PROBLEM SET 3
SIMULATION OF BIOCHEMICAL KINETICS
Part I: The first goal of Part I of this exercise is to compare two approaches to the simulation of biochemical reaction kinetics and to see how error may propagate is such system. The Michaelis-Menten enzyme reaction will constitute the experimental system. The reaction mechanism for this system is:
k1 k3
[E] + [S] ===== [ES] ===== [E] +[P]
k2 k4
where [E] is an enzyme concentration, [S] is a substrate concentration, [P] is the product concentration, [ES] is the concentration of the enzyme-substrate complex, and k1, k2, k3, and k4, are rate constants.
The two approaches to simulation are based on either analytical or numerical solution of rate equations. The analytical approach for a simple Michaelis-Menten reaction system is based on a pseudo steady-state assumption and, strictly speaking, is only valid during initial phase of the reaction. The derivation of the analytical solution depends on solving the change rules for each of the four reactants:
1. d[E]/dt = [ES] (k2+k3) - k1[E][S] - k4[E][P]
2. d[S]/dt = k2[ES] - k1[E][S]
3. d[ES]/dt = k1[E][S] + k4[E][P] - [ES](k2 + k3)
4. d[P]/dt = k3[ES] - k4[E][P]
Assuming that equation 3 is zero (i.e. that the enzyme substrate complex is at steady state), the change rules (eqs. 1 to 4 ) may by simplified to:
, (5)
where Vm = k3*ET, ET=[E] + [ES], and Km = (k2 + k3)/k1. This particular simplification also requires that either k4 = 0 or [P] is zero or very near zero (as it would be during the initial part of the reaction).
The numerical solution, of course, solves the change rules explicitly using some numerical solution procedure (e.g. Euler or Runge-Kutta). In theory, it yields the time course of any single reaction, but it both is slower than the pseudo steady-state analysis and also is subject to numerical errors. Your work in this exercise is to explore the differences and compatibility of these two approaches.
Part II: The second goal of this exercise is to compare various approaches to estimation of parameters of the Michaelis-Menten model of enzyme kinetics. The approaches that you will examine here are in two classes. The first, and the most traditional class, is based on transformations of the pseudo steady-state rate law used in Part I. The second class of approach uses the time course data, which you generate in Part I, with the numerical solution of rate equations. Methods of estimating parameters are described below.
1. Methods of Parameter Estimation
1.a. The Michaelis-Menten model specifies a relationship between a substrate concentration and the reaction rate:
(6)
where parameters are defined in Part I. The two parameters in this model (Km and Vm) are often measured graphically from plots of various linearizations of equation 6.
Three possible linearizations of equation 1 are:
1. Lineweaver-Burke
2. Hanes
3. Eddie-Hofstee
These three are the most popular in the biochemical literature. These are not only convenient for plotting the data, but they are also compatible with simple linear regression. The parameters can thus be estimated by least squares procedures.
For the simple regression model:
y = bo + b1* x
parameters and variables for each transformation are:
Transformation y x bo b1
intercept slope
1 1/V 1/S 1/Vm Km/Vm
2 S/V S Km/Vm 1/Vm
3 V V/S Vm -Km
Part III: The purpose of this part III of the exercise is to explore ways of improving precision and accuracy of parameter estimation. Two ways considered here are alternatives regression procedures and variation of experimental design.
1. Weighted Linear Regression and a Non-Linear Regression Procedures.
The results of Part I should have convinced you that methods of estimating Michael-Menten parameters vary considerably in accuracy. One way to improve accuracy of parameter estimation is to use alternative regression procedures. You will try two alternatives in this exercise: a weighted linear regression procedure and a non-linear regression procedure. These procedures are contained in two files, WTREG.XLS and NOLINEAR.XLS, on the Biology 315 Homepage.
1.a. The weighted linear regression procedure, WTREG.XLS, allows you to select one of two transformations. Run for transformations 1 and 2, on the three data set for the three VE/VP error ratios (0.1, 1, 10).
1.b. The non-linear regression program, NOLINEAR.XLS, uses a solver tool in EXCEL. You will have to import your eight S,V pairs for each replicate. It should be run for all three VE/VP error ratios files.
2. Design of Experiments:
The data sets generated in Part II were based on an implicit experimental design. Eight substrate levels were selected and were evenly distributed over a range 5 to 40 mM. At each substrate level, furthermore, the velocity measurement was replicate. Experimental design, therefore, consisted of the range of substrate concentration, spacing of substrate levels over the range, and the number of replicates at each substrate level. In this Part III, you are to explore the influence of experimental design on accuracy and precision of the parameters estimates by varying range and spacing of substrate levels.
Work Part I :
Ia. For the analytical model of a simple biochemical reaction system, you must write a program that uses equation 5 to simulate the relation between initial reaction velocity and substrate concentration. Your program must generate at least 20 pairs of substrate, velocity data over the substrate range 0 to 40 mM. Assume Km is 21 mM and Vm is 0.01 mM/min. Create a X-Y plot of velocity vs substrate.
Ib. To compare numerical approaches to the same basic reaction system, substitute the change rules in equations 1 to 4 into the Lotka-Volterra Population model that you developed for problem set 2. Be sure to evaluate all change rules before updating concentrations. Assume the following rate constants: k1=0.005, k2=0.005, k3=0.1 and k4=0, the initial values of [P] and [ES] should be zero, and the value of ET should be 1.0. Allow [S] to vary in each simulation. Next, run the simulation model using these new parameter estimates for three starting substrate concentrations (0.5 Km, Km, and 1.5 Km). for each simulation, you must save at least 20 set of data (using a time step of 1 minute). Your output must include the concentration of the four reactants and the velocity of the reaction (d[P]/dt). Generate five separate x-y plots for reaction velocity (d[P]/dt), [P], [ES], [E], and [S].
Ic. The final part of this part I of the Problem Set 3 requires modification of your previous program to a Monte-Carlo form. Error will be assumed to be N(0,s^2), where s is the standard deviation of the error, and it will affect the initial enzyme concentration and/or the measured reaction product. Assuming a 10% error on product (i.e. s/P=0.1) and 5% error on starting enzyme concentration (i.e. s/Vm=0.05), obtain 10 replicate sets of t, Pt for a starting substrate concentration equal to the Km of the reaction. The reactions should run long enough to deplete at least 20% of the initial substrate concentration, and you should have at least 20 pairs of t, Pt for each of the 10 replicates. Produce on scatter plot of P versus t.
Work Part II
To explore errors you will take advantage of Monte-Carlo simulations to generate hypothetical data sets from known models and parameters. Based on the error sources discussed in part 3 of Problem Set 3 Part I, equation 5 becomes:
(7)
where N(,2) is a Normally distributed random variable with mean and variance 2, VE is the variance of the initial enzyme concentration, and VP is the variance of the product.
You must write a program, that will generate pairs of substrate and observed velocity.
Remember to use a Z-distributed random variable for the error terms in equation 4.
r = 2*RND-1
Z = log{(1+r)/(1-r)}/1.82
IIa. Precision and Accuracy using Method 1.a. Create 12 different data sets containing <S,V> data pairs. Initial substrate concentrations should be {5, 10, 15, 20, 25, 30, 35, and 40 mM}, Km = 21 mM, and Vm = 0.1 mM/min. The 12 data sets should be three replicates of four different ratios of VE/VP (0, 0.1, 1.0, and 10.0). Where for VE/VP= 0, VE=0 and VP=0. In last three cases, total variance (VE + VP) should be 0.012. Next, obtain estimates of Vm and Km by linear regression with the three transformations. Finally, for each of the ratios of VE/VP obtain mean and variance or standard deviation of Km and Vm from the three replicates.
Work Part III
IIIa Modify your module in Part II to generate nine new data sets containing substrate and velocity data pairs. Assume that VE/VP = 1 and VE+VP = 0.012 for all cases. These new data sets will contain three replicates according to the following schedule:
Replicates Substrate Concentrations
1 to 3 2.5, 5.0, 10.0, 20.0, 40.0
4 to 6 4.3, 9.1, 13.8, 18.6, 23.3
7 to 9 4.3, 6.6, 10.0, 15.3, 23.3
IIIb Obtain estimates of Km and Vm for transformation 1, 2 and 3 using the linear regression procedures you employed in Problem Set 3 Part II. From the three replicates, calculate the mean and variance or standard deviation of Km and Vm for each of the three experimental designs and transformations.
IIIc. Using WTREG.XLS also obtain estimates of Km and Vm for transformation 1 and 2. Including all three replicates.
IIId. Finally, use solver (from NOLINEAR.XLS template) to obtain 3 replicate estimates of Km and Vm from each of the experimental designs. From the three replicates, calculate the mean and variance or standard deviation of Km and Vm each of the three experimental designs and transformations. Prepare a table showing values of Km and Vm obtained from these analyses.
SUBMIT:
Part I:
1) An excel file with parts Ia to Ic, including the 7 x-y plots (15 points)
2) Your discussion should include (15 points):
a. Discuss of all your results from Part I
b. Suggest a procedure by which to compare output
from models for Ia and Ib above;
c. Using this procedure, compare the output at the three
substrate levels in Ib, and you comment on three differences
d. Suggest a way to incorporate error observed in Ic into
a model like the analytical model in Part Ia.
Part II:
1 ) a) Your program (10 points)
b) For the four ratios of VE/VP and three transformations prepare a table showing mean and variance or standard deviation of Km and Vm of the replicates of each ratio/transformation combinations
2 ) Discussion addressing the following questions (15 points):
a) Based on the analysis in IIa, which of the transformations yields better estimates? Does the ratio of VE/VP affect the relative performance of the various transformations?
PART III:
1) Provide a grand summary table that summarizes the results of Part IIa and IIIb, IIIc and IIId). The table should include Km, variance of Km, Vm and variance or standard deviation of Vm where appropriate (10 points).
2) Discussion of the effect of experimental design and methods of parameter estimation on the accuracy and precision of estimated parameters (15 points).
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