AAN ENZYMATIC HORMESIS BOX
MICHAEL GRINFELD
Abstract.
We present a simple enzymatic system that is capable of a biphasic response undercompetitive inhibition. This is arguably the simplest system that can be said to be hormetic.
Keywords: competitive inhibition, hormesis, Gr¨obner bases Introduction
Though 21st century biology is excellent in collecting data, and very good at translating these datainto therapeutical interventions, it is less good at identifying principles of biological organisation.It has long been suggested (see for example [3]) that hormesis is a principle of biological organisation.Whether it is or is not, which is a highly controversial matter as we discuss below, it would beuseful to present simple mechanisms at any level of biological organisation that allow for hormesis.The present paper takes no position in the heated debate about hormesis and restricts itself tosuggesting an enzymatic “hormesis box”, which might be of independent interest to students ofenzymology. 2.
An introduction to hormesis
To quote Wikipedia [9],
Hormesis “is any process in a cell or organism that exhibits a biphasicresponse to exposure to increasing amounts of a substance or condition.”We have chosen this definition as it is “ethically neutral”: it makes no claim about the substanceor condition being beneficial or noxious (injurious).A typical hormesis dose-response curve which is not so ethically neutral is shown in Figure 1.Here the agent is assumed to be injurious and by a hormetic response in such a situation one(contentiously) means a response in which an injurious agent in small concentration confers benefitson the organism; we chose to label the axes in this way to alert the reader to the crux of thecontroversy.In Figure 1 LNT stands for “Linear-No-Threshold”, the basis of much health and safety legisla-tion. If instead of “noxious” one puts “beneficial” (as in anti-cancer drugs), one should reflect thehormetic and the LNT responses around the abscissa, both, again, debatable statementsTo see the fervour with which the merits and demerits of hormesis as a general principle of or-ganisation are discussed, consult, for example [7]. In our opinion, the clearest (though partisan)analysis of the difficulties with hormesis is in [8][Ch. 3]; that chapter is entitled “Hormesis Harms:the emperor has no biochemistry clothes” . a r X i v : . [ q - b i o . B M ] J a n NTHormetic intensity of noxious agentbenefit toorganism responsedamage
Figure 1.
A hormetic dose response curve..Shrader-Frechette distinguishes three different assertions: [H], [HG] and [HD], and says that [H]is trivially true, [HG] is demonstrably false, and [HD] is ethically and scientifically questionable.Here • [H]: (In a particular biological context) there exists an endpoint (an observable) for whicha noxious agent exhibits a beneficial effect at low concentrations; • [HG]: [H] is generalizable across different biological contexts, endpoints measured, andclasses of chemicals; • [HD]: [H] should be the default assumption in the risk-assessment [and hence in risk-regulation] processes.We comment that methodologically it is not clear by what criteria [HG] can be derived frominstances of [H]: what does “generalizable” really mean? and that [HD] cannot follow from [HG],no matter how much money it saves to a particular industry; LNT will always be much betterequipped to withstand legal challenges.However, the impetus for the present work is Shrader-Frechette’s statement that [H] is “triviallytrue”. While Shrader-Frechette admits that [H] is well documented across plant, fungal and animalkingdoms, there must arise the question why this should be the case.Our tentative answer is that hormesis must hitch a ride on some very general principle (or principles)of biological organisation and does not make this principle (or these principles) fitness-reducing.3. A simple enzymatic model of hormesis
We consider a system comprised of an enzyme E , a substrate S , a product P and an inhibitor I .By hormesis in such a system we mean that the rate of production of P should increase on adding ufficiently small amounts of the inhibitor. Obviously, arbitrarily complex mechanisms leading tohormesis can be devised, involving, for example, additional transcription of genes encoding theenzyme E . We aim here for maximal simplicity and ask what frequently encountered mechanismcan be superimposed on a single enzyme confronted with a (competitive) inhibitor so that thesystem is hormetic.We start with the indisputable fact that many proteins exist in a dimeric form (there is evena special BTB “born-to bind” domain!) [5]. The intuition behind the mechanism that we areproposing is as follows: in the absence of the inhibitor, most molecules of the enzyme E are to befound in the inert, dimer form E . In addition to combining with the enzyme at its active site, theinhibitor also causes the dissociation of the dimer, this making more active enzyme available forthe production of P .Therefore the reactions we must consider are: E + E k (cid:10) k E , (1a) E + I k (cid:10) k EI, (1b) E + S k (cid:10) k ES k (cid:42) E + P, (1c) E + I k (cid:10) k E I k (cid:42) E + I. (1d)We assume that the substrate S is in constant supply. We would like to understand under whatconditions on the the rate of production of P , i.e. the quasi-steady state concentration of thecomplex ES increases as we add a small amount of the inhibitor.Below we denote concentrations of species by square brackets enclosing the symbol of the species.The system (1) and the theory of enzyme kinetics [1] implies that we need to find the expression forthe concentration of ES from the following system of two conservation laws for the total amountof the enzyme and the inhibitor,[ E ] = [ E ] + 2[ E ] + [ EI ] + [ ES ] + 2[ E I ] , (2a)[ I ] = [ I ] + [ EI ] + [ E I ] , (2b)and the four equations we get by making quasi-steady state assumptions for ES , EI and E I : k [ E ][ I ] = k [ EI ] , (3a) k [ E ][ S ] = ( k + k )[ ES ] , (3b) k [ E ][ I ] = ( k + k )[ E I ] , (3c) k [ E ] = 2 k [ E ] + 2 k [ E I ] . (3d)Solving the system of equations (2)–(3) purely symbolically is a formidable task as the scheme(1) involves ten kinetic constants k , . . . , k , and we also have to take into account stoichiometricconstraints. Of course, 4 of the kinetic constants can be absorbed and non-dimensionalisationwill get rid of one of the stoichiometric constraints. The remaining system will still have eightparameters, and as the goal of the paper is not an exhaustive analysis of (2)–(3), we make rathercrude simplifications in order to exhibit the possibility of hormesis in this system. Intuitively, toget a hormetic response, we need both k and k to be “large”, so that without inhibitor mostenzyme is in dimeric form E , and, once inhibitor is added, it causes enough release of the activeenzyme E from the pool sequestered in the dimer to raise the level of production of P . To exhibit he relation between the strengths of the two processes that results in a hormetic response, wesimply put k = k = · · · = k = 1 , and take [ E ] = 1, i.e. set the total concentration of enzyme equal to one, and assume that theconcentration of the substrate S is constant, and satisfies [ S ] (cid:28) [ E ]; in our computations belowwe use [ S ] = 10.Thus we have a system of six equations in the variables [ E ], [ ES ], [ EI ], [ E ], E I ] and [ I ] withsymbolic parameters k , k and [ I ]. Since d [ P ] dt = k [ ES ] , we are really only interested in [ ES ] as a function of these parameters. Obtaining an explicitformula for [ ES ] (satisfying 0 < [ ES ] < [ E ] = 1), is still nontrivial, but a univariate polynomialsatisfied by [ ES ] with coefficients depending on k , k and [ I ], P ([ ES ] , [ I ] , k , k ) can be easilyobtained in MAPLE [4] using the Groebner package (it is denoted by poly in the code in AppendixA. For more information on Gr¨obner bases that have proved to be very useful in solving polynomialequations of enzyme kinetics, the reader is referred to [2].Solving P ([ ES ] , [ I ] , k , k ) = 0, we have that the rate of production of P in the absence of theinhibitor, i.e. with [ I ] = 0, which we denote by r is given by r = 5( − √ k ) k . Now we use a regular perturbation expansion: we write[ ES ] = r + r [ I ] + O ([ I ] ) . Hence hormesis is equivalent to r >
0. Computing r shows that the condition for hormesis canbe written in a very elegant way:(4) k > k ,crit := 7 k + 2 k − k − , k > . From (4) it is clear that hormesis is not possible for any value of k is k ≤
7. In the simulationbelow we choose k = 40, k = 50 which clearly falls in the hormetic regime of (4), and from P ([ ES ] , [ I ] , ,
50) = 0 find the unique value of [ ES ] in [0 ,
1] as a function of [ I ]. The results arepresented in Figure 2, and clearly show biphasic response.4. Discussion
As observed in the Introduction, in Figure 1 the ordinate is in units of “benefit” to the organismas most emotionally charged discussions of ethics and philosophy of hormesis are couched in theseterms. We do not commit to interpret the (counterintuitive) increase in the rate of production of P in our model system as the competitive inhibitor is added to a benefit, being entitled to do soby our ethically neutral definition of hormesis.In this short paper we presented a plausibly general mechanism by which a hormetic biphasicresponse could be generated in a simple enzymatic system; what made the biphasic response possiblewas sequestration of most enzyme in a dimer and the inhibitor-regulated release of the monomersfrom the dimer. It would be interesting to know if systems where this mechanism is operationalexist, and if they do, how frequent are they. P r odu c ti on r a t e Total inhibitor concentration
Figure 2.
The hormetic response in (1); see text for parameter values.The claim [HG] in ethically neutral terms states that biphasic effects are the norm. If true, thisis in fact as astonishing claim as, in our context, it does not mention the inhibitor ever beingencountered before in the history of a species; it could be a chemical that had been synthesized forthe first time in history just before the exposure experiment.Many of R. Rosen’s examples of anticipatory systems [6] can be explained away by recourse to anevolutionary argument: the system behaves in such and such a way (e.g. the product of the firstin a chain of enzymatic reactions leads to an activation of transcription of the last enzyme in thechain) because such behaviour is “fitter” than its absence (as, in the above example, absence of suchan “anticipatory” effect would lead to a buildup of an intermediate in the chain of reactions), inwhich case the description of such a system as “anticipatory” is redundant and has no explanatorypower. Hence an ethically neutral form of [HG], if true, would indicate the existence of intrinsicallyanticipatory systems.
Appendix A. Maple code for computations used in this paper
Here we show how to compute the polynomial in [ ES ], poly and the relation between k and k ( k1crit below). Below ES corresponds to [ ES ]m etc. with(Groebner): conservation lawse1:= E0-E-2*E2-EI-ES-2*E2I:e2 := I0-EI-E2I-I1: eferences [1] A. Cornish–Bowden, Fundamentals of Enzyme Kinetics , John Wiley and Sons, Weinheim 2013.[2] D. Cox, J. Little, and D. O’Shea,
Ideals, Varieties, and Algorithms: an Introduction to Computational AlgebraicGeometry and Commutative Algebra , Springer, New York 2013.[3] V. E. Forbes, Is hormesis an evolutionary expectation? Funct. Ecology (2000), 14–24.[4] Maplesoft, a division of Waterloo Maple Inc., Maple , Waterloo, Ontario 2019.[5] R. Perez-Torrado, D. Yamada, and P.-A. Defossez, Born to bind: the BTB protein-protein interaction domain,Bioessays (2006), 1194-1202.[6] R. Rosen, Anticipatory Systems , Springer, New York 2012.[7] B. Sacks, G. Meyerson, and J. A. Siegel, Epidemiology without biology: false paradigms, unfounded assumptions,and specious statistics in radiation science, Biol. Theory (2016), 69–101.[8] K. Shrader-Frechette, Tainted , Oxford University Press, Oxford 2014.[9] https://en.wikipedia.org/wiki/Hormesis
Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, GlasgowG1 1XH, UK
Email address : [email protected]@strath.ac.uk