Control-theory models of body-weight regulation and body-weight-regulatory appetite

Appetite 144 (2020) 104440

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Control-theory models of body-weight regulation and body-weightregulatory appetite

T

Nori Geary1,∗ Department of Psychiatry, Weill Medical College of Cornell University, New York, NY, USA

ABSTRACT

Human body weight (BW), or some variable related to it, is physiologically regulated. That is, negative feedback from changes in BW elicits compensatory influences on appetite, which may be called BW-regulatory appetite, and a component of energy expenditure (EE) called adaptive thermogenesis (AdEE). BW-regulatory appetite is of general significance because it appears to be related to a variety of aspects of human appetite beyond just energy intake. BW regulation, BW-regulatory appetite and AdEE are frequently discussed using concepts derived from control theory, which is the mathematical description of dynamic systems involving negative feedback. The aim of this review is to critically assess these discussions. Two general types of negative-feedback control have been invoked to describe BW regulation, set-point control and simple negative-feedback control, often called settling-point control in the BW literature. The distinguishing feature of set-point systems is the existence of an externally controlled target level of regulation, the set point. The performance of almost any negative-feedback regulatory system, however, can be modeled on the basis of feedback gain without including a set point. In both set-point and simple negative-feedback models of BW regulation, the precision of regulation is usually determined mainly by feedback gain, which refers to the transformations of feedback into compensatory changes in BW-regulatory appetite and AdEE. Stable BW most probably represents equilibria shaped by feedback gain and tonic open-loop challenges, especially obesogenic environments. Data indicate that simple negative-feedback control accurately models human BW regulation and that the set-point concept is superfluous unless its neuroendocrine representation is found in the brain. Additional research aimed at testing control-theory models in humans and non-human animals is warranted.

1. Aims and scope This review compares mathematical models of the compensatory changes in appetite and energy expenditure (EE) that arise in response to increases or decreases in body weight (BW). The review considers the capacity of two general models, simple negative-feedback control, also known as settling-point control, and set-point control, to accommodate the principal phenomena of BW regulation, especially phenomena related to the obesity pandemic. Appetite, defined here to include conscious and unconscious urges to eat and eating itself, is usually affected much more by preceding meals, environment and learning than by BW (Rogers & Brunstrom, 2016). Nevertheless, changes in BW often provoke compensatory changes in appetite. This component of appetite may be called BWregulatory appetite. It is not synonymous with the concept of homeostatic hunger because homeostatic hunger also includes changes in appetite in response to acute changes in the availability of glucose and other energy metabolites that occur without changes in BW (Levin, Magnan, Dunn-Meynell, & Le Foll, 2011; Ritter & Li, 2019). The BWregulatory system also causes changes in EE, which synergize with BWregulatory appetite to regulate BW. These changes in EE, called adaptive thermogenesis (AdEE), are introduced in §3.5. Research on BW-

regulatory appetite typically focuses only on metabolizable energy intake (EI). As discussed in §2, this is shortsighted. BW-regulatory appetite affects, and is affected by, many facets of human appetite. The models reviewed are negative-feedback control models. Control theory is a specific mathematical approach to the characterization of dynamic systems regulated by feedback (DiStefano, Stubberud, & Williams, 2012; Åström & Murray, 2016). Control theory and the two models reviewed are introduced in §3 and 4, and the models are compared in §5. Two further models are briefly described in §6. Because control-theory models are mathematical, critical discussion of them must include at least a sense of the mathematics. To prevent control-theory quantitative from reducing the accessibility of the review, its rudiments are presented separately, in text boxes. Control theory also has its own jargon, which is not always transparent. Therefore, key control theory terms are defined in Box 1. Mathematical control theory began with James Clerk Maxwell’s (1868) analyses of mechanical governors that limit the speed of machine components, such as the centrifugal governor for steam engines invented by James Watt. Control-theory concepts were introduced to physiology by Norbert Wiener, a pioneer in control theory, and Arturo Rosenblueth, a physiologist working with Walter B. Cannon, who earlier coined the term homeostasis (Cannon, 1929; Cooper, 2008;

Corresponding author. 61 McClellan Farm Road, Underhill, VT, 05489, USA. E-mail address: [email protected]. 1 (Retired) ∗

https://doi.org/10.1016/j.appet.2019.104440 Received 8 January 2019; Received in revised form 8 August 2019; Accepted 2 September 2019 Available online 05 September 2019 0195-6663/ © 2019 Elsevier Ltd. All rights reserved.

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Box 1 Definitions of key control-theory terms Active control or regulation – control or regulation that includes negative-feedback mechanisms that cause responses (in controlled variables) that defend the output or regulated variable. Closed loop – the organization of negative-feedback control; specifically, the linkage from the regulated variable, to feedback, to feedback gain, to the controlled variables, to their integration with inputs, and finally again to the regulated variable (Figs. 2A and 3 and 6A). Closed-loop variable – a variable that the regulatory system controls and that affects the regulated variable, such as EI or EE in the BWregulatory system. Controlled variable - a variable that is actively altered in response to feedback. Error signal – in a set-point control system, the output of the process that combines the setpoint and negative feedbacks (Fig. 6A). The error signal is an input to processes that produce feedback gain and change the controlled variables. Error signals are often transformed to yield better system performance (see PID control). Feedback - information related to the state of an output or regulated variable that is delivered to a system and that can cause a response that affects the state of the regulated variable. Feedback gain - the transformation of the feedback signal into an intermediate control signal effected by the feedback process; in a setpoint system, the intermediate control signal is the error signal (Fig. 6); feedback gain usually the most critical component in defining system performance. Negative feedback – information related to the system's output (i.e., to the regulated variable) that causes the system to react so as to oppose changes in the state of that variable. Forward gain - the transformation from the controlled variables and input into the output of the system. Passive regulation – a static regulatory system that does not include a negative-feedback mechanism that causes responses. PID control – proportional-integral-derivative control, referring to the commonest transformations of error signals in set-point control systems; set-point control systems may include any combination of P, I and D transformations (§4.2.1, Fig. 6B and Box 4). Positive feedback - feedback that causes the system to react so as to magnify the changes in the state of that variable that produce positive feedback. Precision – Several aspects of negative-feedback control can be used to index precision of regulation; this review uses the accuracy of steady-state output, as provisionally defined in §5.1. Open-loop variable – a variable in the system that is not subject to negative feedback; open-loop variables produce disturbances or challenges to a regulatory system. Regulated variable - a variable whose level is maintained to some degree, i.e., prevented from changing as much as it otherwise would, by the regulatory system. Set point – An external input to a set-point system that determines the desired level of the regulated variable; also known as the reference level or Sollwert (German for should-be value). See error signal. Settling point – Wirtshafter and Davis (1977) described a hypothetical simple negative-feedback control system for BW and defined the settling point as the predicted equilibrium or steady-state BW produced by this system. 2. BW-regulatory appetite in context

Goldstein, 2019; Rosenblueth, Wiener, & Bigelow, 1943; Wiener, 1948). The role of negative-feedback control in physiological regulation has been a matter of debate ever since (Booth, 2008; Goldstein, 2019; Goldstein & Kopin, 2017; Gray, Kogan, Marrocco, & McEwen, 2017; Peters & McEwen, 2015; Ramsay & Woods, 2016). This controversy is beyond the scope of the present review. By the 1970s several control-theory models of the regulation of food intake and BW had appeared (Booth, 1978). Control-theory concepts also began to appear regularly in qualitative, heuristic discussions of BW regulation (some references are provided in §5.2). This literature has propagated an unfortunate degree of confusion concerning the characteristics and performance of different control-theory models of BW regulation. One aim of this review is to clarify these points. The review does not treat the physiology of BW-regulatory appetite. Many data support the view that signals related to both adipose-tissue mass or fat mass (FM) and lean or fat-free body mass (FFM) contribute to BW-regulatory appetite (Blundell et al., 2012; Dulloo, Jacquet, MilesChan, & Schutz, 2017; Hopkins & Blundell, 2017; Hopkins et al., 2019; Weise, Hohenadel, Krakoff, & Votruba, 2014). One recent study indicates that a BW-regulatory signal arises from the gravitational force exerted by body mass on the long bones in rats and mice, i.e., that BW per se is regulated (Jansson et al., 2018). Possibly, multiple simultaneous regulatory systems operate. None of this, however, would materially change the conclusions reached here. In addition, the review does not treat neurological and genetic models of obesity. Discussion of these topics would simply parallel the analyses of tonic challenges to BW regulation presented in §5.4 and 5.4. Finally, developmental aspects of BW regulation are not considered. Although these may be a profitable avenue for understanding obesity (Levin, 2010), they have not yet contributed to the issues discussed here.

2.1. Energy homeostasis Energy homeostasis is the concept that the body maintains appropriate levels of metabolic fuels in the blood and stored in tissues (Asarian, Gloy, & Geary, 2012; Berthoud, Münzberg, & Morrison, 2017; Leibel, 2008; Müller, Geisler, Heymsfield, & Bosy-Westphal, 2018) The metabolic fuel most relevant to BW is lipid, especially triacylglyceride, stored in the adipose tissue. Changes in adult BW are determined mainly by energy balance, i.e., the balance or imbalance between net changes in EI and EE, and to a lesser extent by shifts in the relative amounts of FM and FFM, which have different energy densities. The effect of BW changes on amounts of FM and FFM is determined by a host of variables (Heymsfield, Gonzalez, Shen, Redman, & Thomas, 2014), which are not considered here. The quantification of energy balance is introduced in Box 2. In most people, BW regulation is quite precise. Even now, in the midst of the obesity pandemic, most adults display slowly increasing BW trajectories that indicate that EI and EE are matched within ~1% (Block, Subramanian, Christakis, & O’Malley, 2013; Kahn & Cheng, 2008; Norberg et al., 2011; Rosenbaum & Leibel, 2010, 2016; Schwartz et al., 2018; Sheehan, DuBrava, DeChello, & ZFangw, 2003; Speakman et al., 2011; Van Wye, Dublin, Blair, & DiPietro, 2007). This suggests that BW-regulatory appetite and AdEE contribute to every-day EI and EE. Control-theory models of this are considered in §5.3.1. Compensatory changes in BW-regulatory appetite and AdEE following forced changes in BW provide further evidence for BW regulation. In non-human mammals, such compensation typically occurs following either forced BW loss or forced BW gain and leads to near recovery of the initial BW (§5.1). In healthy-weight humans, it usually 2

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Box 2 Body composition and energetics. Body mass is the sum of fat mass (FM; sometimes measured as adipose-tissue mass and sometimes as body lipid content) and fat-free mass (FFM; the rest; more sophisticated models distinguish blood, bone and extracellular fluids from muscle and other organs). FM and FFM can be transformed to metabolizable-energy equivalents (metabolizable energy is the energy produced when a molecule is oxidized to end products that are eliminated from the body, such as water, carbon dioxide, urea, etc.). The sum if FM and FFM in energy units yields body metabolizable-energy content (BE). Further, if and 1 2 is the change in one of these variable during the interval from time 1to 2 , then: 1

2

BE =

1

2

FM +

1

2

(1)

FFM.

Because the metabolizable energy content of ingested food (EI) is the only source of BE, and if EE is total metabolizable energy expenditure, the first law of thermodynamics implies: 1

2

BE =

1

2

EI

1

2

EE.

This is the energy-balance equation. An organism is in energy balance from 1 to 2 if: 1

2

(3)

BE = 0,

which implies that: 1

2

EI =

1

2

(2)

(4)

EE.

BW is only a rough approximation of BE, but suffices for many purposes, including those of the present review. Precise studies of the physiology of energy balance require determination of the energy contents of separate tissues and total FM and FFM. Finally, note that EI and EE in Eqs. (2)–(4) are total EI and EE. In contrast, in discussions of the regulation of BE or BW, the focus is on only the fractions of the totals that are controlled by the regulatory system, i.e., BW-regulatory EI and AdEE. occurs following forced BW loss (§5.1 and Fig. 1). As discussed in §5.4, this regulatory response may explain the frequently observed BW regain following BW loss in persons with obesity. In contrast to forced weight loss, forced BW gain rarely elicits compensatory responses and BW loss. The control-theory implications of such failures are discussed in §5.1 and 5.4.

2.3. The hedonic system The energy-homeostasis system functions in tandem with a hedonic system that mediates the effects of flavor hedonics on eating (Berthoud et al., 2017; Gibbons, Finlayson, Dalton, Caudwell, & Blundell, 2014; Hall, Hammond, & Rahmandad, 2014; Rossi & Stuber, 2018; Schultz, 2015). Flavor hedonics are mediated by a widely distributed neural network (Berridge, 2018; Berthoud et al., 2017; Rossi & Stuber, 2018). The system does not produce negative-valence hunger, but modulates the salience and attractiveness of food-related stimuli so as to elicit approach and ingestion, a process called incentive motivation (Berridge, 2018). Incentive motivation functions with food stimuli (i) that are innately preferred, such as sweet taste, (ii) that are preferred only in particular physiological states, such as salt appetite in a state of sodium-depletion, or (iii) that are associated with stimuli of either of the first two types. Energy homeostasis and hedonics are often characterized as mutually antagonistic. This may be so in the overweight/obese state. It is important to recall, however, that energy homeostasis and hedonics operate synergistically in a state of low BW. This suggests that the two systems co-evolved. The extensive overlap between the neural mechanisms of energy homeostasis system and the hedonic system further supports the view that they are synergistic controllers of appetite (Berthoud et al., 2017; Hall et al., 2014; Sternson & Eiselt, 2017).

2.2. Dieting, restraint, and disinhibition A classic description of the quality of appetite elicited by BW loss comes from the Minnesota starvation experiment, in which healthyweight volunteers were restricted to 75% or less of their normal EI until they lost ~24% of their initial BW (Keys, Brozek, Henschel, Mickelsen, & Taylor, 1950, Fig. 1). As the participants lost BW, they became increasingly hungry and focused on food to the exclusion of other interests; they also became increasingly apathetic, lethargic and irritable (Franklin, Schiele, Brozek, & Keys, 1948; Franklin, Schiele, Brozek & Keys, 1948). Thus, BW-regulatory appetite in response to BW loss seems to be an aversive state that fits the classical drive-reduction theory of negative-valence hunger (Berridge, 2018). Studies of the neural basis for food-restriction induced appetite in mice, however, have provided only mixed support for this view (Betley et al., 2015; Chen, Lin, Zimmerman, Essner, & Knight, 2016). When the food-restriction period of the Minnesota starvation experiment ended, the men often engaged in bouts of voracious eating, although they were warned not to, and they reported feeling out of control of their eating; i.e., they binged (Franklin, Schiele, Brozek & Keys, 1948). Polivy (1996) noted that a very similar constellation of psychological changes and a tendency to binge occurs in people who chronically diet to lose weight. This trait-like syndrome is known as restricted eating (Herman & Mack, 1975; Polivy, Herman, & Warsh, 1978). Remarkably, it appeared to be produced by chronic dieting rather than BW loss per se. Components of the syndrome are now usually measured separately with the three-factor eating questionnaire (Stunkard & Messick, 1985). The disinhibition factor is related to the tendency to binge and is strongly associated with food intake and with body mass index (BMI; BW [kg]/height2 [m2]) (French, Epstein, Jeffery, Blundell, & Wardle, 2012, 2014). The implication is that BW-regulatory appetite plays an important role in many circumstances.

2.4. Cognitive controls of appetite Emerging research indicates that cognitive processes critically influence the penetration of weight-regulatory processes to eating outcomes (e.g., Brewer et al., 2018; Hall et al., 2014; Horstmann et al., 2015; O'Reilly et al., 2014). For example, several studies have documented associations between memory function and BMI (reviewed in Higgs & Spetter, 2018). Such work seems to have the potential to lead to better understanding of why BW-regulatory appetite often seems to be ineffective in current environments (§5.4.2) and to the development of improved strategies for BW-therapy (Buscemi et al., 2013; Forman et al., 2018; Rogers, 2017). 3. Some background concepts §3.1–3.4 develop some control-theory concepts germane to the review. §3.5 introduces AdEE and models of BW dynamics in response to tonic changes in energy balance. 3

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Fig. 2. A. A system diagram for a simple negative-feedback control system. Each block represents a process element of the system, and arrows represent the processes inputs and outputs. Each process transforms its inputs into outputs. For simplicity, all inputs and outputs are considered to be the same units, such as voltage in a Black amplifier (Åström & Murray, 2016). A fraction of the system's output, the feedback signal, is diverted into the feedback process, whose output is the feedback-gain signal. Feedback gain is led to an integrator process (Σ), which subtracts it from the system's input, thus making it a negative-feedback process. The integrated signal is the system's output. The ratio of output to input is the system's forward gain. A well tuned system stabilizes and smooths the output when the input changes; a poorly tuned system may produce uncontrolled oscillations in output. Note (i) that the negative-feedback process closes a loop, shown in red, between the output and input; and (ii) the input is external to the closed loop and is therefore considered an open-loop influence. The system is active because changes in output cause a dynamic modulation of the input, controlled by negative feedback. B. Passive control in a dam-reservoir system. The reservoir has one input, spring meltwater, and two outputs, usage by the village served by the dam and diversion of excess reservoir fill into a spillway. The arrangement provides a degree of regulation of water supply; i.e., prevention of excess reservoir fill. This is passive or static regulation because outputs do not feedback to affect the input. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 1. Active regulation of body weight (BW) in humans by BW-regulatory appetite. A. BW in 32 healthy young men who were maintained in energy balance for two month (baseline; B); then fed an inadequate energy ration for six months (SS), during which they lost 24% of their BW, then re-fed for three months with various controlled feeding regimens (CR); and, finally, one subgroup of 12 men was followed while they ate ad libitum for two months (ALR, right of red line). The BW model predictions are based on Hall's (2006) model of body composition dynamics when nutrient intake is known, introduced in §3.5. B. nutrient intake. Note the intakes of fats (FI), carbohydrates (CI) and protein (PI) during ALR all exceeded baseline intakes while BW was reduced, indicative of active regulation of BW-regulatory appetite. The original data are from the classic Minnesota starvation experiment (Keys, Brozek, Henschel, Mickelsen & Taylor, 1950). The figures are reprinted with permission from Hall (2006). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.).Reprinted from K.D. Hall. (2006) Computational model of in vivo human energy metabolism during semistarvation and refeeding. Amercian Journal of Physiology, 29, E23-E37, with permisssion from the American Physiological Society.

in any input dynamically affect the performance of the entire system. As a consequence of the closed-loop structure, simple causal reasoning is difficult to apply to feedback control (Åström & Murray, 2016). Rather, the characteristics of control systems are better understood mathematically.

3.1. Negative-feedback control and closed-loop variables Classical control-theory regulation depends on negative feedback. Indeed, Åström and Murray (2016) define control as the use of algorithms and feedback in engineered systems. The regulation produced by negative-feedback control is mediated by its closed-loop organization. As shown in Fig. 2A, changes in the system output or regulated variable produce feedback that elicits changes in controlled variables, which in turn act to return the regulated variable to (or toward) its initial level. Thus, the regulated variable is said to be defended. The feedback is negative in that it opposes change in the regulated variable. Each element in a control-theory system diagram such as Fig. 2A represents a mathematical process that transforms the element's input (s) signal into an output. The process acting on negative-feedback signals is called the feedback-gain process. The stronger the feedback gain, the greater the effect of feedback on the controlled variables and system output and, usually, the more precise the regulation. The integrator process combines the system's inputs with the feedback gain, and the controller process transforms the resultant signal into the system's output. This is called the forward gain of the system. Note that because of the interconnected, or closed, structure of the feedback loop, changes

3.2. Linearity As mentioned above, each closed-loop element in a control system represents a mathematical process that transforms the element's input (s) into an output. Most engineered feedback systems use linear functions with 0 y-intercepts and addition to model these processes (“linear” indicates 0 y-intercept linear below). Fig. 3A schematizes a linear transformation of feedbacks from BW into the feedback-gain signal. Note: that (i) feedback gain increases as BW increases, and (ii) the increases can be greater (dotted line) or lessor (solid line), depending on the slope of the linear transformation. Stronger feedback gain more effectively opposes open-loop challenges, but may introduce instability in the system. One advantage of a linear system is that control can be modeled in terms of changes in BW, feedback, etc., from a steady-state baseline rather than in terms of absolute levels. The final outcome of the system 4

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system, such as the input in Fig. 2A. Others are disturbances that pose challenges to regulation. The obesity pandemic (Hales, Carroll, Fryar, & Ogden, 2017; NCD Risk Factor Collaboration, 2016, 2017; Ogden, Carroll, Kit, & Flegal, 2014) is thought to be caused in part by increased EI due to increased availability of palatable, energy-dense foods, increased portion size, etc., and decreases in habitual physical-activity EE due to increased automation, increased television and computer use, etc. These changes are collectively known as the obesogenic environment (Ravussin & Bouchard, 2010; Swinburn, Egger, & Raza, 1999), although which elements of modern environments actually are obesogenic remains unclear (Elgaard Jensen et al., 2019; Mackenbach et al., 2014; Mattes & Foster, 2014). In terms of control theory, the point is that these factors are external influences on EI and EE that are not affected by negative feedback, i.e., are open-loop variables that challenge regulation. When a control system is challenged by strong, tonic open-loop inputs, the final level of the system output or regulated variable is often an equilibrium determined by the relative strength of the controlled variables and the open-loop variables. The strength of the controlled variables is usually mainly dependent on the feedback gain. Such equilibria are discussed more in §4 and §5. 3.4. Active and passive regulation Negative-feedback control produces active regulation in that the feedback process causes dynamic responses in control mechanisms that defend, or resist change in, the output or regulated variable, as diagrammed in Fig. 2A (DiStefano et al., 2012). In contrast, in passive regulatory systems the static structure of the system produces some degree of regulation without feedback linked to response mechanisms. An example of a passive regulation is provided by a water reservoir that is fed by mountain streams and contained by a dam with an overflow spillway, as diagrammed in Fig. 2B. In winter, there is no liquid water and the streams are dry, so the water level in the reservoir decreases as the villagers use stored water. As the spring warmth melts snow in the mountains, streams swell, the reservoir fills, and the water supply can be used by the villagers as long as it lasts. A spillway near the top of the dam allows water to flow out of the reservoir if it exceeds the maximum safe level. If the dam and spillway are designed well, the reservoir level is effectively regulated; normal winters provide enough water for the village through the year until the next spring melt re-fills the reservoir. In control theory, this is a passive regulation because it has no closed loop including dynamic components capable of generating responses that affect the regulated variable (DiStefano et al., 2012). Regulation is achieved solely by the static, fixed structure of the system. The example also illustrates that stability and regulation can be achieved without feedback (Engelberg, 1966; Goldstein, 2019; Goldstein & Kopin, 2017; Gray et al., 2017; Ramsay & Woods, 2016). A physiological example of passive regulation is glycosuria caused by hyperglycemia. Blood glucose levels exceeding about 10 mmol/L (180 mg/dL) produce glucose concentrations that exceed the resorption capacity of the proximal tubules of the kidney, so that glucose is excreted into the urine. The system is structural, without an active, feedback-controlled response. In contrast, the regulation of glycemia by changes in insulin and glucagon secretion is an active negative-feedback control system because the system output, blood glucose level, feeds back to the Islets of Langerhans, where it controls the hormonal responses.

Fig. 3. Schematic diagrams of feedback-gain functions, applicable to either simple negative-feedback control or set-point control. A. Linear feedback-gain functions. The shallow slope of the solid line indicates relatively weak feedback gain, with modest increases or decreases in feedback gain as body weight (BW) diverges from the baseline level. The steeper slope of the dotted line indicates stronger feedback gain. Note that as the feedback gain increases, EI becomes progressively less and EE becomes progressively more than at baseline. B. A non-linear feedback-gain function. The dynamic strength of feedback is indicated by the slope of the function. Note that (i) the slope is relatively flat around the present BW or neutral point, indicating that changes in BW in that range will produce little response; (ii) the slope is very steep for moderate or large decreases in BW, indicating that decreases in BW in that range will produce robust negative feedback; and (iii) the slope is less steep and reaches an asymptote for moderate or large increases in BW, indicating that increases in BW in that range will produce comparatively less robust negative feedback than similar magnitude decreases in BW.

is the change plus the initial value. This strategy simplifies the mathematics and is done often. An example is developed in §4.1 and Box 3. Biological control systems, of course, are not constrained to linearity. Fig. 3B schematizes a non-linear transformation of feedbacks from BW into a feedback-gain signal. Note that in the function shown: (i) Modest increases or decrease in BW produce little change in feedback. (ii) Larger gains in BW increase feedback gain much less than comparable losses in BW decrease feedback gain; i.e., BW loss is resisted more strongly than is BW gain. This seems to be a characteristic of BW regulation in both humans and rodents (e.g., Schwartz et al., 2003; Speakman, 2007, 2018). In engineered systems, feedback gain is usually tuned so that the regulated variable returns to the near its baseline or set-point level under usual operating conditions, i.e., the influence of usual variations in open-loop variables. It seems likely that BW regulation was similarly tuned by natural selection. Some data discussed in §5.4.2 suggest that the human BW-regulatory system dynamically changes its feedbackgain process or set point so that it re-tunes to any tonically maintained level of elevated BW.

3.5. Models of BW dynamics There are several quantitative models of BW dynamics in response to tonic changes in EI (e.g., Antonetti, 1973; Hall, 2006, 2010; Hall et al., 2011; Hall & Jordan, 2008; Navarro-Barrientos, Rivera, & Collins, 2011; Speakman & Westerterp, 2013; Thomas & Antonetti, 2017; Thomas et al., 2011; Westerterp, Donkers, Fredrix,

3.3. Open-loop variables and the obesogenic environment Inputs that enter a control system outside the closed loop are called open-loop variables. Some open-loop variables may be integral to the 5

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Box 3 Linear simple negative-feedback control and its transfer functions. In the most commonly applied simple negative-feedback control system, each process function is 0 y-intercept linear; i.e., each process simply multiplies its input times a constant. Such a system can be called linear simple negative-feedback control. The quantification of linear simple negative-feedback control can be illustrated using the system diagrammed in Fig. 2A. If (i) the output (O) provides the feedback signal, F is the feedback gain, is the summation signal, G is the forward gain, (ii) all these are shorthand for a differential equation expressing the dynamics of the process in the form dF / dt , etc., where t is time, (iii) the processes are linear defined by the constants kF , k , kT and kO , and finally the open loop input (i ) is a constant, then the process functions would be: (5)

F = kF O, = k (i

(6)

F)

G = kG ,

(7)

O = kO G,

(8)

Note that the equations are linked in a closed loop. Therefore, (i) a change in one affects the rest, and (ii) by substitution, solutions for one variable can be expressed in terms of others; such equations are called transfer functions. To further simplify the calculations, changes in levels from some baseline state are often considered. This can be accomplished by setting i and O to 0 at time 0. If this is done and k is set to 1, then transfer function for the dynamic change in system output O following a change in the input (i ) to a new constant level is:

d O = ikF e kG kF t dt where e is Euler's number, the base of natural logarithms. Integrating the equation yields the equation for O at time t:

O (t ) =

i (1 kF

e

kG kF t

).

(9)

(10)

The important point is that as time goes by, the exponential term (e kG kF t ) approaches 0, and the system output O approaches a final stable value whose level is given by i kF , the ratio of the input and the strength, or gain, of the feedback process. Thus, if the change in input is doubled, the change in steady-state output is doubled, and if the feedback gain parameter, kF , is doubled, the change in output produced by a change in input will be halved, as shown in Fig. 5. & Boekhoudt, 1995). Although these models differ in the number of physiological processes modeled and in their computational ease, most predict BW dynamics with similar accuracy (Thomas & Antonetti, 2017; Thomas, Gonzalez, Pereira, Redman, & Heymsfield, 2014). Fig. 1A shows an example. A key aspect of BW-dynamic models is that they distinguish two dynamic components of EE: (i) One component comprises the automatic changes in EE associated with altered BW or body composition. EE changes automatically in response to altered BW or composition because changes in the numbers or type of cells in the body change EE related to cellular energetics, because changes in BW affect physical activity-related EE, and because changes in EI required for maintenance of BW lead to changes in EE associated with processing food (Joosen & Westerterp, 2006; Leibel, Rosenbaum, & Hirsch, 1995; Rosenbaum & Leibel, 2010). These changes require no feedback and can be considered passive regulatory processes, as defined above. (ii) The second component is AdEE. As mentioned in §1.1, AdEE arises in response to feedback from the altered BW of body composition and together with WR-appetite actively regulates BW. In humans who lose ~10% BW, AdEE leads to a decrease of ~10–15% of total EE before BW loss, which greatly slows the rate of BW loss (Leibel et al., 1995; Rosenbaum & Leibel, 2010). A variety of feedback and effector mechanisms appear to contribute to AdEE in underweight (Rosenbaum & Leibel, 2010). AdEE also contributes to regulation of BW in response to BW gain (Leibel et al., 1995; discussed in §5.4.2), although the magnitude of the response appears to vary across individuals and test situations (Joosen & Westerterp, 2006). BW-dynamic models are not control-theory models because, although they model AdEE, they do not model the feedbacks from BW that control BW-regulatory appetite. They can be used, however, to model feedback control of BW-regulatory appetite in experiments in which BW change is tracked (e.g., Thomas et al., 2010; Sanghvi, Redman, Martin, Ravussin, & Hall, 2015). Examples of this are described in §5.3.

4. Features of simple negative-feedback and set-point control 4.1. Simple negative-feedback control Fig. 4 displays a system diagram modeling BW regulation by simple negative-feedback control of BW-regulatory appetite and AdEE. As described in §3.1 and 3.3, the myriad factors influencing appetite and EE are divided into two categories: controlled or closed-loop variables that are influenced by feedback from BW, and uncontrolled or open-loop variables that are not. Box 3 introduces the mathematical formalization of linear simple negative-feedback control. Three important points are: (i) if the openloop inputs are suddenly changed to a new tonic level, BW dynamics follow an exponential curve. (ii) BW usually reaches an equilibrium or steady-state BW. (iii) This equilibrium is affected by the relative strengths of the open-loop inputs and the feedback gain. Fig. 5 shows each of these characteristics. 4.1.1. The settling point In the BW-regulation literature, simple negative-feedback control is often known as settling-point control, after a term introduced by Wirtshafter and Davis (1977) and Davis and Wirtshafter (1978). The settling point is simply the steady-state equilibrium reached by the regulated variable. Wirtshafter and Davis (1977) and Davis and Wirtshafter (1978) offered simple negative-feedback control as a thought experiment to counter arguments that BW-regulation phenomena suggested the operation of a set-point system (e.g., Cabanac, 1971; Keesey, Boyle, Kemnitz, & Mitchell, 1970; Keesey & Powley, 1986, 2008; Mrosovsky & Powley, 1977; Nisbett, 1972; Panksepp, 1974; Powley & Keesey, 1970). The controversy is briefly reviewed in §5.1. Unfortunately, much current literature uses the settling point concept differently than originally defined. Speakman et al. (2011) described settling-point control of BW as a passive regulatory system 6

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because it does not include a set point or “regulated level.” Rosenbaum and Leibel (2016) follow in this. Speakman et al.’s (2011) criterion for passive control is not ideal because the presence of a steady-state BW does not distinguish simple, settling-point and set-point negative-feedback control, as discussed in §4.2. Hall et al. (2012) state that a settling point system has no active feedback. In fact, the settling point model fulfils the standard control-theory criterion for active control; namely, the existence of dynamic control of system response elicited by feedback from the output (DiStefano et al., 2012). Müller et al. (2018) suggest that settling-point control of AdEE does not involve feedback. But most mechanisms of AdEE do involve neuroendocrine feedback (Rosenbaum & Leibel, 2010, 2016). Finally, Hall and Guo (2017) propose a settling-point model that includes negative-feedback control of AdEE, but no BW-regulatory appetite. This is possible, but odd given the emphasis on eating in Wirtshafter and Davis’s (1977) original model and in the data discussed in §5.3).

Fig. 4. A system diagram for simple negative-feedback regulation of body weight (BW) by control of BW-regulatory appetite (BW-R appetite) and adaptive thermogenesis (AdEE). The regulated variable, BW, generates afferent feedback signals that the feedback process transforms into feedback-gain signals. These are transformed into modify the controlled variables, BW-R appetite and AdEE, by the Closed-Loop Controls process. The summator process Σ1 integrates the controlled variables with external signals or open-loop controls (shown in red) and produces the final total levels of appetite and EE. Total appetite and EE modify BW, thus closing the negative-feedback loop. Data indicate that there are many feedback signals and many different controls of BWR appetite and AdEE; in a full model, these would be represented by additional, parallel loops beginning from BW. The arrow linking total appetite and EE to BW simplifies a complex process in which neuroendocrine control signals have consequences outside the body (the organism interacts with the environment to obtain food), which then affect the body (processing of ingesta in part into altered BW); see Booth (2008) for discussion. Note that open-loop variables may include normal controls of appetite and EE as well as pathological influences, such as those thought to be provided by the “obesogenic environment” described in §3.3 and 4.5.2. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

4.2. Set-point or PID control Fig. 6A displays a system diagram modeling BW regulation by setpoint negative-feedback control of weight-regulatory appetite and AdEE. Two components are added to Fig. 6A in comparison with Fig. 3:

Fig. 6. A. A system diagram for set-point regulation of body weight (BW) by control of BW-regulatory appetite (BW-R appetite) and adaptive thermogenesis (AdEE). Two components are added in comparison with Fig. 4: an external input r, the set-point, and a second integrator (Σ2), which subtracts r from the feedback signal to produce an error signal , which is the input to the feedbackgain processes, labelled PID Controls, which may transform into proportional (P), integral (I), or derivative (D) control signals, as explained in Fig. 6B, Box 4 and the text. Note that the Closed-Loop Controls process shown in Fig. 4 is integrated into the PID Controls process for simplicity. PID controls sum to form BW-R appetite and AdEE. As in simple negative-feedback control, the summator process Σ1 integrates the controlled variables with external signals or open-loop controls (shown in red) and produces the final total levels of appetite and EE, which, modify BW and close the negative-feedback loop. B. The mathematical transformations for P, I and D feedback gain, as explained in Box 4 and the text; see Box 4 for definition of the symbols. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 5. Dynamic and steady-state changes in body weight (BW; arbitrary units relative to the initial value, 0) reached by the linear simple negative-feedback control system described in Fig. 3A and Box 3. As shown in Box 3, Equations system output dynamics are described by exponential functions, with final steady-state outputs are given by i kF , the ratio of the open-loop input and the strength, or gain, of the feedback process. X-axis, arbitrary time scale and changes in input and feedback gain. A. The effects of doubling the open-loop input from 1 to 2 and then returning to 1. B. The effect of doubling the feedback gain from 1 to 2 and then returning to 1.

7

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(i) an external input r, the set-point, and (ii) a second integrator (Σ2), which subtracts r from the feedback signal to produce an error signal , which is the input to the feedback-gain process. Each addition confers an advantage: (i) the set point r is an independently adjustable external signal not influenced by the operation of the system; thus, the target level of the regulated variable can be changed; and, (ii) manipulating how is transformed into a control signal (i.e., feedback gain) is a convenient way to make regulation more powerful or precise, as explained in the next section. These features make set-point control preferable to simple negative-feedback control in many engineering applications (Åström & Murray, 2016). The set point is also known as the reference-value, “should-be” value, or Sollwert (German; should-be value). Should-be value is an apt term as it captures the fact that the steady-state body output is often not identical to the set point, as explained below.

inappropriate in many engineering applications. Total feedback gain is the sum of the terms. Linear PI control, i.e., the combination of linear P control and linear I control, is most common in engineering applications (Åström & Murray, 2016). As mentioned above, P set-point control is similar to simple negative-feedback control. A close analog of I set-point control also can be achieved in simple negative-feedback control. This is done by considering some fixed interval ending at the present time and computing changes in the regulated variable (or feedback signal) relative to its value at the beginning of the interval, as explained in Box 5. Finally, D control is identical in set-point control and simple negative-feedback control, as the rate of change in the feedback signal is identical to the rate of change of , which equals the feedback signal minus a constant. Thus, how closely a system maintains the regulated variable at its initial (or set-point level) does not reliably distinguish simple negative-feedback control system from set-point control systems.

4.2.1. PID control Set-point control is often referred to as PID control. PID refers to three common feedback-gain transformations of : proportional (P), integral (I), and derivative (D). The transforms are defined in Fig. 6B and Box 4. P, I and D control can occur in any combination. Most engineered PID control systems are linear: i.e., the transformations of are each multiplied by a constant to result in the control signal. Fig. 7 illustrates how the performances of linear P, PI and PID control depend on the magnitude of these constants. In P control, feedback gain is a function of the present value of . This is similar to simple negative-feedback control and produces similar regulation (Figs. 5 and 7A); i.e., exponential trajectories of the regulated variable and a steady-state error in response to a tonic challenge. In I control, feedback gain is a function of the change in over some interval; i.e., the incremental area under the curve of . This can produce more precise regulation than P control (Fig. 7B). Obviously, however, the interval of integration has to be limited if the system is to escape the influence of long-past errors that have been corrected. Finally, in D control (Fig. 7C), feedback gain is a function of the present rate of change of , which can be understood as a prediction of future errors. D control usually produces the smallest steady-state error, but is also prone to large transient variations before reaching equilibrium, so is

5. Simple negative-feedback and set-point models of BW regulation compared §5 assesses whether simple negative-feedback (settling-point) control and set-point control models better fit the principal phenomena of BW regulation. §5.1 develops provisional criteria for the precision of steady-state regulation. §5.2 briefly summarizes some classic tests of phasic or transient challenges to BW regulation. As has been amply reviewed before, these challenges fail to establish the nature of the BW regulatory system. §5.2 and §5.3 review tonic challenges favoring negative and positive energy balance, respectively. 5.1. Precision of regulation There are many quantitative and graphical approaches to characterizing the stability, transient performance and steady-state performance of control systems (DiStefano et al., 2012; Åström & Murray, 2016). In the present context, steady-state precision is most relevant. For phasic challenges (i.e., phasic changes in open-loop inputs), this may be defined as difference between the steady-state BW before and after the challenge. For tonic challenges, it may be defined as the

Box. 4 Set-point or PID control As described §4.2, the key differences between simple negative-feedback control and set-point control are (i) the addition of an open-loop input that serves as the reference or target value for steady-state regulations, the set point (r ), (ii) the generation of an error signal, , and (iii) the transformations of by up to three feedback-gain functions. is formed by comparing the set-point signal r and the feedback signal FS at time t:

(t ) = fr (FS (t )

r)

Usually with. fr = 1. Set-point control may involve any combination of proportional (P), integral (I) or derivative (D) transformations of FG(t). The general formula for FG(t) in a system involving all three transformations is: 0

FG (t ) = fFG (P ) (t ) + fFG (I )

(t ) dt + fFG (D) d dt ,

(11) into feedback gain,

(12)

where fFG (P ) , fFG (I ) , and fFG (D) are separate feedback-gain functions acting on the P, I and D transformations of , respectively. The integration term spans some interval from some previous time τ to the present time 0. In engineering applications, the transformations alone usually provide sufficient power and flexibility, so that the three feedback-gain functions are simply linear. In this case, the formula for FG(t) becomes: 0

FG (t ) = kP (t ) + kI

(t ) dt + kD d dt ,

(13)

where kP , kI , and kD are constants. This could be called linear proportional, linear integral and linear derivative set-point control. Engineering systems are usually ordinarily assumed to be linear, however, so that is left unsaid. Whether biological systems are similarly linear is uncertain. The final step is the transformation of FG(t) into control signals that produce BW-regulatory appetite and AdEE, as shown in Fig. 6. Even in the case of linear systems, the transfer equations for the steady state of PID controllers are more difficult to solve than those for simple negative-feedback systems shown in Box 2. 8

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Fig. 7. Responses of a variable (Y ) regulated by set-point control in response to increases in the set point r from 0 to 1 at time 0, for A., proportional (P) control, B., proportional-integral (PI) control, and C., proportional-integral-derivative (PID) control. The control system is linear, as described by Box 4, Equation (13). The P controller in A. has kP = 1 (dashed line), 2 (solid), or 5 (dash-dotted). The PI controller in B. has kP = 1 and kI = 0 (dashed), 0.2 (lower solid), 0.5 (upper solid), or 1 (dash-dotted). The PID controller in C. has kP = 2.5, kI = 1.5, and kD = 0 (dashed), 1 (upper solid), 2 (lower solid), or 4 (dash-dotted). Note that in this model, with P control the regulated variable never reaches the set point, with PI control it usually, but not always, does, and with PID control it always does. The oscillations in output could be damped by reducing the magnitude of the feedback gain of the D process or slowing the feedback or closed-loop processes. See (Åström & Murray, 2016) for further details. Reprinted with permission from K.J. Åström & R.M. Murray, Feedback Systems, 2nd edition. Princeton NJ, USA: Princeton University Press, 2016. URL: 995 cds.caltech.edu/~murray/amwiki/index.php/Second_Edition.

Box. 5 Generation of an integral term in a simple negative-feedback control system Consider the body weight (BW) time course shown in the graph below. The control system tracks change in BW beginning at time and integrates these changes from some time τ in the past to the present time 0. If the feedback signal is modeled as BW, the incremental area under the BW curve (A) between times τ and 0 is given by: 0

A=

(BW (t ) dt

BW (0)).

(14)

The integral A can serve to generate an additional feedback-gain term, so that the feedback gain (FG(t)) of simple negative-feedback control becomes: 0

FG (t ) = fFG (P ) BW (t ) + fFG (I )

(BW (t ) dt

BW (0))

(15)

where fFG (P ) and fFG (I ) are separate feedback-gain functions as in Box 4. Eq. (15) is analogous to Box 4, Eqs. (12) and (13), except that current BW is compared with BW at time 0 rather than with a fixed set point. That is, Eq. (14) produces a form of integral control without a fixed reference value. This might achieve strong regulation in a simple negative-feedback control system, as discussed in Box 6.

difference between the steady-state BW before and during the challenge. To facilitate comparisons, the following arbitrary quantitative criteria are used where possible in the next sections: (i) The precision of regulation is strong if steady-state it BW change is < 10 kg; this would maintain the BMI of an average-height person with BMI 22 kg/m2 within the boundaries considered healthy, ~19–25 kg/m2. (ii) For persons with overweight or obesity, the precision of regulation is moderate if steady-state BW change is ≥ 10 kg and < 25 kg; and increase or 25 kg in the normative person would lead to a BMI of ~30 kg/ m2, the usual criterion for obesity. (iii) For persons with overweight or obesity, the precision of regulation is weak if steady-state BW change is ≥ 25. Note, however, that these criteria depend on the strength of the open-loop challenges to regulation. Any regulatory system can be overwhelmed.

5.2. Phasic challenges to BW regulation Forced BW loss elicits strong compensatory responses. BW-regulatory appetite increases, AdEE decreases, and BW and body composition return toward the initial values (Dulloo, 2017; Dulloo, Jacquet, & Girardier, 1997; Heyman et al., 1992; Keys et al., 1950; Leibel et al., 1995; Roberts et al., 1994; Rosenbaum, Kissileff, Mayer, Hirsch, & Leibel, 2010; Winkels et al., 2011). Both simple negative-feedback control and set-point control, whether strong or weak, predict this. That is, after the end of a forced BW-loss period, only BW feedback differs from the initial conditions. Thus, feedback-activated responses should operate until BW and feedback from BW reach the same steady-state levels they had before the challenge. In many cases, however, steady state BW following phasic forced BW loss exceeds the pre-challenge 9

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Box. 6 Modeling the effect of tonic energy loss Polidori et al. (2016) compared the fit of two models of the control of EI by BW using data obtained in a trial in which patients with type 2 diabetes mellitus were treated with 300 mg/d canagliflozin, a sodium-glucose co-transporter-2 antagonist that increased urinary glucose excretion by an estimated 90 g/d (~350 kcal/d). The first model, linear proportional control, was:

EI (t ) = kp

(16)

BW (t ),

where EI (t ) and BW (t ) are the changes in EI and BW, respectively, at time t and kp is the constant describing the linear feedback process. Note that Eq. (14) parallels the equations in Box 3, but solved for EI rather than BW. Thus, as in Box 3, the solution is a negative exponential. The models also included the Hall-model equations to account for the effects of BW loss on EE. This model (i) provided an excellent fit to the observed ~4 kg BW loss and (ii) predicted that the urinary energy loos would be compensated by a ~350 kcal/d compensatory increase in EI, which also fit the observed data (Fig. 9). The steady-state BW reached was about 5% less than the BW at onset. This can be termed weak or moderately strong regulation, depending on one's the criterion for precision. The second model examined by Polidori et al. (2016) was a linear proportional-integral control, as follows:

EI (t ) = kp BW (t ) + kI

BW (t ) dt .

(17)

0

Eq. (17) parallels Box 4, Eq. (13). The authors found that the addition of linear integral control with kI = 1 was not helpful (Fig. 9). That is, the model with I control predicted greater BW-regulatory appetite than was observed and no steady-state error, which was not observed (Fig. 9). In summary, simple linear proportional negative-feedback control seems sufficient to account for the effects of tonic energy loss of ~350 kcal/d on EI and EE. Whether this also would be the case for greater challenges is unclear.

accounted for ~70% of the variance in EI outcomes. The authors noted that this fit might be improved if some of the more potent drugs were assumed to increase the slope of the linear feedback process (i.e., increase feedback gain) or if the special pharmacological properties of some drugs were considered, for example, if orlistat, which inhibits intestinal fat absorption, were assumed to have a selective effect on fat intake. The authors that that the effect of obesity pharmacotherapy consists of a relatively constant drug effect countered by linear proportional negative-feedback BW regulatory system. As explained in §4.2.1, the regulatory system could be simple negative-feedback control or setpoint control. Regulation was strong in 24 of the 28 test groups according to the criteria developed in §5.1. Regulation was moderate only in four groups tested with high doses of phentermine plus fenfluramine or topiramate. Presumably these regimens constituted extremely strong open-loop challenges.

level (e.g., Fig. 1A; Keys et al., 1950). This phenomenon, known as collateral fattening (Dulloo, 2017), is accounted for by an increase in FM, with FFM after the challenge nearly identical with FFM before the challenge. In the Keys et al. (1950) experiment, collateral fattening may have been due to relatively greater increases in fat and carbohydrate intake than protein intake during the recovery (Fig. 1B). Taken together, these data suggest that FFM rather than BW is regulated. In laboratory rodents, forced BW gain typically results in compensatory responses and a return to near the initial levels (e.g., Cohn & Joseph, 1962; Gloy, Lutz, Langhans, Geary, & Hillebrand, 2010; Harris, Kasser, & Martin, 1986; Ravussin et al., 2018). These are predicted by either type of control, as described above. In contrast, in humans, compensatory responses after BW gain often fail to reproduce the initial levels (e.g., Jebb et al., 2006; Levitsky, Obarzanek, Mrdjenovic, & Strupp, 2005; Roberts et al., 1990; Rosenbaum & Leibel, 2010; Siervo et al., 2008). From a control-theory perspective, this could result from an asymmetry in the strength of feedback signals or of feedback gain elicited by decreased vs. increased BW, as shown in Fig. 5B. Finally, many non-human animal species display sustained periods of endogenous or voluntary BW gain or loss before hibernation, migration, during the mating season, etc. (Mrosovsky & Powley, 1977; Mrosovsky & Sherry, 1980). Again, this could result either from a changed set point or from a change in feedback gain (Davis & Wirtshafter, 1978; Wirtshafter & Davis, 1977).

5.3.2. Diabetes pharmacotherapy Human appetite results from complex interactions among environmental, cognitive, hedonic and regulatory influences. Consequently, regulatory influences would be expected to be clearest in situations that minimize cognitive and hedonic influences, such as chronic treatment with agents that lead to insensible changes in energy balance. An example of this is treatment of type 2 diabetes mellitus (T2DM) with sodium-glucose co-transporter-2 (SGLT2) antagonists, which reduce renal glucose re-absorption, leading to substantial glycosuria and clinically relevant reductions in fasting blood glucose. Ferrannini et al. (2015) followed a group of patients with T2DM and mean initial BMI of 30 kg/m2 who were treated for 90 wk with a daily dose of the SGLT2 antagonist empagliflozin that led to a median urinary glucose loss of 51 g/d (~200 kcal/d). According to the Hall model of BW and EE dynamics (§3.3.4), if EI had not changed, this would lead to a BW loss of ~11 kg, whereas observed BW loss was only ~3 kg, indicating that BW loss was opposed by an increase in EI of ~270 kcal/d; i.e., BW-regulatory appetite. According to the criteria developed above, the regulatory system was strong. Polidori, Sanghvi, Seeley, and Hall (2016) fit data from a similar trial to a linear proportional negative-feedback model, as described in Box 6. Patients with T2DM and overweight or obesity (mean BMI, 32 kg/m2) were treated for 52 wk with a daily dose of the SGLT2 antagonist canagliflozin for 52 weeks. Daily urinary glucose loss was assumed to be ~90 g/d (~350 kcal/d) on the basis of previous data. As

5.3. Tonic challenges favoring negative energy balance 5.3.1. Obesity pharmacotherapy Göbel, Sanghvi, and Hall (2014) and Hall, Sanghvi, and Göbel (2017) modeled BW loss in 28 groups of patients with obesity who were treated for at least one year, long enough to reach a steady state, with different drugs or drug combinations, some in several in different dosages. Changes in EE were predicted using measured BW and the Hall model (§3.5), permitting estimation of EI. EI and BW during the dynamic and steady-states of BW change were fit to negative exponential functions, as predicted from linear proportional negative-feedback regulatory system (§4.1 and Box 3; note that in the Hall model of BW dynamics, AdEE is linearly related to EI and thus consistent with a linear model). Steady-state, placebo-controlled BW loss ranged from ~0 to ~12 kg. Initial placebo-controlled decreases in EI ranged from ~250 to ~1100 kcal/d, and steady-state, placebo-controlled EI ranged from an increase of ~75 kcal/d to a decreases of ~ -600 kcal/d. The model 10

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slow increase in the potency of some open-loop stimulatory control of EI; (iii) an increase in the BW set point if the regulatory system is a setpoint system. In addition to BW gain, aging is associated with loss of FFM, especially skeletal muscle mass (Hull et al., 2011; Westerterp, 2018). In women, this is especially clear across the menopausal transition and is associate with an increase in FM also increases (Leeners, Geary, Tobler, & Asarian, 2017). If FFM is regulated separately from BW, this could result from reductions in FFM feedback gain, set point, or a decrease in some open-loop variables that tend to maintain youngadult levels of muscle mass. Such regulation of FFM might stimulate weight-regulatory appetite or decrease AdEE and thereby lead to an increase in BW. 5.4.2. Obesity The last decades have seen the growth of an obesity pandemic (NCD Risk Factor Collaboration, 2016, 2017). For example, in the USA, in 1923 mean adult BMI was ~25 kg/m2 and ~10% of adults had BMI > 30 kg/m2 (Nuttal, 2015), whereas 2016 mean age-adjusted adult BMI was 29.1 in men and 29.6 in women and ~40% of adults had BMI > 30 kg/m2 (Fryar, Kruszon-Moran, Gu, & Ogden, 2018; Hales et al., 2017). Because obesity risk is highly heritable (Elks et al., 2012; Llewellyn, Trzaskowski, Plomin, & Wardle, 2013; O’Rahilly & Farooqi, 2008), it is likely that genetic or epigenetic influences predisposing to obesity act by increasing the effect of modern environmental factors that favor positive energy balance and BW gain (Schwartz et al., 2017). As mentioned in §3.3, it is commonly accepted that one such factor is obesogenic environments, which favor tonically increased EI and tonically decreased EE (Ravussin & Bouchard, 2010; Swinburn et al., 1999). Other possibilities include changes in toxin exposure and changes in the microbiome (Schwartz et al., 2017). In control-theory terms, increased obesity prevalence suggests that the feedback gain of the regulatory system is unable to counter currently prevailing open-loop challenges, leading to new, higher BW equilibria; i.e., according to the provisional criteria proposed in §5.1, regulatory precision in response to tonic challenges favoring positive energy balance is weak. As discussed above (§4.1–4.2) and shown in Figs. 5 and 7), such equilibria could result from either simple negativefeedback control or set-point control. Thus, rapid changes in population BMI do not mean (i) that BW is not regulated by a set-point system, as sometimes concluded (Speakman et al., 2011; Tam, Fukumura, & Jain, 2009), or (ii) that set-point theory is unable to explain the effects of television viewing, etc., on BW (p. 735) and “denies a role for socioeconomic and environmental factors in the etiology of obesity” (p. 736). Even in a set-point system, tonic open-loop challenges can make BW regulation less precise. Obesity does not seem to result from a complete breakdown of the BW-regulatory system. In a landmark study, Leibel et al. (1995) studied the components of EE in persons maintained by controlled feeding at steady BW that were equal to, 10% above or 10% below their usual BW. Subjects with healthy BW or obesity had similar non-resting EE (NREE) when tested at their usual BW, displayed similar increases in NREE following 10% BW gain, and displayed similar decreases in NREE following 10% BW loss (Fig. 9). Thus, under these conditions AdEE contributed to BW regulation in persons with healthy BW and persons with obesity. In a follow-up, persons with obesity who lost 10% BW were found to maintain the decrease in NREE for at least a year (Rosenbaum, Hirsch, Gallagher, & Leibel, 2008). Increased NREE after BW gain is mainly due to decreased efficiently of skeletal muscle and constitutes a major component of AdEE (Rosenbaum & Leibel, 2010). A parallel change in BW-regulatory appetite was documented in another similarly designed study (Kissileff et al., 2012): persons with obesity ate similar sized laboratory test meals before and after 10% BW loss, which represented increased intake relative to daily EE, and endorsed decreased feelings of fullness in response to fixed amounts of food. Others reported that BW loss elicited similar increases in subjective appetite as well as changes in some of the psychological features

Fig. 8. Characterization of feedback control of energy intake in patients with type 2 diabetes who were treated with a SGLT-2 antagonist that increases urinary glucose excretion. A. The patients' body weight change (closed circles, means and 95% confidence intervals) was well fit by a linear proportional negative-feedback control system (solid line), but not by an integral control system (dashed line). The feedback modeled was based on changes from the beginning weight; as described in the text, this could be either a simple negative-feedback system or a set-point system. B. The proportional control system together with measured energy expenditure predicted energy intakes (solid line) that approximated the observed intakes (closed circles, means and 95% confidence intervals); again, integral control did not improve the fit. See text for further discussion. Reprinted from D. Polidori A, Sanghvi, R.J. Seeley & K.D. Hall (2016). How strongly does appetite counter weight loss? Quantification of the feedback control of human energy intake. Obesity (Silver Spring), 24, 22892295, with permisson from John Wiley and Sons, Inc.

shown in Fig. 8, the model provided good fits to the observed BW and EI dynamics. Steady-state BW loss was ~4 kg, suggestive of a strong regulatory system. Polidori et al. (2016) also tested the effect of adding an integral control component to the model. This resulted in a worse fit (Fig. 8). The lack of integral control is to be expected given the persistent steadystate BW loss of ~4 kg. Finally, Polidori et al. (2016) also found that estimated EI increased markedly more than EE decreased during the initial ~6 mo of the trial, suggesting that BW-regulatory appetite contributed more to BW regulation than did changes in EE during the dynamic phase of the BW-regulatory response. Taken together, the data indicate that in patients with T2DM and overweight or obesity, (i) pharmacological treatments that increased EE ~200–350 kcal/d (i) activated marked BW-regulatory appetite, and (ii) this appetite together with AdEE maintained BW ~3–4 kg below the pre-challenge level; that is, the regulatory system, whether simple negative-feedback control or set-point control, was strong according to the criteria developed above. 5.4. Tonic challenges favoring positive energy balance 5.4.1. Aging As mentioned in §2.1, aging is associated with slow BW gain in most individuals. Such BW gain might result from any of several changes in the system: (i) a slow deterioration of the feedback-gain process; (ii) a 11

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uncertain. This is a critical gap given the quantitative importance of BW-regulatory appetite in situatins favoring negative energy balance reviewed in §5.3. The apparent similarities in steady-state NREE and compensatory AdEE and BW-regulatory appetite in healthy-weight persons and persons with obesity has been interpreted to indicate that stable maintained levels of BW, whether healthy-weight or obese, reflect set points, “defended levels” or “defended weights” (Leibel, 2008; Leibel et al., 2015; Schwartz et al., 2017). It would be better to term them equilibrium levels of BW. (i) Even if the stable BW of persons with healthy BMI reflects a set point, the malleability of BW indicates that the regulation is weak; if it were not weak, obesogenic environments would not affect BW so much. Therefore, once in an obesogenic environment, BW increases until reaching a new equilibrium with tonic open- and closed-loop controls balanced. There is no need to posit a new, higher set point. (ii) The criterion for a set point offered - that both increases and decreases in BW elicit compensatory changes - is inadequate. Such compensatory changes say more about feedback gain. If feedback gain is an increasing function of BW, as shown in Fig. 3, any change in tonic open-loop challenges would be countered by compensatory AdEE and BW-regulatory appetite responses, and BW would reach a new equilibrium. The particular BW reached has no significance. What the similarity of baseline NREE and responses to 10% changes in BW in persons with healthy-weight or with obesity does suggest is that the feedbackgain functions of persons with obesity operate well, but are parallel and right shifted in comparison with the feedback-gain functions of persons with healthy BW (Fig. 9) (note that a similar right shift is shown in Ferrannini et al. (2014), Fig. 4, although interpreted diferently by them). Such a right shift is consistent with the possibility that sustained obesity re-tunes the feedback-gain funtion of the BW-regulatory system. Box 7 illustrates how this could be modeled and shows that such retuning does not require a set-point model. Persons with obesity typically find it exceedingly difficult to maintain lowered BW (Butryn, Webb, & Wadden, 2011; Franz et al., 2007; Heymsfield & Wadden, 2017; Leibel et al., 2015; Schwartz et al., 2017).

Fig. 9. Adaptive thermogenesis (AdEE) contributes to regulation of both healthy body weight (BW) and obesity. Non-resting energy expenditure (NREE) contributes importantly to AdEE (although some of the change in NREE is due to changes in BW per se, as described in §3.5]). Leibel et al. (1995) measured EE in young adult (age, 30 y) men and women with steady-state healthy BW or obesity first at their usual BW (~68 and ~130 kg, respectively; dashed red lines) and then either increased or decreased their BW by 10% and repeated the EE measurements Total EE and resting EE were measured by indirect calorimetry, and NREE was calculated by subtraction. The graph shows NREE expressed as kcal/d·kg fat-free mass (FFM). Data reproduced with permission from Leibel et al. (1995). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) Data are from R.L. Leibel, M.Rosenbaum & J.Hirsch (1995). Changes in energy expenditure resulting from altered body weight. New England Journal of Medicine, 332, 621-628, and are. used with the permission of the Massachusetss Medical Society.

of BW-regulatory appetite discussed in §2, including loss of control over eating, dietary restraint, and food cravings (Chaput et al., 2007; Drapeau, Jacob, Panahi, & Tremblay, 2019; Rosenbaum et al., 2010). The effect of BW gain on BW-regulatory appetite, however, remains Box 7 Adaptation to chronically elevated BW.

Adaptation may re-tune a control system by changing the set point or changing the feedback-gain process in response to a change in steadystate BW. In the latter case, this is clearest if one assumes a linear proportional system and calculates change in feedback gain. Re-tuning may occur if the BW trajectory meets a steady-state stability criterion, as follows: Let BW x be the average BW in the interval from days in the past to the present time t = x, and let the stability criterion be a constant c. Then x

if

BW (x )

BW

x

dt

c, do not adapt < c, adapt .

(18) x sum to < c will adaptation occur. The integral in Eq. (16) is continuously That is, only if absolute BW changes during the interval updated, so the system re-tunes to any new steady-state BW. A set point is adapted by re-tuning r to BW x if the stability criterion is met. Feedback gain is adapted by changing the baseline steady-state BW shown in Fig. 7 to BW x if the stability criterion is met; i.e., shifting the feedback gain function left or right by the difference between the present steady state and the initial steady state. The effect of this is clearest if one assumes a linear system and considers changes in feedback gain. In the system shown in Box 2, the feedback-gain function is:

F (t ) = kF BW (t ). Thus, changes in feedback gain, F (t ), from the steady state BW, BWSS , are:

F (t ) = kF [BW (t )

(19) (20)

BWSS ]

This changes following adaptation to:

BW x ] (21) Subtracting from Eq. (18) shows that adaptation reduces the change in feedback gain by a constant, BWSS BW x . In a linear system adapting the set point and adapting the feedback gain in this way have identical effects. That is, if r = BWSS , then the system will return BW toward BWSS before adaptation and toward BW x after adaptation. The effect of adaptation may be understood by noting that in the absence of adaptation, feedback from a body weight below BW x but above the initial steady state level will tend to decrease BW, whereas following adaptation, feedback from the same body weight will tend to increase BW.

F (t ) = kF [BW (t )

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The data reviewed above indicate that such BW regain is due in part to compensatory physiological regulation – i.e., AdEE and BW-regulatory appetite (Hall, 2010: Hall et al., 2011, 2012; Heymsfield & Wadden, 2017; Leibel, 2008; Leibel et al., 1995, 2015). In control-theory terms, if persons return to the same environment after BW loss, then the changed feedback-gain due to changed BW would elicit compensatory AdEE and BW-regulatory appetite until the pre-BW-loss equilibrium between open and closed-loop controls is reestablished (similar to the explanations in §5.2). Again, this occurs whether or not there is a set point. As explained above, re-tuning the system around a chronically elevated BW would strengthen this tendency. Finally, it is important to recognize that several other factors may contribute. For example, compared with persons who maintain BW loss, those who regain BW may live in more strongly obesogenic micro-environments, may have less successful training in behavioral or cognitive strategies to counter BW gain (or less capacity for such training) (Castelnuovo et al., 2017; Tronieri et al., 2019; Wadden, Butryn, Hong, & Tsai, 2014), or may have more strongly ingrained habits that interfere with BW control, akin to the persistent cravings of persons with substance abuse histories (Boswell & Kober, 2016; Pleger, 2018).

6.1. The general model of intake regulation This simple negative-feedback control model focuses on regulation of appetite by feedbacks generated by eating rather than by BW (de Castro, 2010; de Castro & Plunkett, 2002). Controls of eating were identified by associational analyses of diet diaries completed by people living in their everyday environments. Two closed-loop or “compensated factors” were identified, estimated pre-meal gastric fill and premeal subjective hunger. Open-loop, “uncompensated factors” included social facilitation, dietary restraint, biorhythms, food cost and availability, and hedonics. Computer simulations indicated that it was necessary to add negative feedback from BW to achieve a steady-state BW (de Castro and Plucket, 2002). Thus, this version of the model essentially adds novel feedbacks from eating that generate closed-loop controls of appetite to a simple negative-feedback control of BW based on feedbacks from BW. Steady-state levels of BW were achieved that represented equlibria between open- and closed-loop controls, essentially identical to simple negative-feedback regulation of BW, as discussed in §4.1. The major innovation the “general model of intake regulation” is the integration of environmental, psychological, physiological and genetic influences on eating into a control model of eating and BW regulation. This should be useful in integrating BW-regulatory appetite with the many other control of appetite.

5.4.3. Persistence diet-induced obesity in rats Under some conditions, rats made obese by offering palatable, highenergy diets (“diet-induced obese” or DIO rats) maintained their obesity even after being switched back to standard chow diets (Levin, 2010; Levin, Triscari, Hogan, & Sullivan, 1987; Rolls, Rowe, & Turner, 1980). Furthermore, following forced BW loss, they returned to the elevated level when ad libitum chow access was reinstated (Levin & Keesey, 1998). Levin, Dunn-Meynell, Balkan, and Keesey (1997) found that this occurs in ~1/3 of Sprague Dawley rats and that the trait apparently results from a polygenic mode of inheritance. These phenomena pose a conundrum for either control-theory model of BW regulation. Levin (2010) and Levin and Keesey (1998) suggest that DIO rats possess an adjustable set point that diet-induced obesity increases. Box 7 illustrates how this could be modeled in control-theory terms. There are, however, difficulties with this hypothesis. (i) The set-point control would have to be strong in order to prevent a decrease in BW when obese DIO rats are switched from the palatable, high-energy diet to chow, but if it were strong it should not have allowed the dietary obesity to develop in the first place. This suggests that the original set point is weak, but the adjusted set point is strong. (ii) The adjustable set-point must have a maximum. That is because BW increased over the level of produced by the original palatable, highenergy diet when an even more palatable diet was offered, but in this case decreased to the DIO level when chow was re-instated (Levin & Dunn-Meynell, 2002). A simple negative-feedback system requires similar ad hoc adjustments to fit these data. The development of dietary obesity is explained simply as an increase in the strength of open-loop variables. The lack of decrease in BW when the DIO rats are switched back to chow, however, seems to require a complex explanation. (i) Because the diet switch had no effect, the open-loop effect of the palatable, high-energy diet seems to have decreased to the strength of the open-loop effect of chow; i.e., to have adapted as described in Box 7. (ii) But to maintain obesity prior to the switch with a weakened open-loop challenge, the feedback gain of the system seems to have decreased as well. As DIO rats may provide insights relevant to the difficulty of many obese patients to maintain weight loss, it is important to further investigate their intriguing BW-regulatory behavior.

6.2. The dual intervention-point model This model posits two partial set-point mechanisms: (i) one operating below the lower border of a “zone of indifference” of BW and generating error signals that oppose further reductions in BW, and (ii) one operating above the upper border of this zone and generating error signals that oppose further increases in BW (Herman & Polivy, 1984; Levitsky, 2002; Speakman, 2007; Speakman et al., 2011). The “zone of indifference” reflects the recognition that homeostatic regulation does not always maintain the regulated variable at an ideal value, but rather maintains it within an envelope of regulation. As a physiological concept, this goes back to Cannon (1929). Cooper (2008) points out that Cannon (i) chose the prefix homeo for homeostasis because it means similar or like, not same or fixed (homo), and (ii) emphasized that homeostatic variables are regulated within “narrow limits,” not at a constant level. The existence of an envelope of regulation, however, does not require the existence of a set point, much less two. An envelope of regulation would occur with or without a set point in several ways: (i) if there were a threshold in the feedback process (or in a downstream processes) so that smaller perturbations produce no change in feedback; (ii) if a non-linear feedback process resulted in little or no signal for small perturbations, as diagrammed in Fig. 3B; (iii) if the physiological dynamics of the feedbacks, of BW-regulatory appetite and AdEE, or of the regulated variable were simply sluggish, as argued by Chow and Hall (2014). Thus, unless neuroendocrine representations of upper and lower set points are discovered, this model seems unnecessarily complex. 7. Discussion The control-theory approach to regulation is based on negative feedback. Negative feedback is a simple but powerful concept with almost unlimited design and engineering applications (Åström & Murray, 2016). It can also be used to model biological regulatory processes at molecular, cellular and organismic levels (Aoki et al., 2019; Goldstein, 2019; Kammash, 2016; Vodovotz, An, & Androulakis, 2013; Wolkenhauer, Ullah, Wellstead, & Cho, 2005; Åström & Murray, 2016). Control theory describes some phenomena of BW regulation well. For these reasons, it seems worthwhile to understand its application to BW regulation and determine the extent to which it is useful or true. This

6. Other models Two further control models of BW regulation are worthy of mention, the “general model of intake regulation” and the “dual-intervention model.” 13

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review aimed to facilitate such these efforts. Four points are worth emphasizing.

regulatory appetite and related hedonic and cognitive influences, as described in §2. If BW regain, the bane of obesity therapy, is a regulatory response, perhaps it can be understood and turned it against itself.

7.1. Control theory is mathematical modeling Because control theory is a mathematical approach to understanding and designing dynamic systems regulated by feedback (§3 and 4), the success of control-theory models to the physiology of human BW must be judged quantitatively. Although few, quantitative controltheory models of BW-regulatory appetite and AdEE give reason for optimism. For example, Polidori et al. (2016) found that linear, proportional simple negative-feedback control provided good fits to the trajectories of both BW and BW-regulatory appetite in persons with obesity and T2DM who were treated with drugs leading to glycosuria, that BW dynamics were shaped more by BW-regulatory appetite than by AdEE, and that BW regulation was remarkably precise (§5.3.2). Quantitative control-theory models have also been applied to BW regulation Several physiological variables thought to contribute to the control of appetite have been included in control-theory models in nonhuman animals (e.g., Booth, 1978; Jacquier, Crauste, Soulage, & Soula, 2014; Tam et al., 2009). These studies provide a solid basis for further efforts to model BW regulation with control theory and should inform efforts to understand the neuroendocrine mechanisms of this regulation. The central role of feedback gain in such modeling deserves emphasis. Fitting a control-theory model to regulatory phenomena is essentially modeling the feedback-gain functions that determine how changes in BW determine BW-regulatory appetite and AdEE.

7.3. The BW set point is probably a fata morgana A major theme of this review is that set-point control and simple negative-feedback control are mathematically fungible, so that characterization of the performance of a control system alone does not serve to identify the underlying type of system. In particular, systems cannot be distinguished on the basis of regulatory precision. In either set-point control or simple negative-feedback control, steady-state precision (§5.1) can be increased as desired by manipulation of feedback gain. More precise regulation is typically produced in engineered set-point systems by I or PI control, but close analogs to set-point I and D control can be designed in simple negative-feedback control (§4.2.1). Although not reviewed here, use of multiple negative-feedback loops can also increase precision. For example, the performance of a system with two or more balanced simple negative-feedback loops, such as controls of the flows into and out of a reservoir, has been shown to be mathematically equivalent to that of a set-point system (Booth, 1980, 2008; Booth, Toates, & Platt, 1976; Peck, 1976). Thus, none of the data considered here establish which type of control model regulates human BW, and debating the likelihood of the existence of a BW set point seems idle. Characterizing feedback-gain functions is more useful. 7.4. Control-theory predictions are testable

7.2. Control theory may be involved in obesity

This review considers control theory models of components of appetite and EE related to BW. The hypothalamus seems to be a critical node in the neural network mediating such functions (Berthoud et al., 2017; Leibel, 2008; Rossi & Stuber, 2018; Sternson & Eiselt, 2017). Thus, in a sense, this review descends from Eliot Stellar's (1954) classic hypothalamic theory of motivation, with his excitatory and inhibitory centers replaced by negative-feedback control. Therefore, it is appropriate to close, as did he and many of his followers, by emphasizing that the theory generates experimentally accessible hypotheses. Advances in neuroscience methodology now make feasible research aimed at testing control-theory models of BW regulation in the brains of mice and rats. Perhaps the near future will see identification of potential neuroendocrine representations of functioning components of control-theory models of BW regulation, setting the stage for tests of their necessity and sufficiency for the control of BW-regulatory appetite and AdEE. Such work is likely to have translational importance for efforts to discover more effective treatments for disordered appetite and BW.

The obesity pandemic is a massive public-health problem (Wolfenden, Ezzati, Larijani, & Dietz, 2019) and research challenge (Schwarz et al., 2017). In terms of control theory, there are two questions. First, does the BW-regulatory system respond with different efficacy to challenges favoring positive energy balance vs. challenges favoring negative energy balance, thus predisposing us to obsity? Second, can the BW-regulatory system be leveraged to combat obesity? Leibel et al.’s (1995) demonstration of similar compensatory changes in NREE in response to 10% BW gain or loss in persons with healthy-weight or obesity (§5.4.2) indicates that AdEE in response to altered BW is largely intact in persons with obesity. BW-regulatory appetite also appears to effectively oppose BW loss in persons with obesity (§5.3). Whether BW-regulatory appetite opposes BW gain in persons with healthy-weight or obesity, however, is unknown and an important research question. Speakman (2007, 2018 ab) has argued that reduced predation risk in human evolution reduced the selective disadvantage of higher BW. Perhaps this changed the feedback-gain function or set-point threshold eliciting BW-regulatory appetite in overweight, thus predisposing modern humans to obesity. Leibel et al. (1995) also found that NREE was similar in persons with obesity and persons with healthy BW when they were maintained at their usual BW. This suggests that the BW regulatory system is tuned differently in the two groups of persons; i.e., feedback-gain functions are right shifted in those with obesity. Whether this is a pre-existing phenotype that contributes to obesity pathogenesis or is an adaptation to obesity is an important research issue. Consistent with the former possibility, lasting changes have been found in the hypothalamic of rodents with diet-induced obesity before the onset of obesity and in rodents simply exposed to high-fat diets (e.g., Bouret et al., 2008; Chhabra et al., 2016; Horvath et al., 2010; Ravussin et al., 2011; Thaler et al., 2012; Valdearcos et al., 2014). With respect to obesity therapy, the possibilities discussed here provide, if not encouragement, at least some focus for the development of obesity therapy. That is, if the BW-regulatory system operates in obesity, it is important to identify and target those regulatory mechanisms that oppose BW loss, including mechanisms of both BW-

Disclosures The author declares no conflicts. Funding No funding supported this work. Acknowledgment Parts of this review were presented during a Symposium in honor of Prof. Barry Levin sponsored by the Columbia University Appetitive Seminar held in New York, NY USA, May 2017, and during a Mathematical Sciences and Obesity Short Course sponsored by the Indiana University-Bloomington School of Public Health held in Baltimore, MD USA, June 2019. I am grateful to Prof. Levin and Prof. David Booth for many helpful discussions on the issues presented here. I also gratefully acknowledge the helpful comments on the initial version of the review made by Prof. Suzanne Higgs, of Editor-in-Chief of 14

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Appetite, and by the anonymous reviewers. I dedicate the review to the memory of Prof. John D. Davis (1928–2018).

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