we enterred the reaction as irreversible.
4. At the end of the products, type semicolon (“;”) character and Add modifers with their names enclosed within double qoutes. All the modifiers are separated from each other with a space character.
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Since we entered an irreversible reaction, Reversible check box “ ” is presented as unchecked.
Therefore check it manually, to make it a reversible reaction.
Now, open the “Rate Law” dropdown list
and select Reversible Michaelis-Menten, as shown next:
Now click commit button at the bottom of working space. The mathematical formula for Reversible Michaelis-Menten can be seen by clicking “Edit Rate Law” button, this will display the Function, as shown next:
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The Reversible Michaelis-Menten rate law is selected because the Parameters values required for Reversible Michaelis-Menten Modeling are published in the Table 1 of reference literature, already downloaded and shown next:
Now set the model parameters as shown next,
Double click “Parameter Overview” node in “navigation tree” panel to display the values for initial species values and kinetic parameters. To change any value, click on the numerical value for any species or parameter and enter the value, followed by pressing “Enter” Key.
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Click Commit button at the bottom left corner of working space panel. User is now ready to simulate Glucose-6-phsphate catalysed enzyme reaction following Reversible Michaelis-Menten Kinetic law.
However, first enter the format for output to display time course of the reaction. Expand “Output specification” node in navigation tree panel and select Plots(0) by double click, followed by double clicking
“New Plot” in “working space” panel.
This will open new plot_1.
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Click on “New Curve” Button. This will open a dialog box.
Select “Model Time” in left panel and [Fructose 6-phosphate]/(t) and [Glucose 6-phosphate]/(t) in the right panel under Species node of the right panel and click OK button. This will insert a new plot for display of the transient concentrations of selected species on y-axis with model time on x-axis of the plot_1.
Now open the “Tasks” node in “navigation tree” panel and click “Time Course”. Set the Time Course parameter values in the working space as shown.
Now click Run Button at the left bottom corner of working panel. This will display the time course for reversible equilibration of fructose-6-phosphate and glucose-6-phosphate, as shown next.
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This shows that equilibrium is attained at about 300 minutes.
Save the modelled enzyme using “Save As…” command from the file menu by Giving Name “GPI-Human”.
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Enter the name “GPI-Human” and click ‘Save Button’.
Back to Concept Map 3. Summary
In this module, students learnt about irreversible and reversible Michaelis-Menten kinetic laws and SABIO- RK reaction kinetics database, a database of experimentally determined reaction kinetic parameters values. This shows that Enzyme catalyzed reactions have their associated reaction kinetic parameters such as Vmax, Km, Inhibition constants, which can be used to study enzyme kinetics. These biochemical reaction kinetic parameters are stored in the databases, such as the SABIO-RK, presented in this module. These kinetic parameters can be downloaded in SBML and used in modeling enzyme catalysed biochemical reactions by simulation software directly. In addition, SABIO-RK database provides a hyperlink to the actual published research paper and can be seen to find out the details of the experiment as well as further information needed for kinetic modelling of the enzyme.
Students downloaded the reaction kinetics data from SABIO-RK reaction kinetics database and used for simulating enzyme kinetics using COPASI software. Students also learned about entering reactions manually using COPASI, selecting the required rate law for simulating the reaction using downloaded kinetic parameters. The only requirement to be met is the availability kinetic parameters data values to be used in the allowed kinetic law, such as irreversible or reversible Michaelis-Menten rate laws. This will enable the students to enter more reactions, say next reactions in a pathway to simulate complete pathways using collected parameters from SABIO-RK or literature data. In the next module, Students will see examples of coupling two enzyme catalysed reaction to construct complete pathways or even to couple two enzyme catalysed reaction from two different biochemical pathways for constructing a complete biochemical process.
