“You Can't Always Get What You Want” — Linearity as the Golden Ratio of Toxicology

Abstract

Referring to the Golden Ratio (i.e. expressed in the Fibonacci sequence) in nature and art, we conclude that toxicology knows its own Golden Ration, namely linearity. The latter seems imposed on pharmaco-toxicological processes that in fact show far more complexity than simple linearity could hope to elucidate. Understanding physiological and pharmaco-toxicological processes as primarily linear is challenged in this contribution based on very straightforward principles and examples.

INTRODUCTION

The most important aspect of science is finding order in nature; that is unearthing the hidden structure of reality. Imposing order on said found structure similarly is a process well known in the history of science. One of the most famous examples is the Golden Ratio (sectio aurea or section divina), specifically expressed in the so-called Fibonacci sequence. The latter is a sequence wherein any number in the sequence (larger than 3) divided by its predecessor has an approximate value of 1.618. This value is usually denoted as phi (φ). The sequence itself (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc.) comprises of an addition of the two previous numbers in the series.

The Golden Ratio purportedly is found in many natural environments. For example, the nautilus shell usually is put forward as thé example that expresses the Golden Ratio, whereby the Fibonacci sequence is used to graph an infinite logarithmic spiral based on units equated to each number in the sequence (Fig. 1).

FIGURE 1.

The spiral of the nautilus shell is frequently suggested to follow the Fibonacci series and is described as the Golden spiral. In reality, it might, at best, be described as a rough approximation. (see text).

Optimal seedpacking in sunflower heads seem to follow the Golden Ratio. The ordering of leaves of plants seems to follow the Golden Ratio as well, and so on and so forth. Vitruvian man of Leonardo da Vinci is drawn according to the Ratio. Indeed, even a man's arm seems to be structured following phi.

We even impose phi on ratios of the human face most of us would find attractive, that is to say aesthetically pleasing (Prokopakis et al. 2013). Subsequently, we state that such a face having this ratio is attractive. Thereby, phi is equated with beauty. Moreover, we apply phi in architecture and subsequently coin buildings accordingly as pleasing.

But is it true that the Golden Ratio is an unwavering mathematical characteristic of our world? As far as we are concerned, no! (Apart from the fact that that would be a metaphysical question which the empirical sciences could not possibly answer.) More to the point, we seem to impose the Golden Ratio on many aspects of life and forms. For example, looking more closely at the famous nautilus shell, it does not conform to this ratio (Sharp, 2002).

Concisely, occasionally we seem to impose order rather than discover it in the structure of reality. What about our sciences? Do we impose some form of order without paying attention to the order that in fact is? In this contribution, we will enquire into the linear models in toxicology, and will dissect our way of thinking and our ways of imposing order. Our contention is that linearity is the Golden Ratio of current toxicology.

MICHAELIS-MENTEN

A hyperbolic relation between substrate concentration and the enzymatic rate usually describes enzymatic reactions (Figure 2A). From this graph, a maximal rate (v_max) and a K_m (the concentration needed to reach half v_max) can be obtained. These parameters describe the characteristics of the enzymatic reaction. This is usually done by means of a linearization of the hyperbolic relation: the Lineweaver-Burke transformation. Herein we plot the inverse substrate concentration versus the inverse rate. Having done this, it is easy to read off the v_max on the y-axis and the K_m on the x-axis (Figure 1B). Interestingly, the lowest substrate concentrations (where the lowest rates are measured) are the most difficuly to acquire, yet carry the most weight in determining the slope of the line, and thereby the enzymatic parameters (Nelson and Cox, 2012).

FIGURE 2.

In enzyme kinetics (panel A) the substrate concentration is plotted versus the reaction rate, which results in a hyperbolic relationship. Usually the maximal rate of the reaction is determined via a linear transformation (the Lineweaver-Burk transformation) plotting the inverse of the substrate concentration versus the inverse of the rate of the enzyme reaction. Historically the hyperbolic relation between drug concentration and receptor binding of that drug (panel B) is transformed into a linear transformation (between 20 and 80%) via a semi-logarithmic scale of the drug concentration. Similar semi-logarithmic plots are used for receptor responses.

A comparable linearization of a hyperbolic relationship -e.g. the concentration of a drug versus the percentage of receptors occupied by the drug or the concentration of a drug versus the effect caused by receptor occupancy- is found in pharmacology (Figure 2B). However, in these cases, linearization is obtained via plotting the logarithm of the concentration of the drug to either the receptor occupancy or the effect thereof. Linearity is found between 20 and 80% of occupancy or effect. Linearity facilitates the read out of the 50% effects/response (Rang et al., 2007).

The most fundamental processes -i. e. enzyme kinetics and receptor functions- are reduced to linear relationships, albeit with discipline-specific differences. Thereby linearity seems to not only have a reductionist function but is subsequently understood, either inadvertently or explicitly, as the underlying rule of pharmacology and toxicology. This, we will show, is a prime example of the reification fallacy.

LINEARITY — THE BINARY CONSEQUENCES

Linearity imposes its order on the understanding of chemical compounds as either good or bad. Bad thus can only be described as a ‘graded bad’. Nitrate is a prime example we discussed earlier in this Journal (Hanekamp, et al., 2012). Cyanosis in infants-infantile methaemoglobinaemia also known as blue-baby syndrome (Comly, 1945; Walton, 1951) and cancer of the digestive tract (Mirvish, 1995) are the two main health aspects related to nitrate exposure that have been and still are the focus of much research in roughly the past half a century.

The nitrate limit in water is 50 mg of NO₃ ⁻/L in the EU and 44 mg/L in the USA (equivalent to 11.3 and 10 mg of nitrate-N per litre, respectively). These limits are in conformity with WHO recommendations established in 1970 and recently reviewed and reconfirmed in 2008 (WHO, 2008). The limits were originally set, and remain so, on the basis of human health considerations: ‘50 mg/litre to protect against methaemoglobinaemia in bottle-fed infants (short-term exposure)’.

This binary understanding of the toxicology of nitrate was rudely disturbed by the discovery of nitrous oxide as a key second messenger in human physiology (Furchgott and Zawadzki, 1980). This discovery created and creates a multifaceted image on the role of nitrate, but also nitrite, in the human physiology. NO production has been shown to be vital to maintain normal blood circulation and defence against infection (Moncada et al., 1991). Nitric oxide, subsequently, is oxidised from nitrite to nitrate, which is conserved by the kidneys and concentrated in the saliva (McKnight et al., 1999).

The discovery of NO as a vital physiological chemical explains the common knowledge that mammals produce nitrate de novo (Mayerhofer, 1913). Infections yield the most noticeable instance of nitrate-biosynthesis, explaining methaemoglobinaemia as a result of intestinal infections that reduce nitrate to the deleterious nitrite (Hegesh and Shiloah, 1982; Cornblath and Hartmann, 1948) and not exposure to exogenous nitrate as such (L'hirondel et al., 2006).

Furthermore, the use of nitroderivatives such as nitroglycerine to dilate smooth muscle of blood vessels, whereby blood pressure is decreased, is an expression of the fact that NO forming pharmacons have an important function in pharmacotherapy. Nitroglycerine is already used for over 130 years to treat angina pectoris. Another example of nitrous oxide forming compound is the iron complex nitroprusside. Other nitrous oxide forming pharmacons are used nowadays such isosorbide dinitrate or isosorbide mononitrate.

The nitrate case shows that it does not follow linearity of ‘graded badness’; it has many faceted actions that defy any kind of linearity. Vegetables containing high concentrations of nitrate, for instance, are known to reduce blood pressure (John et al., 2002). Nitrate is a likely candidate for this effect, as now becomes increasingly clear (Weitzberg and Lundberg, 2013). The next case exemplifying complexity beyond linearity is oxygen.

OXYGEN — DEEP HISTORY BEYOND LINEARITY

Evolutionary adaptation to the slow appearance and increasing atmospheric concentration of oxygen is impressive. Anaerobic life forms had to adjust to this toxic compound, whereby oxygen started to play a key role in developing aerobic life.

The toxicity of essential oxygen arises from the ease by which it is reduced to water. The intermediate reactive oxygen species, by which biological damage is generated, are contained by enzymes like superoxide dismutase (SOD). This ancient enzyme that converts a toxic intermediate of oxygen (superoxide anion radical), exists in different forms, containing various transition metals (Cu/Zn or Mn in mammals and Fe in bacteria). What SOD does is best described in the following equation, in which SOD-Cu²⁺ is used as an example:

$\begin{array}{c}\text{SOD}-{\text{CU}}^{2+}+{\text{O}}_2\to \text{SOD}-{\text{Cu}}^{+}+{\text{O}}_2\\ {\text{SOD}-\text{Cu}}^{+}+{\text{O}}_2+2{\text{H}}^{+}\to \text{SOD}-{\text{Cu}}^{2+}+{\text{H}}_2{\text{O}}_2\\ \text{Overall}:\hspace{0.17em}2{\text{O}}_2+2{\text{H}}^{+}\to {\text{O}}_2+{\text{H}}_2{\text{O}}_2\end{array}$

As can be easily surmised from the above equations, a tradeoff is struck between the capture of the superoxide radical with the formation of hydrogen peroxide (den Hartog et al., 2003). The latter, however, is further processed by enzymes like glutathione peroxidase and catalase.

This, however, is not the end of the story. The same enzyme responsible for the capture of the superoxide radical generates very reactive hydroxyl radical (Singh et al., 1998):

$\text{SOD}-{\text{Cu}}^{+}+{\text{H}}_2{\text{O}}_2\to \text{SOD}-{\text{Cu}}^{2+}+•\text{OH}+{\text{OH}}^{-}$

Incubation of SOD with the spin traps DMPO or DEMPO results in the formation of respectively the DMPO/•OH and DEMPO/•OH adducts which are clearly detectable by Electron Spin Resonance (Singh et al., 1998).

Overall, adaptation to the essential oxygen, through protective enzymes such as SOD, has counter-consequences with respect to the incontrovertible formation of hydroxyl radicals. Adaptation, thus, has the potential to develop its own intrinsic deleterious consequences. Linearity, again, by default cannot capture this picture of reality (Fig. 3).

FIGURE 3.

Superoxide radical scavenging (left side of figure) and hydroxyl radical formation by SOD1 (right side of figure). The system which is used to show that SOD1 has both radical scavenging and radical producing capacity, consists of xanthine and xanthine oxidase and detector molecules for superoxide anion radicals (nitro blue tetrazolium reduction, abbreviated as NBT plotted on left x-axis) and hydroxyl radicals (coumarin-3-carboxylic acid) hydroxylation, abbreviated as 3-CCA plotted on right x-axis). At low concentrations SOD scavenges radicals and and high concentration the generation of radicals comes up.

AMPLIFYING OXYGEN PHYSIOLOGY

Oxygen, and the physiological protection thereagainst, has its own homeostasis that can be disrupted. When the balance is out of sync and in favour of the oxidant formation, this is called oxidative stress. The consequences of oxidative stress are manifold. Damage to proteins, lipids or DNA occur. Also activation of transcription factors come into play. As depicted in Fig. 4, oxidative stress results in phosphorylation of the inbitor I-κB of the transcription factor NF-κB.

FIGURE 4.

NF-κB and oxidative stress. Activation of the trannscription factor NF-κB (P65-P50) occurs via phosphorylation and subsequent ubiquitination of the inhibitor I-κB. Translocation of NF-κB into the nucleus leads to the production of pro-inflammatory cytokines and amplifies the oxidative response.

Phosphorylation of I-κB is followed by ubiquitination and its subsequent break down. As a result NF-κB becomes activated and can translocate to the nucleus where it activates the transcription of DNA leading to pro-inflammatory mediators (van de Berg et al., 2001). This attracts and activates neutrophils and macrophages that enhance the generation of reactive oxygen species (Fig. 5). Thus, the oxygen radicals primarily formed, function as a primer for the inflammation through NF-κB, whereby oxidative stress is amplified (Hazewindus et al., 2012).

FIGURE 5.

The vicious circle of oxidative stress and inflammation. It shows that the adaptation itself can result in a run-away physiology. Non-linearity in the pathophysiological response seems inevitable, resetting the ‘health state’.

The process described here cannot be represented in a linear fashion, as we are dealing with systems crosstalking, whereby oxidative stress is propagated.

LINEARITY AS MISPLACED CONCRETENESS — SOME CONCLUSIONS

In this short overview, we have made it quite clear that linearity is hardly a descriptive reality in toxicology and physiology at all. Nevertheless, it functions as a prespective reduction of both. It thus seems that we have our own Golden Ratio that we regard erroneous in those examples we have given at the beginning of this paper. We seemingly cannot do without prescriptive structures, despite the severe limiations those engender subsequently, both theoretically and experimentally.

Therefore, linear approaches may easily generate the reification fallacy. Abstractions are very useful-understanding certain physiological processes as linear is an example of abstraction- and are of themselves harmless when we keep in mind that we are abstracting. However, we run into severe problems when we think of abstractions as if they were actual realities themselves, thereby ‘reifying’ (objectifying) them. Worse, and this we have seen in policies based on linearity (e.g. the European REACH policy), when we think of the abstractions as somehow more real than the concrete realities from which they have been abstracted, when we find ourselves in a real pickle.

Alternatives may not be easy to find. This conclusion is more than just finding the right one-size-fits-all dose-response curve. It refers to the complexity of biological systems and the inevitability of discovering new pathways that may open different venues of research and understanding. This latter remark seems gratuituous, but nevertheless underlines the fact that the more interlinking pathways we wil find, the more absurd linearity will seem.

And obviously, there is a more thoroughgoing outlook within toxicology and pharmacology, which attempts to incorporate the complexity we could only touch on here concisely. The very raison d'être of this Journal is to explore that outlook, and the biphasic character thereof seems to gather momentum. What remains for now can only be described insipidly as awareness: even if adaptation might result into damage, the damage itself will give rise to further adaptation, which makes linearity by default moot. That is the essentiality of life.