Models of body weight and fatness regulation

Abstract

Body weight and fatness appear to be regulated phenomena. Several different theoretical models are available to capture the essence of this idea. These include the set-point, dynamic equilibrium, adiposity force, control theory-settling point, Hall–Guo, operation point and dual intervention point (DIP) models. The set-point model posits a single reference point around which levels of fat are regulated. The dynamic equilibrium model suggests that the apparent regulation of body fat around a reference point is an illusion owing to the necessary impacts of weight change on energy expenditure. Control theory focuses on the importance of feedback gain and suggests set-point and dynamic equilibrium are ends of a continuum of feedback gain. Control theory models have also been called ‘settling point’ models. The Hall–Guo, operation point and DIP models also bring together the set-point and dynamic equilibrium ideas into a single framework. The DIP proposes a zone of indifference where dynamic equilibrium ‘regulation’ predominates, bounded by upper and lower intervention points beyond which physiological mechanisms are activated. The drifty gene hypothesis is an idea explaining where this individual variation in the upper intervention point might come from. We conclude that further experiments to test between the models are sorely required.

This article is part of a discussion meeting issue ‘Causes of obesity: theories, conjectures and evidence (Part II)’.

1. Introduction—why do we have body fat?

Lipid storage is highly conserved across the tree of animal life. All animals, from simple prokaryotes to endothermic mammals and birds, have stored body lipids [1]. Only a small group of prokaryotes that live in nutrient dense environments lack the ability to accumulate intracellular lipids [2]. In metazoans, lipids are generally stored in specialized cells (adipocytes). These can be located in different areas of the body. The nematode worm Caenorhabditis elegans, for example, stores lipids mostly in the intestine [3]. Many arthropods, especially insects, have a dedicated ‘fat body’ in the abdomen [4]. Most vertebrates also have dedicated tissues for fat storage (adipose tissue) but in many ectotherms the liver is an additional a major storage organ for lipids [1], and in these animals the adipose tissue depots maybe relatively small. Adipose tissue only consists about 50–60% adipocytes by number [5,6] reflecting its other roles as an endocrine and immune organ [7,8]. Animals store fat in part because energy intake is discontinuous, while energy expenditure is always happening. Most vertebrate animals cannot feed continuously because they have been selected by evolution to do lots of other vital things for survival and reproduction, like attracting a mate, mating, raising their offspring, drinking and sleeping. Systems evolved, therefore, to enable animals to harvest food from the environment at a much faster rate than it is used, to give time for these other essential activities. However, this temporal discrepancy creates the need for a mechanism to store the ingested energy between meals. For terrestrial animals fat is good for this job because it has a high energy density, and can be stored without water, and so it is not heavy to carry around. Yet, if animals only stored enough fat to get them from meal to meal they would be susceptible to running out of energy, if for some reason the food supply failed. Hence, stored fat is also a hedge against food shortage, consistent with the absence of lipid storage in prokaryotes that live in nutrient dense environments.

An important question is how much should be stored to protect against the possibility of food supply failure? Clearly, the more fat an individual stores the longer it could survive when the food supply fails [9]. Hence, if that was all there was to it then animals should in theory store as much as they possibly can limited only by the physical constraints of their bodies. No animals do that, for several reasons. First, there is a diminishing return for the investment once a certain level has been stored, because the probability of a starvation event is inversely related to its duration. In other words, there is no point storing enough fat to last 10 years, if the longest starvation event is only going to be a week [9]. Second, evolution has no foresight. It can only work on past events. So, if starvation periods historically only lasted a few weeks there would be no selective pressure to store more than sufficient fat to last that long. Individuals storing enough for eight weeks would have no selective advantage over those storing enough for four weeks, even if theoretically storing enough for six weeks enables longer survival. Even a lean person of normal weight carries enough fat to survive more than a month [10,11]. In other words, the force to deposit fat to survive periods of energy shortfall probably explains why our ‘normal’ body mass index (BMI) is 20–25 kg m⁻², and not 15–18 kg m⁻². It does not explain why some people have BMIs greater than 30 kg m⁻². There is no pressure to store more than necessary to survive common events because storing fat also has some negative downsides [12]. It costs energy to find the food to make the fat, excess stored lipid costs more energy to move around, so fatter individuals have to eat more to remain fat, it makes animals slower and less manoeuvrable and it reduces opportunities to hide in small places. These three factors may have direct impacts on survival because agility and hiding in small places may be critical for escaping attempted predation events [13–15]. At higher levels it may reduce fecundity levels [16]. Hence storing the maximum physically possible amount of fat given the body size and shape does not maximize overall fitness. The level of fat animals should carry is, therefore, a compromise between having enough to get through periodic food shortages, but not so much that it greatly increases the risk of predation [8,12,17–19] (figure 1).

Figure 1.

The contrasting impacts of changes in stored fat in a hypothetical small mammal on the mortality impacts owing to starvation during food shortage and predation. The starvation risk (black dotted) follows a negative exponential while the predation risk (red dotted) follows a positive exponential. The combined mortality (solid black) has a minimum level at 5 g body fat indicating an optimum level of fatness minimizing mortality. From Speakman [12].

These arguments have been largely developed with terrestrial animals and humans in mind. For aquatic animals the balance of selective forces may be rather different, because fat may provide additional advantages for animals living in water. First, fat is positively buoyant, being 90% less dense than water, while muscle is 5% more dense and bone 85% more dense than water. Storing fat may, therefore, contribute to making aquatic animals neutrally buoyant. This is shown by the fact fish which store more fat reduce the size of their swim bladders to attain neutral buoyancy [20]. Storing fat and achieving neutral buoyancy may provide swimming efficiency savings for diving animals [21]. Second, for aquatic endotherms fat provides an insulation layer against heat loss [22] which is more important in water than in air because water has 23 times the thermal conductivity of air and, therefore, promotes considerable heat loss. Most aquatic mammals have extensive subcutaneous fat stores [23] to provide such insulation, rather than as a hedge against starvation risk.

If the trade-off idea in terrestrial animals is correct the level of storage should be sensitive to the level of energy demands. This is because if there is a given risk of food supply failure individuals with greater demands will run out of stored fat faster than individuals with lower demands [24]. This has been suggested to potentially be the reason female animals (including humans) store more body fat than males [24–26]. The argument is that reproduction by females is very high cost compared to reproduction in males and hence they may need more fat to get them through periods when food supply fails while reproducing. This is not consistent, however, with observations that fat storage actually increases during the menopause [27,28]. Moreover, male reproduction while not involving direct investment into offspring, may be very costly in terms of energy and time spent on territoriality and aggression [29]. In the wild many male rodents deposit large amounts of fat in spring to support their reproductive efforts in summer (e.g. voles [30]). Fat also often serves as a reservoir that allows female animals to increase energy delivery to their offspring above that available from the environment during any particular reproductive event, thereby maximizing reproductive output. Such animals which support reproduction predominantly from fat stores are known as capital breeders [31,32]. The most spectacular of which are probably hooded seals (Cytophora cristata) where the females deposit over 3 kg of high fat milk containing about 79.5 MJ of energy per day into their pups, over a period of just 4 days [33]. Hence, fat is also stored in anticipation of stochastic unpredictable failure of food supply, but also in anticipation of periods where future demands will certainly exceed future supply. Typical examples, in addition to reproduction, are fattening prior to hibernation and migration [34–37].

Ecosystem wide food shortages must be quite rare events, but individuals may face food shortages for other reasons such as infection with pathogens [12]. Hence, there does not need to be periodic failure in the food supply that affects all individuals in a given population, for this trade-off argument for fat storage levels to hold water. When infected, individuals often stop feeding (infection induced anorexia) for a while to ride out the infection [38]. Individuals storing more fat survive longer [39–41]. Infection induced anorexia probably evolved because it would be disadvantageous to go and forage in this condition because the animals might become prey themselves if they are conspicuously sick, and their foraging efficiency might be so reduced they might not recover the energy they are spending as intake, and might hence run out of reserves faster [12]. Moreover, it may be an integral active part of host defence against the pathogen [42–44]. For example, in a study of infected mice individuals that were force fed to the same level as uninfected controls had greater mortality and shortened survival times [44]. Infection induced anorexia is something most people have experienced. You get infected and all you want to do is curl into a ball until it is all over. If you did not have stored body fat, you could not do that. The level of fat storage in many small animals seems to be flexible in the face of changes in the risk of predation [45–50]. Studies changing the risk of starvation have been successful in birds [51] but have proved less successful in small mammals (e.g. [52]). However, this may be because the starvation is imposed directly, when what animals might instead only be sensitive to is the changing risk of infection. Overall then, animals including humans probably evolved to store fat as a trade-off between these opposing selective forces. Because of similarities between animals and humans in the fact they store fat, and they probably evolved under similar selection pressures regarding levels of their fat stores, a goal for many decades has been to derive theoretical models that can be used to understand fat storage and thereby understand the origins of the obesity epidemic.

2. Theoretical models for body fat regulation

Theoretical models provide an intellectual framework for integrating existing data, help delineate contrasting interpretations of data, and suggest the design of new studies to critically test their veracity and discriminate between competing models. Below, we describe several competing theoretical models for body fat regulation.

(a) . The set-point model

The model in figure 1 has a point in the middle where the benefits of storing more fat as a hedge against periodic infection or other cause of food shortage, are exactly balanced by the disadvantages in terms of predation risk. One could imagine then that a regulation system might evolve with this point as its central feature. When body fatness rises above the point then countervailing changes in intake and expenditure would be enabled reducing body fatness, while decreases in fatness below the point would enable the opposite responses. Because there is a point around which regulation happens with this model it is known as the ‘set-point’ model [25,26,53]. Kennedy [54] called this system the ‘lipostat’, some later studies called it the ‘ponderostat’ [55] or ‘adipostat’ [56].

A starting point for many discussions regarding support for the set-point ‘lipostatic’ regulation of body weight and body fatness in humans is the overall apparent stability of human energy storage in the face of large fluxes in the levels of intake and expenditure. This stability then is taken to imply an ability to match intake and expenditure extremely precisely, and the existence of such a system by extension, implies very precise regulation of weight/fatness. For example, over the course of a year an individual adult human might consume about 5000 MJ of energy [57]. Yet, the change in body weight over the same period might on average be only 0.3–0.5 kg [58–61]—which, assuming it is all fat, would be approximately 20 MJ (or about 0.4% of the intake) [62]. If the accumulated weight is not all fat then the amount of energy stored is considerably lower. For instance, if fat comprised 50% of the weight change and the balance is lean tissue comprising 73% water, then the 0.5 kg of weight change would be only about 11 MJ. The suggestion then is that we must have a sophisticated system regulating the level of stored body fat that balances our daily intake to expenditure with a discrepancy of only about 55 kJ d⁻¹—around 13 kcal d⁻¹. If the weight is a mix of lean and fat tissue as suggested above then the discrepancy might be only 7 kcal d⁻¹—less than a small square of chocolate. It is often suggested that this provides prima facie evidence for the existence of a set-point regulation system for our body weight/fatness.

These lines of reasoning, however, have three major flaws. The first, is that the stability can be generated by models other than a set-point system [62,63], most notably passive feedback owing to the weight dependency of energy expenditure (see below). Second, it is implied that the discrepancy between intake and expenditure accumulates slowly on a day-to-day basis and the hypothetical person slowly gains just 1–1.5 g d⁻¹ until at the end of the year when the 0.3–0.5 kg has been accreted. This pattern, however, does not accord with common experience or actual data. Over the course of a year people may gain and lose enormous amounts of weight. During the holiday season weight gain averages 0.4–0.9 kg [64]. However, the highest 5% weight gainers increase by over 3 kg in a period of just a few weeks [65–67]. After this gain, individuals may be in negative energy balance and losing weight for much of the remainder of the year [68]. Moreover, many individuals show a weekly cycle in body weight, reflecting the change in food intake between weekends and weekdays [67–71]. Average accumulation over the weekend (Friday morning to Monday morning) is about the same as the average over a whole year! A third issue is that the average 0.5 kg weight change is exactly that—a population average. Behind that average, however, lie enormous individual differences. For example, a study of 192 individuals who had previously had obesity and were followed for 6 years, the average weight change over 6 years was 0.5 kg yr⁻¹. But the standard deviation was 2.6 kg yr⁻¹ hence the top 5% of individuals were gaining about 5.7 kg yr⁻¹ and the bottom 5% losing 4.7 kg yr⁻¹. For 569 individuals that had never had obesity the average change was 0.33 kg yr⁻¹, but the standard deviation was 1.5 kg yr⁻¹. Indicating the top 5% gained 3.3 kg yr⁻¹ while the lowest 5% were losing 2.7 kg yr⁻¹ [59]. Hence, while the population average is impressively stable, it does not provide any evidence for a set-point at the individual level, which is impressively variable.

There are, however, much stronger lines of evidence supporting the existence of a body fatness set-point, particularly from controlled animal studies. Many animal studies have been performed where individuals are calorie restricted leading to a depletion in the size of their fat stores. The level of store is, therefore, moved away from the supposed set-point. The set-point model predicts that on release from this restriction the animals should show a period of hyperphagia and/or hypometabolism. Many studies have demonstrated such an effect. One such study in mice [72] is shown in figure 2, where the level of restriction is at two levels, and the generated hyperphagia is greater when the restriction and fat depletion is greater. Such studies have also been performed in normal weight humans, with similar outcomes [73–76], although such responses are not always observed. Compared with underfeeding studies, manipulating normal weight individuals in the opposite way by overfeeding them to generate elevated fat stores has been performed much more frequently [76–81]. The main focus of such studies, however, has been the reaction to the overfeeding rather than following individuals after overfeeding stopped to evaluate hypophagic responses [82]. The few studies that have been performed indicate that hypophagia may be observed in young men [83] but is absent in older men [84]. Unfortunately, you cannot just tell an animal to eat more in the same way as a human. However, such manipulations have also been performed in animals by feeding them via a gastric tube rather than allowing them to free-feed [85–89]. In all these studies the rodents show the anticipated period of hypophagia when released from the period of overfeeding, consistent with the set-point model. However, the hypophagia is not always sufficient to return the level of fatness to that prior to the intervention, as a set-point model might predict [89].

Figure 2.

Post-restriction hyperphagia in mice following 25 days of calorie restriction at two different levels (50% and 65%). Mice at 50% restriction were given half the food of controls and those at 65% restriction were given 65% of controls. Hyperphagia when the mice were returned to ad libitum feeding lasted for several days relative to the pre-restriction and control intake and was greater after greater restriction. Redrawn from Hambly et al. [72].

In addition to monitoring what happens after under or overfeeding work has also been undertaken to explore responses during the period of manipulation. If there is a set-point one would anticipate not only a response after release from the manipulation but also while it is happening. For example, including compensatory increases in metabolic rate to counteract overfeeding and decrease in metabolic rate to counteract underfeeding. Very many studies have explored such effects in both rodents and humans with the changes in metabolism following the expectation [76]. These responses will be discussed in more depth below.

Finally, there has been considerable work to try and elucidate the molecular basis of the set-point system. Initial ideas in this respect were focused on steroid hormone signalling [44,90] and lipid extracts [91], but the discovery of leptin [91,92] which is produced in adipocytes in circulates in rough proportion to the level of body fatness provided a candidate for the signal by which the fatness of the body is communicated to the brain and compared to the set-point [93–95]. Interestingly, those vertebrate animals that store fat mostly in the liver also produce leptin in the liver in proportion to its fat content [96–98] supporting the idea that its role is to signal the level of the fat stores to the brain. There is much evidence that lowered leptin leads to increased food intake. This includes loss of function mutations in both mice and humans in both leptin and its receptor [99,100]. Such animals and individuals have voracious appetites and deposit enormous fat stores, which can be reversed by treating with leptin. The signalling pathways in the brain whereby lowered leptin (or defective reception of the leptin signal) exerts its effects have been partially resolved [101–103]. In brief, leptin receptors are found on several neuronal populations in the hypothalamus, in particular cells expressing Agouti-related protein (AgRP), neuropeptide Y, cells expressing pro-opiomelanocortin (POMC), and cocaine and amphetamine regulated transcript. Low leptin stimulates AgRP neurons and inhibits POMC neurons. These two neuron populations project to a third type of cell in the paraventricular nucleus expressing Melanocortin receptor 4 (MC4R) where the AgRP expressing neurons secrete AgRP which is an antagonist of this receptor and POMC neurons secrete alphaMSH which is an agonist. Thus lowered leptin inhibits MC4R leading to downstream elevation of food intake.

Surprisingly, however, elevating leptin in individuals without loss of function mutations did not provoke a similar response [104,105]. These observations suggest that the role of leptin in the set-point model is more complex than simply signalling the level of fatness and to accommodate these observations a new condition was invented called ‘leptin resistance’ [106] to capture the idea that both animals and humans are resistant to the effects of leptin once it is above the level defined by the set-point. In this situation many other pathways are involved in the regulation of food intake, largely independent of the effects of leptin. It should be noted that the molecular basis of the set-point itself has never been resolved, but from brain lesioning studies it was suggested that it resides in the lateral hypothalamus [19,107,108].

The main argument against the set-point idea is the observation that apart from periods of food restriction that lead to compensatory hyperphagia, regulation of body weight is not very good. Our food intake and body weight/fatness seem remarkably malleable and responsive to our environments and social situations. The fact we over-eat enormously at certain times of year with little opposition, and often follow a weekly cycle of weekend indulgence followed by weekday periods of austerity [56–60] does not indicate any strong level of control over energy intake. These observations, particularly at the weekly level, might be because the timescale over which regulation happens is quite long. Moreover, the fact that we have an obesity epidemic, clearly driven by environmental changes stimulating our intake [109–111] and reducing our expenditure [112] seems incompatible with the idea of a set-point system regulating our body weight. This could be explained if for some reason the set-points in different individuals have expanded upwards during the last 70 years or so. However, given we know nothing of the molecular basis of the set-point, the reason why that might have happened is unclear.

There are two other issues related to how such a set-point system evolves that lead to questions about its existence. The first is that if we assume the set-point evolved owing to natural selection because of to predation and starvation risk, and we then represent these effects by increasing and decreasing curves of mortality risk versus stored fat, it turns out that there is a range of fatness levels where the positive effects on starvation risk get balanced out by the negative effects of predation risk. In most realistic scenarios for the effects of starvation and predation in relation to fatness there is a broad region where there is actually no selective pressure [12]. It is only when the impact of fat storage on predation and starvation risk become enormous that there is a narrow enough range to lead to evolution of a set-point. The second aspect is that the relations of starvation and predation risk to stored fat probably vary enormously in both time and space, so the ‘optimum’ at any one place and time is very likely to be different from the optimum elsewhere at a different time. Thus, selection on a single set-point would be highly unlikely because such a point would almost certainly be in the wrong place in most situations [113].

(b) . The dynamic equilibrium model

A defining aspect of the set-point model, as presented above, is that body weight is regulated around a set-point via active control of energy intake and expenditure. The dynamic equilibrium model [114] suggests that the regulation of body weight is just an illusion, and there is no active control of intake at all. This is because changes in body weight in either direction lead to changes in energy expenditure that will to an extent resist the change. Consequently, while it is often stated that you only need to eat say 50 kcal d⁻¹ more than your needs, to develop massive obesity over 25 years [60] in reality if you did eat 50 kcal more than your current requirements you would only gain 2 kg because that additional weight would then increase expenditure 50 kcal and you would be back in energy balance. Similarly, that is why you cannot cure a massive obesity problem by just eating 50 kcal less per day. The reality then is that if you eat any level of food intake your weight will change until your increased expenditure matches the increased intake. If these conditions remained constant your body weight would eventually reach a dynamic equilibrium at a new level, although the time to reach equilibrium may actually be several years. We go through periods of accumulation when we are in positive energy balance—the holidays and weekends, and periods of depletion during the rest of the year. Another interesting example is children, whose BMI z-scores increase during the summer holidays but remain stable during the rest of the year [115]. During the periods of average accumulation, some people gain much more than others. Some lose weight at the same time. Factors driving obesity are largely independent of biology and include social factors, poverty and education [116–118] and the built environment [119,120]. It is like we are all just aimlessly drifting around nudged this way and that by a string of environmental factors, with a slow upward trajectory that is the obesity epidemic. This pattern is also strong evidence against different people having different set-points with some having higher points than others. This is because, like the mouse after a period of restriction where its actual fat is below its set-point, individuals in the same state should experience profound hyperphagia driving them up to their individual set-points. That generally is not happening. Fat is increasing slowly and sporadically and people find a balance at the point where their expenditure balances their intake, depending on their own particular social conditions; and that dynamic equilibrium can change radically in either direction over time.

This slow slide upwards is probably stimulated mostly by greater food intake and the food environment [109–111]. There has been debate about whether it has been contributed to by reduced physical activity. Data on this are equivocal. Worktime physical activity appears to have declined [121] but leisure time physical activity has increased [122]. We know people with obesity are less active [123,124], but a key question is whether this reduced activity leads them to expend less energy. That is far from obvious because the cost of moving around is greater at higher body weights. Measures of total daily energy expenditure dating back to the late 1980s show that indeed we are expending about 5–7% less energy now than we were 35 years ago [112]. However, surprisingly this seems to be owing to reduced basal metabolism rather than reduced expenditure on activity. The causes of this decrease are currently unknown.

The dynamic equilibrium model is unable to explain the hyper- and hypophagia that follow from under- and overfeeding experiments. Moreover, in practice the time taken to reach equilibrium may mean that individuals are never in stable conditions long enough to actually be at an equilibrium point. It is important to correct here an error in the literature pertaining to the set-point and dynamic equilibrium models. In some previous papers by the present authors the dynamic equilibrium model was incorrectly called the ‘settling point’ model and was characterized by the lack of active control of food intake in response to alterations in body weight or fat [9,12,62,125]. However, the original description of a settling point model, and use of that term, was by Wirtshafter & Davis [126], and this was a model that included active control of food intake.

3. Reconciling the set-point and dynamic equilibrium models

Given the problems with the set-point and dynamic equilibrium models there have been attempts to try and reconcile them.

(a) . Control theory-settling point

The classical set-point model is a negative feedback system as defined by the following equations. That is there is a reference set-point (SP). As the controlled or process variable (PV) departs from that value, features are activated that aim to return the regulated variable to the set-point. The regulation is active. The intensity of the control processes is presumed proportional to the deviation from the set-point and possibly the accumulated deviations over some duration. If we define an error term e(t) varying in time as the difference between SP and the current value of the PV related to body fat:

$$\mathrm{i}.\mathrm{e}.\hspace{0.17em}e(t)=\mathrm{SP}\text{–}\mathrm{PV},$$

then the behaviour of the system is defined by the level of feedback control u(t) also varying in time on the PV:

$$\mathrm{where}\hspace{0.17em}u(t)\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}{K}_1\cdot e(t)\hspace{0.17em}+K_2/T_i\hspace{0.17em}\hspace{0.17em}{\int }^t\hspace{0.17em}e(t),$$

where K₁ and K₂ are the gains of the control feedback loops and T_i is the integral time constant [127–130].

Negative feedback has been known since the 1800s and the mathematics concerning such systems is known as ‘control theory’ (reviewed in [128,129]). The key aspect of such systems is the coupling of the feedback signal to the system under control defined by the magnitude of the ‘gain’ parameters (K₁ and K₂) and the mathematical properties of the feedback signal. This is important because if the feedback gain is relatively weak, and there is no integral control term in the feedback signal, then it is possible for external perturbations of the system to result in a steady state that is quite distant from the actual set-point. These different steady state conditions have been called ‘settling points’ [126]. Hence, control theory permits multiple steady states depending on variations in both feedback gain and the strength of external perturbations even though there is only one defined set-point. In the limiting condition where the feedback gain parameters are zero then the system shows no regulatory feedback and hence behaves as the dynamic equilibrium point model does, while with strong feedback the set-point dominates the system's behaviour, especially when an integral control term amplifies the feedback signal based on the duration that the system has been perturbed away from the set-point. Thus, rather than two conflicting models, control theory suggests these are just ends of a spectrum that is defined by the feedback gains (K₁ and K₂). In humans, it appears that feedback gain parameter values of K₁ ∼ 100 kcal kg⁻¹ d⁻¹ and K₂ = 0 kcal kg⁻¹ d⁻² are most consistent with the data from people in response to drugs that increase urinary glucose excretion [131].

(b) . The Hall–Guo ‘set-point’ model

Hall & Guo [125] formulated a new way of looking at the ‘set-point’ model. They started with the observation that energy expenditure increases in approximate proportion to body weight—the expenditure line. In situations where an individual is in energy balance, intake will by definition follow the same line (figure 3a). Individuals at a given body weight then exist at different positions along this line. They then define a second line (called here the intake response line) which captures the intake responses when weight is perturbed away from steady state. In this situation, the intake response line is a negative function of the deviation of body weight from weight at the intersection of the two lines. This intersection is, therefore, an equilibrium point that Hall and Guo called the ‘set-point’. Reactions to over or underfeeding interventions are captured by movement of the ‘intake response line’ which results in stability at a new intersection point of the two lines. The model predicts reactive hyper- and hypophagic responses to perturbation of intake in the same way that the set-point model does. However, the advantage of visualizing the model in this way is that it is possible to reconcile the set-point and dynamic equilibrium models by envisaging that the ‘set-point’ is not fixed but able to move up and down the expenditure line. That is the intake response line provides the ‘set-point’ nature of the model, while the moving position of intersection on the expenditure line, provides the dynamic equilibrium aspects. As the authors note, the impact of factors such as the food environment and other non-homeostatic factors can be incorporated into this model by altering the position or slope of the ‘intake response line’, which alters the intersection point on the expenditure line. The mechanisms linking these environmental factors to the changes in the intake response line are not fully known, but recent work in mouse models has shown that different foods have strong influences on the brain circuits controlling energy intake [133–138].

Figure 3.

Two different models for regulation of body weight. (a) The Hall–Guo model [125] and (b) the operating point model [132]. In the Hall–Guo model expenditure or intake is plotted as a function of body weight. There is a positive relationship between expenditure and weight (expenditure line: orange line). The intake response line (blue) shows the opposite trend. Where they intersect (black dot) defines the set-point of the system (orange dotted line). In the operating point model there is an increasing level of fat storage at any given weight called the ‘diet line’ (red line). There is also a negative line crossing this called the appetite line. Where they intersect (black dot) is called the operating point (dotted orange line).

(c) . The ‘operating point’ model

Bar et al. [132] presented a model that is very similar to the Hall–Guo model [125] that they called the ‘operating point’ model specifically to distinguish it from the set-point idea (figure 3b). The model is designed primarily to explain animal studies where rodents are under and over fed in a controlled way, and then their reaction is reported after these levels of intervention. The formulation of the model indeed appears different to the set-point shown in figure 1, but is similar to the set-point model as envisaged by Hall & Guo [125] (figure 3a). They first suggest that there is a positive linear relationship between the level of fat storage at a given level of ‘controlled food intake’. They call this the ‘diet line’. This is the line pertaining to the period when the animals are given a fixed level of food that they themselves cannot control. In rodent experiments this might be performed for example by feeding the animals via a tube into their stomachs and providing them with no other food source. They then define a second line called the appetite line which they state is the intake when the animal is fed ad libitum and can choose what it eats. It is important to note that this line pertains to the post intervention feeding period. Normally, one would anticipate that under free feeding there would be a positive relationship between intake and body fat. However, the line in the Bar model has a negative relationship, similar to the ‘intake response line’ in [125]. They draw this as a curve rather than a linear relationship as given to the diet line based on some empirical data from rodent studies. The intersection of the diet line and the appetite line is called the ‘operating point’. It is assumed that leptin and leptin resistance provide the necessary molecular mechanisms that underpin the model (although also stated other factors may be important).

Although the manner in which the lines are formulated appear different to the set-point model (figure 1) and the ‘Hall–Guo’ model, the operating point model is indeed a set-point model. Reactions to deviations from the set-point during over or underfeeding are captured by the appetite line which always returns the body weight to the operating point. Food intake on the appetite line is driven by the level of fatness relative to the fatness at the operating point. The model predicts reactive hyper and hypophagic responses following release from over- and underfeeding in the same way that the set-point model does.

Like the ‘Hall–Guo’ model the advantage of visualizing the model in this way is that it is possible to reconcile the set-point and dynamic equilibrium models by envisaging that the operating point is not fixed but able to slide up and down the diet line. That is, the appetite line provides the ‘set-point’ nature of the model, while the diet line, provides the dynamic equilibrium aspects, because the operating point is sliding up and down the diet line. The authors suggest that the main factor pushing up the position of the operating point is leptin resistance. This might then also explain why individuals differ in their levels of body fatness—individual variation in leptin resistance leading to shifts in the operating point along the diet line. Where these individual differences in leptin resistance come from, or why they exist is not discussed in the operating point paper.

It should be noted that the concept of a sliding set-point, largely based on changes in leptin sensitivity and resistance, has been invoked previously to explain photoperiod induced changes in body fatness in photoperiod sensitive rodents, following the pioneering work of Mrosovsky & Fisher [139]. It has been observed in many small rodents that a switch in photoperiod leads to a profound change in body weight and fatness. This includes ground squirrels, lemmings, hamsters and voles [30,140–144]. The change involves a period of leptin insensitivity when weight is changing compared to the periods when weight is stable, caused by constitutive changes in the activity of suppressor of cytokine signalling 3 [145,146] suggesting that during the period of change there might be an abandonment of regulation. The direction of the change varies between species—some get fat under short photoperiods, while others get fat under long photoperiods. A critical experiment was performed showing the sliding set-point nature of the change. When hamsters are flipped from long to short days they lose weight. They can be forced to lose weight more rapidly by restricting their intake. If this is done, and then they are allowed to free-feed they actually increase their weight back up to the same level as unmanipulated hamsters responding normally to the photoperiod switch [147,148]. This clearly demonstrates the phase of weight loss is not owing to them flipping from one regulated level to a different one under the changed photoperiod, but owing to a sliding level of body weight regulation. Leptin infusion during recovery blunted the response suggesting that while sensitivity to leptin was reduced it still forms part of the sliding set-point mechanism. Photoperiod impacts on weight regulation have been recently reviewed [149]. Seasonal changes in body fat can also be explained using the control theory model.

(d) . The dual intervention point model

An alternative attempt to reconcile the set-point and dynamic equilibrium models is provided by the dual intervention point (DIP) model [9,12,150–152]. The DIP model suggests that there is no set-point but rather there are two intervention points (IPs) between which there is an unregulated zone, or zone of indifference. The lower intervention point (LIP) defines a weight below which active physiological feedback mechanisms kick into action to preserve or restore body weight, in much the same way as the lower side of the set-point. The upper intervention point (UIP) defines a body weight above which active feedback kicks in to limit weight gain and bring it back down to the zone of indifference. It is important to appreciate that these are suggested to be completely separate and independently regulated phenomena [10]. By this model the dynamic equilibrium type behaviour occurs between the two independent limits (figure 4a). This model differs from the ‘Hall–Guo’ model and ‘operation point’ models where the set-point is moving around, in suggesting there is no set-point at all within the zone of indifference.

Figure 4.

The dual intervention point (DIP) model. (a) The body weight trajectories of two individuals in time that differ in their upper intervention points (UIPs). The orange line for the individual with UIP1 and the blue line for the individual with UIP2. Between the lower intervention point (LIP) and UIPs the weights drift around at random (based on recent energy balance). However, when they hit the LIP or UIPs there are physiological responses (represented by the orange and blue arrows) that prevent further loss or gain. The two individuals with different UIPs reach different maximal body weights. (b) The Hall–Guo model replotted to be consistent with the DIP model. Instead of the intake response line crossing the expenditure line at a single point the upper and lower parts of the intake response diverge in different locations. Between these intake tracks expenditure and there is no regulation. The divergence points define the LIP and UIPs. (c) The operative point model replotted to be consistent with the DIP. Here, the appetite line is also split leading to a zone where there is no appetite line. The divergence points define the LIP and UIP, respectively.

This model resolves many issues in understanding the phenomena connected with body weight regulation. In particular, it can be used to understand how it is possible for weight to be ‘regulated’, while at the same time, there has also been an obesity epidemic. That is there have been changes in the environment that promote over consumption of energy, and or reductions in energy expenditure, which put people into positive energy balance and hence susceptibility to weight gain. Individuals gain weight, without any physiological resistance while they are in the zone of indifference. Individuals vary in their weight gain because they differ in the position of the UIP. This also explains why individuals have an unopposed weekly cycle of weight gain in response to changes in energy balance between weekdays and weekends, and also have a largely unopposed annual cycle of weight gain during the ‘holiday season’, and subsequent loss, which generally does not mirror the gains. Yet it also explains why individuals who are exposed to extreme weight loss interventions such as the Minnesota starvation experiment [61] show rebound hyperphagia after release from restriction—interpreted that they were reduced below their LIPs. Conversely, hypophagic reactions to overfeeding [70,71] might be considered as generated by pushing people above their UIPs. The model also explains why the response to leptin is asymmetrical without any need to invoke some mechanisms like ‘leptin resistance’ to explain it [9].

As pointed out by Geary [129] the control theory model can generate a similar pattern of response if there is a set-point in the middle of the supposed zone of indifference, and a nonlinear feedback gain surrounding this point (fig. 2 in [129]). Geary has suggested that the control theory model is preferable to the DIP model because of its parsimony and mathematical tractability because it is rooted in a long history of engineering applications. We disagree for two reasons. First, it implies similar biology at the upper and lower IPs because they are the product of a single control system. Engineering systems based on negative feedback generally have the same control mechanism on the upper and lower side of the regulation point. However, there is no reason why biological systems need to match that, nor any advantage to constraining biology to fit the engineering model. Indeed, the biology at the LIP and UIP is currently far from clear and what little evidence we have suggests the upper and lower points are probably controlled by very different systems [153–155]. Second it also implies a set-point at the middle of the zone of indifference (the 0 value in fig. 2 of [129]) for which there is no biological explanation, or biological reality. The ‘Hall–Guo’ and ‘operating point’ models can also be formulated to generate behaviour similar to the DIP model. For example, if the intake response line (Hall–Guo) or the appetite line (operation point) is assumed to diverge from the diet line in two separate places. Between these points the body weight is free to slide up and down without provoking any response, but once it hits the points where the appetite lines diverge then feedback kicks in to force weight up or down accordingly.

One question that arises is why individuals might vary in the location of their UIPs thereby generating individual variation in levels of obesity among individuals living in the modern ‘obesogenic environment’. One idea to explain this has been called the ‘drifty gene’ hypothesis [156]. The idea is that early in our evolutionary history when we were Australopithecines we probably had IPs that were quite close together and in consequence regulated our body weight quite tightly. This is because at this phase of our evolution we were prey for many large predators that were common at the time. About 2 Ma, however, we massively reduced the risk of predation because we invented tools, weapons and fire that we could use to defend ourselves, and also became more social enabling more sophisticated defence strategies. Once we virtually eliminated predation there was no longer a selective pressure sustaining the UIP and probably over time random mutations in genes defining this point would change its position, without selective consequences [152,156]. No other explanation has been posited to explain for example why the control theory-settling point model, ‘Hall–Guo’ model or operational model versions that mimic DIP behaviour may have evolved.

4. Problems with the dual intervention point/drifty gene idea

(a) . Why did we only get an obesity epidemic recently?

The obesity epidemic is a recent phenomenon. Although there was an indication that levels of adiposity rose during the1930s [157] it is only since the 1960s that there has been a widespread increase in levels of stored fat. This increase has now spread to almost all countries across the world leading the epidemic to be described as a pandemic [158]. The drifty gene hypothesis posits that from about 2 Ma the genes that define the UIP started to accumulate random mutations which under an absence of selection were not purged but drifted. If that process took 2 Myr to happen then it would have been 99% complete by 20 000 years ago. The genetic architecture of pre-agricultural revolution individuals would have been virtually identical to that of modern humans. So if there were scant controls on the upper limit of fat accumulation then why did we only start to get a pandemic recently?

The most often used argument for this situation is that food supplies were historically limited. Hence people may have had elevated UIPs but there was just insufficient food around for people to gain weight and rise up to these levels (e.g. [159]). There are some data consistent with the idea that food supply is a permissive factor. Early in the development of obesity in different countries it is actually the most wealthy individuals who show elevated fat storage first [160]. That might suggest that increasing weight was only feasible for those individuals who had access to sufficient food to make it possible to become obese. As nations develop, wealth spreads and food supplies become more accessible and widespread [93,94] then more and more people have access to sufficient food to gain weight. At that stage the link between obesity prevalence and wealth reverses as the issue potentially starts to become the quality of available food rather than its quantity.

This argument, however, raises some issues. The first is that if food supply has historically always been insufficient to allow individuals to gain large amounts of body weight, then why would we need any sort of mechanisms to evolve to regulate it? It is hard to construct how much food was available to ancestral hominoid populations, particularly for Australopithecenes, which is presumably when the twin controls on body weight evolved. We do know, however, that these populations were very heavily predated by a large community of large carnivores [152,156]. So even if food was not massively abundant there was presumably enough around and the predation risk so large it was sufficient for an upper limiting control to evolve.

One scenario is that in the early phase of our history 4–2 Ma there was an enormous risk of predation and presumably a sufficient amount of food that there was a risk of weight gain sufficient to cause a predation selection pressure [156] A system, therefore, evolved, or was at least retained from pre-hominid ancestors, that involved UIPs and LIPs that were quite close together. Two million years ago, we massively reduced the predation risk, the UIP started to fall apart but the food supply was insufficient for us to develop obesity. That pertained until the agricultural revolution. Thereafter wealthy individuals had sufficient access to food that some of them (with high UIPs) developed obesity, and then in modern post world war II societies general wealth increased until everyone had sufficient access to food to enable obesity in those with high UIPs. Some societies are still in this transition.

There are two inconsistencies with this argument. First, populations of humans increased dramatically between 2 Ma and 200 000 years ago [161], this would suggest they were not food limited. Moreover, one phenomenon that is clear in the history of human movement across the globe since 40 000 years ago is that as they expanded within and then out of Africa, wherever humans went it was accompanied by a rapid decline in the populations of large herbivores [162–164]. Whether humans were causally responsible for these shifts is debated [165,166]. One idea is that climate change was a factor that drove megaherbivore population crashes and this also stimulated human migrations. Direct killing and extirpation, however, have also been suggested to be important. Indeed, there is an argument that humans hunted not only for food but for pleasure, or ritual, and that overkilling on slowly reproducing megaherbivores was unsustainable and drove populations relatively rapidly to extinction (e.g. [162,167,168]). If this was the case the massive overkilling would be incompatible with the argument that food was always in short supply. Moreover, this excess killing all happened in the interval between 12 000 and 40 000 years ago, when genetically humans were almost identical to us nowadays. Bendor and colleagues have argued that our ancestral diet was largely fat based [169]. Was there an obesity epidemic in the late Pleistocene as humans gorged on the fat from carcasses of megaherbivores? We assume not, but actually we do not know. However, we do know that there are figurines (so-called ‘Venus figurines’) dating back 14–38 000 years ago that depict individuals living with obesity, for example the Venus of Willendorf [170]. These have exaggerated sexual features and been interpreted as representing fertility deities [171]. This interpretation, however, seems dubious given the negative association between extreme obesity and fertility [16]. Another explanation is that the figurines represent survival symbols carved during a period of climate change reflecting advantages of fat storage to survive periods of food shortage [172]. This interpretation, however, is at odds with the complete lack of evidence that people storing more fat have greater survival chances during periods of food shortage, and the fact the people who do die in famines are mostly the very young and very old, not the lean [173]. Perhaps then these are just carvings of people (maybe self-carved [174]) with a phenotype that was not particularly rare at this time in our history. This raises the question, did people stop carving figurines when the megaherbivores all became extinct and the plentiful food supply ended, and hence the numbers of people living with obesity became much lower?

(b) . What about animals in zoos and pets getting fat while they have a very different selection background

The drifty gene explanation [156] for the diversity in the UIP is a uniquely human selection scenario. Other animals did not invent fire and weapons and hence did not release themselves from predation. Yet it has been observed that when we bring animals into captivity, in zoos and as pets, then at least some of them become obese. We know that in some cases obesity is driven by recent mutations that have been introduced during the breeding and selection process. Mutations in the MC4R gene, for example, make beagles and other dogs susceptible to obesity [175,176]. Moreover, some Labrador dogs have a mutation in the POMC gene that makes them susceptible to over-eating and obesity [177]. This seems an exceptional situation though. One scenario is that those species that become obese in captivity are also those species that also have low predation risk in the wild. They may then have lost the upper regulatory point in a similar fashion to the suggestion it has drifted in humans. We do not for example observe that wild small rodents when brought into captivity develop obesity. Indeed when exposed to high fat diets they show remarkable resistance to body fat gain [178,179].

Pontzer [180] reviewed the fatness of captive primates relative to wild individuals and found enormous diversity in the patterns for both males and females. Most, however, became fatter than their counterparts in the wild. This was in part linked to diet with those having more fruit in the diet in the wild showing greater captive fatness. Being in captivity involves large changes in the type and quantity of food available. Moreover, although it is often suggested that wild and captive primates have similar levels of expenditure [181] there are no actual measures of energy expenditure in free-living primates on which to base this. It could then be a combination of altered diet and lowered expenditure that makes them more likely to go into positive energy balance. This still, however, raises the question of why they do not hit an UIP that stops them developing obesity if the DIP model is correct. Pontzer [180] concluded that humans are not exceptional compared to other primates in their propensity to gain weight when given high energy density foods in combination with an environment where the effort to get the food is minimal.

Another interpretation, however, is that the UIP is not rigidly fixed but sensitive to local predation risk conditions. Indeed, we assume such flexibility in experimental studies where the predation risk is manipulated and a body weight response predicted [48–51]. Primates may be able to perceive that in captivity their predation risks are very much reduced and the corresponding UIP may move upwards in response. By contrast, small rodents like mice and voles may not be able to make this distinction and do not, therefore, show the same responses. Perhaps a rodent exception is pet rats that seem able to form bonds with their owners and presumably do not, therefore, see them as predators. Correspondingly, rats in captivity can become extremely obese.

(c) . Resistance to weight loss when inside the zone of indifference—defending a new set-point?

A major problem with the DIP is that it suggests between the IPs the body weight and fatness should just adjust to the level of intake in the same way as a dynamic equilibrium model. That is, unlike a set-point model individuals should not show any corrective reaction to weight loss until they cross the LIP. Yet, when people with obesity (and hence unlikely to be near their LIP) go on calorie restricted diets, they show profound changes in their metabolism and behaviour that look like they are defending their adiposity. Specifically, they get hungry, when food is available they eat food to break their diet, and they show metabolic compensation—that is they reduce their energy expenditure. Are these observations reconcilable within the DIP model?

One idea for why individuals may show behaviours to defend a high attained level of adiposity is that as adiposity increases this causes a change in the position of the LIP. Specifically, the LIP gets dragged up towards the UIP [182]. This might explain the reactive responses to restriction because once the individual had lost only a small amount of weight they would then cross the elevated LIP. While this might provide a suitable mechanistic explanation for the immediate reactions to weight loss, it makes little evolutionary sense. If the role of the LIP is to provide a fat buffer that protects against periods of infection induced anorexia (or catastrophic failure of food supply) then why would it move up when food supply is sufficiently abundant to have led to fat accumulation across the zone of indifference between the LIP and UIP? If the LIP is at all phenotypically flexible we might expect it to move upwards if there is a perception of elevated starvation or infection risk, but to be independent of food supply/intake. A flexible LIP dragged up by rising adiposity does not seem a realistic explanation.

An alternative idea is that it is set-point like, but not actually defence of an attained level of adiposity. What appears to be ‘set-point behaviour’ defending an attained level of adiposity is actually something very different with a completely different goal. When put under energy restriction at any body weight/fatness the body has an immediate problem. All physiological processes require energy, all enzymatic reactions require energy, all muscle contractions like the beating heart and breathing require energy, all nervous responses require energy. When there is a shortfall in exogenous supply, we do not have enough energy coming in to meet these multiple demands. Consequently, there has to be an immediate response to the shortfall which aims to restore the balance between energy demand and supply. In this circumstance there are three (not mutually exclusive) available options: (i) cut back demand by lowering metabolic rate and activity levels; (ii) increase food intake—which may mean a paradoxical elevation in activity to go feeding; and (iii) withdraw stored energy from glycogen, fat and protein stores. Under energy restriction people get hungry, which is the brain encouraging them to find food to make up the deficit between what they are spending and what is coming in as food. They also withdraw energy from reserves and so lose weight and fat. Plus they show reactive changes in metabolism. The elevated intake and reduced metabolism might look like the body is defending the attained level of fatness—i.e. set-point behaviour. However, that is not necessarily the aim of what they are doing. They may be just trying to offset the energy imbalance, and this just looks like they are trying to defend the attained fatness, because the two responses look very similar.

How could we separate ‘trying to offset the energy imbalance’ from ‘attempting to preserve the level of body fat’—i.e. a set-point. One way to distinguish if this behaviour is defending a set-point, or offsetting energy imbalance, is to consider the situation when food is not restricted in supply. In this situation, individuals on diets must show cognitive restraint in their intake to comply with the prescribed diet. The individuals feel impetus to eat and this challenges their ability to maintain the lower level of intake. The drive to eat derives from the shortfall between intake and expenditure. This is clear because it occurs prior to any significant weight loss. A decline in leptin that is independent of the level of body fat is probably an important signal causing this drive to consume [183,184]. As fat loss occurs then there may be additional signals of reduced adiposity and these may potentially add to the drive to eat. The distinguishing point between offsetting the energy imbalance, and responding to an attained level of fatness is the level of intake displayed when people finally give in to their physiology. The imbalance model suggests that they would only elevate their intake sufficiently to come back into balance. That would be back to the level they started at, or somewhat lower if some reduction in metabolism had already occurred. The defence of an attained fatness, however, predicts there should be a period of reactive hyperphagia, where individuals consume more than their original intake, until the levels of body fat have been restored.

To evaluate whether individuals show reactive hyperphagia after a period of weight loss we used a validated mathematical model [60,131,185] to analyse the energy balance dynamics in response to treatment with semaglutide, a glucagon-like peptide 1 receptor agonist that suppresses appetite, and its withdrawal. In one study [186], patients with obesity lost weight for 17 months on semaglutide and then the drug was withdrawn. Figure 5a illustrates that weight regain ensued after removal of the drug and the simulated energy balance dynamics (figure 5b) indicate a substantial degree of reactive hyperphagia. This is consistent with the defence of a set-point. By contrast, another study [187] provided semaglutide to patients with obesity, for five months, but they were then randomized to either switch to placebo or continue semaglutide and followed for another 12 months. Figure 5c illustrates that the group who switched to placebo gradually gained weight, whereas the group continuing semaglutide lost additional weight. The simulated energy intake and expenditure dynamics (figure 5d) illustrate that the placebo group increased energy intake to baseline levels without showing hyperphagia. These data are more consistent with the DIP model. Although the group that showed hyperphagia had lost more weight than those who were switched to placebo it seems unlikely that the weight loss of the group that were treated for longer had reduced them to below their LIP thereby explaining their hyperphagia. The contrasting results of these two experiments do not support either the DIP or set-point interpretations, but point to important psychological aspects that may drive these phenomena that neither model accommodates easily.

Figure 5.

Modelled responses to treatment and withdrawal of semaglutide to evaluate if there is reactive hyperphagia following weight loss. (a) and (c) show the weight loss data from the original studies, while (b) and (d) show the modelled changes in energy expenditure and intake from these data. Reactive hyperphagia is predicted by the set-point model but it is not predicted by the dual intervention point model. The study in (a) and (b) [186] shows reactive hyperphagia while the study in (c) and (d) [187] does not. (Online version in colour.)

A second distinguishing feature between the two interpretations is whether the individuals on restriction go back up to their newly defended level of fatness, or whether they just come back to a new balance between intake and expenditure that may be at a lower level of fatness than they previously attained. The problem here is always one of time course. That is how rapidly would we expect individuals defending a set-point to return to their original weight and fatness. In the case of the placebo group in figure 5c,d, model simulations predict that restoration to within 1% of the original body weight would require approximately 4 years (not shown). Does that indicate there is a ‘set-point’ driving them back up, or is the required time such that we might argue there is no such drive?

Overall, it is often interpreted that what happens when people with obesity lose weight is that they show responses consistent with defending a set-point. However, separating that interpretation from them just attempting to make an energy balance in the face of an exogenous shortfall in energy supply is difficult. The behaviour of individuals when they come off restriction currently provides little strong evidence to support or refute either the DIP model or the set-point idea.

5. Conclusion

Set-point and dynamic equilibrium conceptualizations both share strengths and weaknesses when applied to understand the control of body weight. There has been a range of models that attempt to reconcile these problems. The different reconciling models—control theory-settling point, Hall–Guo, operative point and DIP models as described here actually all propose similar but subtly different solutions. No one model explains the totality of the available information—suggesting we maybe need new models and tests to explain what is happening. We focused in the latter part of the paper in particular on problems with the DIP model. A key observation that is inconsistent with this model is that when individuals with obesity go on calorie restricted intake diets to lose weight, they show reactions that suggest they are defending their original body weight. We suggest that what we interpret as ‘defence of the original body weight’ may in fact be the individuals enabling responses simply to make an immediate energy balance. The defence of the original weight is simply an interpretation we place onto the data because the set-point idea is so ingrained in our psyche. Testing between these interpretations may provide evidence to support or disprove the alternative ideas. Such experimental data are sorely needed.

Data accessibility

This article has no additional data.

Authors' contributions

J.R.S.: conceptualization, writing—original draft, writing—review and editing; K.D.H.: conceptualization, writing—original draft, writing—review and editing.

Both authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

We declare we have no competing interests.

Funding

J.R.S. was supported by the National Key R&D Program (grant no. 2018YFA0801000) and a PIFI professorial Fellowship.