non-linear regression in the biological chemistry laboratory
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michaelis-menten kinetics of invertase
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last revision: 13-03-2017

linear and nonlinear regression in laboratorial biological chemistry education

Table 1 shows the four reaction equations that have to be considered in a weak diprotic acid (H2A) titration with strong base (MOH)

experimental data acquisition

Prepare a sucrose solution with known concentration and stabilize its temperature. Add an invertase solution to it and simultaneously start the clock. Take samples of the reaction (discontinuous assay) and transfer them, at precise times, to test tubes containing DNS (3,5‑dinitrosalicylic acid). This stops the reaction in the samples by inactivating the enzyme and also assays the reducing products, glucose and fructose, by measuring absorbance at 540 nm. Repeat the procedure for several more sucrose concentrations keeping the total enzyme concentration constant and low, but not so low as to become saturated by the substrate.

(Prepare a DNS absorbance standard curve in advance using glucose standards.)

model prescription

The Michaelis-Menten (MM) enzymatic process [1,2], Reactions 1 and 2, is the simplest description of the action of a biological catalyst. It considers the reversible formation of an enzyme-substrate complex, ES, from which irreversibly results a product P. The overall process is S → P. The MM initial rate equation, Equation 3, is derived by assuming that the process goes through a steady-state concentration of ES. It predicts that the initial rate, v0, has a limit when substrate concentration approaches infinity (catalytic sites saturation), known as maximum rate: vmax = kcat [E]total. [E]total is the total enzyme concentration (or total catalytic sites concentration) and KM is the Michaelis‑Menten constant which can be expressed as (kcat + kr)/kf. Application of the MM rate equation is restricted to conditions of negligible reverse rate of Reaction 2, that is, when product concentration is zero or very low and/or the reverse rate constant of Reaction 2 is much lower than the forward rate constant (large equilibrium constant). Rates should therefore be measured from time zero and using the smallest possible time interval, starting with only substrate and enzyme. Generally, product concentration will increase non-linearly with time but if a small assay time interval is used, the length of which depends on the particular reaction rate, the product increase will be apparently linear and the initial MM process rate is easily obtained by linear regression, as illustrated in Figure 1.

Non-linear regression of Equation 3 over {reaction rate, [substrate]} data allows to estimate values for parameters KM and kcat at the experimental temperature.

There are alternative linear rate functions for the MM process. For example, the inversion of Equation 3 results in Equation 4, known as Lineweaver-Burk’s. In this new arrangement, the linear variables are 1/v0 and 1/[S] and the linear parameters are combinations of the original non-linear parameters:

slope = KM/(kcat [E]total)                        intercept = 1/(kcat [E]total)

Obtaining kcat values at different temperatures is a means to estimate the activation entropy, ΔS‡, enthalpy, ΔH‡, and Gibbs energy, ΔG‡, of the forward direction of Reaction 2 by regression of the Eyring‑Polanyi equation [3], Equation 5, over {temperature, kcat} data. In this equation, k is a rate constant, kB is the Boltzmann constant, h is the Planck constant, R is the ideal gas constant and T is temperature. It is derived from transition state kinetic theory and can be linearized as Equation 6.

Figure 2 summarizes all the experimental and calculation procedures.

computational function

Matlab files can be downloaded using the links at the bottom of this page. Function kenz.m is a simple implementation of the Matlab function nlinfit.m which estimates the parameters of Equation 3 by nonlinear least squares fitting. Also, Matlab functions nlparci.m and ttest.m are used to calculate, respectively, the uncertainty associated with the estimated parameter values and the Student’s t-statistic for the residue sample against a normal distribution of samples with mean zero. The code can be inspected with any word processor and run in a Matlab console window using a command like:

kenz(datavariablename, [kcat ,KM], 'namestring', o )

where datavariablename is the name of a variable in the Matlab workspace containing values for the experimental variables (see sample file below). The second argument of kenz.m is a vector of initial values for the adjustable parameters (example: [150 0.03]) and the last argument, o, is set to 1 to perform regression or 0 to just plot data points and the function curve calculated using the initial parameters. This last option is useful to manually adjust the initial parameter values and see the effects on the theoretical function. After a good initial parameter vector is found by trial and error, the nonlinear fitting process is much more likely to converge.



Picture

Picture
Table 1 shows the four reaction equations that have to be considered in a weak diprotic acid (H2A) titration with strong base (MOH)

Figure 1 – The concentration of glucose increases in an apparently linear fashion as a function of time, at least for the first five minutes. The linear regressions do not extrapolate well to the origin, as it should, due to bad absorbance reference setting by the students. However, the slopes are independent of that setting and still give good estimates of the initial MM process rates. These rates are used in Figure 3 (top).


Picture

Picture
Table 1 shows the four reaction equations that have to be considered in a weak diprotic acid (H2A) titration with strong base (MOH)

Figure 2 – Diagram summarizing experimental and calculation procedures. The experimental measurements are coded with brown text and the calculations with green text. Coded with black text are all the calculated quantities.


Picture
Table 1 shows the four reaction equations that have to be considered in a weak diprotic acid (H2A) titration with strong base (MOH)

examples and discussion

The enzyme invertase catalyses the almost complete reaction of sucrose with water (hydrolysis) to produce glucose and fructose, Reaction 7. This reaction is not formally similar to the theoretic MM enzymatic process nor to the overall MM reaction S → P. But since water concentration is essentially constant (it is the solvent), and the number of products is irrelevant, because theoretically there is no reversal of reaction 2, it is adequate to test the MM process description on invertase’s kinetic data with S as sucrose.

Figure 3 shows the best fit of Equation 3 over two experimental data sets obtained in class, one with low random errors and another with high random errors. It is instructive to point out how non-linear functions and its linearizations “interpret” noise differently. In the low-error experiment, values obtained for kcat and KM by non-linear and linear regressions are identical as expected (99/99 and 0.021/0.021, respectively) while in the high-error experiment the two types of regression output very different values (88/73 and 0.033/0.021, respectively). This means that 73/0.021 will not make a good non-linear data fit while 88/0.033 will not make a good linear data fit [4]. When noisy data is unavoidable then it all depends on the nature of the noise sources. The reasonable option would be to go with the most natural functional description of the target system, in this case the non-linear function [5,6].

Notice that error estimates of linearized functions parameter values also have to be correctly propagated to the original parameters (slope and intercept to kcat and KM in this case) taking into account the mathematical operations required for conversion.

It is also important to note that this experiment is an accumulation of a series of interdependent procedures, starting with the DNS/glucose absorbance standard curve and ending with the Gibbs energy of activation (Figure 2). The best way to ensure low error on parameter values, besides careful control and execution of procedures, is to acquire the largest possible experimental data samples.


Table 1 shows the four reaction equations that have to be considered in a weak diprotic acid (H2A) titration with strong base (MOH)

Figure 3 – Results from non-linear regression of Equation 3 with more precise (top) and imprecise (bottom) experimental data.

Picture
Picture

Table 1 shows the four reaction equations that have to be considered in a weak diprotic acid (H2A) titration with strong base (MOH)

references

[1] https://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics (Jan-2017)

[2] Johnson, K. A., Goody, R. S. (2011) The Original Michaelis Constant: Translation of the 1913 Michaelis–Menten Paper. Biochemistry, 50, 8264-8269. DOI: 10.1021/bi201284u

[3] https://en.wikipedia.org/wiki/Eyring_equation (Jan-2017)

[4] https://www.mathworks.com/help/stats/examples/pitfalls-in-fitting-nonlinear-models-by-transforming-to-linearity.html (Jan-2017)

[5] Lente, G., Fábián, I., Poë, A. J. (2005) A common misconception about the Eyring equation. New J. Chem., 29, 759-760.

[6] Espenson, J. H., in Chemical Kinetics and Reaction Mechanisms, McGraw-Hill, New York, 1995, 2nd edn., p. 158

MatlabInvertaseKineticData.mat
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kenz.m
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