Obesity, weight gain and a cookie

Many books and articles on dieting, weight gain, and obesity often quote the rule of “3500 Calories is a pound”. By this, they mean that for every 3500 “extra” Calories consumed, you will gain one pound. So if you ate 100 extra Calories a day (e.g. a cookie), you would gain a pound in 35 days. In metric units, this is equivalent to 7700 Calories per kilogram. As I will show below, this rule is confusing and wrong on two accounts, although by sheer coincidence, it can be used as a mnemonic to estimate how extra food will lead to increased body weight. The basis of this rule is that the metabolizable energy density of fat is about 9300 Calories per kilogram and protein is about 4000 Calories per kilogram, and if most of our weight gain is in fat then you can persuade yourself that 7700 Calories per kilogram is a reasonable number. I should note that the dietary Calorie is equivalent to one kilocalorie or about 4.2 kilojoules of energy.

The first reason this rule is wrong is that the energy density of deposited tissue is not fixed at 3500 Calories per pound because the composition of that tissue is not fixed. It depends on how much you weigh and possibly diet composition. Kevin Hall has a nice article on this. The second problem is that this rule completely ignores the fact that as you gain weight your energetic needs change so not all of the extra 100 Calories can go to new tissue. If you apply the rule, it says that if you eat an extra 100 Calories per day, or one cookie, then in 35 days you’ll have gained a pound and in one year you’ll have gained 10 pounds. This is actually reasonable and close to being correct. However, if you continue to apply the rule then you’ll also predict that in 10 years you’ll gain 100 pounds and your weight will increase linearly forever. This is clearly not correct. What really happens is that as you gain weight you also burn more energy and eventually you’ll reach a new steady state where you burn what you eat.

The rule is also confusing because there are at least three ways of interpreting “eating an extra cookie per day”. What it was originally intended to address is the situation where everything else in your life is constant (i.e. eat the same thing, do the same amount of physical activity, and have no change in health), and then you eat an extra cookie. The presumption is that previously you were burning everything you eat and now you have an extra 100 Calories per day that you store as new body tissue. A second interpretation is that you are always exactly one cookie or 100 Calories out of energy balance every day. So, whatever amount of energy you are burning at this moment, you eat 100 Calories extra. This would imply that you are always out of energy balance and your weight would indeed increase linearly in time. The third interpretation is that you eat an extra cookie per day over what you ate the previous day. This would imply that you eat one cookie today, two tomorrow, three the next day and so forth. Because, the rule is inherently wrong and that it can be interpreted in multiple ways, it has led to great confusion and myths about dieting and how much you need to eat to lose or maintain weight. Below, I will try to make all of these concepts precise.

At the simplest level, body weight change is about conservation of energy. We take in energy in the form of food and we burn energy to maintain bodily functions and perform physical work. Any excess energy we take in gets stored in new body tissue. The confounding part of modeling this process is that our bodies are not homogeneous so how much weight you gain will depend on what form of tissue that excess energy was stored in. In my paper with Kevin Hall, I showed that you can model this process with two differential equations and for small changes in weight it reduces to a simple linear differential equation which can be written as

$\rho\frac{dW}{dt}=I-a(W-W_b)$ (1)

where $W$ is body weight in kilograms, $\rho$ is the energy density of added body tissue measured in Calories per kilogram, $I$ is the metabolizable food intake rate in Calories per day, and $a (W-W_b)=E$ is the energy expenditure rate in Calories per day, which to first approximation is linear in the body weight. The slope parameter $a$ measured in Calories per kilogram per day is a measure of how much extra energy is burned for an increase in body weight. The parameter $W_b$ is just some constant that can be measured. The equation is considered to be valid on time scales of a week or longer so the fluctuations in intake and expenditure rates are averaged over, i.e. daily and weekly variations are averaged out.

We can now consider the consequences of an extra cookie per day. The body is in steady state $\rho \frac{dW}{dt} = 0$ , when it is in a state of energy balance $I= E$ , (everything that is eaten is burned or excreted). The first interpretation of an extra cookie per day is that you are in a steady state with a steady lifestyle and your average daily intake rate is fixed at some constant like 2000 Calories per day. You now eat that exact amount plus one extra cookie, e.g. 2100 Calories per day. So what will happen is that you will be in positive energy balance and your weight will increase and will approach a new steady state exponentially. The exact time course can be written down by solving equation (1) but for our purposes that is not necessary. We know that at steady state $W=I/a+W_b$ , so if you eat an extra $\Delta I$ calories then you will eventually reach a new steady state of $W=(I+\Delta I)/a+W_b$ and your weight gain will simply be $\Delta W = \Delta I/ a$ . In our recent paper in PLoS One, we estimated the average slope parameter $a$ for the US population to be 22 Calories per kilogram per day. However, this value will vary from person to person and depends on their metabolic rate and physical activity level. If you exercise a lot the slope will be bigger. What this implies is that an extra 100 Calorie cookie per day will lead to a weight gain of about 4.5 kilograms or 10 pounds.

The second interpretation is that you are always one cookie per day out of energy balance. In that case, equation (1) takes the form of $\rho \frac{dW}{dt} = C$ , where C is the caloric content of a cookie per day. This can be easily solved and you’ll find that the weight will then increase linearly as $W=(C/\rho) t + W_0$ , where $W_0$ is the weight at time $t=0$ . In our paper for the US population, we estimated that on average $\rho= 9100$ Calories per kilogram, which is slightly larger than 3500 Calories per pound. Hence, being one cookie out of energy balance leads to a weight gain of $100/9100 \times 365 = 4$ kilograms per year. The amount extra you eat corresponding to this linear weight gain is found by solving equation (1) for the intake, which gives $I = C+a((C/\rho) t +W_0-W_b)$ . The change in food intake per day (or intake acceleration) is then given by $a C/\rho = 22\times 100/9100 = 0.2$ Calories per day per day. So we eat 100 Calories extra the first day and then add a miniscule 0.2 Calories per day for each day after that.

Finally, the third interpretation is that you eat an extra cookie per day, so you eat one cookie today, two cookies tomorrow and so forth. In this case the equation (1) is $\rho \frac{dW}{dt} = I + c t - a(W-W_b)$ , where c is now 100 Calories per day per day, i.e. it is the acceleration of food intake. This equation can also be solved exactly and has the solution $W = (c/a) t +((I-a W_b)/a- c\rho/a^2)(1-\exp(-(a/\rho)t)+W_0$ , which basically means that after the transients decay, the weight will increase at an astonishing rate of $c/a = 100/22 = 4.5$ kilograms per day. Obviously, this could never occur because we could never eat that much.

Let’s return to scenario one where we have a fixed diet and we increase it by one cookie. The 3500 Calories is a pound rule amounts to ignoring the increase in energy expenditure so that equation (1) becomes $\rho \frac{dW}{dt} = I - E$ , where $E$ is now some fixed constant. From this you would assume that $W = ((I-E)/\rho) t$ and your weight would increase linearly forever. If you looked at just the change in weight by increasing your food intake by $\Delta I$ then this is equivalent to saying that $\Delta W = (\Delta I/\rho) t$ . The real equation from above is $\Delta W = \Delta I/a$ . So, if we set the two equal to each other we find $(\Delta I/\rho) t = \Delta I/a$ and solving for time $t$ we get $t =\rho/a = 7700/22 =350$ , days. Hence, if we applied the 3500 Calories is a pound rule for about a year then we get the correct result for the new steady state weight. The reason this works is because of a complete numerical coincidence that the time constant for weight change $\rho/a$ is 350 days or nearly a year.