Monday, January 30, 2012

Kleiber's law and its possible implications for obesity

Kleiber's law () is one of those “laws” of nature that is both derived from, and seems to fit quite well with, empirical data. It applies to most animals, including humans. The law is roughly summarized through the equation below, where E = energy expenditure at rest per day, and M = body weight in kilograms.

Because of various assumptions made in the original formulation of the law, the values of E do not translate very well to calories as measured today. What is important is the exponent, and what it means in terms of relative increases in weight. Since the exponent in the equation is 3/4, which is lower than 1, the law essentially states that as body weight increases animals become more efficient from an energy expenditure perspective. For example, the energy expenditure at rest of an elephant, per unit of body weight, is significantly lower than that of a mouse.

The difference in weight does not have to be as large as that between an elephant and a mouse for a clear difference in energy expenditure to be noticed. Moreover, the increase in energy efficiency predicted by the law is independent of what makes up the weight; whether it is more or less lean body mass, for example. And the law is very generic, also applying to different animals of the same species, and even the same animal at different developmental stages.

Extrapolating the law to humans is quite interesting. Let us consider a person weighing 68 kg (about 150 lbs). According to Kleiber's law, and using a constant multiplied to M to make it consistent with current calorie measurement assumptions (see Notes at the end of this post), this person’s energy expenditure at rest per day would be about 1,847 calories.

A person weighing 95 kg (about 210 lbs) would spend 2,374 calories at rest per day according to Kleiber's law. However, if we were to assume a linear increase based on the daily calorie expenditure at a weight of 68 kg, this person weighing 95 kg would spend 2,508 calories per day at rest. The difference of approximately 206 calories per day is a reflection of Kleiber's law.

This difference of 206 calories per day would translate into about 23 g of extra body fat being stored per day. Per month this would be about 688 g, a little more than 1.5 lbs. Not a negligible amount. So, as you become obese, your body becomes even more efficient on a weight-adjusted basis, from an energy expenditure perspective.

One more roadblock to go from obese to lean.

Now, here is the interesting part. It is unreasonable to assume that the extra mass itself has a significantly lower metabolic rate, with this fully accounting for the relative increase in efficiency. It makes more sense to think that the extra mass leads to systemic adaptations, which in turn lead to whole-body economies of scale (). In existing bodies, these adaptations should happen over time, as long-term compensatory adaptations ().

The implications are fascinating. One implication is that, if the compensatory adaptations that lead to a lower metabolic rate are long term, they should also take some time to undo. This is what some call having a “broken metabolism”; which may turn out not to be “broken”, but having some inertia to overcome before it comes back to a former state. Thus, lower metabolic rates should generally be observed in the formerly obese, with reductions compatible with Kleiber's law. Those reductions themselves should be positively correlated with the ratio of time spent in the obese and lean states.

Someone who was obese at 95 kg should have a metabolic rate approximately 5.6 percent lower than a never obese person, soon after reaching a weight of 68 kg (5.6 percent = [2,508 – 2,374] / 2,374). If the compensatory adaptation can be reversed, as I believe it can, we should see slightly lower percentage reductions in studies including formerly obese participants who had been lean for a while. This expectation is consistent with empirical evidence. For example, a study by Astrup and colleagues () concluded that: “Formerly obese subjects had a 3–5% lower mean relative RMR than control subjects”.

Another implication, which is related to the one above, is that someone who becomes obese and goes right back to lean should not see that kind of inertia. That is, that person should go right back to his or her lean resting metabolic rate. Perhaps Drew Manning’s Fit-2-Fat-2-Fit experiment () will shed some light on this possible implication.

A person becoming obese and going right back to lean is not a very common occurrence. Sometimes this is done on purpose, for professional reasons, such as before and after photos for diet products. Believed it or not, there is a market for this!


- Calorie expenditure estimation varies a lot depending on the equation used. The multiplier used here was 78,  based on Cunningham’s equation, and assuming 10 percent body fat. The calorie expenditure for the same 68 kg person using Katch-McArdle’s equation, also assuming 10 percent body fat, would be about 1,692 calories. That would lead to a different multiplier.

- The really important thing to keep in mind, for the purposes of the discussion presented here, is the relative decrease in energy expenditure at rest, per unit of weight, as weight goes up. So we stuck with the 78 multiplier for illustration purposes.

- There is a lot of variation across individuals in energy expenditure at rest due to other factors such as nonexercise activity thermogenesis ().


Paul Jaminet said...

Hi Ned,

I would have thought a major factor in Kleiber's law would be the surface area to volume ratio, which would lead to something close to a 2/3 exponent, not far from 3/4. Then I could see other factors, like compensatory adaptations, moving the exponent to what is observed.

If that's the case, then most of the change in energy expenditure in the obese should be recovered as the weight is lost, not with a long adjustment period. So it looks to me like the process of weight loss, rather than obesity itself, is more like to be inducing a reduced metabolic rate.

Beth@WeightMaven said...

I would also wonder how the composition of the body weight affects metabolism post weight loss. Going from obese to lean often involves loss of LBM (especially if bad diet and/or exercise -- read cortisol-inducing chronic cardio a la Biggest Loser is the mechanism).

It may just be wishful thinking on my part, but I'm hoping that eating the right diet and doing the right kind of exercise not only minimizes LBM but also sends the right kinds of signals to actually help improve metabolism.

Exceptionally Brash said...

Oh, I am done trying to model it! As I get older, I do notice more and more: thyroid function = f(energy expenditure). I can almost hear the machinery grinding up or down, the correlation very high. So, reducing diets (or increasing diets, as long as I say "fixed" from the leptin reset) are toast.

Chase Saunders said...

Good point, Paul, and I think it's correct re: 2/3 of the 3/4. Much is written in biology about the surface area to volume ratio, so someone may have quantified head loss, but I would have to guess it's proportionate.

Still, the gap is 8%. From a metabolic perspective this is still huge and may well fit with the narrative you describe. It's fascinating to think this could quantify the adaptive response of an organism to it's particular point in the surface area to volume continuum.

Chase Saunders said...

Since my initial comment was directed at Paul, I should have said "the narrative that Ned describes".

LeonRover said...

Hey Ned.

Mike Eades had a blog on this Sept 2009

He reproduces Kleiber's hand drawn data on log-log paper. There are two other annotated lines Weight and Surface bounding the data, but with no indication as to the significance.

My copy of McArdle,Katch^2 (pub. 2000) contains reference to Cunningham FFM (fat free mass), Harris & Benedict (body mass, height and age). There is, in addition, a nomogram which calculate BSA ( body surface area) from an equation which has the form

BSA= const X height^0.725 X body-mass^0.425

multiply this by a emissivity rate which varies by age and gender to derive an RDEE (BMR).

This is a mess! Particularly as nowhere is a mention of standard errors of estimation, or any of those nice things which data nerds with a background in AOV like to see. (Which reminds me that Staffan Lindeberg's AHS presentation on Youtube includes some of the best graphical data plots I have come across. McArdle,Katch^2 have a lot to learn.)

One intriguing question springs to my thermodynamically educated mind - have any studies been done on the basal temperatures of these mites and giants? I am minded to ask as Petro HyperVet has recently mentioned that "little furry rodents" are inordinately warm!

Mind you, once one discusses emissivity rates & temperatures, one is into Thermo Law 2 and Entropy . . . . Hmmmm


Marie Curious said...

Hi Ned, your posts often bring concepts into the nutrition discussion that I don't see anywhere else and I thoroughly enjoy them. Could you help me better understand your reasoning in this one? It seems critically based on your statement that "It is unreasonable to assume that the extra mass itself has a significantly lower metabolic rate,...". Why do you think it unreasonable? Or rather, why unreasonable if we're talking about the same individual at two weights? I though that the mass an obese person accumulates is Mostly Fat (not Mostly muscle, or organ or bone...) or is that just a common fallacy? If however it Is indeed mostly fat, could you calculate if the energy expenditure difference per pound of fat vs.pound of muscle is enough to account for the apparently better efficiency of the same person in the obese state? Is that even possible?
Thanks for considering this!

Ned Kock said...

Hi Paul. Yes, much of the thinking behind the law relies on the surface to volume ratio, but that doesn’t agree as well with the data as the 3/4. There are exceptions to the latter as well. As for the inertia, many studies suggest that it exists, compounding the lower metabolic rate associated with lower weight.

Ned Kock said...

Hi Leon. Smaller bodies tend to dissipate more energy than bigger ones, which is in part due to the higher surface-volume ratio that Paul and Chase referred to.

Ned Kock said...

Hi Marie, thanks. Well, there was a lot of talk a while ago about muscle burning a lot of energy, way more than fat. It turned out not to be true.

“Marie Curious” eh? I like that.

Ned Kock said...

By the way, bigger animals generally tend to live longer than smaller ones, which some have taken as evidence that a lower metabolic rate increases longevity.

But I don’t think this is the cause of the overweight-longevity paradox:

LeonRover said...

"evidence that a lower metabolic rate increases longevity"

or even

"that longevity lowers metabolic rate"

Either way it might be conjectured that mouse-seconds tick faster than whale-seconds in terms of how many caesium atom vibrations contribute to either!!

Ahh - the mysterious milieu interieur.


Sam Knox said...


Not to pick on Stephan Guyenet, but he once told me in a comment reply that people stop getting fat when energy expenditure rises to equal intake:

"As body mass increases, energy expenditure increases, so you will not keep gaining fat mass indefinitely. A person will reach a plateau when energy intake matches expenditure."

Kleiber's Law clearly precludes this from happening by an increase in BMR, and I don't know of any evidence that the overweight or obese spontaneously engage in increased levels of physical activity, although they may do so as a part of a weight-loss program.

Your (or anyone else's) thoughts?

Morris said...

Application of Kleiber’s law to fat gain (implied in your analysis) leaves me blurry-eyed. Nick Lane presents a very good and interesting analysis in Power, Sex, Suicide. According to Lane, the ¾ exponent only holds across a wide range of species but not within species, where it varies widely; ie ¾ is an artefact. Original work done with dogs supports a 2/3 scaling. Lane analyses the question in terms of metabolism not networks (per the Sante Fe thinking) and indicates variation of the scaling with the metabolic activity of particular tissue eg active tissue like muscle scales more closely to 1. Overall it seems that ¾ is a reasonable number but for the wrong reasons. Extrapolation to fat gain seems like quite a stretch.

johnnyv said...

Re: Sam Nox

Seems like the "plateau" would more likely be due to filling current fat cells to maximum capacity so that they leak triglycerides despite of high insulin levels.

Suddenly there is more fat available for energy production until the number of fat cells increases again.

Then when you lose all that weight back to your original(if at all possible) you have more fat cells with less fat per cell so way less leptin production.

If only it was as easy to have those extra fat cells die off as it was to produce new ones.

Marie Curious said...

Thanks for liking that Ned ;-) My area is, oh so coincidentally, physical chemistry. However right now, I'm just a dog with a bone, can you help me out? I remember the 'bust' about how much more energy muscle burns vs.fat and that no, you don't burn a whole lot more calories a day just by building up muscle. Yet, bear with me here : a lb.of fat at rest burns ~3 calories per day, while a lb.of muscle ~6 calories (some texts say 2 and 5, but I'm choosing the easier ratio!). Now, if someone gains weight, they change their fat % quite a bit (eg. at normal weight they may be 25% fat, but when obese they'd be 45% fat, or 50% or more...). Now they have More of their body (eg.45%) burning at the lower calorie rate (3 kC/lb/day). So their total expenditure would not scale linearly with their weight. Right? This of course is at rest. Once they move, things change dynamically because it takes more energy to move the extra weight.
I was just wondering if it's possible to model this? And does it seem like this difference in body composition may be a significant factor in the change in energy expenditure of the obese? In which case there wouldn't be a long adjustment period after weight loss (looking to give my overweight cousin some hope, she saw that recent NY Times article!)

Ned Kock said...

Hi Sam. Stephan is right about that, as he is about many things. The main reasons are those mentioned by johnnyv, along with others, such as a limit on what someone can eat. (Even that guy in the Monty Python movie exploded after too much eating; clear scientific proof that there such a limit!)

In fact, the megafat are somewhat rare, and often quite resistant to complications that are associated with lipotoxicity, including type 2 diabetes. Unlike most of us, the fat cells of the megafat take much longer to become insulin resistant.

Ned Kock said...

Hi MC.

> a lb.of fat at rest burns ~3 calories per day, while a lb.of muscle ~6 calories (some texts say 2 and 5, but I'm choosing the easier ratio!)

Well, with a difference of 3 calories per lb, times 60 (210-150), would give 180 calories per day. That is a bit less than the 206 calories per day from Kleiber's law.

It is possible, but we still have studies like Astrup et al.’s suggesting that the metabolism of the formerly obese is slower.

Marie Curious said...

I see ! and that's an interesting study, thanks again Ned.
Spurs thought> I wonder if we know whether a formerly obese person's composition has changed yet again on losing weight, that is, Not back to the original composition. Vague recollections of 'slow twitch muscle fibers' more efficient than 'fast' ones and building up slow ones to help carry/move extra weight. But then, that circles back to some degree of 'system adaptation' I guess :-) That would take time indeed to readjust (maybe with HIIT or Tabata for fast twitch -I wonder...).

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