A case of serendipity*

An account is given of how a sensitive bioassay system for measurement of the neurotransmitter acetylcholine serendipitously led to the identification of adenosine triphosphate (ATP) released in vitro from active skeletal muscle. Subsequent application of the identification procedures to exercising human muscle in vivo, cardiac muscle cells in vitro, and human erythrocytes exposed to hypoxia gave rise to the general concept of ATP as a molecule that could influence cell function from the extracellular direction. Mechanisms of ATP release from cells in terms of “trigger” events such as mechanical distortion of the membrane, depolarization of the membrane, and exposure to hypoxia are discussed. Potential therapeutic uses of extracellular ATP in cancer therapy, radiation therapy, and a possible influence upon aging are discussed. Possible roles (distant and local) of extracellular ATP released from muscle during whole body exercise are discussed.
Introduction
In 1961 Boyd and Eadie [5] developed a highly sensitive bioassay for acetylcholine using an in situ perfusion technique of frog heart. Based on the great sensitivity of the frog heart to acetylcholine (Fig. 1) a project was suggested by B. Katz to I. A. Boyd; they thought it possible that the amounts of acetylcholine released from stimulated motor nerve endings in frog skeletal muscle could be directly measured. The skeletal muscle to be used was the frog sartorius, about which a lot was known regarding the distribution and population of the motor nerve terminals. At that time the association of miniature end-plate potentials (MEPPs) with the release of acetylcholine in packets (“quanta”) together with the anatomical existence of synaptic vesicles at the motor nerve terminals was highly plausible. Any accurate measurement of acetylcholine release from motor nerve stimulation would provide valuable evidence for (or against) the association and the establishment of the “quantal” theory of neuromuscular transmission. In essence the experimental approach mimicked the classic experiment of Otto Loewi, where fluid perfusing a first frog heart was used to perfuse a second heart. When the vagus nerve supplying the first heart was stimulated cardiac arrest occurred, and shortly after this the second heart also stopped beating. Loewi termed the vagal inhibitory transmitter Vagusstoff. An isolated frog sartorius was to be stimulated via the motor nerve in a bathing solution and the solution then perfused through a frog heart. When a stimulated muscle solution was perfused a surprising stimulatory response was elicited, quite unlike the anticipated inhibitory effect of acetylcholine (Fig. 1). Clearly this response would preclude any accurate assay for acetylcholine (as well as any successful conclusion to a thesis project aimed at substantiating the quantal theory of neuromuscular transmission!).
Fig. 1The response of a very sensitive frog heart to graded concentrations of acetylcholine (g.ml−1). A concentration of 2.5 × 10−13 g.ml−1 (1.54 × 10−12 mmol.ml−1) produced a threshold response [19]
It was evident that for successful measurement of acetylcholine, the identification of the interfering substance, and its removal from the solution, had to be achieved. At this stage the basic question became: what substance(s) released from active skeletal muscle could possibly stimulate the frog heart? A “blunderbuss” approach was adopted, by perfusing through the heart as many available substances, known to exist in muscle, to see if any of them produced the stimulatory effect. The purine monophosphates and adenosine were eliminated since they did not stimulate the heart. Electrolyte changes, including an elevated potassium level, were also ruled out. Changes in calcium levels were hardly detectable in the bathing solution.
One obvious candidate as a stimulatory substance was catecholamine material. However, when the frog heart response to adrenaline was blocked by an ergot/pronethalol perfusion, the stimulatory effect remained (Fig. 3).
Fig. 2Effect of graded concentrations of ATP (g.ml−1) perfused through a frog heart [6]. Note that low concentrations have a pure inotropic effect, while a triphasic response was produced by higher concentrations. Further analysis of this triphasic response has since been made [17]
Indirect versus direct muscle stimulation The distribution of motor end plates in the sartorius muscle (there are no end plate regions at the pelvic end of the frog sartorius) allowed muscle fibers to be stimulated directly, without involvement of the neuromuscular junction in the activation process. Solutions bathing muscles stimulated directly produced the same stimulatory effect as the solutions from muscles stimulated via the motor nerve (indirect stimulation). This indicated that it was highly unlikely that the stimulatory substance had as its major source the activated motor nerve terminal.
Identification procedures
Identification of the stimulatory substance was established using three procedures: gel filtration, enzyme action of apyrase, and firefly tail luminescence.
Gel filtration chromatography The use of a gel filtration chromatography technique (Sephadex G-25, fine grade) enabled removal of protein from the stimulated muscle solution, allowing spectral absorbance analysis. An absorbance peak obtained at 265λ was the first hint that a purine compound was present in the solution. The gel filtration technique also established a molecular weight of the stimulatory substance close to that of ATP.In the face of prevailing dogma that ATP could not cross cell membranes, perfusion with ATP was reluctantly performed. A stimulatory effect was obtained—the author had become a citizen of the Republic of Serendip! A dose response of the frog heart to ATP is shown in Fig. 2.
Fig. 3Comparison of the stimulatory action of a stimulated muscle solution (‘X’) on a frog heart before and after adrenergic blockade. Vertical interrupted line, 2-h period during which heart was perfused with pronethalol hydrochloride (10−6 g.ml−1) and ergotamine tartrate (10−6 g.ml−1). The stimulatory action of the stimulated muscle solution was unaffected by adrenergic blockade. Reproduced from Boyd and Forrester [6], with permission of the Physiological Society
Incubation with apyrase In 1962 Traverso-Cori and Cori [61] demonstrated that an enzyme purified from potatoes (“apyrase”) could selectively cleave the terminal phosphate of ATP. The stimulated muscle solution was incubated with apyrase, as were the stimulatory fractions from the gel filtration technique, and it was shown that the stimulatory effect was abolished, indicating that a phosphorylated compound was responsible for the stimulatory effect.
The luciferin/luciferase test for ATP As long ago as 1947 McElroy [38] showed that when a crude extract of lampyrid beetles (Photinus pyralis) was exposed to ATP a light signal was generated. When the stimulated muscle solution was applied to luciferin extract a light signal was produced (see [18]).
Procedures applied to perfused frog musculature
As this work developed the question “does ATP appear extracellularly as a result of tissue damage?” was frequently asked. Dissection damage to the sartorius muscle was avoided by the use of the perfused hindlimb preparation of the frog, where instrument trauma was limited to the insertion of cannulae. The identification procedures for ATP were applied to the perfusate from stimulated (indirect) hindlimb musculature. Table 1 shows the relationship of ATP output to the frequency of motor nerve impulses [23]. When these amounts (0.24–1.8 nmol 100 g−1 min−1) are compared to the amounts of ATP released from isolated sartorius (50 nmol 100 g−1 min−1, [6]), it is seen that there is a great reduction detected in the perfusate sample, presumably as a result of exposure to greater ecto-ATPase activity that was not encountered in solutions bathing isolated muscles.
Table 1Relationship of ATP output to frequency of stimulation of perfused frog limbaHzpmol volley−1 100 g−1pmol 100 g−1 min−114.024021.518051.4420103.01,800aData from Forrester and Hassan [23]
Procedures applied to human muscle in vivo
The appearance of ATP in the perfusate from active frog hindlimb musculature raised the possibility that ATP might be released from active muscle into the circulating blood in vivo. The identification procedures were applied to human plasma obtained from resting and exercising human forearm muscle [20, 21]. Figure 4 gives an example of the combined use of gel filtration chromatography, firefly luciferase, and apyrase for the identification and quantification of ATP in human plasma from exercising human forearm muscle [22]. Estimates of plasma ATP levels can only be approximate, since the accumulation in the blood is continuously offset by ATPase activity of both plasma and the surfaces of the erythrocyte and endothelial cells.
Fig. 4Elution of ATP and plasma samples from exercising human forearm through a Sephadex column. Elution fractions were estimated with firefly extract. a ATP, 10−6 g.ml−1. b Post-occlusion human sample from a subject during forearm exercise. c Samples from another subject exercising forearm muscle, fraction at 9 ml was incubated with apyrase and retested on firefly extract. Fraction at 10 ml, incubation control. Reproduced from Forrester [22], with permission of the Physiological Society
Sequelae
The present-day knowledge and extensive classification of purine receptors (see review by [48]) has vindicated the concept of an extracellular system of ATP action influencing many tissues of the body. However, the mode of ATP release from cells remains a formidable challenge (see review by [56]). In the cases of skeletal muscle, cardiac muscle, and erythrocyte some specific circumstances leading to ATP release, e.g. membrane depolarization, hypoxia, and mechanical distortion [25], should be addressed further.
Association of ATP release from depolarizing membrane  In 1962 Abood et al. [1] assessed outflux of various phosphates during membrane depolarization of excitable tissues. Prior to that Holton [32] detected ATP release from sensory nerves stimulated antidromically. An elegant experiment by Israel et al. [33] showed discrete release of ATP closely related to membrane depolarization in the electric organ of Torpedo. In 1973 Ohta et al. [40] studied and compared the mode of action of several veratrum alkaloid compounds on axonal membranes (squid giant axon). Some of them exerted a highly specific action on the resting sodium permeability, with veratridine having the most potent effect. Two results led them to conclude that the depolarization evoked by veratridine was caused by specifically increasing the membrane permeability to sodium ions: (1) application of tetrodotoxin (TTX), which specifically blocks sodium channels, restored the resting membrane potential after depolarization by veratridine and (2) removal of sodium ions from both external and internal phases of the axon, thus eliminating the sodium term from the Goldman-Hodgkin-Katz constant field equation describing the membrane potential [31], had the result that veratridine had no depolarizing action after sodium removal. In 1977 White [63] demonstrated ATP release from depolarizing isolated synaptosomes prepared from rat brain. The synaptosomes were suspended in a solution containing firefly extract. Depolarization was achieved by (1) elevation of extracellular potassium and (2) with application of veratridine. A light signal was evoked in both circumstances. Prior exposure of the synaptosomes to TTX, thus blocking the sodium channels, also blocked the light signal evoked by veratridine. This suggested that the release of ATP in response to veratridine in the absence of TTX was due to membrane depolarization following the opening of sodium channels. A record of one such experiment by Tom White is shown in a previous publication ([25] Fig. 16b).
Hypoxia The potency of hypoxia in evoking ATP release from heart cells and erythrocytes has been demonstrated [4, 24]. In the case of human erythrocyte, ATP release in response to hypoxia was very effectively blocked by the nucleoside transport blocker nitrobenzylthioinosine (NBTI). The nucleoside transporter has now been cloned [26] and is thought to be a member of a group of unclassified integral membrane proteins (see review by [8]). The connection between blockade of a nucleoside transporter and inhibition of nucleotide release (ATP) in response to hypoxia is unclear.Some headway has been made toward unraveling oxygen-sensing mechanisms in certain oxygen-sensitive cells. In 1988 Lopez-Barneo et al. [36], studying chemoreceptive carotid body cells, demonstrated that lowered oxygen tension inhibited potassium channels, resulting in cell depolarization. The presence of oxygen-sensing potassium channels has been reported in many different cell types, for example, in pulmonary vascular myocytes, which constrict due to membrane depolarization following hypoxia [44] and in rat pheochromocytoma cells [12]. Does ATP release associated with membrane depolarization, as reported above [63], occur in these cells? Perhaps release of ATP from heart cells and erythrocytes in response to hypoxia involves the presence of oxygen-sensitive potassium channels, the sequence of events being hypoxia, leading to inhibition of oxygen-sensitive channels, followed by membrane depolarization and ATP release. To date, these potassium channels have not been identified in heart cells or erythrocytes.
Mechanical distortion A broad spectrum of mechanical distortion exists, ranging from gross distortion of whole cells to the impact of suction electrodes in the patch-clamp techniques for recording of currents passing through single ion channels in the membrane. There is little doubt that mechanical distortion of the whole cell membrane can lead to the release of ATP into the immediate environment. Sprague et al. [59] showed a close relationship between the extent of erythrocyte deformation and ATP release. Increases in shear stress applied to endothelial cells have been shown to liberate “vasoactive” substances from vascular endothelial cells [10, 45, 47, 54]. ATP release into the pulmonary vascular bed occurs within seconds after an increase in flow rate [30]. Grygoczyk and Hanrahan [27] clearly showed that minimum mechanical disturbance, such as changing the solution bathing the preparation, enhanced ATP release within seconds. Schwiebert [55] emphasized that the degree of suction pressure in patch-clamp techniques may determine the rate of ATP appearance. A further factor complicating physical distortion of the cell membrane is the possible involvement of the underlying cytoskeletal system. Frequency of membrane channel opening is influenced by exposure to cytochalasin B, which disrupts the cytoskeleton [62]. A comprehensive review by Morris [39] outlining the widespread occurrence of mechanosensitive ion channels in a variety of tissues is highly recommended.
Perspectives
Towards the search for an ATP channel in the plasma membrane Braunstein et al. [7], investigating the mechanism of cell volume regulation, showed that the cystic fibrosis transmembrane conductance regulator (CFTR) restored cell volume (following hypotonic challenge) by a mechanism involving ATP release. This release could be blocked by DIDS and gadolinium, suggesting that ATP might be released through a channel separate from the CFTR. Their working hypothesis was that cell volume is sensed and transduced by an ABC transporter (ATP binding cassette) enhancing ATP release through activation of a separate but associated ATP channel. The earlier observation [4] that hypoxic-induced ATP release from erythrocyte was associated with “band 3” membrane prompted the speculation that band 3 could be equated with the CFTR protein moiety. However, these proteins have been cloned and sequenced and are quite dissimilar in character [34, 52].ATP passage across the inner mitochondrial membrane has long been known. Thinnes et al. [60] have identified a voltage–dependent anion channel (VDAC, “porin”) which acts as a conduit for newly synthesized ATP in and out of the mitochondrion. VDAC is present in a multitude of tissue types, including human skeletal muscle [51] (Table 1).
Exercise and circulating purines An early example of the appearance of purines in the circulating blood in response to graduated whole body exercise was obtained by Peter Parkinson [43]. It is known that a large proportion of circulating ATP is degraded in passage through the lung [15], perhaps, along with widespread ecto-ATPase activity, minimizing the effect of widespread vasodilatation and offsetting unwanted hypotension and collapse during exercise! The products of purine degradation are probably distributed generally, especially to tissues that cannot synthesize the purine ring moiety (see “Discussion,” [11]). This phenomenon could appropriately be included in the concept of “ATP expansion” proposed by Abraham et al. [2]. This begs the question: is one of the beneficial effects of an exercise regime due to the intermittent “internal transfusion” of purines, including ATP, from exercising muscle to sustain “ATP expansion” in the tissues?
Use of ATP in cancer therapy Encouraging results have been obtained in the field of cancer therapy using extracellular ATP. A direct effect of ATP on tumor cells was demonstrated by Rapaport and coworkers [49, 50]. It was shown in various human cell lines that an increase in cellular ATP pools resulted in inhibition of DNA replication in the synthesis phase of the cell cycle, with subsequent arrest of cell growth. Inhibition of cancer growth in Ehrlich tumor cells by extracellular ATP has also been found [14, 35]. The dramatic weight loss seen in some forms of advanced cancer (cachexia) can be slowed with a regime of intravenous ATP [3]. As shown in Fig. 5 maintenance of muscle strength was also a benefit.
Fig. 5Changes in body weight and muscle strength in advanced non-small-cell lung cancer after ATP administration. a Weight change. b Elbow flexor performance (top); knee extensor performance (bottom). Reproduced from Agteresch et al. [3], with permission
Protective effect of ATP against ischemia Improved recovery of ischemic liver, kidney, and myocardial tissues after perfusion of ATP-MgCl has been shown [9, 37, 41, 58].
Protective effect of ATP in radiation therapy Impairment of wound healing following radiation therapy has been shown to be ameliorated with use of ATP-MgCl perfusion regimes [57].Although little understood, these protective effects may be associated with general elevation of tissue ATP levels, restoring the total “energy pool” of the body (see “ATP expansion” in review by [2]).
Aging and extracellular ATP levels The release of ATP from vascular endothelial cells in rat caudal artery is reduced with advancing age. Both spontaneous ATP release and release induced by methoxamine, an α-adrenoceptor agonist, were reduced in aged vascular endothelium [28]. It is interesting that enhanced release of adenyl purines from these cells was invoked by hypotension induced by exercise [29]. A clear example of the effect of age on the ATP content of human erythrocytes is shown in Table 1 [46].Aging has also been shown to diminish phosphorus metabolites in left ventricular hypertrophy as measured by 31P magnetic resonance spectroscopy [42].If aging is associated with lower tissue levels of ATP, and applied exogenous ATP can replenish these levels (“ATP expansion” see above), could aging reversal be brought about by increasing the exogenous ATP levels? At this juncture it should be noted that extracellular ATP can act as a powerful trigger of programmed cell death (apoptosis) in endothelial cells [13, 53]. Perhaps the phenomenon of apoptosis is designed to prevent the Malthusian nightmare of universal immortality!
Epilogos
And so what began as a project for the assessment of quantal release of acetylcholine from motor nerve terminals1 resulted in the finding that ATP was released from active skeletal muscle, leading to the establishment of a significant role for ATP in the hyperemia of exercising skeletal and heart muscle, as well as an appreciation of the widespread effects of extracellular ATP.
Intrinsic to the concept of serendipity in science is the element of surprise, which, although conferring excitement to investigators, only reveals our frailty of prediction in scientific matters. Inspirata accident magis saepe quam quae speres—What you do not expect happens more frequently than what you do expect (Plautus).

serendipity
hypoxia
atp release from cells
extracellular atp
exercise hyperemia


Energy metabolism, altered proteins, sirtuins and ageing: converging mechanisms?

The predominant molecular symptom of ageing is the accumulation of altered gene products. Nutritional studies show that ageing in animals can be significantly influenced by dietary restriction. Genetics has revealed that ageing may be controlled by changes in intracellular NAD/NADH ratio regulating sirtuin activity. Physiological and other approaches indicate that mitochondria may also regulate ageing. A mechanism is proposed which links diet, exercise and mitochondria-dependent changes in NAD/NADH ratio to intracellular generation of altered proteins. It is suggested that ad libitum feeding conditions decrease NAD availability which also decreases metabolism of the triose phosphate glycolytic intermediates, glyceraldehyde-3-phosphate and dihydroxyacetone-phosphate, which can spontaneously decompose into methylglyoxal (MG). MG is a highly toxic glycating agent and a major source of protein advanced-glycosylation end-products (AGEs). MG and AGEs can induce mitochondrial dysfunction and formation of reactive oxygen species (ROS), as well as affect gene expression and intracellular signalling. In dietary restriction–induced fasting, NADH would be oxidised and NAD regenerated via mitochondrial action. This would not only activate sirtuins and extend lifespan but also suppress MG formation. This proposal can also explain the apparent paradox whereby increased aerobic activity suppresses formation of glycoxidized proteins and extends lifespan. Variation in mitochondrial DNA composition and consequent mutation rate, arising from dietary-controlled differences in DNA precursor ratios, could also contribute to tissue differences in age-related mitochondrial dysfunction.
NAD and life-span
Genetic studies using a range of organisms have indicated that enzymes called sirtuins are linked to the control of ageing and life-span (Longo and Kennedy 2006; Leibiger and Berggren 2006; Lin and Guarente 2003). Sirtuins catalyse NAD-dependent deacetylation of histones (and other proteins) with the concomitant release of nicotimanide and O-acetyl-ADP-ribose (Howitz et al. 2003). Other studies have suggested that metabolism of the redox couple NAD/NADH provides a link between sirtuin activity and the control of cell senescence and organism life-span (Denu 2003, 2007; Belenky et al. 2007; Bordone and Guarente 2005): NAD-dependent protein deacetylation helps maintain the juvenile phenotype, whereas inhibition of deacetylation activity by NADH or nicotinamide, or by NAD unavailability, promote the onset of cellular aging and decrease organism lifespan.
Ageing, dietary restriction and NAD
Ageing can be delayed in various organisms by dietary restriction (DR) induced by a permanent decrease in calorie intake (called caloric restriction—CR). Recent observations have shown that an intermittent feeding (IF) protocol, which need not involve any overall reduction in calorie intake, can also delay ageing (Martin et al. 2006; Masternak et al. 2005; Mattson and Wan 2005). The mechanisms by which DR delays ageing and increases life-span are far from completely understood (Sinclair 2005). It is likely, however, that both CR and IF promote similar effects on the frequency of glycolysis and subsequent fasting periods (Hipkiss 2006a and 2007), i.e. glycolysis would be discontinuous, only operating post-prandialy. In contrast, glycolysis would be almost continuous under ad libitum (AL) feeding conditions. It is suggested that during the periods of fasting (induced by either CR or IF) the NAD/NADH ratio would differ from that prevailing in the AL case where fasting would be unlikely or negligible. In the AL condition, continuous glycolytic throughput would tend to provoke an accumulation of NADH and lower NAD availability, whereas the CR- and IF-induced fasting would decrease glycolytic NAD demand and increase NADH oxidation and NAD regeneration.
Ageing and accumulation of altered proteins
At the biochemical level ageing is characterized by the accumulation of altered protein molecules. The changes in protein structure result from intrinsic polypeptide instability as well as the actions of deleterious endogenous and exogenous agents (see Hipkiss 2006b; Schoneich 2006 and refs. therein). As yet it is unclear how changes in NAD metabolism might induce generation of altered proteins which characterise the aged phenotype.
Protein glycation and ageing
Formation of protein advanced glycation end-products (AGEs) is an important consequence of ageing and is increased particularly under conditions of uncontrolled glucose metabolism (e.g. hyperglycaemia) (see Ahmed and Thornally 2007; Thornalley 2007 and refs. therein for recent reviews). Protein AGEs can themselves induce inflammatory conditions and provoke production of reactive oxygen species (ROS) which can further compromise cell function. Indeed recent studies have shown that decreasing dietary AGE intake preserves defence functions against oxidative stress and decreases tissue damage in humans, and extends lifespan in mice, while increasing dietary AGE intake is correspondingly deleterious and accelerates ageing and decreases life-span (Cai et al. 2007; Uribarri et al. 2007a, b). Hence it is at least conceivable that decreasing metabolically-generated protein AGEs could help decrease the overall AGE load and could have beneficial effects by suppressing ageing and extending lifespan.
NAD and accumulation of methylglyoxal, an endogenous glycating agent
NAD is essential for the metabolism of the glycolytic intermediate glyceraldehyde-3-phosphate (G3P) via the action of glyceraldehyde-3-phosphate dehydrogenase (GAPDH), generating 1,3-diphosphoglycerate (1,3DPG) and NADH as products. It is argued above that in the AL condition, glycolysis would be continuous, which would tend to lower NAD levels and raise NADH levels. This would occur especially should mitochondrial-mediated NADH re-oxidation to NAD be correspondingly lowered to compensate for the extra ATP synthesised via glycolysis, assuming cellular ATP demand remains unchanged. Limitation of NAD availability would lower GAPDH activity and promote an accumulation of G3P. The immediate precursor of G3P is dihydroxyacetone phosphate (DHAP); both of these trioses can glycate proteins. More importantly, however, is the fact that both G3P and DHAP can spontaneously decompose into methylglyoxal (MG), a highly toxic and very reactive glycating agent. It is likely therefore that changes in NAD availability could strongly influence MG production.
It has previously been proposed that differences in glycolytic frequency could help explain why dietary restriction delays cellular and organism aging, possibly due to decreased MG generation during DR conditions (Hipkiss 2006a). MG is a highly active glycating agent which is thought to be responsible for the increased protein/lipid glycation detected during hyperglycaemic conditions and for much of the protein/lipid glycation associated with diabetic complications (Ahmed and Thornalley 2007; Thornalley 2007). Although MG is a normal cellular constituent, its excessive production is deleterious (plasma MG levels are raised to around 800 nmol/l in young diabetics compared to about 400 nmol/l in young non-diabetic subjects, (Han et al. 2007)). Importantly, MG can induce many of the deleterious physiological and biochemical changes characteristic of the aged phenotype, including increased ROS generation, mitochondrial dysfunction, apoptosis and inhibition of cell division (see Hipkiss 2006a and refs therein).
A number of studies, some very recent, reinforce the notion that changes in cellular MG content are important determinants of the formation of altered protein that characterise senescence (Gomes et al. 2006). Even at non-toxic concentrations, MG can influence cell proliferation by forming adducts with growth factor receptors (Cantero et al. 2007). MG can also inhibit the activity of GAPDH (Lee et al. 2005), causing triose phosphate accumulation and thereby increasing MG generation, and so inducing a highly deleterious cycle. MG can induce apoptosis (Nicolay et al. 2006) and also affect gene expression and signal transduction, at least in cultured cells (Du et al. 2003; Yao et al. 2006; Ramasamy et al. 2006). Two studies in Drosophila have shown that mutation in triosephosphate isomerase (the enzyme which converts DHAP into G3P, preceding GAPDH in the glycolytic pathway, and which is known to undergo age-related post-synthetic modification in vivo (Gracy et al. 1990)) is highly deleterious, causing paralysis, neurodegeneration and decreases life-span (Gnerer et al. 2006; Celotto et al. 2006), possibly because of MG accumulation. Human studies have shown that a deficiency in triosephosphate isomerase activity, causes increases in the levels of both DHAP (up to 20-fold (Schneider 2000)) and MG (Ahmed et al. 2003), and induces neuromuscular degeneration and early death (Schneider et al. 1965; Valentine 1966). Other studies have shown that MG induces apoptosis in neutrophils (Gawlowski et al. 2007), inhibits extracellular matrix remodelling (Chong et al. 2007) and can interfere with the stress response (Oya-Ito et al. 2006) by suppressing NF-kappaB-responsive gene activation (Laga et al. 2007).
Hence it is reasonable to suggest that any increased MG generated in AL fed animals, compared to animals subjected to CR or IF, could make a significant contribution to cellular dysfunction. During the fasting periods in DR animals, NADH generated during glycolysis would be oxidized mitochondrially for ATP production, and NAD would be regenerated thereby allowing continued G3P metabolism, and preventing triose phosphate accumulation and consequently suppressing MG generation. This condition would decrease MG-induced macromolecular glycoxidation, mitochondrial damage, dysfunctional signalling and gene expression, as described above. Such a scenario is consistent with the findings that raising NAD levels, or lowering NADH levels by increasing its oxidation, also promote sirtuin activation, with concomitant beneficial effects on cell survival etc. Table 1 illustrates the interrelationship and overlap between sirtuin regulation, generation of altered proteins and mitochondrial activity, exerted by metabolic effects on NAD and NADH levels.
Table 1Predicted effects of aerobic exercise and fasting induced by caloric restriction or intermittent feeding, on NAD and NADH levels, methylglyoxal (MG) levels, mitochondrial (mito) activity and sirtuin activityConditionsNADNADH MGMito activitySirtuin activityAd libitum fedLowHighHigh LoweredLoweredFastingHighLowLowIncreasedIncreasedAerobic exercise HighLowLowIncreasedIncreasedIncreased MG levels are partly responsible for the increased generation of altered proteins that accompanies ageing
Any situation such as fasting which maintains NAD levels, either via regeneration from NADH, or by synthesis de novo or via a scavenging pathway, would facilitate metabolism of the MG precursors G3P and DHAP, and so decrease the incidence of MG-induced macromolecular damage. The increase in free-radical-mediated damage which occurs during AL feeding, compared to the CR and IF conditions, might occur as a result of not only MG-induced generation of ROS following its reaction with proteins etc., but also via plasma membrane-mediated NAD(P)H-oxidase activity. Furthermore, because less ATP is required from mitochondrial function due to continuous ATP synthesis via glycolysis in the AL-fed state, the decreased supply of electrons (as acetyl-CoA or from NADH) to the electron transport chain would tend to produce more incompletely reduced oxygen moieties i.e. oxygen free-radicals. Any increased intra-mitochondrial ROS production could also increase the probability of mitochondrial dysfunction.
Protection against MG is afforded by the glyoxalase system which consists of two enzymes; glyoxalase I (GLX I), which uses glutathione to convert MG to a D-lactoyl-glutathione, and glyoxalase II (GLX II), which completes the detoxification by generating D-lactate and reduced glutathione. Over-expression of GLX I can inhibit formation of hyperglycaemia-induced AGEs (Shinohara et al. 1998), while a deficiency in GLX I in humans is associated with increased protein glycation (Miyata et al. 2001). GLX II activity may be rate-limiting in MG detoxification; GLX II over-expression is protective against MG-induced cell death, whilst its deficiency promotes MG-induced cell death (Xu and Chen 2006). It is also interesting that tumour necrosis factor can induce phosphorylation of GLX I which also results in substantial increase in cellular MG (van Herreweghe et al. 2002).
Tissue differences in ageing susceptibility
Tissues appear to age at different rates as shown by the varied incidence of dysfunctional mitochondria between tissues in the same organism. Variation in tissue susceptibility to MG may reside partly in differing levels of the glyoxalase system together with those molecules (glutathione, polyamines, carnosine, creatine, pyridoxamine) which normally exert protective carbonyl scavenging activity towards glycating agents such as MG (Hipkiss 2005; de Arriba et al. 2006).
Dietary-induced effects on metabolism could conceivably also affect mitochondrial DNA composition and hence mitochondrial protein structure and function. It has been found that mitochondrial DNA mutation rate may vary up to three-fold according to the relative concentrations of the four deoxyribonucleoside triphosphates in the nucleotide pool (Song et al. 2005; Mathews and Song 2007). It is possible that dietary changes could affect the composition of the nucleotide pool and thereby affect mitochondrial DNA composition during its synthesis. Pool composition could vary between tissues, and any consequent differences in mitochondrial DNA mutation rate would contribute to tissue-specific age-related mitochondrial change. Thus mitochondrial dysfunction could be either a cause or a consequence of ageing (Hipkiss 2003), depending on the prevailing circumstances.
The beneficial effects of functional mitochondria on NAD regeneration
The recent observations (i) by Belenky et al. (2007) showing that life-span extension in yeast is dependent upon NAD synthesis, (ii) that efficient mitochondrial function was necessary for maximal longevity in yeast (Piper et al. 2006), and (iii) that mitochondrial uncoupling, which increases NADH oxidation, decreases telomere damage and delays senescence in cultured human fibroblasts (Passos et al. 2007), are observations entirely consistent with the above proposal. The proposed beneficial effects of NADH oxidation to regenerate NAD via mitochondrial function would also help explain how aerobic exercise may delay development of the aged phenotype including production of altered proteins, as well as resolve the apparent paradox that increased oxygen utilization suppresses age-related change. The efficient regeneration of NAD via effective mitochondrial function is also consistent with mitochondrial ageing theories which postulate that mitochondrial dysfunction is key to the onset of ageing.
Also consistent with the present proposal are the very recent findings of Smith et al. (2007) who concluded that, in the yeast Saccharomyces cerevisiae at least, elevated respiration is an important determinant of chronological longevity. They observed that growth on non-fermentable carbon sources, which forced the cells to employ respiration exclusively, extended lifespan, but which caloric restriction did not further enhance. This again illustrates, simplistically perhaps, the potential anti-ageing functions of aerobic respiration and the deleterious effects of glycolysis, both possibly mediated via changes NAD and NADH levels, which in turn regulate MG generation. Controversially, however, these authors also found that caloric restriction-mediated lifespan extension occurred independently of sirtuin activity in Saccharomyces cerevisiae.
Other functions induced by DR
Ageing is a complex phenomenon. It is likely that the rate-limiting event which increases cellular and hence organism vulnerability to death may vary according to circumstances. For example anti-oxidant functions may not be limiting in conditions where oxidative stress is not involved. There are an increasing number of findings suggesting that proteolytic dysfunction involving either proteasomes or autophagy cause altered protein to accumulate and compromise cell survival and which can be affected by dietary restriction (Bergamini et al. 2003). Conversely activation of autophagy by inhibiting the target of rapamycin (TOR) signalling pathway can increase lifespan, at least in yeast (Bonawitz et al. 2007) and a nematode worm (Henderson et al. 2006). The recent observation that the sirtuin-like activity, histone deacetylase 6 (HDAC6), may provide a mechanistic link between the autophagic and ubiquitin-proteasome proteolytic systems in Drosophila (Pandey et al. 2007), and the observation that up-regulation of neuronal sirtuin1 activity elevates the activity of the α-protease and prevents accumulation of the amyloid peptide (Qin et al. 2006), support the idea that both formation and degradation of aberrant proteins are important for control of ageing and related disorders.
Conclusion
It is proposed that dietary-induced changes in NAD and NADH levels, as revealed by their regulation of sirtuin activity, may also control the concentration of deleterious glycolytic intermediates G3P and DHAP, and thereby also control formation of MG and generation of protein AGEs. The accumulation of MG and protein AGEs may compromise tissue function including mitochondrial activity and thereby contribute to organism ageing. Conversely, conditions that stimulate mitochondrial function will help regenerate NAD, maintain sirtuin activity and decrease formation of protein AGEs, intra- and extra-mitochondrial ROS can thereby delay ageing onset.

altered proteins
ageing
dietary restriction
nad
nadh
methylglyoxal
glycolysis
deacetylases


Power Calculations Using Exact Data Simulation: A Useful Tool for Genetic Study Designs

Statistical power calculations constitute an essential first step in the planning of scientific studies. If sufficient summary statistics are available, power calculations are in principle straightforward and computationally light. In designs, which comprise distinct groups (e.g., MZ & DZ twins), sufficient statistics can be calculated within each group, and analyzed in a multi-group model. However, when the number of possible groups is prohibitively large (say, in the hundreds), power calculations on the basis of the summary statistics become impractical. In that case, researchers may resort to Monte Carlo based power studies, which involve the simulation of hundreds or thousands of replicate samples for each specified set of population parameters. Here we present exact data simulation as a third method of power calculation. Exact data simulation involves a transformation of raw data so that the data fit the hypothesized model exactly. As in power calculation with summary statistics, exact data simulation is computationally light, while the number of groups in the analysis has little bearing on the practicality of the method. The method is applied to three genetic designs for illustrative purposes.
Introduction
The importance of statistical power in (behavior) genetic analyses is evident in the number of articles devoted to power calculations. Power has been studied in virtually all research designs, ranging from the classical twin design (Martin et al. 1978; Neale et al. 1994), to extended family designs (e.g., Heath et al. 1985; Heath and Eaves 1985; Posthuma and Boomsma 2000), to sibpair and family linkage and association designs, either in- or excluding gene by environment interaction (Abecasis et al. 2000a, b; Boomsma and Dolan 1998; Dolan et al. 1999; Fulker and Cherny 1996; Purcell 2002; Purcell and Sham 2002; Sham et al. 2000; Sham and Hewitt 1999; Sham et al. 2002; Van den Oord 1999). For a wide range of genetic designs, the Genetic Power Calculator1 (Purcell et al. 2003) can be used to calculate power. However, for customized designs and specific research questions, researchers may have to resort to their own procedure to carry out power calculations.
Power calculation based on the likelihood with the general Pearson–Nyman statistical decision theory takes two forms. First, the non-centrality parameter λ of the non-null χ2-distribution can be calculated in the analysis of exact sufficient statistics (e.g., Dolan et al. 1999). If the distribution of the data is multivariate normal, the expected variance covariance matrix Σ and the means vector μ are sufficient statistics, as they define the likelihood of the data up to an arbitrary constant (Azzelini 1996). Second, when Σ and μ are not sufficient statistics, the non-centrality parameter λ of the non-null χ2-distribution can be estimated on the basis of the analysis of simulated data using Monte Carlo simulation methods (e.g., Fulker and Cherny 1996; Abecasis et al. 2002a, b; Purcell 2002; van den Oord 1999). The latter is computationally intensive, but does not require the presence of sufficient summary statistics, whereas the former is computationally light, but does require sufficient summary statistics.
The aim of the present note is to discuss a third method of power calculation, which we refer to as exact data simulation. This method is suitable when data are multivariate normal, and sufficient summary statistics are in principle available, but the number of possible groups is prohibitively large (say, in the hundreds). The large number of distinct groups renders power calculations on the basis of the summary statistics impractical. Usually, researchers resort to Monte Carlo based power studies under such circumstances. However, exact data simulation, in combination with the definition variable facilities in packages like Mplus (fourth edition, Muthén and Muthén 1998–1997) or the freely available Mx program2 (Neale et al. 2003), is also applicable, and is more efficient than raw data simulation. Exact data simulation was used by Dolan et al. (2005) to evaluate the effects of missing data on the power in structural equation modeling, and by Van der Sluis et al. (Under revision) to evaluate the power to detect gene by environment interaction in sib-pair association studies. Although the technique of exact simulation is in itself not new (Bollen and Stine provided the basics in 1993, and the technique of exact data simulation has recently been integrated as a distinct function in the freely available R-program3), we wish to bring it to the attention of geneticists since this method of calculating power has general value in the field of genetic modeling. Below, we shortly recapitulate the basics of power analysis, and then outline the procedure of exact simulation, which may be implemented readily (we use the freely available R program). The method is illustrated in three genetic designs. Although we confine ourselves in this paper to illustrations in the context of genetic designs, we stress that this form of simulation can be used for power calculations in a wide range of other designs such as random effects models, growth curve and simplex models, and structural models.
Power calculation
The concept of power is closely related to the two types of statistical errors: the Type I error (i.e., the probability of rejecting a true hypothesis, α), and the Type II error (i.e., the probability of accepting a false hypothesis, β). Power is defined as 1 − β, i.e., the probability of rejecting a false hypothesis, or the probability of not making a Type II error. The basic aim of a power study is to determine the sample size N, which is required to achieve adequate power, given chosen α and a particular effect size.
For example, suppose that we want to fit a classical univariate ACE twin-model (see Fig. 1), and we expect that additive genetic effects (a) account for 40% of the phenotypic variance, besides effects of shared (c) and unique (e) environment. We denote this model 1, or hypothesis 1, H1. Under model 1, four parameters are estimated: the path coefficient for the additive genetic effects A (a), the path coefficient for the shared environment C (c), the path coefficient for the unique environment E (e, which includes measurement error), and the means of the twins (μ), which are usually set to be equal within twin pairs and across MZ and DZ twins. The parameter vector θ for this model with df1 degrees of freedom, is θ1 = {a, c, e, μ}, from which the covariance matrices Σ1mz and Σ1dz and mean vector μ1 can be derived. Now consider a second model, which we denote model 0, or H0, in which the additive genetic effects are assumed to account for only 5% of the phenotypic variance. This alternative model will be characterized by parameter vector θ0 = {c, e, μ}. Note that in model 0, parameter a is not estimated (and thus not part of θ) but fixed at a value that corresponds exactly to 5% of the phenotypic variation being explained by additive genetic effects. This model has df0 degrees of freedom, covariance matrices Σ0mz & Σ0dz, and mean vector μ0. This alternative model H0 is nested in the null-model H1, because the parameters in θ0 represent a subset of the parameters in θ1 (e.g., Bollen 1989; Satorra and Saris 1985). The study of power is subsequently concerned with the probability of rejecting the false model H0 in favor of the true model H1, given α, the true value of parameter a (the effect size), and sample size N.
Fig. 1Classical univariate ACE-twin model
To calculate power, we adopt the method of Satorra and Saris (1985, see also Saris and Satorra 1993), which is based on the normal theory log-likelihood ratio test statistic T. In a single group, T is calculated as follows: where N is the sample size, p denotes the number of variables in the analysis, and Σ and μ, and S and m represent the theoretical and observed variance covariance matrix and the means vector, respectively. Given that the assumptions of normal theory maximum likelihood are met (e.g., multivariate normality, and a large sample of independently and identically distributed cases), and under the assumption that Σ and μ represent the true model (Σ1 and μ1), the test statistic T follows a χ2 distribution with df1 degrees of freedom, i.e., T ∼ χ2(df1) (Azzeline 1996; Bollen 1989). If Σ and μ do not represent the true model but the alternative model (Σ0 and μ0), and given regularity conditions are satisfied (practically amounting to multivariate normality, limited misfit and large sample size N), the test statistic T follows a non-central χ2 distribution with df0 degrees of freedom and non-centrality parameter λ, i.e., T ∼ χ2(df0, λ), where λ > 0.
Given the significance level of the test α, and the difference in degrees of freedom between the true model and the alternative model, df1 − df0, the criterion level cα can be obtained from a χ2 table. If the test statistic T exceeds this criterion level, i.e., T > cα, then the alternative model is rejected in favor of the true model (i.e., the fit of the alternative model to the data is significantly worse than the fit of the true model). The aim of power studies is to determine the probability of observing T > cα, i.e., P(χ2(df0, λ) > cα), given Σ0, μ0, Σ1, μ1, N, and α, i.e., the probability of rejecting the alternative model in favor of the true model.
The non-centrality parameter λ can be obtained by fitting the alternative model to the true Σ1 and μ1, whereby λ equals the difference in the χ2 fit statistic of the model H1 and the χ2 fit statistic of the alternative model H0. That is (again in a single group), and the non-null distribution of this test statistic is χ2(df1 − df0,λ). A variety of programs can subsequently be used to integrate the non-null distribution to obtain the power (e.g., R, Mx; see also Hewitt and Heath 1988). Note that some packages, such as the Mx program, also compute the total sample size that would be required (given the reported proportion of subjects in each group) to reject the hypothesis at various power levels.
As stated in the introduction, power calculations usually take on one of two forms. First, one may be in the position that all information present in the raw data can be summarized in the covariance matrix Σ and the means vector μ, in which case Σ and μ are sufficient statistics, because they define the likelihood of the data up to an arbitrary constant (Azzelini 1996). In that case, one can derive the expected population statistics Σ and μ for every group in the study design under H1 and H0, and base the power calculations on these summary statistics.
The second method of power calculation is applied when sufficient summary statistics are not available. For example, the population statistics Σ and μ do not summarize all information present in the raw data when the continuous data are a mixture (i.e., a convex combination of different distributions; McLachlan and Peel 2002), or when data are missing at random (MAR; Shafer and Graham 2002). In family studies, gene by environment interaction may render the summary statistics insufficient. For instance, Purcell (2002) showed how environmental moderation on the means and variances can be modeled (see Fig. 2). For both MZ and DZ twins, the variance of twin i is calculated as: while the for MZ twins, covariance between twin i and twin j is calculated as: and for DZ twins as: The expected values of means for all twins are calculated as: Fig. 2Univariate ACE-twin model including moderation on the variances and the means
If the environmental moderator is categorical or ordinal (e.g., gender or affection status), the sufficient statistics (Σ and μ) are available, assuming that the data are normally distributed conditional on the levels of the moderator. In that instance, the twins in a pair may be concordant with respect to the moderator (i.e., both twins score 0, or both twins score 1), or the twins may be discordant with respect to the moderator (i.e., scoring 0 and 1, respectively). For both MZ and DZ twins, Σ and μ can be formulated for all possible combinations, such that all information present in the raw data is summarized with 6 different variance covariance matrixes Σ (3 for the MZ twins, and 3 for the DZ twins) and 3 different means vectors μ (assuming no relation between zygosity and mean), as such distinguishing 6 different groups. By comparing the fit of the model including moderator effects on both the means and the variances, to a model in which the moderator only affects the means (say), one can obtain an estimate of non-centrality parameter λ, from which the power to detect the effect of the moderator can be derived.
However, if the moderator is continuously distributed, sufficient statistics cannot be calculated. In the absence of sufficient statistics, power calculation may be conducted by means of Monte Carlo simulation. This design implies determination of the values of the parameters of interest (e.g., based on previous studies or corresponding to realistic effect sizes), and subsequent (quasi-) random data generation according to the true model H1, with realistic sample size N. By fitting the false model H0 to the data simulated according to the true model H1, an indication of the power is obtained. However, in contrast to the situation in which sufficient summary statistics are available, parameter values are not recovered exactly when the H1 model is fitted, as the random data are the outcome of a stochastic sample process. Therefore, the difference in the χ2 statistic of the model H0 and the model H1 cannot be taken as an exact estimate of the non-centrality parameter λ. To solve this, a large number of datasets are usually generated, and λ is estimated as the mean of the difference in χ2 obtained in these data sets minus the number of degrees of freedom (df1 − df0). Since power studies often concern multiple parameters with multiple values, such Monte Carlo simulation studies can be prohibitively intensive. As an alternative, simulated sample sizes may be chosen very large to induce asymptotic behavior of the χ2 statistic. However, how large a sample size should be chosen depends on the study design in question, and very large sample sizes also render the analyses computationally intensive.
Exact data simulation
Power calculations based on sufficient summary statistics are computationally relatively efficient to carry out. However, the actual feasibility of this type of power calculation depends on the number of distinct groups. If the number of groups is large (i.e., >100), it may be more convenient to carry out Monte Carlo based power calculations. We now introduce the concept of exact data simulation, which shares the virtues of the power studies based on summary statistics, but is more practicable given a large number of distinct groups.
The idea of exact data simulation is that data, which are randomly generated to begin with, can subsequently be transformed to fit the null-model H0 exactly. That is, first a data file is generated using a normal distribution quasi-random number generator. These data are then transformed, using a transformation proposed by Bollen and Stine (1993), so that the variance covariance matrix and means are exactly as specified under the model H0.
Assume a total sample size of N, and k distinct groups with known probability pk. Let Y denote the Nk × q data matrix for group k, where Nk is N*pk (possibly rounded to the nearest integer) and q is the number of variables. Let m denote the q × 1 vector of observed means and S = YtY/(N − 1)−mmt be the observed covariance matrix, Σ the expected covariance matrix implied by model H1, and μ the expected means vector implied by the model H1. Let S1/2 and Σ1/2 then denote the square root factorization of the positive definite matrices S and Σ such as given by a Cholesky factorization. It can then be shown that the covariance matrix and mean vector of data matrix Z, which is obtained through the following transformation of Yequals Σ and μ, exactly (Bollen and Stine 1993). In Eq. 7, J is a unit vector of length q, and ⊗ denotes the Kronecker product. This transformation allows one to create raw data for numerous groups that fit the null-model exactly. Consequently, when the null-model used for the generation of the data is fitted to these transformed data, all parameter values used for the simulation are recovered exactly. Let Log L0 and Log LA denote the maximum values of log-likelihood functions. The difference 2Log LA − 2Log L0 equals the non-centrality parameter λ.
Compared to multi-group power calculation with summary statistics, which becomes unpractical when the number of groups is large, the practicality of exact data simulation is unaffected by the number of groups in the analysis. However, one issue does require attention. When the number of groups is large (e.g., 256, see Illustration 1 below), the probability pk that a subject belongs to a group k may be relatively small. In order to calculate the Cholesky decomposition of the observed variance covariance matrix S, the number of observations in a group Nk should equal at least q + 1 (i.e., to ensure that S is positive definite and the Cholesky decomposition is possible). There are two ways to handle this problem. First, one can choose N to be sufficiently large that all groups, even those with small probabilities, by choosing a very large overall sample size N for the simulation. Power analyses based on this very large sample size produce non-centrality parameters which can subsequently be used to calculate power for other, more realistic sample sizes. Second, one can choose a smaller overall sample size, and accept that not all possible groups will be represented in the power calculations. This choice is usually justified since very small groups (e.g., including 2 subjects out of a possible 10,000) do not contribute much to the power. However, power calculations are more precise when all groups are represented in the simulation, i.e., overall N is large. Furthermore, it is in principle possible that the presence of all groups is required for model identification.
Note that packages like Mx and R will estimate the number of data vectors required for a power of e.g. 80%, given the proportion of subjects in each group. So while Nk ≥ q + 1 is required for exact simulation, Mx will return an overall sample size N, in which many groups may represented by fewer than q + 1 observations, which is in line with what one would expect to observe in research practice. Using the exact data simulation script subsequently to simulate data with the sample size advertised by Mx would not result in a power of 80% since the groups with Nk < q observations are not represented in the simulated data and thus do not contribute to the power.
Having discussed the concept of power, and the exact data simulation procedure, we will now illustrate the virtues of exact data simulation with three behavior genetics examples. We chose Mx to analyze the simulated data because of the program’s inbuilt option to calculate the sample size required for different power levels given the non-centrality parameter. However, the non-centrality parameter can also be obtained through other software (e.g., LISREL, Mplus). The calculation of sample sizes required for different power levels can then be done using other programs like R. The R-scripts used to simulate the data, and the Mx scripts used to analyze the data are available in the Mx scripts library.4 A small R-script for power computations based on non-centrality parameters can be downloaded from the library as well.
Illustration 1: multivariate ACE-model with data MCAR
Let us consider a four-variate ACE-model with data obtained from MZ and DZ twins (no additional family members). We assume a model with one common genetic factor, one common shared environmental factor, and specifics for A, C and E for all four traits. The model is illustrated in Fig. 3. Parameter values are chosen such that additive genetic influences, shared environmental influences, and non-shared environmental influences explain 50%, 30% and 20% of the total variance, respectively. Of the additive genetic variance, 60% is attributable to the common genetic factor (i.e., 30% of the total variance), and 40% to the specifics of A (i.e., 20% of the total variance). Of the shared environmental variance, 50% is attributable to the common shared environmental factor (i.e., 15% of the total variance), and 50% to the specifics of C (i.e., 15% of the total variance). Table 1 contains the correlation matrices for the MZ and DZ twins in the case that data are not missing. Means for all traits are equal to zero. We consider the power to reject the alternative hypothesis that the genetic specifics for all four traits explain not 20% but only 5% of the total variance,5 i.e., one common genetic factor is (almost) sufficient to explain all genetic variance and covariance in the four traits. If there are no missing data (situation S1), then one could simply use summary statistics to obtain power information, as this is a 2-group analysis. However, suppose we want to study the influence of data missing completely at random (MCAR) on the power to reject the hypothesis that all genetic specifics are zero. Here we consider two scenarios. First, we study the case that the probability of data being MCAR is 20% for all variables (situation S2). Given that we have q = 2 × 4 observations per family, this kind of missingness could yield 28 − 1 = 255 possible data patterns, i.e., 255 different groups (we discard the group in which all data are missing). In that case, power calculations using summary statistics are impractical, whereas exact data simulation is feasible. Note that some data patterns are rather unlikely, e.g., the probability of observing a valid observation for the first trait of the first twin only, while all other observations in the family are missing, is .8*(.27) = 1.024−05. Remember that the Cholesky decomposition cannot be calculated if the number of observations in a group Nk does not equal at least q + 1, so the simulated N needs to be very large if one wants all data patterns to be present in the simulated data set (about (q + 1)/1.024−05 ≈ 900,000). Yet, as very rare observations will hardly contribute to the power (5 of the 900,000 cases in the present example), one can just as well adopt a smaller sample size, and accept the fact that some groups (i.e., patterns of observations) will not be represented, and calculate power given the most likely patterns of observations.
Fig. 3Four-variate ACE-model with common factors for additive genetic and shared environmental effects, and specifics for A, C and ETable 1Illustration 1: Four-variate cross-trait-cross-twin MZ correlations (below diagonal) and DZ correlation (above diagonal) for data without missingnessTwin 1Twin 2Trait1Trait2Trait3Trait4Trait1Trait2Trait3Trait4Twin 1Trait11.00.45.45.45.55.30.30.30Trait2.451.00.45.45.30.55.30.30Trait3.45.451.00.45.30.30.55.30Trait4.45.45.451.00.30.30.30.55Twin 2Trait1.80.45.45.451.00.45.45.45Trait2.45.80.45.45.451.00.45.45Trait3.45.45.80.45.45.451.00.45Trait4.45.45.45.80.45.45.451.00Note: Sample size N is not reported; as the simulations are exact, this correlation matrix should result independent of the sample size chosen for the simulations when data are not missing
Second, we study the case where the probability of data being MCAR is 40% for the first two variables, and zero for the second two (situation S3). This could be a realistic scenario in practice, for example, when a questionnaire study that measures two traits is extended during the data collection to include two additional traits, or when data from different studies are combined (one study in which all four traits were measured, while another study only included measurements of two traits for, e.g., economical reasons). Given that we have q = 2 × 4 observations per family of which only four variables show missingness, this would yield 24 = 16 possible data patterns, i.e., 16 different groups. With only 16 groups, power calculations using summary statistics would be feasible. However, one efficiently setup exact data simulation script can handle both this simple pattern of missingness, and more complex.
For the simulations we chose an overall sample size of 50,000 families (1/3 MZ, 2/3 DZ), which means that for situations S1 and S3, all groups (data patterns) are represented (50,000 and 49,999 cases simulated, respectively), while for situation S2, only the 163 most likely groups of the possible 255 are represented (49,999 cases simulated).
The three simulated data sets were subsequently analyzed in Mx. In the Mx-script, we specify different groups for the MZ and DZ twins. Because we use full information maximum likelihood to accommodate the missingness, we do not need to specify different groups for all possible missing data patterns. The Mx command ‘option power’ (α = .05, df = 4) was used to obtain an estimation of the total sample size that would be required for a power of 80%, given the current proportions of subjects in each group.
We find that for situation S1 (no missingness), 302 families are required for 80% power to reject the alternative hypothesis that the genetic specifics for all four traits explain 5% rather than 20% of the variance each, while for situations S2 (20% missingness for all variables) and S3 (40% missingness for only variable 1 and 2) the number of families required to obtain 80% power is estimated at 494 and 474, respectively. We hasten to note that these results are not informative for the case that data are missing at random (MAR), rather than MCAR (see Schafer and Graham 2002, for a comprehensive review on missingness and statistical procedures for handling missing data).
These power calculations took about 2 min for each situation S. Within the Monte Carlo framework, acquisition of similar power results would take at least T times as long for each situation S (where T is the number of replications one chooses to do). Given that the time required to write the data simulation script is equal for Monte Carlo simulation and exact simulation, it is clear that exact simulation saves a lot of time.
Illustration 2: gene by environment interaction with latent G and measured, categorical E
Gene by Environment (G × E) interaction is an important issue. From the perspective of the power study, a problem with the presence of G × E when the E is continuously distributed is that it renders single summary statistics insufficient; in the presence of G × E, (co)variances and means depend on the level of the environmental moderator, as we have seen in Eqs. 3–6. Purcell (2002) showed how G × E on the means and variances can be modeled if G is latent, and E is measured.
In power calculations in the G × E context, one can adopt a multi-group design, if the environmental moderator is categorical. For example, consider a classical ACE-twin design. If the environmental moderator is dichotomous (e.g., males versus females, young versus old, smoking versus non-smoking), the sample consisting of MZ and DZ twin pairs can be split up into twin pairs who are concordant with respect to the moderator (e.g., both twins do, or do not, smoke), and twin pairs who are discordant with respect to the moderator (only one of the twins in a pair smokes). With a dichotomous moderator and only two subjects per family, power calculations using summary statistics are feasible as there are only three distinguishable groups (not accounting for the distinction between MZ and DZ twins). However, suppose that you have measured an environmental moderator with 4 levels (coded 0, 1, 2, 3) in twin-pairs and their parents. With four persons per family and four possible moderator levels, there are 44 = 256 possible family configurations. In that case, multi-group analysis with summary statistics is impractical, and exact data simulation may be used instead.
For this illustration, parameters a, c, and e were all set to 1, such that the total variance equaled 3 (excluding moderating and main effects). The moderator, which was assumed independent of genotype in this illustration, was coded 0 to 3, such that the group with 0 on the moderator can be considered the baseline condition. The probability for all moderation levels was set to .25, and moderation levels were modeled as independent across family members (i.e., the probability for each family member’s moderation level was independent of the moderation levels of the other family members). Moderation in C, E and the means was fixed to 0, but the regression weight of the moderator was set to .2 for the additive genetic effects, such that the moderator explained 20% of the variance in the total population (i.e., 0%, 13%, 24%, and 34% of the variance, respectively, depending on the level of the moderator). The model is illustrated in Fig. 4. For the simulation, we chose an overall sample size N of 10,000 (1/3 MZ and 2/3 DZ twins). Fig. 4Univariate ACE-model for parents and twin-offspring, including moderation on the variances and the means
Given these simulated data, we want to estimate the power to reject the alternative hypothesis that all moderator effects on the variances are zero (i.e., no G × E, or C × E, or E × E). In practice, one would fix all regression weights concerning the moderating effects on the variances (βa, βc, and βe) to zero at once, resulting in a test with 3 degrees of freedom.
The data were analyzed in Mx: different groups were specified for the MZ and DZ twins, and the moderator featured as a so-called definition variable. The ‘option power’ command (α = .05, df = 3) was again used to obtain an estimation of the total sample size that would be required for a power of 80%, given the current proportions of subjects in each group.
With the probability for all moderation levels fixed to .25, all 256 groups were represented in both the MZ and the DZ twins in the simulated data file (9,984 cases simulated), and for the chosen values, the analysis shows that we would need 165 families for a power of 80%. If we were to change the moderator level probabilities from .25 for every level to .4, .3, .2, and .1 for levels 0, 1, 2 and 3 respectively, then 179 and 225 groups would be represented in MZ and DZ twins, respectively (9,705 cases simulated). In that case, 206 families would be required for a power of 80% even though the moderator effect (βa) is unchanged.
These power calculations took at most 1 minute in total. Again, acquisition of similar power results would take at least T times as long within the Monte Carlo framework (where T is the number of replications). Assembling the data simulation script takes equally long for both types of simulation, so overall, exact simulation saves time.
Illustration 3: association for a tri-allelic locus with different allele frequencies
The aim of association studies is to determine whether genetic variation is associated with the risk for disease or the expression of a continuously distributed trait. Association studies may produce false positives, i.e., significant association in the absence of any true genetic effects. Population stratification is one source of false positives, i.e., the mixture of two populations with different allele frequencies and different phenotypic means. Fulker et al.
(1999) showed that this type of spurious association can be avoided in a family-based study design. In this illustration, we focus on the situation in which data are available for pairs of siblings. Although this design allows for the simultaneous modeling of linkage and association, we limit the analysis to the association, but note that linkage information (i.e., IBD sharing estimation) could be included in exact data simulation scripts. For the present illustration, however, we assume that the locus under study is the QTL itself and not a marker in linkage disequilibrium with the QTL.
If a locus is diallelic, 22 = 4 genotypes can be distinguished: AA, AB, BA and BB (of course, in practice, there are only 2 + 1 = 3 distinguishable groups as AB and BA are the same, but when simulating the data exactly, it is convenient simply to treat them as different groups). These 22 genotypes give rise to (22)2 = 16 possible combinations of siblings (not accounting order), i.e., a 16 group analysis. Note that this is the simplest case: with 3 alleles, the number of possible sib-pairs is already (32)2 = 81, and when the locus under study is a polymorphic marker, with, say, 15 possible alleles, the number of distinguishable sib-pairs is (152)2 = 50,625. Clearly, multi-group analyses with sufficient summary statistics quickly become impractical as the number of alleles––or loci––increases.
We illustrate the use of exact data simulation in the context of the sib-pair association design, for a tri-allelic locus with alleles A, B, and C, with frequencies p, q, and r, respectively. The aim of this particular power calculation is to determine the influence of the allele frequencies on the power to detect a QTL. The biometrical model for a tri-allelic locus is summarized in Table 2. As with the more familiar diallelic case, the expected genotypic value  is assumed zero, so that everything is scaled in terms of deviations. In the case of three alleles, 2 genotypic values are distinguished, which were both fixed to .206, so that AA was associated with an increase of .206, BB with an increase of .206, and CC with a decrease of−.206 to .206 = −.412. Dominance was assumed to be absent, so the genotypic effect for the heterozygous genotypes AB, AC and BC was calculated as the mean of the effects of the homozygous groups. In the case of equal allele frequencies (p = q = r = 1/3), this QTL explains 2.5% of the variance (as determined using regular regression with the phenotype as dependent variable and genotype as predictor). Note that the variance explained by the QTL depends on the allele frequencies, so even though the genotypic values remain the same across all simulations, varying the allele frequencies affects the effect size of the QTL effect. For all simulations, background variance was decomposed such that additive genetic effects explained 30%, and unique environmental influences (E) explained 70% of the variance that remained after the QTL-effects was taken into account. Overall sample size N was fixed to 10,000 (note that because of rounding, the actual N modeled will not be equal to the overall sample size N of 10,000; see Table 3).Table 2Expectations for a tri-allelic locus following the standard biometric model when dominance is assumed absentGenotypeAAABBBACBCCCGenotype frequency fijp22pqq22pr2qrr2Genotypic value gijx(x + y)/2yx + z/2 = −y/2y + z/2 = −x/2zNote: p, q, and r denote the frequencies of alleles A, B and C, respectively; x is the genotypic value associated with genotype AA, y the genotypic value associated with genotype BB. As  the genotypic value for genotype CC is z = −x − y (i.e., x + y + z = 0). μqtl denotes the expected contribution of the QTL to the population mean, and σ2qtl denotes the expected contribution of the QTL to the population variance (adapted from Falconer and Mackay 1996)Table 3Results for illustration 3: Power calculations for sib-pair association with a tri-allelic locus with fixed genotypic valuesFrequencies alleles A, B, CEffect size (%)Actual NNr of groups represented χ2(6)Observed power N required for power of 80%.33/.33/.332.59,63981837.8241157.25/.5/.251.79,98581734.3571185.45/.45/.1.69,99381352.9631386.1/.45/.452.89,99280966.6981141Note: Effect size is defined as % of variance explained by QTL; Actual N refers to actual number of sib-pairs in the analysis; Nr of groups represented refers to the number of groups, of the possible 81, that were represented in the analysis; χ2(6): the χ2-value of the test for association when the genotypic effects for all 6 distinguishable genotypes are fixed to zero; Observed power refers to the power observed for the modeled sample size N
Note that between and within effects were exactly equal (i.e., B = W) as we did not model population stratification; all between and within parameters can thus be fixed to be equal without loss of fit. The overall test for genetic association then involves fixing the genotypic effects for all 6 distinguishable genotypes (AA, AB, AC, BB, BC, and CC) to zero, i.e., 6 degrees of freedom.
The simulated sib-pair data were analyzed in Mx, using the ‘option power’ command (α = .05,df = 6) to obtain the sample size required for a power of 80%, given the current proportions of subjects in each group. The results presented in Table 3 show that the power to detect a QTL with certain genotypic values depends on the allele frequencies. As expected, the power is greatest when the frequency for the allele with the largest genotypic value (allele C) is highest.
With exact data simulation, these power calculations took about 2 min for each choice of allele frequencies. Again, it would take at least T times as long to obtain similar power results within the Monte Carlo framework (where T is the number of replications), while the time required to write the data simulation script takes equally long for both types of simulation.
Conclusion
In this paper we discussed a third method of power calculation, which can be useful when sufficient summary statistics are available in principle, but the number of possible groups is so large to render a multi-group analysis impractical. The illustrations presented in this paper represent only a few of the possible (behavior genetics) designs in which exact data simulation may prove useful. Other models for which exact data simulation can be used include random-effects models, latent growth curve models, simplex models, and (hierarchical) structural models, either or not in the context of genetics, just to name a few. Exact data simulation does not require more programming skills, or programming time, than Monte Carlo simulation, but one may save a lot of time analyzing the simulated data and calculating power, especially when one wishes to construct graphs of power vs. effect size.
In this paper, we used the Mx program to analyze the simulated data because of its inbuilt power calculation function. Another useful option of Mx in this context is the possibility to output individual likelihood statistics for each raw data group. This information can be used to identify the groups that contribute most to the power to detect the effects of interest. Of course, various other statistical software packages (e.g., QTDT, LISREL, MPlus, R) can also be used in combination with exact data simulation to obtain the non-centrality parameters required for power calculations.
We emphasize that the power results obtained through exact data simulation are exactly similar to power results obtained through the analysis of summary statistics, and, just like power calculation using summary statistics, asymptotically similar to results obtained through Monte Carlo simulation (depending on the number of runs used in Monte Carlo). Differences between those two customary method of power calculation and exact data simulation only occur when subgroups have very low probabilities and the simulated overall sample size is not large enough to include all possible groups to sufficient extent; these small groups may then not be represented in the exact power simulation, while they may be (more or less) represented in other methods. However, as stated previously, the ensuing differences with respect to the power results, are very small as such small groups hardly contribute to the power anyway. Even so, to avoid the exclusion of small groups, one should choose a sufficiently large overall sample size in exact simulation, such that all groups are represented. This is perfectly doable, and does not alter the practicability of the method as it still involves analyzing a single (yet larger) dataset. The non-centrality parameter obtained in the analysis of the large simulated data set can subsequently be used to calculate the power for smaller, more realistic sample sizes. Alternatively, one may decide to accept the absence of certain groups, and the implied slight underestimation of power. Happily, the discrepancy between the intended N and the realized N is simple to calculate (as demonstrated in the R script available in the Mx scripts library), so that one can readily obtain an impression of the implications of this decision.
Throughout the paper, we have assumed that the data, conditional on group, are normally distributed, so that sufficient statistics are in principle available. With respect to situations that preclude sufficient statistics, the present method may still have some use. For instance, a continuous moderator in a G × E model, as discussed by Purcell (2002), might be approximated by a 5 point or 7 point Likert scale, which would render exact simulation possible in principle (see illustration 2).
Finally we note that the extension of this method to discrete data would obviously be very useful, and does seem feasible.