BCHM2972 ELMA Simulation
page 1
Simulating the ELMA reaction.
The Mathematics behind the reaction.
The ELMA reaction is a pseudo first order reaction. In other words, everything is in vast excess
except the glucose. The rate of the reaction is therefore dependent upon the [glucose]. This can be
written mathematically as:
-d[S]/dt is proportional to [S] where S (=substrate) is, say, glucose.
Inverting and integrating (and fiddling around) we get…
[S] = Ae-kt………(1).
Now, to find the [S] at the beginning of the reaction, (lets call that [S0]) we find [S] at t = 0 by
substituting into equation (1)  [S0] = Ae0 or A (as e0 = 1)
Now the expression looks like this:
[S] = [S0]e-kt ………..(3)
Now also, [P] = [So] – [S]…….(2). where [S0] is the starting substrate concentration.
Therefore [P]/[So] = 1 – [S]/[So]………..(4)
and rearranging equation (3) we get [S]/[S0] = e-kt ………….(5).
So plug equation (5) into equation (4)  [P]/[S0] = 1 - e-kt
Now we can solve this equation to find [P] when we know t (time), k (rate constant) and [S0]
Half-life (t½) and rate constant
Half life is the constant of a first order reaction. It is the time it takes for half the starting substrate, S0
to be converted to product. Determine this value using the optimized timecourse you did in the first
week. Irrespective of the starting concentration, the half-life, t ½ for a particular reaction is constant
(Convince yourself of this by determining t ½ at different [S0] ).
We can also then express k (the rate constant) in terms of the half-life:
The half-life mathematically is when [S]/[S0] = ½.
So equation (4) becomes ½ = e-kt  -ln0.5/t ½ = k
BCHM2972 ELMA Simulation
page 2
Setting up the time course simulation
On the top row of one Excel worksheet set up a series of cells: the Extinction Coefficient; the rate
constant, k; the fold-enzyme and the half-life, t½.

(μM-1cm-1)

0.0053
Rate
constant,
k
1. This number
came from the
gradient of the
standard curve.
Take care with
units!
0.11552453
[enzyme]
-fold
Half-life,
t½ (min)
1
2.Determine the half life from
your timecourse. It is the time
at which the absorbance = 1/2
the maximum. It is
independent of [S0]
Name this cell half_life (see
below if you are unsure how
to name cells)
3. The rate constant
cell has the formula
=LN(0.5)/Half_life
inserted
as k = -ln0.5/t ½
Now we have the parameters in place and we are ready to simulate the standard curve.

-1
-1
(mM cm )
[So]
(nmol/ml)
Time
(min)
0
5
10
15
20
25
30
0
0.053
50
Rate
[enzyme]
constant, 0.11552453
-fold
k
Absorbance
100
150
200
Half-life,
t½ (min)
1
250
P/So
0
0.438769
0.68502
0.823223
0.900787
0.944319
0.96875
35 0.982462
40 0.990157
45 0.994476
Formula entered:
=1-EXP(-$F$1*A5)
This is 1 - e-kt
as [P]/[S0] = 1 - e-kt
6
The [glucose] used in
your standard curve
6
BCHM2972 ELMA Simulation
page 3
The next step involves converting this ratio to absorbances. This is quite easy. Multiply the [P]/[S0]
ratio by the [S0] (to get [P]) and the extinction coefficient and you have the absorbances. Use fixed
rows, columns or both to allow you to drag the formula across and down the whole table.

0.0053
(mM-1cm-1)
[So]
(nmol/ml)
Time
(min)
0
50
0
0
0
0
0
0
0
0
0.116274
0.18153
0.218154
0.238709
0.250244
0.256719
Rate
[enzyme]
constant, 0.11552453
-fold
k
Absorbance
100
150
Half-life,
t½ (min)
1
200
6
250
P/So
0
5
10
15
20
25
30
0
0.438769
0.68502
0.823223
0.900787
0.944319
0.96875
35 0.982462
40 0.990157
45 0.994476
0
0
0
0
0.232548 0.348821336 0.465095 0.581369
0.36306 0.544590691 0.726121 0.907651
0.436308 0.654462527 0.872617 1.090771
0.477417 0.71612601 0.954835 1.193543
0.500489 0.75073347 1.000978 1.251222
0.513438 0.77015625 1.026875 1.283594
0 0.260352 0.520705 0.781056917 1.041409 1.301762
0 0.262392 0.524783 0.787174709 1.049566 1.311958
0 0.263536 0.527072 0.790608204 1.054144 1.31768
You can now plot the graphs; time course and standard curve.
Simulating a change in [enzyme]
To set this up simply multiply k (the rate constant) by the [enzyme], expressed as fold change. You
are now in the position to change the amount of enzyme you add by adjusting the fold change cell.
The effect of the half life of the reaction can also be seen. Here are the graphs if the half life is 6 min
and there is 1-fold [enzyme].
1.4
250 M
1.2
200 M
1.0
150 M
0.8
0.6
100 M
0.4
50 M
0.2
Absorbance 500 nm
Absorbance 500 nm
1.4
y = 0.0053x
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0
10
20
30
Tim e (m in)
40
50
0
50
100
150
[Glucose] (M)
200
250
BCHM2972 ELMA Simulation
page 4
If the amount of enzyme decreases…
If I drop the [enzyme] to 0.1 then the reaction is not completed in the 45 min.
Time Course
Standard Curve: Glucose
0.6
0.5
Absorbance 500 nm
Absorbance 500 nm
0.6
0.4
0.3
0.2
0.1
0
-5
y = 0.0021x
R2 = 1
0.5
0.4
0.3
0.2
0.1
0
5
15
25
35
45
0
50
Time (min)
100
150
200
250
[glucose] (nmol/mL)
The standard curve is still linear but it does not reach the desired final absorbance. The time course is
a fan rather than a series of plateaux. Does this matter if the standard curve is still linear?
Increasing the amount of enzyme…
If I add 5 times the enzyme the reaction just reaches the plateau quicker. The standard curve is the
same, it is just more expensive as the enzyme is usually the most expensive component of an ELMA.
You do finish the reaction quicker, in this case the reaction is all over in 5 min.
Time Course
Standard Curve: Glucose
1.4
1.4
Absorbance 500 nm
Absorbance 500 nm
1.2
1
0.8
0.6
0.4
0.2
0
-5
y = 0.0053x
R2 = 1
1.2
1
0.8
0.6
0.4
0.2
0
5
15
25
Time (min)
35
45
0
50
100
150
[glucose] (nmol/mL)
200
250
BCHM2972 ELMA Simulation
page 5
Simulating impurities in the standard glucose.
Now let’s try to simulate the effect of an impure standard glucose solution. To do this you will need to
add a cell with the % impurity at the top of the spreadsheet with the other parameters. You may like to
name this cell. Then you need to apply this %impurity to the [S0] values as this will alter the effective
[S0]. This is done by multiplying the [S0] by (100-%impurity)/100 and then making sure the
absorbances are calculated from the effective [S0] rather than the [S0]. If you set the impurity at 50 %
the graph should look like this:
Time Course
Standard Curve: Glucose
0.7
0.7
Absorbance 500 nm
Absorbance 500 nm
0.6
0.5
0.4
0.3
0.2
0.1
0
-5
y = 0.0026x
R2 = 1
0.6
0.5
0.4
0.3
0.2
0.1
0
5
15
25
Time (min)
35
45
0
50
100
150
200
250
[glucose] (nmol/mL)
Note that the effect of an impurity is to lower the final absorbances of all the standards but the time
course still plateaus. The standard curve is linear; the gradient is just lower. Can you distinguish
between denatured enzyme and impure standard glucose from the standard curve?
Limiting Reagents
A second scenario is when you have a limiting amount of one of the reagents. In the case of the
glucose assay this will be the 4-AAP. To set up this simulation you have to be aware of the
stoichiometry of the reaction. Two glucose molecules require 1 4-AAP, so the standard curve
becomes limited at [4-AAP] < 125 M. At concentrations above 125 M the reaction will proceed,
albeit not under truly first order conditions (the reaction rate may be more dependent on the [AAP]
than the [glucose]). To simulate this situation you need to use the “IF” statements in Excel. Set up a
cell which will have the [AAP] (see the final screen shot). To this cell you can enter any numerical
value. Then into another row add your IF statement. We want the IF statement to imply “If the value
of [glucose] is less than 2x value of [4AAP], then the ‘usable’ [substrate] will be the same as the
actual [substrate], BUT if not (ie if [glucose] is > 2x[4AAP], then assay is limited by [4AAP]…and
the usable [S0] = 2x [4AAP]”
IF Statement:
IF the [S0] < [AAP]*2 then use [S0]. IF [S0]>[AAP]*2, use [AAP]*2 as the [S0]. While this gives an
over-simplified representation of the effect of reagent limitation, it does show the bunch-up nicely.
The standard curve becomes very bent!!
BCHM2972 ELMA Simulation
page 6
Time Course
Standard Curve: Glucose
0.3
0.2
Absorbance 500 nm
Absorbance 500 nm
0.25
0.15
0.1
0.05
0
-5
y = 0.0011x
R2 = 0.3923
0.25
0.2
0.15
0.1
0.05
0
5
15
25
35
45
0
Time (min)
50
100
150
200
250
[glucose] (nmol/mL)
The additional Excel skills which would make this simulation really work well would be:



Naming cells
Macros
Protecting your worksheet once you have finished.
I have included a short set of instructions on how to do these 3 tasks at the end of this section.
Record macros for the increase and decrease in [enzyme], the limiting and excess
reagent, impurities in the standard glucose, increasing and decreasing the half-life of
the reaction, maybe even new chromophores with higher or lower extinction
coefficients.
Finally to make the simulation very user-friendly move the graphs up to cover the absorbance
calculations. Your users don’t need to see the data, just the graphs!
BCHM2972 ELMA Simulation
page 7
Present all the simulations on ONE page: The whole page would look
something like this:
The standard [4-AAP] is
2.5 mM or 2500 mM.
Enter the %
impurity here
% Impurity
0

(mM-1cm-1)
Effective
[S0]
Reagent
Limitations
0.0053
Rate
[enzyme]
constant, 0.11552453
-fold
k
Half-life,
t½ (min)
1
0
50
100
150
200
250
0
50
100
150
200
250
200
250
Time (min)
0
5
10
15
20
25
30
0
50
0
0
0
0
0
0
0
0
0.116274
0.18153
0.218154
0.238709
0.250244
0.256719
100
150
35 0.982462
40 0.990157
45 0.994476
2500
This takes the
limiting reagent
into account.
P/So
0
0.438769
0.68502
0.823223
0.900787
0.944319
0.96875
[AAP]
(uM)
The effective [S0] is
shown here. This takes
into account the impurity
of the standard glucose.
Absorbance
[So]
(nmol/ml)
6
0
0
0
0
0.232548 0.348821336 0.465095 0.581369
0.36306 0.544590691 0.726121 0.907651
0.436308 0.654462527 0.872617 1.090771
0.477417 0.71612601 0.954835 1.193543
0.500489 0.75073347 1.000978 1.251222
0.513438 0.77015625 1.026875 1.283594
0 0.260352 0.520705 0.781056917 1.041409 1.301762
0 0.262392 0.524783 0.787174709 1.049566 1.311958
0 0.263536 0.527072 0.790608204 1.054144 1.31768
Time Course
Standard Curve: Glucose
1.4
1.4
Absorbance 500 nm
Absorbance 500 nm
1.2
1
0.8
0.6
0.4
0.2
Move the
graphs
up to
cover the
data.
R2 = 1
1
0.8
0.6
0.4
0.2
0
0
-5
y = 0.0053x
1.2
5
15
25
Time (min)
35
45
0
50
100
150
200
250
[glucose] (nmol/mL)
Naming cells.
It is sometimes helpful to name cells, rather than absolutely referencing them with $ everywhere. To
name a cell, click on the cell  insert name  define and then give it an informative
title. Whenever you click in that cell (to incorporate it into an equation) the cell will appear as its
name in the equation and it will be absolutely referenced (ie fixed).
BCHM2972 ELMA Simulation
page 8
Recording Macros.
What is a Macro? It is a section of code (written in visual basic), which performs a common task.
Don’t panic: you don’t have to know visual basic to do this….I don’t! The Excel program has a builtin process, which can record your task and convert it to visual basic for you. You must ensure that you
know EXACTLY what you need to do before recording your macro. It is a good idea to practice the
task a couple of times before starting to record. You should also finish the whole spreadsheet before
recording macros, as changing things later can upset the macro!
Once you are confident you know what to do go to Tools  Macros  record new macro.
First record a macro to “reset” the graphs at the default position. This is often a great idea when you
have many alternative options to try (which you will later). Set the half-life to 6, the [enzyme] to 1,
the impurity to 0 and the [4-AAP] to 2500. These are the default conditions. Every time you record a
macro, you will need to type in the value for EVERY parameter that may change in other macros
(even if it doesn’t change in the particular macro you are recording). Otherwise, that value will not
return to the original value after you run a different macro.
Be sure to give your macro a name and assign it a hotkey. Record your hotkeys as you go along.
When you have finished the actions click the stop (blue square) in the record box. Make sure you have
worked out which cell you want to finish in before recording.
We now want to assign this macro to a button. The buttons are found in the FORMS toolbar
(view  toolbars  forms). Click on the button option and then place the button
appropriately somewhere near the graphs. Give the button a name etc. You will be asked which macro
you want to assign to the button. Click on the appropriate macro and you are all set. If you want to
change the size of your button or edit the text once you have located it and clicked elsewhere, simply
right click on the button.
Protecting your worksheet.
When you have completed your tutorial you need to lock all the cells except for the ones your macro
(or the user of your spreadsheet) will want to change. You don’t want someone to come in and alter
all the other settings. This process is very handy to know how to do in Excel and is actually done in a
reverse process. You go into your spreadsheet unlock the cells you need to change then protect the
whole sheet. To unlock the relevant cells click on the cells and go to Format  cells 
protection  then untick the locked box. Then go to Tools  protection  protect
sheet. It will ask you for a password (not a bad idea). Make sure you write your password down
somewhere!