Enzyme kinetics is the study of factors that determine the speed of enzyme-catalysed reactions. It utilizes some mathematical equations that can be confusing to students when they first encounter them. However, the theory of kinetics is both logical and simple, and it is essential to develop an understanding of this subject in order to be able to appreciate the role of enzymes both in metabolism and in biotechnology.
Assays (measurements) of enzyme activity can be performed in either a discontinuous or continuous fashion. Discontinuous methods involve mixing the substrate and enzyme together and measuring the product formed after a set period of time, so these methods are generally easy and quick to perform. In general we would use such discontinuous assays when we know little about the system (and are making preliminary investigations), or alternatively when we know a great deal about the system and are certain that the time interval we are choosing is appropriate.
In continuous enzyme assays we would generally study the rate of an enzyme-catalysed reaction by mixing the enzyme with the substrate and continuously measuring the appearance of product over time. Of course we could equally well measure the rate of the reaction by measuring the disappearance of substrate over time. Apart from the actual direction (one increasing and one decreasing), the two values would be identical. In enzyme kinetics experiments, for convenience we very often use an artificial substrate called a chromogen that yields a brightly coloured product, making the reaction easy to follow using a colorimeter or a spectrophotometer. However, we could in fact use any available analytical equipment that has the capacity to measure the concentration of either the product or the substrate.
In almost all cases we would also add a buffer solution to the mixture. As we shall see, enzyme activity is strongly influenced by pH, so it is important to set the pH at a specific value and keep it constant throughout the experiment.
Our first enzyme kinetics experiment may therefore involve mixing a substrate solution (chromogen) with a buffer solution and adding the enzyme. This mixture would then be placed in a spectrophotometer and the appearance of the coloured product would be measured. This would enable us to follow a rapid reaction which, after a few seconds or minutes, might start to slow down, as shown in Figure 4.
A common reason for this slowing down of the speed (rate) of the reaction is that the substrate within the mixture is being used up and thus becoming limiting. Alternatively, it may be that the enzyme is unstable and is denaturing over the course of the experiment, or it could be that the pH of the mixture is changing, as many reactions either consume or release protons. For these reasons, when we are asked to specify the rate of a reaction we do so early on, as soon as the enzyme has been added, and when none of the above-mentioned limitations apply. We refer to this initial rapid rate as the initial velocity (v0). Measurement of the reaction rate at this early stage is also quite straightforward, as the rate is effectively linear, so we can simply draw a straight line and measure the gradient (by dividing the concentration change by the time interval) in order to evaluate the reaction rate over this period.
We may now perform a range of similar enzyme assays to evaluate how the initial velocity changes when the substrate or enzyme concentration is altered, or when the pH is changed. These studies will help us to characterize the properties of the enzyme under study.
The relationship between enzyme concentration and the rate of the reaction is usually a simple one. If we repeat the experiment just described, but add 10% more enzyme, the reaction will be 10% faster, and if we double the enzyme concentration the reaction will proceed twice as fast. Thus there is a simple linear relationship between the reaction rate and the amount of enzyme available to catalyse the reaction (Figure 5).
This relationship applies both to enzymes in vivo and to those used in biotechnological applications, where regulation of the amount of enzyme present may control reaction rates.
When we perform a series of enzyme assays using the same enzyme concentration, but with a range of different substrate concentrations, a slightly more complex relationship emerges, as shown in Figure 6. Initially, when the substrate concentration is increased, the rate of reaction increases considerably. However, as the substrate concentration is increased further the effects on the reaction rate start to decline, until a stage is reached where increasing the substrate concentration has little further effect on the reaction rate. At this point the enzyme is considered to be coming close to saturation with substrate, and demonstrating its maximal velocity (Vmax). Note that this maximal velocity is in fact a theoretical limit that will not be truly achieved in any experiment, although we might come very close to it.
The relationship described here is a fairly common one, which a mathematician would immediately identify as a rectangular hyperbola. The equation that describes such a relationship is as follows:
The two constants a and b thus allow us to describe this hyperbolic relationship, just as with a linear relationship (y = mx + c), which can be expressed by the two constants m (the slope) and c (the intercept).
We have in fact already defined the constant a — it is Vmax. The constant b is a little more complex, as it is the value on the x-axis that gives half of the maximal value of y. In enzymology we refer to this as the Michaelis constant (Km), which is defined as the substrate concentration that gives half-maximal velocity.
Our final equation, usually called the Michaelis–Menten equation, therefore becomes:
In 1913, Leonor Michaelis and Maud Menten first showed that it was in fact possible to derive this equation mathematically from first principles, with some simple assumptions about the way in which an enzyme reacts with a substrate to form a product. Central to their derivation is the concept that the reaction takes place via the formation of an ES complex which, once formed, can either dissociate (productively) to release product, or else dissociate in the reverse direction without any formation of product. Thus the reaction can be represented as follows, with k1, k−1 and k2 being the rate constants of the three individual reaction steps:
The Michaelis–Menten derivation requires two important assumptions. The first assumption is that we are considering the initial velocity of the reaction (v0), when the product concentration will be negligibly small (i.e. [S] ≫ [P]), such that we can ignore the possibility of any product reverting to substrate. The second assumption is that the concentration of substrate greatly exceeds the concentration of enzyme (i.e. [S]≫[E]).
The derivation begins with an equation for the expression of the initial rate, the rate of formation of product, as the rate at which the ES complex dissociates to form product. This is based upon the rate constant k2 and the concentration of the ES complex, as follows: 1
Since ES is an intermediate, its concentration is unknown, but we can express it in terms of known values. In a steady-state approximation we can assume that although the concentration of substrate and product changes, the concentration of the ES complex itself remains constant. The rate of formation of the ES complex and the rate of its breakdown must therefore balance, where: and
Hence, at steady state:
This equation can be rearranged to yield [ES] as follows: 2
The Michaelis constant Km can be defined as follows:
Equation 2 may thus be simplified to: 3
Since the concentration of substrate greatly exceeds the concentration of enzyme (i.e. [S] ≫ [E]), the concentration of uncombined substrate [S] is almost equal to the total concentration of substrate. The concentration of uncombined enzyme [E] is equal to the total enzyme concentration [E]T minus that combined with substrate [ES]. Introducing these terms to Equation 3 and solving for ES gives us the following: 4
We can then introduce this term into Equation 1 to give: 5
The term k2[E]T in fact represents Vmax, the maximal velocity. Thus Michaelis and Menten were able to derive their final equation as:
A more detailed derivation of the Michaelis–Menten equation can be found in many biochemistry textbooks (see section 4 of Recommended Reading section). There are also some very helpful web-based tutorials available on the subject.
Michaelis constants have been determined for many commonly used enzymes, and are typically in the lower millimolar range (Table 5).
It should be noted that enzymes which catalyse the same reaction, but which are derived from different organisms, can have widely differing Km values. Furthermore, an enzyme with multiple substrates can have quite different Km values for each substrate.
A low Km value indicates that the enzyme requires only a small amount of substrate in order to become saturated. Therefore the maximum velocity is reached at relatively low substrate concentrations. A high Km value indicates the need for high substrate concentrations in order to achieve maximum reaction velocity. Thus we generally refer to Km as a measure of the affinity of the enzyme for its substrate—in fact it is an inverse measure, where a high Km indicates a low affinity, and vice versa.
The Km value tells us several important things about a particular enzyme.
An enzyme with a low Km value relative to the physiological concentration of substrate will probably always be saturated with substrate, and will therefore act at a constant rate, regardless of variations in the concentration of substrate within the physiological range.
An enzyme with a high Km value relative to the physiological concentration of substrate will not be saturated with substrate, and its activity will therefore vary according to the concentration of substrate, so the rate of formation of product will depend on the availability of substrate.
If an enzyme acts on several substrates, the substrate with the lowest Km value is frequently assumed to be that enzyme's ‘natural’ substrate, although this may not be true in all cases.
If two enzymes (with similar Vmax) in different metabolic pathways compete for the same substrate, then if we know the Km values for the two enzymes we can predict the relative activity of the two pathways. Essentially the pathway that has the enzyme with the lower Km value is likely to be the ‘preferred pathway’, and more substrate will flow through that pathway under most conditions. For example, phosphofructokinase (PFK) is the enzyme that catalyses the first committed step in the glycolytic pathway, which generates energy in the form of ATP for the cell, whereas glucose-1-phosphate uridylyltransferase (GUT) is an enzyme early in the pathway leading to the synthesis of glycogen (an energy storage molecule). Both enzymes use hexose monophosphates as substrates, but the Km
potassium and sodium phosphates(po-tas-e-um/soe-dee-um foss-fates) ,
K-Phos M.F(trade name),
K-Phos Neutral(trade name),
K-Phos No. 2(trade name),
Uro-KP Neutral(trade name)
Pregnancy Category: C
Treatment and prevention of phosphate depletion in patients who are unable to ingest adequate dietary phosphate.Adjunct therapy of urinary tract infections with methenamine hippurate or mandelate.Prevention of calcium urinary stones.Phosphate salts of potassium may be used in hypokalemic patients with metabolic acidosis or coexisting phosphorus deficiency.
Phosphate is present in bone and is involved in energy transfer and carbohydrate metabolism.
Serves as a buffer for the excretion of hydrogen ions by the kidneys.
Dibasic potassium phosphate is converted in renal tubule to monobasic salt, resulting in urinary acidification, which is required for methenamine hippurate or mandelate to be active as urinary anti-infectives.
Acidification of urine increases solubility of calcium, decreasing calcium stone formation.
Replacement of phosphorus in deficiency states.
Increased efficacy of methenamine.
Decreased formation of calcium urinary tract stones.
Absorption: Well absorbed following oral administration. Vitamin D promotes GI absorption of phosphates.
Distribution: Phosphates enter extracellular fluids and are then actively transported to sites of action.
Metabolism and Excretion: Excreted mainly (>90%) by the kidneys.
Time/action profile (effects on serum phosphate levels)
Contraindicated in: Hyperkalemia (potassium salts);Hyperphosphatemia;Hypocalcemia;Severe renal impairment;Untreated Addison’s disease (potassium salts).
Use Cautiously in: Hyperparathyroidism;Cardiac disease;Hypernatremia (sodium phosphate only);Hypertension (sodium phosphate only);Renal impairment.
Adverse Reactions/Side EffectsRelated to hyperphosphatemia, unless otherwise indicated
Central nervous system
- arrhythmias (life-threatening)
- cardiac arrest (life-threatening)
- ECG changes (absent P waves, widening of the QRS complex with biphasic curve, peaked T waves)
- diarrhea (most frequent)
- abdominal pain
Fluid and Electrolyte
- hypocalcemia, hyperkalemia:
- muscle cramps
- flaccid paralysis
- heaviness of legs
Drug-Drug interactionConcurrent use of potassium-sparing, diuretics, ACE inhibitors, or angiotensin II receptor blockers with potassium phosphates may result in hyperkalemia.Concurrent use of corticosteroids with sodium phosphate may result in hypernatremia.Concurrent administration of calcium-, magnesium-, or aluminum-containing compounds ↓ absorption of phosphates by formation of insoluble complexes.Vitamin D enhances the absorption of phosphates.Oxalates (in spinach and rhubarb) and phytates (in bran and whole grains) may ↓ absorption of phosphates by binding them in the GI tract.
Oral (Adults and Children >4 yr) 250–500 mg (8–16 mmol) phosphorus (1–2 packets) 4 times daily.
Oral (Children <4 yr) 250 mg (8 mmol) phosphorus (1 packet) 4 times daily.
Oral (Adults) 2 tablets 4 times/day.
Oral (Adults) 50–150 mmol/day in divided doses.
Oral (Children) 2–3 mmol/kg/day in divided doses.
Potassium and Sodium Phosphates
Tablets (K-Phos MF): elemental phosphorus 125.6 mg (4 mmol), sodium 67 mg (2.9 mEq), and potassium 44.5 mg (1.1 mEq)
Tablets (K-Phos Neutral): elemental phosphorus 250 mg (8 mmol), sodium 298 mg (13 mEq), and potassium 45 mg (1.1 mEq)
Tablets (K-Phos No.2): elemental phosphorus 250 mg (8 mmol), sodium 134 mg (5.8 mEq), and potassium 88 mg (2.3 mEq)
Tablets (Uro-KP Neutral): elemental phosphorus 258 mg, sodium 262.4 mg (10.8 mEq), and potassium 49.4 mg (1.3 mEq)
Powder for oral solution (Neutra-Phos): elemental phosphorus 250 mg (8 mmol), sodium 164 mg (7.1 mEq), and potassium 278 mg (7.1 mEq)/packet
- Assess patient for signs and symptoms of hypokalemia (weakness, fatigue, arrhythmias, presence of U waves on ECG, polyuria, polydipsia) and hypophosphatemia (anorexia, weakness, decreased reflexes, bone pain, confusion, blood dyscrasias) throughout therapy.
- Monitor intake and output ratios and daily weight. Report significant discrepancies.
- Lab Test Considerations: Monitor serum phosphate, potassium, sodium, and calcium levels prior to and periodically throughout therapy. Increased phosphate may cause hypocalcemia.
- Monitor renal function studies prior to and periodically throughout therapy.
- Monitor urinary pH in patients receiving potassium and sodium phosphate as a urinary acidifier.
Potential Nursing DiagnosesImbalanced nutrition: less than body requirements (Indications)
- Oral: Tablets should be dissolved in a full glass of water. Allow mixture to stand for 2–5 min to ensure it is fully dissolved. Solutions prepared by pharmacy should not be further diluted.
- Medication should be administered after meals to minimize gastric irritation and laxative effect.
- Do not administer simultaneously with antacids containing aluminum, magnesium, or calcium.
- Explain to the patient the purpose of the medication and the need to take as directed. Take missed doses as soon as remembered unless within 1 or 2 hr of the next dose. Explain that the tablets should not be swallowed whole. Tablets should be dissolved in water.
- Instruct patients in low-sodium diet (see ).
- Advise patient of the importance of maintaining a high fluid intake (drinking at least one 8-oz glass of water each hr) to prevent kidney stones.
- Instruct the patient to promptly report diarrhea, weakness, fatigue, muscle cramps, unexplained weight gain, swelling of lower extremities, shortness of breath, unusual thirst, or tremors.
- Prevention and correction of serum phosphate and potassium deficiencies.
- Maintenance of acid urine.
- Decreased urine calcium, which prevents formation of renal calculi.