## Abstract

The oxygen dissociation curve (ODC) of hemoglobin (Hb) has been widely studied and mathematically described for nearly a century. Numerous mathematical models have been designed to predict with ever-increasing accuracy the behavior of oxygen transport by Hb in differing conditions of pH, carbon dioxide, temperature, Hb levels, and 2,3-diphosphoglycerate concentrations that enable their applications in various clinical situations. The modeling techniques employed in many existing models are notably borrowed from advanced and highly sophisticated mathematics that are likely to surpass the comprehensibility of many medical and bioscience students due to the high level of “mathematical maturity” required. It is, however, a worthy teaching point in physiology lectures to illustrate in simple mathematics the fundamental reason for the crucial sigmoidal configuration of the ODC such that the medical and bioscience undergraduates can readily appreciate it, which is the objective of this basic dissertation.

- sigmoidal curve
- horizontal asymptote
- point of inflexion

the oxygen dissociation curve (ODC) of hemoglobin (Hb) as it is taught in physiology lectures has profound clinical importance, being applicable in numerous situations of health and disease, such as in the neonatal period (16), aging (10), hemorrhage (13), hemoglobinopathies (9), septic shock (22), diabetes mellitus (5), oxygenation of tumor tissues (21), sickle cell disease (6), carbon monoxide intoxication (4), anesthesia (12), open heart surgery (7), and respiratory failure (20), just to highlight a few. A close observation of its sigmoidal shape (Fig. 1) quickly reveals a couple of unique properties, namely, that oxygen saturation (Sa_{O2}) approaches a horizontal asymptote as the oxygen tension exceeds 70 mmHg, while it declines precipitously down the steep slope toward a point of inflexion when the oxygen tension falls off the “shoulder” of the ODC below ∼60 mmHg.

Such clinical implications are undeniably well known to many students and health professionals alike. Because the majority of students can see how the shape of the ODC accounts for the oxygenation behavior of Hb and grasp the physiological and clinical implications of a shift to the left or right of the ODC depending on circumstances, it is dubious if there is any extra value or relevance in trying to impart a deeper understanding into the “mathematical mechanisms” surrounding the ODC. Arguably, though, if our philosophy as educators has always been that of actively promoting academic excellence, then every devoted and ardent teacher should serve the interest of students to advance in higher learning by delving further into abstruse topics through “made simple” pedagogical approaches using elementary yet enlightening alternative explanations with the objective of fostering academic curiosity and a perpetual penetrating passion for challenging subjects in these young and maturing minds. So, it could be beneficial to illustrate with rudimentary mathematical concepts that capture the plain sigmoidal feature of the ODC without venturing into complicated and confusing models for educational purposes.

The present-day mathematical models that describe the ODC of Hb employ highly sophisticated mathematics with resulting complexity that frequently eludes complete understanding by medical students and biological sciences undergraduates. This usually results in little or superficial comprehension of the conundrum behind the crucial sigmoidal shape of the ODC. Hence, the shape of the ODC is frequently assumed as a fact without further question. Nevertheless, it is worthy to simplify and strip down the complex models of the ODC so as to distill only the most basic mathematical concepts from fundamental thermodynamic and biochemical kinetics principles that would still allow any student to grasp the derivation of the sigmoidal essence of the ODC.

My personal experience lecturing medical students, interns, residents, and fellows over the years using only high school mathematics to explain the shape of the Hb ODC was very instructive in itself, as it showed me that many who followed the mathematical reasoning were overwhelmed with a deep sense of satisfaction that came with renewed understanding, and thus well rekindled their appreciation of the ODC much better than they ever did before. The notion that many medical students, perhaps, have chosen to enter medical schools to escape mathematics and the harder sciences remains debatable and yet might have some element of truth. However, in my contacts with students, I found it equally undeniable that a good proportion of them had commented that they could finally figure out how the basic oxygenation process of Hb should naturally lead to an equation of the form that bears a sigmoidal curve following a demonstration of a straightforward mathematical derivation. Interestingly, there had even been anecdotal instances in which students who never truly enjoyed mathematics had paradoxically remarked how they were positively marveled by the mathematical illustration that allowed them to finally “see” why the ODC should be sigmoidal and how that experience itself had inspired them and fueled their interest to consider such approaches for other physiological processes as well.

## A Brief History of Mathematical Models of the ODC of Hb

One of the earliest mathematical descriptions of the ODC of Hb is the well-known Hill's equation, a simple equation proposed in 1910 by Archibald Vivian Hill (1886–1977), a British physiologist and Nobel laureate in Physiology or Medicine (8) (Fig. 2). Following this, other models, such as Adair's equation and Margaria's equation, which took into account the affinity of the heme ferrous moiety for the fourth molecule of oxygen being some 125 times that of the first 3 reactions, emerged and remain very useful today but may appear somewhat unwieldy to nonmathematicians (2,14). Soon after, the original Hill's equation was revisited and modified more accurately by John Severinghaus, who incorporated additional terms that reflected the cooperativity kinetics of virtually simultaneous binding of the last three oxygen molecules through favorable steric conformational changes (17).

Over time, the modeling of the Hb-oxygen interaction has increasingly relied on advanced mathematics to describe the ODC with yet greater accuracy. These include models that employed partial differential equations (15), hyperbolic trigonometrical functions based on known dynamic structural changes during oxygenation and deoxygenation (3,18), and the equations of Ackers and Halvorson (1). Still other formulas, such as the Kelman's equation, have been devised that are also highly precise in the prediction of Sa_{O2} for any given oxygen tension, but which are totally empirical and not based on any physiological principles, and would therefore not offer any additional insights to its special configuration (11). These models are unquestionably more complex, and their comprehensibility by students is correspondingly diminished. As such, it is helpful to introduce students of physiology and biomedicine to the ODC by using only the simplest mathematical principles to achieve better understanding, from which the more discerning and interested students with greater mathematical aptitude may subsequently delve further to more complex models if they so desire.

## Derivation of the Oxyhemoglobin ODC

Consider the oxygenation of a Hb molecule as four sequential steps, given that each of the four heme groups within the two α-globin and two β-globin chains binds to a molecule of oxygen.

Thus, (1) (2) (3) (4) From both a thermodynamic and kinetic perspective, the association constants (*k*_{1}, *k*_{2}, *k*_{3}, and *k*_{4}, respectively) can be represented by *Eqs. 5*–*8* according to the law of mass action as follows: (5) (6) (7) (8) and, therefore, (9) where *K* is the net association constant for the overall reversible reaction of oxygenation of Hb. (10) where *K* is the association constant of the reaction.

For simplicity, let us assume that all oxyhemoglobin in the circulation at any point in time is predominantly in the form of HbO_{8} molecules without any significant contributions from intermediate species of partially oxygenated Hb molecules.

Thus, (11) (12) This expression is readily identifiable as having a form similar to the Hill's equation.

Let *x* = [O_{2}] = Po_{2}, where Po_{2} is the partial pressure of oxygen, and let *y* = Sa_{O2}, such that we can then represent *Eq. 12* algebraically with *y* as a function of *x* as follows: (13) The configuration of this mathematical relation can be solved by determining the presence of any turning points, points of inflexion, and horizontal or vertical asymptotes.

Hence, differentiating *y* with respect to *x* yields: Also, when *x* = 0, d*x*/d*y* = 0, which indicates that a minimum extremum (i.e., minimum turning point) exists at *x* = 0.

When *x* → ∞, d*x*/d*y* → 0, indicating that the curve approaches a horizontal asymptote at large values of *x*. In the actual situation, this horizontal asymptote corresponds to a Sa_{O2} of 100% at standard conditions of temperature, pH, and pressure.

Thus,

We next examine the second derivative of *x* to determine the existence of any points of inflexion along the curve. Hence, differentiating d*y*/d*x* with respect to *x* yields: (14)

Since *K* is positive, *x* is positive when d^{2}*y*/d*x*^{2} = 0.

Thus, when Therefore, when d^{2}*y*/d*x*^{2} = 0, the gradient [i.e., d*y*/d*x* ≅ (K)^{1/4}] at this point of inflexion is positive.

Because d*y*/d*x* > 0 when *x* > 0, this implies that when *x* = 3/(5*K*), a nonstationary rising point of inflexion occurs with the derivative at that point having a positive gradient. In addition, the derivative of *y* is also positive on both sides of this point.

At this point of inflexion, the value of *y* can be shown to be ∼40%, as follows: Also, based on actual data on normal human Hb (HbA), the association constants of oxygen are 2.0 ± 1.1 × 10^{−6} and 2.9 ± 1.4 × 10^{−6} M^{−1} for the α-chain (*K*α) and the β-chain (Kβ) of Hb, respectively (19). The overall association constant is taken as the geometric mean of *K*α and *K*β, that is, Substituting this value of *K* into *Eq. 14*, it can be predicted from this model that the Po_{2} at the point of inflexion is as follows: In human experimental ODC data such as those shown in Fig. 1, the Sa_{O2} when the point of inflexion occurs at a pH of 7.4 lies in the region between ∼30% and 50%, which comes fairly close to our calculated value. Also, the value of the oxygen tension at that point of inflexion falls between ∼20 and 25 mmHg, which agrees quite well with our calculated value of 22.4 mmHg. It is crucial to distinguish the point of inflexion of the ODC from the point at which the oxygen tension corresponds to a Sa_{O2} of 50%. Figure 3 illustrates the shape of the ODC as predicted from the mathematical reasoning and calculations as depicted above.

## Conclusions

Although it is clear that only the most accurate mathematical models that predict the variables of the ODC robustly will find their ways into actual medical applications, such as being utilized clinically in commercial blood gas analyzers for automated computerized calculations of the various blood gas parameters within the austere environments of intensive care units that manage the critically ill, it is still worthwhile exploring the first principles and simple concepts that lead to the derivation of an “oversimplified” model that still reveals the basic sigmoidal shape of the ODC for educational purposes, which is the discourse of this article.

For didactic purposes in this article, meant to illumine students of physiology, medicine, and biological sciences, as many of the intermediate steps of calculations as possible are deliberately shown in this minitreatise to take the students through the process of deriving the equations of interest, as it is a general observation that many students often face difficulties, albeit to varying degrees, in deciphering how an initial mathematical axiom leads to the final concluding formulas without illustrating clearly the process of derivation. Thus, unlike a paper targeted at a mathematical audience, this article is pitched to students of biomedicine who do not major in mathematics and therefore makes no assumptions that such intermediate steps should be understood and simply omitted in the interest of space and time. It is hoped that this basic mathematical proof of the sigmoidal shape of the ODC can be useful to both students and teachers of physiology alike as they embark on their journey in deeper appreciation of the nature of Hb-oxygen interactions.

## Acknowledgments

The author is grateful to Sim Joo Tan for the patient secretarial assistance, to Rachel May-Wern Leow for discussion of the outline and invaluable pointers with regard to simplifying the mathematical derivation to maximize facilitation of teaching, and to Veronica May-Gwen Leow and Abigail May-Shan Leow for the wonderful inputs and encouragements.

- © 2007 American Physiological Society