# 4.8: Enzyme Parameters - Biology

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## (V_{max}) & (K_{cat})

On a plot of Velocity versus Substrate Concentration ( V vs. [S]), the maximum velocity (known as Vmax) is the value on the Y axis that the curve asymptotically approaches. It should be noted that the value of V max depends on the amount of enzyme used in a reaction. Double the amount of enzyme, double the Vmax . If one wanted to compare the velocities of two different enzymes, it would be necessary to use the same amounts of enzyme in the different reactions they catalyze. It is desirable to have a measure of velocity that is independent of enzyme concentration. For this, we define the value Kcat , also known as the turnover number. Mathematically,

[ ext{Kcat} = frac{V_{max}}{ [Enzyme]} ag{4.7.1}]

To determine Kcat, one must obviously know the Vmax at a particular concentration of enzyme, but the beauty of the term is that it is a measure of velocity independent of enzyme concentration, thanks to the term in the denominator. Kcat is thus a constant for an enzyme under given conditions. The units of K cat are ( ext{time}^{-1}). An example would be 35/second. This would mean that each molecule of enzyme is catalyzing the formation of 35 molecules of product every second. While that might seem like a high value, there are enzymes known (carbonic anhydrase, for example) that have Kcat values of (10^6)/second. This astonishing number illustrates clearly why enzymes seem almost magical in their action.

## (K_M)

Another parameter of an enzyme that is useful is known as KM , the Michaelis constant. What it measures, in simple terms, is the affinity an enzyme has for its substrate. Affinities of enzymes for substrates vary considerably, so knowing KM helps us to understand how well an enzyme is suited to the substrate being used. Measurement of KM depends on the measurement of Vmax. On a V vs. [S] plot, KM is determined as the x value that give Vmax/2. A common mistake students make in describing V max is saying that KM = Vmax/2. This is, of course not true. KM is a substrate concentration and is the amount of substrate it takes for an enzyme to reach Vmax/2. On the other hand Vmax/2 is a velocity and is nothing more than that. The value of KM is inversely related to the affinity of the enzyme for its substrate. High values of KM correspond to low enzyme affinity for substrate (it takes more substrate to get to Vmax ). Low KM values for an enzyme correspond to high affinity for substrate.

## PCR Cycling Parameters&ndashSix Key Considerations for Success

PCR cycling and running parameters must be set up for efficient amplification, once appropriate amounts of DNA input and PCR components have been determined. The characteristics of the DNA polymerases, the types of PCR buffers, and the complexity of template DNA will all influence setup of these reaction conditions. Sections on this page discuss general considerations for PCR cycling parameters, beginning with an illustration of the key steps of the PCR process (Figure 1).

Figure 1. Illustration of the main steps in PCR─denaturation, annealing, extension─to amplify target sequence from a template DNA.

### Video: Principles of PCR

PCR relies on three thermal cycling steps to amplify a target DNA sequence. This video explains how these three steps work in PCR.

## Normal ranges of some blood chemistry parameters in adult farmed Atlantic salmon, Salmo salar

Blood samples from healthy adult Atlantic salmon fed an optimal diet in net sea pens were collected at intervals from October to May. Haematological determinations and biochemical serum analyses were carried out on 20 fish in each of seven samples. The ranges of haemato-logical values for sample means were: haematocrit 44–49%, haemoglobin 8.9–10.4 g 100ml −1 , red blood cell count 0.85–1.10 × 10 12 l −1 , MCV 441–553 × 10 −15 1, MCH 94–106 × 10 −6 g, MCHC 19.4–21.7 g 100ml −1 and leucocrit 0.43–0.96%. The ranges of enzyme activities in serum, for sample means, were: alkaline phosphatase 647–988Ul −1 , aspartate aminotrans-ferase 202–351 Ul −1 and alanine aminotransferase 4–8 Ul −1 . The ranges of the other parameters analyzed in serum were: total protein 41.6–56.6 gl −1 , albumin 18.3–24.3 gl −1 , albumin/total protein ratio 39.3–44.0%, creatinine 26–46 μmol, triglycerides 2.53–4.98 μmol and cholesterol 9.3–12.8 μmol. These values are considered to be the normal ranges in healthy fish. Variations due to seasonal changes, and the clinical significance of the selected parameters, are discussed. Data showing the reproducibility of the biochemical analyses in serum are presented.

## DESCRIPTION AND DEVELOPMENT OF THE MODEL

The simplest model of an enzyme-catalyzed reaction mechanism is given in Fig. 1. The mechanism consists of two steps the first one is the binding of enzyme to substrate, and the second one includes both the catalysis of the reaction and the release of the product. The binding of the substrate to the enzyme is determined by the affinity constant KA = k1/k−1 the Michaelis constants are KS = (k2 + k−1)/k1 for the substrate and KP = (k2 + k−1)/k−2 for the product. The maximum velocity for the forward reaction is Vmax = k2eT, where eT is the total amount of enzyme, and k2 is the catalytic rate constant, which is also called kcat.

Now let us study the effect of changing the values of certain rate constants to show that the reaction velocity is an optimizable quantity. The model shown in Fig. 1 is really much more complicated than it appears to be, because it involves six independent parameters, four rate constants and S and P concentrations, or five if it deals with a chemical reaction that has a fixed equilibrium constant. Thus, to present a simple case that can be analyzed easily, let us fix the values of some parameters in addition to Keq, say s and p (the substrate and product concentrations), k1 (which is the rate constant of enzyme-substrate binding), and k2 (which involves the catalytic and product release constants). This choice has a clear chemical meaning, as both the binding of the enzyme to the substrate and the catalysis rate have a limit imposed by chemical constraints of the active site and substrate structure. Thus, the evolution of the enzyme toward an optimum set of values of rate constants can be summarized as follows.

Let us imagine a hypothetical sequence of evolutionary steps for an enzyme with fixed values of k1 and k2 (see Fig. 2). In a first stage the enzyme has a very low affinity for the substrate, yielding obviously a very low reaction rate, and then evolution and natural selection work thereafter, improving the affinity of the enzyme for its substrate, by decreasing k−1. Because the equilibrium constant of the reaction Keq = (k1 · k2)/(k−1 · k−2) is fixed, as are k1 and k2 in this case, the decrease of k−1 must be accompanied by an increase of k−2. This last effect is, in principle, unfavorable to the whole activity, but because the former stage yielded a very low reaction rate the positive effect of a decrease in k−1 is more important than the negative one of an increase in k−2, and so the global result is that the second stage (Fig. 2) has a net increase in the reaction velocity. Now let us imagine that the affinity is additionally increased by further decreases of k−1, again involving a corresponding increase of k−2. At this stage the affinity of the enzyme for its substrate has been greatly increased, but instead of enhancing the reaction rate, it now decreases the catalytic efficiency. The price one has to pay for such a high affinity for the substrate is a high affinity for the product, as well the enzyme tightly binds the substrate but also tightly binds the product. Because the product is only slowly released, the system now exhibits a poor net reaction velocity. This simple reasoning demonstrates that the value of the affinity constant has an optimum value to get the maximal activity of the enzyme. In other words, the mathematical function that relates the reaction velocity with the set of rate constant values of the enzyme is an optimizable function and so a clear target of natural selection.

## What Causes “False” Elevation of Biomarkers?

Not all elevations in cardiac biomarkers indicate a heart attack.

Creatine kinase levels can become elevated with any muscle injury, or with damage to the brain or lungs, or with liver or kidney disease.

Elevations in the troponin blood level is really quite specific for cardiac cell damage, so strictly speaking, there is no such thing as a “false” elevation of troponin. However, damage to cardiac cells can occur for reasons other than an acute heart attack. These conditions may include heart failure, myocarditis, rapid atrial fibrillation, sepsis, coronary artery spasm, aortic dissection, stress cardiomyopathy, or severe pulmonary embolus. ﻿ ﻿

The diagnosis of a heart attack relies not on a single blood test, but also on clinical symptoms, ECG changes, and (often) on a pattern of biomarker elevations suggesting acute heart cell injury.

## Template DNA denaturation assessment

The initial denaturation step is carried out at the beginning of PCR to separate the double-stranded template DNA into single strands so that the primers can bind to the target region and initiate extension. Complete denaturation of the input DNA helps ensure efficient amplification of the target sequence during the first amplification cycle. Furthermore, the high temperature at this step helps inactivate heat-labile proteases or nucleases that may be present in the sample, with minimal impact on thermostable DNA polymerases. When using a hot-start DNA polymerase, this step also serves to activate the enzyme, although a separate activation step may be recommended by the enzyme supplier.

The initial denaturation step is commonly performed at 94–98°C for 1–3 minutes. The time and temperature of this step can vary depending on the nature of the template DNA and salt concentrations of buffer. For example, mammalian genomic DNA may require longer incubation periods than plasmids and PCR products, based on DNA complexity and size. Similarly, DNA with high GC content (e.g., >65%) often calls for longer incubation or higher temperature for denaturation (Figure 2). Buffers with high salts (as required by some DNA polymerases) generally need higher denaturation temperatures (e.g., 98°C) to separate double-stranded DNA (Figure 3).

Figure 2. Increasing the initial denaturation time improves the PCR yield of a GC-rich, 0.7 kb fragment amplified from a human gDNA sample. The initial denaturation steps were set to 0, 0.5, 1, 3, and 5 minutes respectively.

Some DNA polymerases such as Taq DNA polymerase can become easily denatured from prolonged incubation above 95°C. To compensate for decreased activity in this scenario, more enzymes may be added after the initial denaturation step, or a higher-than-recommended amount of DNA polymerase can be added at the beginning. Highly thermostable enzymes such as those derived from Archaea are able to withstand prolonged high temperatures and remain active throughout PCR (learn more about DNA polymerase characteristics).

## BRIEF DESCRIPTION OF THE PROGRAM

The program runs under Matlab 6.1 (The MathWorks Inc., Natick, MA) on PC.

### Generation of Pseudo-experimental Data—

The model that generates the pseudo-experimental data uses differential Equation 1, where Km is a continuous function of pH, and kp is a continuous function of pH, temperature, and time to account for the changes of enzyme activity in a diprotic enzyme model with an ionizable substrate and for the time-dependent heat denaturation of the enzyme [ 9 ]. The pseudo-measured data are generated as a result of integration of the Equation 1 for user-defined time intervals using the Matlab function “ode15s.” In addition to the continuous dependence of Km and kp on pH and temperature, the pseudo-real nature of the generated data is ensured by the random incorporation of three types of errors for each repeat of the experiments. The biological variations of the enzyme preparations are simulated by evenly distributed relative deviations of Km and kp. The experimenter's errors are simulated by normally distributed variations of the Et and S0 concentrations, the relative values of which decrease linearly with the increase of Et and S0. The errors in the sampling instrumentation are mimicked by the normally distributed variations in P, the relative values of which decrease linearly with the increase of the logarithm of P. Thus, the pseudo-measured data are generated with acceptable noise.

### Identification of the Kinetic Parameters—

The kinetic parameters are identified with minimization of the square residues in model Equation 1 using the pseudo-measured data and the Matlab function “lsqnonlin.”

### Estimation of the Parameter Distribution—

Two methods for evaluation of the parameter distribution are available. The Monte Carlo procedure builds up the synthetic sample with random drawing of data from a normally distributed set with mean and variance values coinciding with the respective values of the pseudo-measured sample. The bootstrap procedure creates the synthetic sample with random drawing with replacement directly from the pseudo-measured sample (applicable only if the latter contains a sufficiently high number of points). The confidence areas are defined on the basis of the χ 2 discrepancy measure as discussed above in relation to Fig. 4 [ 10 ].

### Graphics and Interface—

The graphics facilities of Matlab are used for the visualization of the experimental simulations. The user interface of the program allows for entry with unique personal identification number, and thus the individual sessions are stored, and if necessary the students can continue their work on a later occasion from the stage that they have reached.

Adjustment of the steady-state experimental conditions. A, an experiment during which the steady-state condition is not fulfilled (the ES decreases in the second half of the evaluated incubation period). B, an experiment that meets the requirements of the steady state. Students can visually check the validity of the empiric “criteria” for the steady-state model (ΔS < 0.1S0, EtKm + S0), which otherwise should be proved with complex mathematical procedures [ 8 ].

General conditions for optimal enzyme action. A, continuous monitoring of the product generation in the course of the enzyme-catalyzed reaction at various temperatures. The continuous assay illustrates also the intrinsic errors of the sampling procedure that impose the necessity for repeated sampling. B, end-point enzyme activity assay for various sampling times. The data are a cross-section of the continuous assay in A for 1- and 10-min incubations. Comparison of the two curves illustrates the apparent nature of the “optimal” temperature caused by the time dependence of the heat denaturation.

Identification of parameters with non-linear regression analysis. A, non-linear regression analysis of the experimental data with the least-squares method. B, model of the experimental errors. The dependence of the standard deviation (σ) is modeled as a function of the mean measured values (μ).

## Acknowledgements

We are grateful to Prof. Patricia Burkhardt-Holm for her continuous support and encouragement. We thank Bernd Egger, Astrid Böhne, Philipp Hirsch, and Patricia Burkhardt-Holm for critically reading the manuscript. We thank Fabio Cortesi for his insightful comments and the Center for Marine Evolutionary Biology for hosting a Blast server and a genome browser. Computational resources were provided by the CESNET LM2015042 and the CERIT Scientific Cloud LM2015085, provided under the program “Projects of Large Research, Development, and Innovations Infrastructures”. We thank Maria Leptin for her helpful comments and advice during annotation of the inflammasome components.

### Permissions

Fish used in this work were caught in accordance with permission 2-3-6-4-1 from the Cantonal Office for Environment and Energy, Basel Stadt.

## Acknowledgements

We are grateful to Alvaro Cuevas of Hamilton Robotics for his examples, guidance, and assistance in making use of the Original Equipment Manufacturer (OEM) interface, along with the rest of Hamilton Robotics. We thank Jason Yang, Stephen Von Stetina, Ethan Alley, Brian Wang, Samantha Shepherd, Timothy Erps, and the three reviewers for thoughtful comments and discussion. EAD was supported by the National Institute for Allergy and Infectious Diseases (F31 AI145181-01). EJC was supported by the Ruth L. Kirschstein NRSA fellowship from the National Cancer Institute (F32 CA247274-01). This work was supported by the MIT Media Lab, an Alfred P. Sloan Research Fellowship (to KME), gifts from the Open Philanthropy Project and the Reid Hoffman Foundation (to K. M. E.), the National Institute of Digestive and Kidney Diseases (R00 DK102669-01 to KME), the Burroughs Wellcome Fund (IRSA 1016432 to KME) and the DARPA Safe Genes Program (N66001-17-2-4054 to KME). The findings, views, and/or opinions expressed are those of the authors and should not be interpreted as representing the official views or policies of the Department of Defense or the U. S. Government.