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Volume 288, Issue 7 p. 2068-2083
Words of Advice
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Words of Advice: teaching enzyme kinetics

Bharath Srinivasan

Corresponding Author

Bharath Srinivasan

Mechanistic Biology and Profiling, Discovery Sciences, R&D, AstraZeneca, Cambridge, UK

Correspondence

B. Srinivasan, Mechanistic Biology and Profiling, Discovery Sciences, R&D, AstraZeneca, The Darwin Building, 310 Milton Rd, Milton, Cambridge CB4 0WG, UK

Tel: +351 920122242

E-mail: [email protected]

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First published: 27 September 2020
Citations: 26

Abstract

Enzymology is concerned with the study of enzyme structure, function, regulation and kinetics. It is an interdisciplinary subject that can be treated as an exclusive sphere of exhaustive inquiry within mathematical, physico-chemical and biological sciences. Hence, teaching of enzymology, in general, and enzyme kinetics, in particular, should be undertaken in an interdisciplinary manner for a holistic appreciation of this subject. Further, analogous examples from everyday life should form an integral component of the teaching for an intuitive grasp of the subject matter. Furthermore, simulation-based appreciation of enzyme kinetics should be preferred over simplifying assumptions and approximations of traditional enzyme kinetics teaching. In this Words of Advice, I outline the domain depth of enzymology across the various disciplines and provide initial ideas on how appropriate analogies can provide firm insights into the subject. Further, I demonstrate how an intuitive feel for the subject can help not only in grasping abstract concepts but also extending it in experimental design and subsequent interpretation. Use of simulations in grasping complex concepts is also advocated given the advantages this medium offers over traditional approaches involving images and molecular models. Furthermore, I discuss the merits of incorporating the historical backdrop of major discoveries in enzymological teaching. We, at AstraZeneca, have experimented with this approach with the desired outcome of generating interest in the subject from people practising diverse disciplines.

Abbreviations

  • DNA
  • deoxyribonucleic acids
  • MO
  • molecular orbital
  • OCW
  • OpenCourseWare
  • PDE
  • partial differential equation
  • PTM
  • post-translational modifications
  • QSSA
  • quasi-steady-state assumptions
  • RNA
  • ribonucleic acid
  • TST
  • transition state theory
  • Introduction

    A practising educator of enzyme kinetics needs constant reinvention of ways to convey the concepts for effective teaching. There are several different models of teaching and learning that help in the appropriate construction and delivery of concepts: they help the instructor by providing them with a framework (both theoretical and instructional) and in creating guidelines for effective teaching of a specific subject, content or process. Further, they also provide insights into the selective reception of some concepts by a subset of audience and not others. There is ample literature on aids and approaches to teach enzyme kinetics because of the difficulty in conveying its concepts in a clear manner. Educators have explored instruction-based, analogy-based, experiment-based and simulation-based approaches to convey an intuitive feel about the subject [[1-6]]. However, despite these approaches, students often do not emerge with a clear idea on the subject and how it influences their day-to-day life. For instance, despite consuming aspirin as a common medication for several different ailments on an almost daily basis, very few would recognize it as an irreversible inhibitor of the enzyme cyclooxygenase, the latter being involved in inflammation and pain. A wholesome appreciation of the subject will educate the next generation of rational scientists (e.g. drug hunters) who will apply the right enzymological knowledge to guide drug discovery.

    Enzymology, with its roots in several different disciplines of scientific thinking, has often been taught in an insular way, and discipline-specific teaching biases have influenced the way students perceive the subject. Depending on the teacher's area of specialization and the speciality in which the student is majoring, it is quite a challenge to teach enzyme kinetics. Filling the blackboard with equations and reaction schemes is the last thing an instructor would want to do when addressing an audience who are predominantly biologists. The farthest one ventures in such a setting is to explain the difference between rate constants and absolute rates and, in some cases, derive the rate equations defining uni-uni reactions. Discussion of concepts such as multisubstrate or product reactions and inhibition schemes, except for a conceptual verbal picture, is normally out of scope. Discussion of dynamic systems is almost unheard of, giving the students a very partial view on the elegance inherent in this subject. At the end, students come back with an array of equilibrium constants (Ks), pseudo-equilibrium constants (Km), rate constants (kon, koff, kcat) and a parameter called Vmax. They start assuming that acquiring the jargon of kinetics is what constitutes kinetics and use the terms association, dissociation, Michaelis–Menten, rapid equilibrium, steady state, pre-steady state, initial velocity, King–Altman, Theorell–Chance and so forth, often out of context.

    Now, let us assume that the enzymology class is administered by a mathematician to students who are majoring in mathematics. Words such as allosteric regulation, multiprotein complexes, protein half-life, feedback loop, post-translational modifications (PTM), signalling cascades and metabolic crosstalk are hardly, if ever, elaborated upon. Concepts in dynamic systems including terms such as ordinary and partial differential equations (ODEs and PDEs), simplification, scaling, bifurcation, stability, steady-state assumptions (SSA) and quasi-steady-state assumptions (QSSA) are often used. Further, discretization, stiffness and relevant numerical methods such as semi-implicit Runge–Kutta schemes and method of lines or finite elements (FE) as well as approaches to parameter estimation and sensitivity analyses would dominate the discussion. However, all of the above ignores the fact that a true understanding of the enzyme system can only be undertaken under an appropriate backdrop of biology that it operates within.

    Further, aspects of geometric complementarity between ligand and enzyme (with their roots in 3-D geometry) coupled to the force fields that dictate the strength of such interactions constitute an extensive field of literature in the physical and chemical sciences. An in-depth understanding in physico-chemical sciences is also required to appreciate the fact that enzymes are polymers of amino acids and possess the traits of polymers that are either stabilized or destabilized by kosmotropes and chaotropes, respectively. It has also been recently demonstrated that the viscoelasticity and nanorheology of the protein polymer and the medium determine the rate of catalysis and that small-molecule ligands perturb these properties substantially. Several enzyme systems show a viscoelastic transition in their dynamics as a function of ligand binding. The pH of the medium, the pKa of the amino acids and ionic strength are additional aspects that fall within the purview of the physico-chemical literature.

    The examples above highlight the problem of insular teaching in enzymology. In all these scenarios, it is clear that a marriage of disciplines is essential for the full appreciation of the subject matter. Thus, the purpose of this Words of Advice is to provide a framework to achieve that objective by (a) helping the reader appreciate how rich enzymology is within the respective disciplines of mathematics, physics/chemistry and biology and emphasizing the necessity to take an interdisciplinary approach to holistically appreciate the subject, (b) providing pointers about the use of intuition and analogies in conveying difficult concepts for ease of comprehension, (c) using simulation as a means of gaining a physical appreciation for the pulsation inherent in enzymology and (d) using history as a backdrop for better comprehension. Further, I also present an example of how this method was applied successfully at AstraZeneca to convey the message across to experts from different disciplines and streams of science.

    Interdisciplinary nature of enzyme kinetic teaching

    Comprehension and appreciation of enzyme kinetics as a quantitative discipline

    Analysis of enzyme kinetics data is a quantitative science (Box 1). The below paragraph, while providing a brief introduction on the derivation of rate equations in enzyme kinetics, emphasizes its deeply quantitative nature and the need for an enumerative appreciation of the discipline.

    Until recently, deterministic rate equations have been successfully used to infer the rate of enzyme-catalysed reactions carried out in vitro. Enzyme binds and transforms the substrate to product as a function of time. Assuming law of mass action and homogenous suspension of substrate and enzyme, analysis of enzyme kinetics can become quickly complex with increasing numbers of intermediates and the potential for reversible reactions. Adrian J. Brown and Victor Henri laid the foundation for quantitative enzymology that was improved substantially by Leonar Michaelis and Maud Menten for real experimental data under rapid equilibrium conditions [[7-10]]. This represents a simplified scaling of a complex dynamic system where the initial fast transient accumulation of the enzyme–substrate (ES) complex is treated as being in equilibrium with free enzyme and free substrate. Subsequently, George Edward Briggs and John Burdon Sanderson Haldane treated the system as ‘quasi-steady-state (QSS)’, where the substrate–enzyme intermediate is quickly accumulated during the initial fast transient of the reaction but is slowly depleted during the slow and long timescale [[11, 12]]. More conventionally, kinetic data obtained under initial velocity conditions (< 5% substrate-to-product conversion) provide insights into enzyme mechanisms based upon derived mechanistic rate expressions. These rate expressions are solutions of nonlinear system of four differential equations (rate of change of substrate, rate of change of enzyme, rate of change of enzyme–substrate complex and rate of change of product) (Fig. 1). Using the law of mass action, the rate equations for the concentration of the various species are derived with ease. Appropriate solutions can be obtained to those equations by solving them either analytically or numerically. In the case of enzymes, a conservation law can be applied for the enzyme in its bound and free form to obtain an analytical solution. Since enzyme is the catalyst which does not change as a function of time except for partitioning between free and bound forms, one differential equation d[E]/dt is assumed, with good reason, as zero. The same holds true for rate of change of product as a function of enzyme–substrate complex formation. However, the expressions for rate of change of enzyme–substrate complex and substrate require numerical solutions with appropriate boundary conditions (initial conditions at t = 0). Under the boundary conditions specified, substituting in the conservation law for the enzyme, the equilibrium dissociation constant for the enzyme–substrate complex (Ks = koff/kon) and the Michaelis–Menten constant (Km = Ks + K = koff/kon + kcat/kon, where K is the Van Slyke–Cullen constant), one can solve for the equations. These would yield us steady-state expression for the concentrations of all the intermediates in an enzyme-catalysed reaction. The above scenario is tractable when one is dealing with uni-uni system of single substrate going to single product.

    Details are in the caption following the image
    Chemical, physical, mathematical and biological facets of enzyme kinetics and the necessity to use an interdisciplinary approach to teach it.

    Based on the above discussion, for a single-enzyme, single-substrate reaction, it is clear that Km must be equal to or greater than KD (Km =[(koff + kcat)/kon] and KD [= koff/kon]). At one extreme is the Michaelis and Menten assumption of rapid equilibrium kcat ≪ koff (Km will be same as KD for all practical purposes). The other extreme is Van Slyke–Cullen behaviour that assumes that kcat ≫ koff. These two extremes are special cases of the Briggs and Haldane kinetics.

    Box 1. Nature of enzymology that makes teaching it challenging

    • Enzymology requires the right balance of cross-domain expertise for effective teaching.
    • It could be an exclusive quantitative exercise understanding enzymology from the perspective of modelling, fitting and parameter estimation.
    • It could be an exclusive physico-chemical exercise understanding aspects such as enzyme–ligand (substrate/modulator) interaction employing force fields or by treating proteins as polymers.
    • It could be an exclusive biological exercise understanding the role of enzyme as players in the metabolic cascades.
    • The abstract quality of enzymology entails use of analogies to convey an intuitive feel for the subject.

    However, when the system under study involves multiple substrates, products and intermediate complexes, derivation of the steady-state rate equations becomes significantly more complex. Graphical methods perfected by King–Altman and their extensions amenable to computer programming have been developed to make the derivation of steady-state rate equations for these systems easy. Cha employed graph theory to make the King–Altman method more powerful [[13, 14]]. The computer implementation of Cha's method incorporating exhaustive search was presented by Athel Cornish-Bowden [[14, 15]]. For the interested reader, Huang has reviewed all the approaches systematically [[16]]. Cooperative systems present yet another challenge in modelling, with linked equilibrium constants in the binding polynomials. Additional aspects of non-Michaelis–Menten behaviour such as allovalent interactions between a receptor and polyvalent ligand [[17]], cellular crowding [[18]], two-dimensional interfacial kinetics [[19]], single-molecule enzyme kinetics that uses chemical master equation (CME) [[20]], systems showing processive and distributive enzyme kinetics, and autocatalytic enzymes all require nonlinear mathematical modelling to gain insights (Box 1). All of the above aspects serve to emphasize the deeply quantitative nature of enzymology.

    Box 2. Approaches for effective teaching of enzymology

    • A cross-domain approach conveying quantitative, physico-chemical and biological know-how in equal measure by a single instructor.
    • A cross-domain approach delivered by a group of collaborating instructors.
    • Analogies should be extensively employed to ensure an intuitive appreciation of the field.
    • Discussions and case studies should be an integral part of the teaching exercise.
    • Laboratory-based hands-on approach should be encouraged with well-understood enzymes to ensure practical appreciation.
    • Use of simulation-based approaches should be encouraged. A list of freely available simulation packages (for educational noncommercial purposes alone) are as follows: KinTek Explorer, Gepasi (superseded by COPASI), KinSim and Dynafit.
    • Discussion of concepts in enzymology against their historical backdrop for ease of comprehension.

    Comprehension and appreciation of enzyme kinetics within biological context

    Enzymes are biological macromolecules that operate within the context of an organism's effort to survive within their respective environments (Box 1). This entails the maintenance of status quo by defying the second law of thermodynamics at the expense of energy that the organism acquires from its surrounding. Enzymes, as part of metabolic networks that are intricately regulated, contribute to this acquisition and dissemination of energy. Comprehending enzyme kinetics is essential for quantitatively understanding and predicting how functional behaviour arises from changes in the dynamic concentration of cell components effected by enzymes. However, the key term here is ‘functional behaviour’. A deep understanding of protein–substrate-product within the context of their biological role will lead the students to appreciate the concept of enzymology at a deeper level. For example, a true appreciation of the regulation in phosphofructokinase can only be obtained within the context of the flux through glycolytic cycle and the behaviour of its various other enzymes. Likewise, an understanding of the enzymes isocitrate dehydrogenase and α-ketoglutarate dehydrogenase can only be obtained within the context of the citric acid cycle. Understanding the chemistry that these enzymes catalyse, without the backdrop of the pathways that these enzymes operate within, will lead to a skewed perception on their role and its appreciation.

    Protein synthesis is facilitated by the process of transcription and translation that involves several protein/enzyme complexes that are regulated in a complex spatio-temporal manner. Once the protein/enzyme is produced, its activity contributes to the biology of the system but is in turn governed by factors such as product inhibition, substrate inhibition at high substrate concentration, positive homotropic cooperativity, negative homotropic cooperativity and allosteric regulation in positive and negative feedback loops. Proteins are also extensively regulated and adopt different folds and functional states as a function of metal ion gradients [[21]]. Further, it is demonstrated with increased frequency that multiprotein complexes are a means of metabolic clustering by sequestering substrate–product cascades and preventing the loss of intermediates. This has been demonstrated with purinosomes [[22]], respirasomes [[23]] and glycolytic metabolon [[24]]. Protein half-lives, compartmentalization and shuttling, mutation-based selection, spatio-temporal regulation and PTM-mediated effects are other aspects of biological regulation of protein function (Fig. 1).

    Comprehension and appreciation of enzyme kinetics as a physico-chemical event

    Often, there is a tendency to cluster enzymology as an exclusively chemical discipline. Most enzymology laboratories and instructors are housed within the chemistry departments of universities. An exclusively chemical (or physico-chemical) perspective on enzymology emphasizes aspects such as rate, acceleration, molecularity, rate constants, time, nanorheology and viscoelasticity with strong roots in classical mechanics. Since enzymes are mostly polymers of amino acids and occasionally of nucleotides, they have properties and scaling functions very similar to conventional synthetic polymers; examples are tacticity (stereochemistry of adjacent chiral centres in a polymer), copolymerization and branching. The extensive physico-chemical toolkit and methods developed to assess and understand polymers can be applied with equal facility to protein and/or nucleic acid polymers. Likewise, an emerging biophysical view of rate enhancements by enzymes espouses an integrated view of incorporating information on conserved networks of residues connecting surface regions to the active site serving as thermodynamical couplers between the hydration shell, bulk solvent and the catalysed reaction at the active site [[25]].

    Box 3. Possible caveats on methods from Box 2

    • In a cross-domain approach delivered by different instructors, care should be exercised to ensure that the styles of the various instructors are harmonious and not discordant.
    • Additionally, none of the domains should dominate the discourse to convey a biased/lopsided perception of enzymology.
    • In a cross-domain approach delivered by a single instructor, the instructor is advised to ensure rigorous homework in order to prevent conveying erroneous impression about the aspects that do not fall under the purview of his core domain expertise.
    • Analogies can be tricky and have the potential to convey a wrong impression about the field given the ease of their comprehension. Analogies should be devised carefully, debated extensively and assessed rigorously before using them in a classroom.
    • A few simulation packages need licensing for enabling work with real experimental data.
    • Too much emphasis on history might skew the conceptual framework within which enzymology operates, giving the impression of a history class.
    Another aspect of enzymes that can be appreciated exclusively by physico-chemical treatment is the way enzymes reduce the activation energy barrier as treated in transition state theory (TST) [[2, 26]]. Aspects like orbital steering of substrates to reduce the activation energy barrier in enzyme-catalysed reactions have a strong basis in molecular orbital (MO) theory (Fig. 1). More recently, attempts have been made to explain rate enhancements by electron tunnelling (also called quantum tunnelling) [[27, 28]] and hydrogen tunnelling [[29]]. The laws of quantum physics allow very small particles (e.g. protons and electrons) to pass classically insurmountable barriers. Although enzymes are electrical insulators, electrons have a small probability of travelling up to approximately 3 × 10−9 m through them. Likewise, hydrogen ions are transferred from substrates to products via tunnelling effect in enzymes where the activation energy barrier is high and, hence, insurmountable [[30]]. However, the probability of H-tunnelling is substantially lower than quantum tunnelling since hydrogen is ~ 1840 times heavier than an electron (i.e. the probability of tunnelling is inversely proportional to mass governed by the equation λ = h/mν, where λ is the wavelength, h is Planck's constant, m is the mass, and ν is the frequency). Understanding these events associated with enzyme catalysis requires strong fundamentals in quantum mechanics.

    Glossary Box

    • Association rate constant: The rate of complex formation between enzyme and substrate/inhibitor per second in a 1 molar solution of enzyme and the substrate/inhibitor. The units of ka are m−1·s−1.
    • Dissociation rate constant: The first-order rate constant signifying the dissociation of the enzyme–substrate/inhibitor complex.
    • Transition state theory (TST): Theory that strives to explain the tremendous rate enhancements brought about by enzymes in their substrate-to-product conversion. The theory assumes chemical equilibrium between reactants and activated transition state complexes.
    • Competitive inhibition: This inhibition signifies that the binding of the inhibitor and substrate is mutually exclusive.
    • Uncompetitive inhibition: This inhibition signifies that the binding of inhibitor is conditional upon binding of substrate.
    • Noncompetitive inhibition: Inhibition types falling between the extremes of competitive and uncompetitive inhibition are called noncompetitive inhibition. In a more traditional sense, inhibitor binding and substrate binding are independent events.
    • Catalytic rate constant: The number of times enzyme active site converts substrate to product per unit time (also known as turnover number) with units of Time−1.
    • Dixon plot: A graphical method for determination of mechanism of inhibition and equilibrium dissociation constant (Ki) of the inhibitor for the enzymes. The plot has reciprocal velocity on the y-axis and inhibitor concentration on the x-axis.
    • Lineweaver–Burk Plot: Double-reciprocal linear transformation of the velocity versus substrate concentration plot for ease of parameter estimation before the advent of computers for nonlinear curve fits.
    • Uni-Uni and Bi-Uni: The molecularity of a reaction indicating how many substrate molecules transform into how many product molecules.
    • Michaelis–Menten constant: It is a pseudo-equilibrium constant signifying the concentration of substrate at half-maximal velocity of the enzyme.
    • Rapid equilibrium: Condition that assumes that the rate at which the enzyme and the substrate associate to form enzyme–substrate complex (k1[E][S]) is the same as the rate at which the latter dissociates back to its individual components (k−1[ES]). Further, k−1 is far greater than k2, the rate at which [ES] dissociates to form products.
    • Initial velocity: Velocity at the beginning of an assay. More realistically, conditions under which less than 5% of the substrate is converted to product.
    • Steady-state: Conditions under which the rate of enzyme–substrate complex formation is equal to the rate of its degradation either back to substrate or to the product.
    • Van Slyke–Cullen mechanism: A special case of Henri–Michaelis–Menten kinetics where there is no reverse reaction from the complex back to the substrates. That is, kcat is a lot greater in magnitude than koff.
    • Kosmotropes: For proteins, factors promoting noncovalent interaction with water molecules leading to stabilization of the native state of the protein.
    • Chaotropes: A chaotropic agent operates by disrupting the hydrogen bonding network between water molecules. This affects the stability of the native state of protein molecules in solution by weakening the hydrophobic effect.

    Enzymes interact with their substrates to form enzyme–substrate binary complexes or with inhibitors to form enzyme–inhibitor binary complexes. Further, binary complexation can lead to ternary complexation in the case of ordered bisubstrate reactions driving the chemistry or uncompetitive inhibition in the case of inhibitors. The way an enzyme interacts with its substrate/s or small molecule/s is dictated by forces that require a preliminary understanding of the underlying physics to enable complete appreciation. The nature of forces that govern such interactions can be broadly classified into covalent and noncovalent (Fig. 1). Covalent interactions are mostly approximated by simple harmonic functions, typified by a spring, having a resting state, a stretched tense state and a compressed tense state. The state of a covalently attached system (bond length, bond angle or dihedral angle) can either be in the resting state with mean distance or can substantially deviate from the resting state as far as afforded by the spring constant. All noncovalent interactions are special cases of electrostatic interactions that are either constant or transient (charge–charge, charge–dipole, dipole–dipole, dipole-induced dipole, charge-induced dipole, hydrogen bonds, van der Waals and dispersion interactions) (Fig. 1). Once again, an understanding of these potentials and forces has a basis in molecular and quantum mechanics that fall within the exclusive domain of physico-chemical knowledge base.

    Developing an integrated interdisciplinary approach to teaching enzyme kinetics

    Above, I highlighted how a domain-specific approach to enzyme kinetics is rich in details within the confines of each discipline. However, the boundaries are artificial and necessitate an interdisciplinary approach to gain a holistic perspective of the field (Fig. 2A, Boxes 1 and 2). Despite individual, and often differing, opinions, combining the aspects of biology, chemistry, physics and mathematics is a must in understanding and teaching enzyme kinetics (Boxes 1 and 2). The most important aspect is to decide on the structure of such an interdisciplinary course. Often, an interdisciplinary approach of teaching poses some challenges that would have to be overcome in attempting its successful implementation in a classroom (Fig. 2B, Box 3). It is incumbent on the instructor to be sufficiently proficient in the varied disciplines for them to feel comfortable in introducing the topic from a multidisciplinary perspective and enable students to appreciate the merits of such an approach (Fig. 2). In the case of enzyme kinetics, the instructor should strive to gain considerable exposure into biochemistry/molecular biology, calculus/three-dimensional geometry and basics of chemical kinetics/thermodynamics. This will facilitate the ease of interdisciplinary approach that the instructor will bring to the teaching of enzyme kinetics. Another challenge is that there are often fundamental differences in the way different disciplines approach a particular problem, both theoretically and methodologically (Fig. 2B). This often makes it difficult to design a coherent framework without systematic effort invested into it. However, it would have to be conveyed that this dichotomy not only adds richness to the field but can also yield novel insights that an insular attempt would fail to provide. The most challenging task in approaching interdisciplinary teaching is to manage an appropriate threshold of complexity that will help students to appreciate the richness without unduly daunting them with unnecessary details (Fig. 2B). Once the right balance is achieved, it has the advantage of imparting both depth and breadth of learning to the students with lasting retention of concepts. Another perceptible challenge that the instructor would have to overcome is not to become territorial about their domain and end-up bringing polarity to their approach.

    Details are in the caption following the image
    (A) Venn diagram representation of the nature of enzymology as a merger of principles from mathematics, physico-chemistry and biology. (B) Schematic representation of the principal challenges in interdisciplinary teaching of a subject or domain.

    Analogies and intuition as means of conveying information

    Analogies have been extensively applied as aids to facilitate the understanding of abstract concepts (Box 2) [[1]]. However, one ought to be careful with the use of analogies as a teaching aid; because of their ease of understanding, they can lead to significant confusion and misinterpretation (Box 3). Orgill and Bodner have delineated the traits of good analogies [[31, 32]]. According to them, good analogies are simple, serve as a mnemonic and are rooted in familiar analogous concept(s). Further, they are usually employed to introduce a difficult concept that does not lend itself to easy visualization. For effectiveness, the analogy must be clear and possess a one-to-one relationship with the actual concept.

    Transition state stabilization, reduction in activation energy and the analogy of mother-assisted child vaccination

    In the absence of enzymes, it takes millions of years for some reactions to happen. To understand why this is the case, assume a bimolecular reaction yielding a single fused product (Bi-Uni). For the reactants to form the product, they would have to find each other and hit each other with correct orientation, with sufficient force and after appropriate desolvation. The probability of this event happening, with molecular crowding, weak thermal energy (KBt) at room temperature tossing around the molecules, the complicated 3D of delocalized electron clouds in molecular orbitals and water surrounding everything, is almost nil. Let us take the analogy of a doctor trying to vaccinate a child running in a field on a cold day. The doctor would have to find the right child among the other children (molecular recognition in the crowded cellular system), remove their jacket (desolvation), position his needle at the arm of the child (orbital steering) and apply appropriate pressure to the needle to pierce the child and immunize them (thermal energy to overcome the activation barrier). This event is very unlikely to happen given that the doctor does not know the child and the child is not a passive subject who will lend himself/herself to get vaccinated even if caught by the doctor. There are several strategies by which enzymes are known to reduce the activation energy and stabilize the transition state. Now assume that the mother of the child was around. She would not take much time in recognizing her child (active site complementarity between enzyme and substrate), would carry him playfully to a warm room, remove his/her jacket (desolvation), hold the child tightly in her lap (constrain molecular motion), invite the doctor with the needle pointed appropriately at their arm (orbital steering) and, as she hums soothing songs in the ears of the child, encourage the doctor to inject the vaccine. In chemical reaction terms, the enzyme plays a role analogous to the mother in the above example. The active site pocket possesses geometric shape and chemical moieties that preferentially filter the substrate from the crowded interiors of the intracellular space, desolvate the molecules to expose their molecular surfaces, orient them within appropriate geometry and distance constraints (orientation effect/orbital steering) and apply strain on them to mimic the transition state (racking effect). Further, by sequestering and juxtaposing the substrates in the active site, enzymes make molecular approach a first-order process rather than the second-order process that it actually is in solution.

    Another very useful analogy that helps visualizing the concept of how, in the absence of enzymes, reactions are unfeasible on the timescales of earth; one can take a stack of puzzle pieces and shake them in a jar. The probability of the puzzle pieces assembling into a puzzle is infinitely small even if the shaking is done with sufficient vigour for prolonged periods of time. This is because random shuffling of the pieces might not result in optimal alignment of pieces to assemble the puzzle. However, with the facility of cognition and pattern recognition that our brain possesses, the pieces can be assembled into the solved puzzle within a reasonable time. Once again, in enzyme kinetic parlance, the mind is synonymous with the enzyme that has acquired its fitness over millions of years of evolutionary time.

    kon and the concept of butterfly catching

    kon is the association rate constant. However, a true appreciation of kon is rarely, if ever, provided by conventional textbooks or lectures. An analogy simplifies its conception: let us assume a uni-uni reaction. Enzyme and substrate combine to form the ES complex that subsequently gets converted to enzyme and product. The concentration of the [ES] complex, which is an indication of the velocity, would be proportional to the concentration of substrate and enzyme (v ∝ [S][E]). In other words, v would be equal to a rate constant time concentration of substrate (v = k[S][E]). Converting the terms to their respective dimensionality gives the following units for the rate constant, k = concentration −1·s−1. This pseudo-first-order association constant with units of concentration inverse time inverse is known as kon. This rate constant is called pseudo first-order because it is substrate concentration-independent at extremely high substrate. To understand this conceptually, let us consider the analogy of a butterfly collector who has a huge collection of butterflies and is aiming to get the golden butterfly for his collection. He carries his butterfly net and walks to a nearby park. He soon realizes that the park is teeming with butterflies of various definitions, colours and aspects. It also dawns upon him that, no matter how hard he tries, his ability to catch the golden butterfly would depend on (a) his ability to spot the golden one from butterflies of other definitions (substrate discrimination), (b) his ability to throw the net and catch it (binding), (c) how fast he can sample the park to catch the butterflies (thermal energy), (d) what are the forces that discourage him from sampling (dielectric constant, viscosity, pH-facilitated correct ionization, etc.), and, (e) lastly and most importantly, the concentration of the golden butterflies, that is, the higher the number of golden butterflies, the more the chance that a random fling of the net can catch one (concentration of the substrate). By increasing the concentration systematically, the probability of netting a butterfly can be increased limited only by how fast the butterfly catcher can fling the net (diffusion limit). Further, at high substrate concentrations, the butterfly collector need not fling his net to catch a butterfly and butterflies would automatically walk into the trap.

    However, it would have to be emphasized here that the above analogy is only a means of gaining preliminary insight into the system. Advanced appreciation of this concept would involve an appreciation of shape complementarity between the enzyme active site and substrate and flexibility of enzyme molecule leading to several distinct subpopulations of the latter. In fact, in a survey of > 1000 different enzymes, it has been shown that the fraction of futile encounters (φ) is ≥ 0.9999. This essentially translates into < 1 productive outcome for 104 encounters [[33]]. The above aspects can be built into the butterfly catching analogy by suggesting that most butterflies, which get trapped in the net, have a nonzero probability of escaping the net.

    Intuitive feel for enzyme inhibition mechanism

    Various textbooks have repeatedly ingrained the belief that reversible enzyme inhibition should be classified based on the effect of the inhibitor on the Km (representing the E + S = ES step) and Vmax (representing the ES = ES*, * indicates the transition state) of the substrate. Km, which represents a pseudo-affinity constant and is the substrate concentration at Vmax/2, is not a dissociation constant and hence not a reliable indicator for assessing the effect of inhibitor. Moreover, Km is just a ratio of the two fundamentals parameters Vmax and Vmax/Km as discussed below [[34]]. For instance, relying on Km and Vmax, competitive (same Vmax and different Km) and noncompetitive (for α = 1, same Km and different Vmax) inhibition fall on the extremes of the spectrum of inhibition. Uncompetitive inhibition is sandwiched between them because of changing Km and Vmax, respectively. However, one fails to appreciate the fact that the change in Km is positive (i.e. decreasing Km and increasing affinity) for uncompetitive inhibition. Moreover, a casual perusal of the scheme for various inhibition types shows that noncompetitive inhibition should be in the middle with flanking competitive and uncompetitive inhibition (Scheme 1).

    Details are in the caption following the image
    Modes of inhibitor interaction along the reaction coordinate. E represents enzyme, S represents substrate, I represents inhibitor, and P represents product formed. Ki is the equilibrium constant between free enzyme, free inhibitor and the enzyme–inhibitor complex ([E][I]/[EI]); Ks is the equilibrium constant between free enzyme, free substrate and the enzyme–substrate complex ([E][S]/[ES]); kcat is the first-order rate constant representing enzyme turnover; and α is the factor that modulates the equilibrium constants Ks and Ki.

    It becomes clear from the scheme that an uncompetitive inhibitor binds to the ES complex alone, while competitive inhibitor binds to the free enzyme alone. Noncompetitive inhibitor, however, binds to both free E and ES complex determined by the α-factor. Numerous commentaries [[35]] have emphasized the importance of moving away from Km- and Vmax-based classification and towards the Vmax/Km- and Vmax-based classification approach since Km is obtained by multiplying the slope of LB plot (Km/Vmax) with Vmax [[36]] and is a secondary parameter [[37]]. The latter has a firmer basis because of the following rationalization.

    When substrate is titrated and velocity is monitored at a fixed enzyme concentration, the shape of the curve is hyperbolic (commonly understood as the Michaelis–Menten plot). The initial region of the curve is highly sensitive to and proportional to minor perturbations of the substrate and is equated by the rate Vmax/Km.
    urn:x-wiley:1742464X:media:febs15537:febs15537-math-0001
    at very less [S] vis-à-vis the Km, the substrate in the denominator can be ignored, altering the expression as follows
    urn:x-wiley:1742464X:media:febs15537:febs15537-math-0002
    where Vmax/Km is the proportionality constant with units of time−1.
    On similar lines, at very high substrate ([S] >>>> Km), the Km term in the denominator can be ignored and the expression becomes
    urn:x-wiley:1742464X:media:febs15537:febs15537-math-0003

    As can be immediately seen, the initial phase of the reaction where the free E form predominates is governed by the rate constant Vmax/Km (representing E + S = ES*), while the latter phase where ES form predominates is directly equated to Vmax (ES = ES*) (Fig. 3).

    Details are in the caption following the image
    An intuitive feel for inhibition. Unlike the traditional view on inhibition assessing the effect of inhibitor on Km and Vmax, it would be more pragmatic to use the parameters Vmax and Vmax/Km, the rate constants that are proportional to the initial first-order phase and zero-order phase vis-à-vis substrate concentration in the Michaelis–Menten hyperbolic plot. This way of looking at inhibition also provides a physical rationalization of the form of enzyme the inhibitor interacts with (i.e. either E or ES).

    Dixon plot [[38]] and Lineweaver–Burk plot [[39]] have been extensively employed to understand and assess enzyme inhibition (Fig. 3). Hence, rather than using Km (x-axis intercept on a double-reciprocal Lineweaver–Burk, LB, plot is −1/Km) and Vmax (y-axis intercept on LB plot is 1/Vmax) as the two parameters that determine the type of inhibition, it makes more sense to use Vmax/Km (reciprocal of slope on LB plot) and Vmax as the two parameters. This will directly correlate with whether an inhibitor preferentially interacts with the free enzyme or the enzyme–substrate complex depending on whether it affects the Vmax/Km or Vmax alone, respectively. This places noncompetitive inhibitor in between the forms that interact with free E (competitive) and ES complex (uncompetitive). This way of looking at inhibition data affords an intuitive feel about inhibition (Fig. 3).

    Importantly, the above section is presented exclusively with the aim of making the pedagogy of inhibition comprehensible to students from the intuitive point of view. It emphasizes the visual aspect of appreciating the spectrum of various linear inhibition modalities. Once the various modes of linear inhibition are appreciated, analysis of inhibition data to derive appropriate parameters should be undertaken using nonlinear regression and global curve fitting methods as elaborated in several textbooks [[39, 40]].

    Simulation as a means of conveying an intuitive feel for enzyme action

    Simulation has been extensively employed as a tool to gain an intuitive feel for enzyme kinetics [[41-43]]. Popular utilities such as KinTek Simulator [[44, 45]], Dynafit [[46]], KinSim [[47]], Gepasi [[48, 49]] and COPASI [[50]] have been developed and widely used in the last decade for gaining a visual feel for enzyme kinetics (from an educational perspective) and as a tool to model complete time-course measurements (Box 2). Changing the value of association (kon) and dissociation (koff) rate constants, the catalytic rate constant (kcat), enzyme and substrate concentration and the time frame of assay for a uni-uni reaction with a single intermediate complex, one gets a visual feel about how the various measurables vary as a function of the above-mentioned parameters. One can tease apart the individual contribution of the various parameters on the overall shape of the progress curves. Manipulating variables and viewing real-time alterations in the build-up of product or depletion of substrate provides an intuitive grasp of enzyme behaviour. Further, simulations provide extended opportunities beyond the classroom, at one's own leisure, to practise concepts in enzyme kinetics in a visually stimulating manner that could aid in the learning process. Figure 4 shows the shape of the progress curve changing as a function of change in parameters such as enzyme concentration (Fig. 4A), association rate constant (kon or k1) (Fig. 4B), turnover number (kcat or k2) (Fig. 4C) and k−2 (Fig. 4D) for the reaction scheme (Scheme 2) shown below. As is evident from the plots, a change in any one of the above parameters immediately provides cues on how it modulates the shape of the velocity curve. This will not only help one appreciate the effect of these parameters on appreciating the kinetics but also will help one design laboratory experiments with maximum information content.

    Details are in the caption following the image
    Appreciation of how various parameters affect the shape of the progress curves using dynamic simulation. (A) Enzyme variation is 0.01 µm, 0.02 µm and 0.03 µm (left to right). The substrate concentration is 20 µm, time is 200 s, kon is 1 µm−1·s−1, koff is 500 s−1, and kcat is 50 000 s−1. (B) kon is 0.1 µm−1·s−1, 0.5 µm−1·s−1 and 1 µm−1·s−1 (left to right). The enzyme concentration is 0.01 µm, substrate concentration is 20 µm, time is 200 s, koff is 500 s−1, and kcat is 50 000 s−1. (C) kcat is 50 s−1, 500 s−1 and 50 000 s−1 (left to right). The enzyme concentration is 0.01 µm, substrate concentration is 20 µm, time is 200 s, kon is 1 µm−1·s−1, and koff is 500 s−1. For panels (A), (B) and (C), k−2 is 0, and (D) k−2 is 10, 100 and 150 s−1 (left to right).The enzyme concentration is 0.01 µm, the substrate concentration is 20 µm, time is 200 s, kon is 1 µm−1·s−1, koff is 500 s−1, and kcat is 50 000 s−1. The green and red lines represent substrate and product, respectively. The plots were generated using KinTek Global Kinetic Explorer version 9.0.200102.
    Details are in the caption following the image
    KinTek Scheme for a uni-uni reaction with various rate constants.

    Though several studies have explored this aspect of teaching at a greater depth [[5, 6, 42-45, 51, 52]], the author believes that this aspect should be brought back to the forefront of teaching and learning enzymology. This is all the more pertinent given the complexity of kinetic mechanisms that multiprotein complexes, post-translationally modified proteins and proteins displaying non-MM kinetics show.

    Putting enzymological teaching within its right historical context

    Enzymology teaching, with its fair share of quantitative, physico-chemical and biological aspects, can come across as an abstract science. We, as human beings, relate to the action and work of other human beings by virtue of our social nature. Continuous emphasis on the abstract can sometimes serve to detract the students into believing that enzymology is a science devoid of humans. It would be highly rewarding and enlightening if the conceptual framework of enzymology is introduced through the trials and tribulations of its proponents (Box 2). This would not only serve to enrich the narrative but would also provide the essential context for the circumstantial compulsion in investigating a phenomenon and instituting the necessary axioms. I have had the honour of experiencing this approach first hand in the teachings of and personal interactions with Professors P. Balaram, M.R.N. Murthy, M. Vijayan and Jeffrey Skolnick. As part of their classroom teachings and personal interactions, when the discussions would seem to wade into too much abstract on the student's or post-doc's comprehension radar, they would launch into a tangential about the personalities who drove innovation within a field and the audience would come back with a far more enlightened view on the subject matter being discussed. The trick is to ensure that the right balance is achieved in using history as a bait in getting the student stimulated while, at the same time, ensuring that the classroom does not become a history lecture. Below, I briefly discuss three examples, the first two involving the protagonists and another emphasizing the role of socioeconomic backdrop for scientific innovation, to emphasize the point of enriching an abstract narrative in enzymology.

    Any narrative on enzymological history would invoke a lot of scientists, most of them belonging to the male gender. Very few would appreciate the fact that three fundamental concepts in enzymology were given to us by women in a predominantly male-dominated field at the time (Fig. 5A). It is common knowledge that Maud Menten, working alongside Leonar Michaelis, contributed to the now famous Michaelis–Menten kinetics that all of us are aware of. However, what very few know is that approximately 45 years before Menten's claim-to-fame equation, Maria Manaseina, a Russian investigator working with Julius Wiesner, published results claiming the discovery of cell-free fermentation [[53]]. It would take another decade for Edward Buchner to lay claim to the same concept and discredit the discovery of his predecessor. Though no conclusive evidence exists either falsifying Buchner's assertion or vindicating Manaseina's claims, it is highly enlightening to understand that experiments refuting vitalism were underway even before Buchner's attempts, setting the stage for his eventual conclusive results. Almost a century predating Manaseina, another exceptional woman, Elizabeth Fulhame, performed experiments that crystallized the concept of catalysis. Though Berzelius was later credited with coining the term catalysis, she was the pioneering figure in investigating the role of water in facilitating catalysis. Her results were published in the book entitled An Essay on Combustion [[54]] (Fig. 5A). This preamble to introducing enzyme studies would serve the dual purpose of entertaining the students and dispelling any gender prototyping in the teaching, learning and innovations in this field.

    Details are in the caption following the image
    History as a backdrop in the effective teaching of enzyme kinetics. (A) Three prominent women who made significant contributions to the field of catalysis and enzyme kinetics. From left to right, the front page of the book published by Elizabeth Fulhame that laid the conceptual framework for what we know as catalysis, Maria Manaseina and Maud Menten. (B) The principal protagonists and the timeline of the evolution of enzyme kinetics as a quantitative discipline. From left to right: Adrian J. Brown, Victor Henri, Leonar Michaelis, Maud Menten, George Edward Briggs and John Burdon Sanderson Haldane. The figures have been hand-drawn with charcoal and white chalk on tinted paper. The front page of Elizabeth Fulhame's book has been reproduced here with permission from the Science History Institute.

    The evolution of quantitative nature in enzymology involved pioneering work by Victor Henri and Adrian J. Brown (initial framework for quantitative analysis of enzyme kinetic reaction) [[55]] (Fig. 5). Adrian Brown, a British scientist working at the University of Birmingham studying malting and brewing, argued that the kinetics of enzyme action is indicative of enzyme–substrate complex formation. Victor Henri, a French Russian physical chemist, introduced the fundamental mathematical formulation that helps comprehend enzyme kinetics [[56]]. He did this in active consultation with Max Bodenstein who is considered as the father of chemical kinetics. This was followed by the landmark work of Leonar Michaelis, a German biochemist, and Maud Menten, a Canadian researcher, for real experimental data under rapid equilibrium condition (rate of ES complex dissociation to E + S is much faster than transformation of ES complex to product [[7, 9, 10, 57]]). This axiom was further improved by two Cambridge researchers, George Edward Briggs and John Burdon Sanderson Haldane, who introduced the quasi-steady-state assumption to the treatment. Work by these six pioneering figures constitutes most of the quantitative framework within which enzyme kinetic data are analysed (Fig. 5). Juxtaposing the history while introducing and discussing the quantitative nature of enzymology would serve to dilute the abstractness of the subject matter at hand and establish the international character of scientific research (Fig. 5B).

    Evolution of enzymology as a science relied very much on the art of brewing and the necessity for inebriation. Without humankind's interest in fermentation and the economic benefits that the exercise of brewing bestowed upon its practitioners, much of the enzymology as we know it now would not exist. Not to mention, some of the first enzymes studied were involved in sugar fermentation, most notably invertase. This emphasizes how the socioeconomic backdrop helps and reinforces a discipline. Narrating the evolution of enzymology in terms of the socioeconomic backdrop makes it rich and entertaining.

    As exemplified in the above three examples, this kind of an intertwined approach juxtaposing people and concepts can serve to enrich the teaching of enzymology by animating it. Further, it will serve to enhance the retention of concepts in enzymology.

    The AstraZeneca experiment

    A course audience expects to gain an intuitive feel for a subject rather than straight-jacketed recitation of the jargon associated with a particular discipline. As a way of assessing the above approach, and to understand whether I can apply this approach to professionals whose times are partitioned between their professional commitments and their desire to learn, I instituted the ‘Enzymology Book Club’ at AstraZeneca to understand and revise the fundamentals of enzymology. The coursework was initially tailored to cater to the members of the Mechanistic Biology and Profiling (MBP) division to enable them to make informed decisions about their results drawing on the toolkit of enzymology. The experiment was whether this approach has the potential of retaining the initial participation while attracting new audience. I tailored my teaching sessions on popular books authored by Robert Copeland [[58]], Irwin Segel [[59]], Paul Cook & W.W. Cleland [[60]] and Athel Cornish-Bowden [[41]]. Further, I also borrowed themes and insights from audio-visual resources made available by MIT OpenCourseWare [[61]]. The whole effort was to convey an intuitive grasp of the subject utilizing the principles espoused in the above sections (interdisciplinary, intuitive and analogy-driven), for lasting retention. The lectures went into considerable details about an appreciation for why enzymology is important and is a thriving arm of drug discovery, and how enzymes function and bring about the tremendous rate enhancements, concepts in transition state stabilization, derivation of rate equations, multisubstrate reaction mechanisms, models of inhibition and so forth. To our satisfaction, not only was the initial attendance considerable, word-of-mouth advertising by the audience about the quality of coursework attracted new attendees from across departmental boundaries with additional participants coming from the assay development unit, biophysics and cell reagents. I am in the process of continuing this experiment as a means of developing mutual appreciation for various streams of scientific inquiry to carry forward informed drug discovery.

    Conclusions and perspective

    This Words of Advice presents my attempt to show enzyme kinetics as a rich and varied field of inquiry involving cross-disciplinary understanding to gain a full appreciation of the subject. It espouses a teaching methodology that can leverage the expertise from the distinct disciplines of mathematics, physico-chemistry and biology to make the science exciting and memorable (Box 2). Further, it presents examples of analogies that can help in the teaching approach (Box 2). As an implementation of the approach, I present my experience with the evolving enzyme book club at AstraZeneca.

    Acknowledgements

    I would like to acknowledge the audience of the enzymology book club who are enthusiastic participants lending themselves to my experimental and atypical teaching methods. I would like to acknowledge the leadership of MBP, especially Rachel Grimley, Derek Barratt and James Robinson, for their constant support and creating this atmosphere conducive to scientific inquiry and outreach. This endeavour would not have succeeded if not for the initiative and diligence of Fatima Ghari, a senior research scientist at MBP. A special word of thanks to Mathew Jackson for experimenting with ways of translating the contents into a visual teaching aid. I would also like to acknowledge Rachel Grimley, Claus Bendtsen, James Robinson, Anne Jackson, Puneet Khurana, Ganesh Kadamur Bhavani, Sanjeev Kumar, Vijay Jayaraman, Craig Hughes and Omar Alkhatib for their inputs on the manuscript that enabled its improvement.

      Conflict of interest

      The author declares no conflict of interest.