### Abstract

The success of the enzymologist's test tube as both symbol and tool of modern biology is unquestioned. If it were not for the reductionist approach that it symbolizes, our understanding of living organisms would not be what it is today; it would have remained a much more superficial and descriptive kind of understanding without the knowledge of underlying mechanisms that this tool has provided. However, there is evidence in nearly every field that the reductionist approach alone is not sufficient to develop a deep understanding of all relevant phenomena. This is perhaps most evident when one considers organizationally complex biochemical systems. There is need of a rigorous integrative approach that will unify our knowledge of the molecular elements and extend our understanding of the intact system. Enzyme kinetics, which is arguably the most quantitative methodology associated with the use of the test tube, provides the common ground where reductionist and integrative approaches meet. This chapter has examined in some detail these two complementary uses of kinetics. The traditional and most successful use to date has been the elucidation of mechanisms for isolated reactions, and for this the Michaelis-Menten Formalism is the accepted paradigm. The major postulates and corresponding practices that have evolved to ensure the success of this program were described above. The purpose was to make explicit what is often tacitly assumed by textbook writers and practicing kineticists. Although there are numerous abuses of classical enzyme kinetics that can be documented when these issues are overlooked, it is sufficient for this part of the critique to note that when the canons of good practice are followed, one tends to obtain valid results. The more problematic implications of these classical assumptions are manifested when one attempts to transfer knowledge from the context of the test tube to that of the living system. The use of kinetics to characterize the behavior of integrated biochemical systems is a more recent and less developed practice. One of the more important issues in this integrative context is the selection of an appropriate formal representation. The most common approach is simply to adopt the Michaelis-Menten Formalism that has served so well for the elucidation of isolated reaction mechanisms. However, as the discussion above showed, there are difficulties in estimating the parameters of this formalism in general, and there is a combinatorial explosion in the amount of data required to characterize the rate law by kinetic means. Thus, even if the Michaelis-Menten Formalism were appropriate in principle, there would be severe practical difficulties with using it to characterize the integrated behavior of complex biochemical systems. This problem is analogous to describing the behavior of a gas using Newton's laws of motion for the individual molecules and keeping track of all their trajectories (i.e., a microscopic approach). It can be conceptualized, but in practice it is impossible. It is much more fruitful to characterize such a system in terms of the simple gas laws of thermodynamics (i.e., a macroscopic approach). The key issue is selection of an appropriate representation for the system. At some level all of our representations in science are approximations. It is important to acknowledge this fact, to seek appropriate approximations for the conditions of interest, and to recognize the limits of our approximations. The characterization of enzyme-catalyzed reactions within the Michaelis-Menten Formalism should be seen in this light. Indeed, some authors have clearly recognized this point. As has noted, "Many important enzyme reactions are being found to be more complex than those systems upon which the [Michaelis-Menten] theory was based, and the complications... make it evident that the basic Michaelis-Menten equation has only a limited range of applicability." However, there are several reasons why Michaelis-Menten kinetics are taken by many to be reality itself, despite evidence suggesting that few, if any, enzymes actually fit it well when examined carefully (). In large part this has to do with the traditional approach to teaching enzyme kinetics. In most textbooks the subject is presented in an oversimplified dogmatic fashion, often justified by decrying the lack of mathematical ability on the part of the students and appealing to pedagogical requirements for simplification of the subject matter. As an antidote to this approach, which he considers responsible for numerous errors and misconceptions found in chemistry (and biochemistry) textbooks, quotes Einstein: "We should make things as simple as possible, but not simpler". If the limitations of the Michaelis-Menten Formalism were based only on these practical issues, one might argue that it could be used at least for those enzymes (and there are many) that it seems to fit reasonably well in vitro, while leaving the others to be dealt with on an ad hoc basis. However, there is a more fundamental critique (see above) demonstrating that the postulates of the Michaelis-Menten Formalism and the canons of good enzymological prractice in vitro, which serve so well for the elucidation of isolated reaction mechanisms, are not appropriate for characterizing the behavior of integrated biochemical systems. Formalisms other than the Michaelis-Menten must be considered for this integrative purpose. The Power-Law Formalism described above provides an attractive alternative. In this formalism, the rates of formation and removal of each elemental component of the system are described by a product of power-law functions, one power function for each variable affecting the rate process in question. These elemental rate laws can be combined according to several general strategies to yield a description of the intact system. The strategy leading to the local S-system representation has received the most attention to date. Given this representation, it has been demonstrated that all the well known growth laws and allometric relationships follow by deduction, and, thus, that the Power-Law Formalism is consistent with known systemic behavior. This is not the case for the other two formalisms commonly used in biochemistry- the Linear Formalism and the Michaelis-Menten Formalism. The Linear Formalism implies linear relationships among the constituents of a system in quasi-steady state, which is inconsistent with the wealth of experimental evidence showing that these relationships are nonlinear in most cases. The Michaelis-Menten Formalism has no known solution in terms of elementary mathematical functions, so it is difficult to determine the extent to which this formalism is consistent with the experimentally observed data. However, it is possible to deduce the systemic behavior of simple specific systems involving a few rational functions and find examples in which the elements do not exhibit allometric relationships. The Power-Law Formalism possesses a number of advantages that recommend it for the analysis of integrated biochemical systems. As discussed above, we saw that estimation of the kinetic parameters that characterize the molecular elements of a system in this representation reduces to the straightforward task of linear regression. Furthermore, the experimental data necessary for this estimation increase only as the number of interactions, not as an exponential function of the number of interactions, as is the case in other formalisms. The mathematical tractability of the local S-system representation is evident in the characterization of the intact system and in the ease with which the systemic behavior can be related to the underlying molecular determinants of the system (see above). Indeed, the mathematical tractability of this representation is the very feature that allowed proof of its consistency with experimentally observed growth laws and allometric relationships. It also allowed the diagnoses of deficiencies in the current model of the TCA cycle in Dictostelium and the prediction of modifications that led to an improved model (see above). These considerations demonstrate that the local S-system representation is an appropriate means to characterize integrated biochemical systems. However, if we are to use this representation intelligently, we must be aware of its valid range of application and of the signs that indicate when this range has been exceeded. This issue is inextricably linked to the quantitative question of accuracy that was considered earlier. The range of accurate representation for well-defined model systems varies from a minimum of about two-fold to a maximum greater than 90-fold, with an average range of 20-fold. Similar results have been obtained from direct measurement of logarithmic gains in the intact system, but in this case ranges as large as 1000-fold have been observed. The range of accurate representation provided by the local S-system representation is broad enough to encompass the typical physiological range of variation seen in humans, and perhaps much of the relevant pathological range as well (see above). Many metabolites have ranges of variation around 2-fold, with hormones tending to have the highest normal ranges (typically 5-fold, but may be as high as 10- to 100-fold) as well as the highest pathological ranges (up to 10,000-fold for some tumors). The average range of variation over a wide variety of metabolites is about 3- to 5-fold. It can be concluded that the actual ranges of variation and the ranges of accurate representation are roughly the same. The Power-Law Formalism provides additional strategies for representing systems that require a higher degree of accuracy (see above). The fundamental representation within the Power-Law Formalism employs products of power-law functions to represent elemental chemical reactions as the basic components of the system.

Original language | English (US) |
---|---|

Pages (from-to) | 93-146 |

Number of pages | 54 |

Journal | Principles of Medical Biology |

Volume | 4 |

Issue number | PART 1 |

DOIs | |

State | Published - 1995 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Biochemistry, Genetics and Molecular Biology(all)
- Medicine(all)

### Cite this

*Principles of Medical Biology*,

*4*(PART 1), 93-146. https://doi.org/10.1016/S1569-2582(06)80007-3

**Chapter 5 Enzyme kinetics in vitro and in vivo : Michaelis-Menten revisited.** / Savageau, Michael A.

Research output: Contribution to journal › Article

*Principles of Medical Biology*, vol. 4, no. PART 1, pp. 93-146. https://doi.org/10.1016/S1569-2582(06)80007-3

}

TY - JOUR

T1 - Chapter 5 Enzyme kinetics in vitro and in vivo

T2 - Michaelis-Menten revisited

AU - Savageau, Michael A.

PY - 1995

Y1 - 1995

N2 - The success of the enzymologist's test tube as both symbol and tool of modern biology is unquestioned. If it were not for the reductionist approach that it symbolizes, our understanding of living organisms would not be what it is today; it would have remained a much more superficial and descriptive kind of understanding without the knowledge of underlying mechanisms that this tool has provided. However, there is evidence in nearly every field that the reductionist approach alone is not sufficient to develop a deep understanding of all relevant phenomena. This is perhaps most evident when one considers organizationally complex biochemical systems. There is need of a rigorous integrative approach that will unify our knowledge of the molecular elements and extend our understanding of the intact system. Enzyme kinetics, which is arguably the most quantitative methodology associated with the use of the test tube, provides the common ground where reductionist and integrative approaches meet. This chapter has examined in some detail these two complementary uses of kinetics. The traditional and most successful use to date has been the elucidation of mechanisms for isolated reactions, and for this the Michaelis-Menten Formalism is the accepted paradigm. The major postulates and corresponding practices that have evolved to ensure the success of this program were described above. The purpose was to make explicit what is often tacitly assumed by textbook writers and practicing kineticists. Although there are numerous abuses of classical enzyme kinetics that can be documented when these issues are overlooked, it is sufficient for this part of the critique to note that when the canons of good practice are followed, one tends to obtain valid results. The more problematic implications of these classical assumptions are manifested when one attempts to transfer knowledge from the context of the test tube to that of the living system. The use of kinetics to characterize the behavior of integrated biochemical systems is a more recent and less developed practice. One of the more important issues in this integrative context is the selection of an appropriate formal representation. The most common approach is simply to adopt the Michaelis-Menten Formalism that has served so well for the elucidation of isolated reaction mechanisms. However, as the discussion above showed, there are difficulties in estimating the parameters of this formalism in general, and there is a combinatorial explosion in the amount of data required to characterize the rate law by kinetic means. Thus, even if the Michaelis-Menten Formalism were appropriate in principle, there would be severe practical difficulties with using it to characterize the integrated behavior of complex biochemical systems. This problem is analogous to describing the behavior of a gas using Newton's laws of motion for the individual molecules and keeping track of all their trajectories (i.e., a microscopic approach). It can be conceptualized, but in practice it is impossible. It is much more fruitful to characterize such a system in terms of the simple gas laws of thermodynamics (i.e., a macroscopic approach). The key issue is selection of an appropriate representation for the system. At some level all of our representations in science are approximations. It is important to acknowledge this fact, to seek appropriate approximations for the conditions of interest, and to recognize the limits of our approximations. The characterization of enzyme-catalyzed reactions within the Michaelis-Menten Formalism should be seen in this light. Indeed, some authors have clearly recognized this point. As has noted, "Many important enzyme reactions are being found to be more complex than those systems upon which the [Michaelis-Menten] theory was based, and the complications... make it evident that the basic Michaelis-Menten equation has only a limited range of applicability." However, there are several reasons why Michaelis-Menten kinetics are taken by many to be reality itself, despite evidence suggesting that few, if any, enzymes actually fit it well when examined carefully (). In large part this has to do with the traditional approach to teaching enzyme kinetics. In most textbooks the subject is presented in an oversimplified dogmatic fashion, often justified by decrying the lack of mathematical ability on the part of the students and appealing to pedagogical requirements for simplification of the subject matter. As an antidote to this approach, which he considers responsible for numerous errors and misconceptions found in chemistry (and biochemistry) textbooks, quotes Einstein: "We should make things as simple as possible, but not simpler". If the limitations of the Michaelis-Menten Formalism were based only on these practical issues, one might argue that it could be used at least for those enzymes (and there are many) that it seems to fit reasonably well in vitro, while leaving the others to be dealt with on an ad hoc basis. However, there is a more fundamental critique (see above) demonstrating that the postulates of the Michaelis-Menten Formalism and the canons of good enzymological prractice in vitro, which serve so well for the elucidation of isolated reaction mechanisms, are not appropriate for characterizing the behavior of integrated biochemical systems. Formalisms other than the Michaelis-Menten must be considered for this integrative purpose. The Power-Law Formalism described above provides an attractive alternative. In this formalism, the rates of formation and removal of each elemental component of the system are described by a product of power-law functions, one power function for each variable affecting the rate process in question. These elemental rate laws can be combined according to several general strategies to yield a description of the intact system. The strategy leading to the local S-system representation has received the most attention to date. Given this representation, it has been demonstrated that all the well known growth laws and allometric relationships follow by deduction, and, thus, that the Power-Law Formalism is consistent with known systemic behavior. This is not the case for the other two formalisms commonly used in biochemistry- the Linear Formalism and the Michaelis-Menten Formalism. The Linear Formalism implies linear relationships among the constituents of a system in quasi-steady state, which is inconsistent with the wealth of experimental evidence showing that these relationships are nonlinear in most cases. The Michaelis-Menten Formalism has no known solution in terms of elementary mathematical functions, so it is difficult to determine the extent to which this formalism is consistent with the experimentally observed data. However, it is possible to deduce the systemic behavior of simple specific systems involving a few rational functions and find examples in which the elements do not exhibit allometric relationships. The Power-Law Formalism possesses a number of advantages that recommend it for the analysis of integrated biochemical systems. As discussed above, we saw that estimation of the kinetic parameters that characterize the molecular elements of a system in this representation reduces to the straightforward task of linear regression. Furthermore, the experimental data necessary for this estimation increase only as the number of interactions, not as an exponential function of the number of interactions, as is the case in other formalisms. The mathematical tractability of the local S-system representation is evident in the characterization of the intact system and in the ease with which the systemic behavior can be related to the underlying molecular determinants of the system (see above). Indeed, the mathematical tractability of this representation is the very feature that allowed proof of its consistency with experimentally observed growth laws and allometric relationships. It also allowed the diagnoses of deficiencies in the current model of the TCA cycle in Dictostelium and the prediction of modifications that led to an improved model (see above). These considerations demonstrate that the local S-system representation is an appropriate means to characterize integrated biochemical systems. However, if we are to use this representation intelligently, we must be aware of its valid range of application and of the signs that indicate when this range has been exceeded. This issue is inextricably linked to the quantitative question of accuracy that was considered earlier. The range of accurate representation for well-defined model systems varies from a minimum of about two-fold to a maximum greater than 90-fold, with an average range of 20-fold. Similar results have been obtained from direct measurement of logarithmic gains in the intact system, but in this case ranges as large as 1000-fold have been observed. The range of accurate representation provided by the local S-system representation is broad enough to encompass the typical physiological range of variation seen in humans, and perhaps much of the relevant pathological range as well (see above). Many metabolites have ranges of variation around 2-fold, with hormones tending to have the highest normal ranges (typically 5-fold, but may be as high as 10- to 100-fold) as well as the highest pathological ranges (up to 10,000-fold for some tumors). The average range of variation over a wide variety of metabolites is about 3- to 5-fold. It can be concluded that the actual ranges of variation and the ranges of accurate representation are roughly the same. The Power-Law Formalism provides additional strategies for representing systems that require a higher degree of accuracy (see above). The fundamental representation within the Power-Law Formalism employs products of power-law functions to represent elemental chemical reactions as the basic components of the system.

AB - The success of the enzymologist's test tube as both symbol and tool of modern biology is unquestioned. If it were not for the reductionist approach that it symbolizes, our understanding of living organisms would not be what it is today; it would have remained a much more superficial and descriptive kind of understanding without the knowledge of underlying mechanisms that this tool has provided. However, there is evidence in nearly every field that the reductionist approach alone is not sufficient to develop a deep understanding of all relevant phenomena. This is perhaps most evident when one considers organizationally complex biochemical systems. There is need of a rigorous integrative approach that will unify our knowledge of the molecular elements and extend our understanding of the intact system. Enzyme kinetics, which is arguably the most quantitative methodology associated with the use of the test tube, provides the common ground where reductionist and integrative approaches meet. This chapter has examined in some detail these two complementary uses of kinetics. The traditional and most successful use to date has been the elucidation of mechanisms for isolated reactions, and for this the Michaelis-Menten Formalism is the accepted paradigm. The major postulates and corresponding practices that have evolved to ensure the success of this program were described above. The purpose was to make explicit what is often tacitly assumed by textbook writers and practicing kineticists. Although there are numerous abuses of classical enzyme kinetics that can be documented when these issues are overlooked, it is sufficient for this part of the critique to note that when the canons of good practice are followed, one tends to obtain valid results. The more problematic implications of these classical assumptions are manifested when one attempts to transfer knowledge from the context of the test tube to that of the living system. The use of kinetics to characterize the behavior of integrated biochemical systems is a more recent and less developed practice. One of the more important issues in this integrative context is the selection of an appropriate formal representation. The most common approach is simply to adopt the Michaelis-Menten Formalism that has served so well for the elucidation of isolated reaction mechanisms. However, as the discussion above showed, there are difficulties in estimating the parameters of this formalism in general, and there is a combinatorial explosion in the amount of data required to characterize the rate law by kinetic means. Thus, even if the Michaelis-Menten Formalism were appropriate in principle, there would be severe practical difficulties with using it to characterize the integrated behavior of complex biochemical systems. This problem is analogous to describing the behavior of a gas using Newton's laws of motion for the individual molecules and keeping track of all their trajectories (i.e., a microscopic approach). It can be conceptualized, but in practice it is impossible. It is much more fruitful to characterize such a system in terms of the simple gas laws of thermodynamics (i.e., a macroscopic approach). The key issue is selection of an appropriate representation for the system. At some level all of our representations in science are approximations. It is important to acknowledge this fact, to seek appropriate approximations for the conditions of interest, and to recognize the limits of our approximations. The characterization of enzyme-catalyzed reactions within the Michaelis-Menten Formalism should be seen in this light. Indeed, some authors have clearly recognized this point. As has noted, "Many important enzyme reactions are being found to be more complex than those systems upon which the [Michaelis-Menten] theory was based, and the complications... make it evident that the basic Michaelis-Menten equation has only a limited range of applicability." However, there are several reasons why Michaelis-Menten kinetics are taken by many to be reality itself, despite evidence suggesting that few, if any, enzymes actually fit it well when examined carefully (). In large part this has to do with the traditional approach to teaching enzyme kinetics. In most textbooks the subject is presented in an oversimplified dogmatic fashion, often justified by decrying the lack of mathematical ability on the part of the students and appealing to pedagogical requirements for simplification of the subject matter. As an antidote to this approach, which he considers responsible for numerous errors and misconceptions found in chemistry (and biochemistry) textbooks, quotes Einstein: "We should make things as simple as possible, but not simpler". If the limitations of the Michaelis-Menten Formalism were based only on these practical issues, one might argue that it could be used at least for those enzymes (and there are many) that it seems to fit reasonably well in vitro, while leaving the others to be dealt with on an ad hoc basis. However, there is a more fundamental critique (see above) demonstrating that the postulates of the Michaelis-Menten Formalism and the canons of good enzymological prractice in vitro, which serve so well for the elucidation of isolated reaction mechanisms, are not appropriate for characterizing the behavior of integrated biochemical systems. Formalisms other than the Michaelis-Menten must be considered for this integrative purpose. The Power-Law Formalism described above provides an attractive alternative. In this formalism, the rates of formation and removal of each elemental component of the system are described by a product of power-law functions, one power function for each variable affecting the rate process in question. These elemental rate laws can be combined according to several general strategies to yield a description of the intact system. The strategy leading to the local S-system representation has received the most attention to date. Given this representation, it has been demonstrated that all the well known growth laws and allometric relationships follow by deduction, and, thus, that the Power-Law Formalism is consistent with known systemic behavior. This is not the case for the other two formalisms commonly used in biochemistry- the Linear Formalism and the Michaelis-Menten Formalism. The Linear Formalism implies linear relationships among the constituents of a system in quasi-steady state, which is inconsistent with the wealth of experimental evidence showing that these relationships are nonlinear in most cases. The Michaelis-Menten Formalism has no known solution in terms of elementary mathematical functions, so it is difficult to determine the extent to which this formalism is consistent with the experimentally observed data. However, it is possible to deduce the systemic behavior of simple specific systems involving a few rational functions and find examples in which the elements do not exhibit allometric relationships. The Power-Law Formalism possesses a number of advantages that recommend it for the analysis of integrated biochemical systems. As discussed above, we saw that estimation of the kinetic parameters that characterize the molecular elements of a system in this representation reduces to the straightforward task of linear regression. Furthermore, the experimental data necessary for this estimation increase only as the number of interactions, not as an exponential function of the number of interactions, as is the case in other formalisms. The mathematical tractability of the local S-system representation is evident in the characterization of the intact system and in the ease with which the systemic behavior can be related to the underlying molecular determinants of the system (see above). Indeed, the mathematical tractability of this representation is the very feature that allowed proof of its consistency with experimentally observed growth laws and allometric relationships. It also allowed the diagnoses of deficiencies in the current model of the TCA cycle in Dictostelium and the prediction of modifications that led to an improved model (see above). These considerations demonstrate that the local S-system representation is an appropriate means to characterize integrated biochemical systems. However, if we are to use this representation intelligently, we must be aware of its valid range of application and of the signs that indicate when this range has been exceeded. This issue is inextricably linked to the quantitative question of accuracy that was considered earlier. The range of accurate representation for well-defined model systems varies from a minimum of about two-fold to a maximum greater than 90-fold, with an average range of 20-fold. Similar results have been obtained from direct measurement of logarithmic gains in the intact system, but in this case ranges as large as 1000-fold have been observed. The range of accurate representation provided by the local S-system representation is broad enough to encompass the typical physiological range of variation seen in humans, and perhaps much of the relevant pathological range as well (see above). Many metabolites have ranges of variation around 2-fold, with hormones tending to have the highest normal ranges (typically 5-fold, but may be as high as 10- to 100-fold) as well as the highest pathological ranges (up to 10,000-fold for some tumors). The average range of variation over a wide variety of metabolites is about 3- to 5-fold. It can be concluded that the actual ranges of variation and the ranges of accurate representation are roughly the same. The Power-Law Formalism provides additional strategies for representing systems that require a higher degree of accuracy (see above). The fundamental representation within the Power-Law Formalism employs products of power-law functions to represent elemental chemical reactions as the basic components of the system.

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