The mouse has become the most popular organism for investigating molecular mechanisms of body weight regulation. But understanding the physiological context by which a molecule exerts its effect on body weight requires knowledge of energy intake, energy expenditure, and fuel selection. Our simple mathematical method calculates the dynamics of energy output and fuel selection over extended time periods using longitudinal measurements of body weight, food intake, and body composition. We showed that our method can detect both transient changes of energy expenditure and net fat oxidation rates as well as longer timescale changes found with weight gain and loss. Similar methodology has been previously developed by our group to relate human body-composition changes with dynamic adaptations of fuel selection in both adults 
and infants 
. The method is especially well-suited for mouse studies because food intake can be accurately measured over the extended time periods required to measure significant changes of body weight and body fat. While we have applied the model to data averaged within groups of mice, it would be also interesting to examine individual mouse trajectories as a way of investigating inter-individual variability.
Our equations extract information about energy output that is already present in the body weight and food intake data. Other than the law of energy conservation, the only assumption was that the relationship between changes of body fat and fat-free mass were described by a well-defined function in accordance with the Forbes theory of body composition change 
. This assumption was confirmed in the present study for mature male C57BL/6 mice () and we hypothesize that genetic manipulations can alter the shape of this function. However, once the function has been determined we showed that it provided accurate estimates of body fat changes in an independent feeding experiment using body weight measurements alone (). Therefore, knowledge of the Forbes curve for a given mouse model eliminates the need for frequent body composition measurements.
To estimate the net fat oxidation rate and RQ, an additional assumption regarding carbohydrate balance was required (see Methods
). We found that the Forbes function () determined the relationship between food intake, body composition change, and net fat oxidation rate 
. While both humans and mice have Forbes functions that increase with body fat mass, the concavity of the curves is opposite 
. Therefore, great caution should be exercised when extrapolating fuel selection results in mice to predict human responses. The physiological reason for this difference is presently unclear. Our research group is actively engaged in developing detailed models of the complex interactions between carbohydrate, fat, and protein metabolism in humans 
to better understand the relationship between the physiological drivers of fuel selection and the Forbes body composition curve. We plan to develop similar models in mice to help understand these relationships and the differences between the species.
In contrast to our method, currently available techniques for estimating energy expenditure are expensive, involve a plethora of assumptions, and can impact the behavior of the mice 
. These factors make it common to find reports of energy expenditure rates that are quantitatively inconsistent with the measured energy intake and body weight changes found in mice that were not subjected to these procedures. As an illustrative example, consider the recent publication by Funato et al. where the energy intake rate of the wild type mice was at least 17 kcal/d and the energy expenditure measured by indirect calorimetry was less than 5 kcal/hr/(kg BW)0.75
. This translates to an absolute expenditure rate of less than 10.7 kcal/d for a mouse that was at most 40 grams at the time of measurement 
. Such a large positive energy balance would translate to a rate of weight change of at least 4.7 g/week (if all excess energy was deposited as fat) versus the measured weight gain which was less than 1 g/week. The purpose of this example is not to criticize the work of Funato et al., but rather to highlight how even careful indirect calorimetry and food intake measurements can lead to estimates of energy imbalance that are inconsistent with the weight gain measurements.
Our own attempt to use indirect calorimetry to validate the model predictions of energy expenditure and fuel selection highlighted two important issues. First, the mice that were consuming the high energy diets lost significant amounts of weight when moved to the indirect calorimetry cages indicating that their behavior was not representative of the mice not subjected to the procedure. Second, the measured energy expenditure rates were unrealistically high compared to the model predictions for all groups of mice. In fact, the measured energy expenditure rate was higher than the measured energy intake in the chow-fed mice that did not lose weight (an impossibility) and greatly exceeded the expenditure required to explain the weight loss in the mice fed the high energy diets. These discrepancies led us to diagnose a technical problem with the indirect calorimetry equipment. Thus, we were unable to validate the model estimates of energy expenditure and fuel selection.
The field of farm animal nutrition has a long and rich history of using mathematical modeling to analyze animal growth and identify nutritional factors that potentially limit growth rate 
. The simplest models describe the efficiencies of various diets in their ability to deposit body energy, often specified in terms of body fat and protein 
. Inputs to such models include energy intake, body weight, and the rates of body fat and protein deposition. The model outputs include the efficiencies of protein and fat deposition as well as the so-called maintenance energy requirement which is roughly defined as the energy intake required when the animal is not growing. An alternative representation uses energy intake, body weight, total energy expenditure (by calorimetry methods), and protein deposition rate (via nitrogen balance) as model inputs and predicts the maintenance energy requirement, fat deposition rate, and body protein and fat deposition efficiencies.
At the next level of complexity, animal growth models prescribe an energy partitioning rule that specifies how body protein will accumulate for a given food intake rate as a function of body weight, age, or body protein. Energy partitioning rules are often complex 
, but can be thought of as similar to the Forbes function that specifies how energy imbalances are partitioned between body fat and fat-free mass. A significant difference is that our approach is applied to mature mice whose overall growth rate was minimal despite their ability to gain and lose fat-free mass in response to the various diets.
Once the partitioning rule is specified, the outputs of animal growth models include body fat mass, maintenance energy requirement, as well as body fat and protein deposition efficiencies given the food intake and body weight as model inputs. In contrast, our model outputs are body fat mass, fuel selection, and total energy expenditure which are more relevant for mouse obesity studies and avoids the known problem of arbitrarily distributing total energy expenditure between tissue deposition costs versus maintenance energy requirements 
. Animal growth models have often used power-law functions of body weight to model the maintenance energy requirements that were previously calculated using the above methods. Once specified, the model of maintenance energy requirements can be used along with the calculated efficiencies of protein and fat deposition and the energy partitioning rule to predict body weight and body fat change as a function of the food intake 
. We are presently developing a model of total energy expenditure in mice that will allow prediction of body weight and composition changes as well as fuel selection when food intake is the only input to the model.
A weakness of our methodology is that it does not distinguish the various components of energy output including resting metabolic rate, thermic effect of feeding, adaptive thermogenesis, physical activity, or any changes of energy excreted in urine and feces that are unaccounted for by the estimates of diet metabolizability. Furthermore, the method does not operate on a within-day time scale and therefore cannot address changes between day versus night or transitions between fed and fasted states. Indirect calorimetry is required to address these issues and would provide important information for the interpretation of our calculated longer-term estimates of energy output and fuel selection. We believe that the combination of our continuous-time methodology with indirect calorimetry measurements at judiciously chosen time points can be applied to various mouse models of obesity as a powerful tool for characterizing the metabolic dynamics underlying experimentally observed body weight changes.