**Reviewer 1**
**(Dr. E. Cabral Balreira)**

The authors propose an interesting model between Energy intake and body weight based on Fick’s second law of diffusion. This is an important are of research and the present article provides an interesting contribution to this research area and can provide new insight into the body weight mechanisms. The work justifying the use of Fick’s second law of diffusion went to a substantial justification and it is well motivated and explained. The referee agrees with the authors that there should be an investigation of the model.

The referee recommends that a major revision of the paper is required in order to be published in the Biology Direct.

The main issues are outlined as follows.

· In the results section, the authors did not fully disclosed their hypotheses that the energy density of body mass is independent of time. They needed this fact to arrive at their model equation (4). This is not supported by the previous discussion and it is big hypotheses that needs more explanation.

Author reply: *Generally, for adult men (20-33y) in the Minnesota human starvation study, the change in body weight is largely due to fat mass (FM), but not fat-free mass(FFM). As we can see from Kyle* et al.*, fat-free mass does not change much at middle age (from 18-34y to 35-59y), especially when compared with fat mass which changes significantly during the same period* [17]*. Considering that the energy density of FM is much higher than that of FFM, the energy change is largely decided by change in FM. Thus, the change in energy intake, d(ρ*V), is approximately the change in energy in fat mass, which can be represented as p*d(V), in which p is the energy density of fat mass. The energy density of fat mass is supposed to be a constant, so we think the formula d(ρ*V) = p*dV is valid and the possible error here won’t affect our conclusion significantly.*

· In the section titled simulation of body weight change using the developed model, it is unclear how the authors obtained the experimental data.

*Author reply: Total Energy Expenditure (TEE) includes two major parts: Resting Energy Expenditure (REE), the amount of calories needed to maintain basic body systems and body temperature at rest; Activity Energy Expenditure (AEE), the amount of calories used during activity* [20]*. Net energy intake is the difference between food intake and TEE. Although TEE was not measured in the Minnesota starvation study, TEE can be obtained through calculating REE and AEE* [19,21]*.*

*REE is calculated from Basal oxygen (cc/min) and kcalorie equivalent per cc/min. The daily energy expenditure at rest converts cc of oxygen/min into liters of oxygen/day, multiplied by the kcalorie equivalent of oxygen. The caloric equivalent of each cc of oxygen consumed in the resting state is calculated on the basis of Thorne Martin Carpenter’s 1921 table* [19]*. The group’s REE of 994.2 kcal/day at S24 equals group oxygen consumption of 139.1 cc/min multiplied by 1.44 (1440 min divided by 1000) and the groups’ caloric equivalent of oxygen of 4.964 kcal/cc.*

*We here give an example to show how AEE is calculated. 22 miles per week of outdoor walking means 3.14 miles walking per day. A man’s normal walking speed is 3 miles per hour or so. When a 54 kg man walks with speed of 3 mph, the energy expenditure is 3.6 kcal/min. At S24, the group’s body weight is 52.57 kg. The group’s energy expenditure is (52.57/54)*(3.14/3)*3.6*60 = 220.1(kcal/day)* [21]*. When a 54 kg subject walks at 3.5 mph for half hour per week on a treadmill, his energy expenditure is 4.2 kcal/min. The group’s energy expenditure is (52.57/54)*4.2*30/7 = 17.52 (kcal/day)* [21]*. These two parts of walking energy expenditure added, we can know AEE is 237.62 kcal/day.*

*So at S24, TEE is 1231.83 kcal/day, net energy intake is 409.8 kcal/day.*

From their work, in page 8 below equation (9), the authors state that they simply generated data from their own model and use that same data to validate the model. Such approach is circular and does not support the model validation. It simply shows that the ISCEM algorithm is working properly.

The authors must validate their model using the actual experimental data which they display in Table 1. Using the data from the Minnesota human starvation study, the authors need to estimate the parameters of their model, plot the actual results against the model predictions and report the R^{2} value.

Author reply: *In fact, we* ac*tually estimated the model parameters using the experimental data from Table*1*. We actually used the experimental data from Table*1*to validate the model. We also plotted the actual experimental results against the model predictions and reported the R*^{2}*value.*

· Finally, the authors need to better explain how the ISCEM algorithm works and how is the SCEM-UA algorithm optimizing the parameters in their nonlinear problem.

Author reply: *Corrected*.

**Reviewer 2**
**(Prof. Yang Kuang)**

This paper address an interesting but potentially controversial modeling problem that due to the quality or simplicity of the data, may be modeled by other simple or simpler models. There seems to be no real difficulties in fitting the data sets used in the three Figures. For example, using the first few weeks' data, we can find a energy and mass conversion rate for each subject and then use their weekly Total Energy Expenditure (TEE) to predict their weekly weight. Maybe the authors can comment on why such a simple and intuitive approach was not explored?

Author reply: *We proposed a molecular diffusion based model to uncover the relationship between energy intake and body weight. We used the data from the Minnesota human starvation study to verify the validity of our molecular diffusion based model. Because the relationship between body weight and energy intake is not linear, to predict body weight simply using the energy and mass conversion rate is not feasible, even if from a pure data fitting purpose.*

**Reviewer 3**
**(Dr. Chao Chen)**

The authors propose a mathematical model in which body weight at time t is a function of linear combination of an error function, erf(#/#t) (a monotonic increasing function), and its complement 1-erf(#/#t)(a monotonic decreasing function), derived from the hypothesis of molecular diffusion following Fick’s second law. The model is found to have a good fit to a set of data taken from the Minnesota human starvation study. However, only data from the second phase of the study during the 24 weeks starvation period are used for model fitting; excluding data of the control and recovery phases from the same study.

Author reply: *In order to make clear how the body weight is affected by energy intake, we chose the data of starvation period from the Minnesota human starvation study*.

The authors claim: “This model provides valuable insights into the neural basis of behavioral decisions and their resulting effects”. It is difficult to see, on the basis of the presentation, any mechanistic connection as claimed. This article is just a data fitting exercise because similar models that are linear combination of two monotonic functions of opposing trends can also adequately fit the data.

Author reply: *This sentence, “This model provides valuable insights into the neural basis of behavioral decisions and their resulting effects”, is deleted.*

*We considered that molecular diffusion (of, for example, neuropeptides) plays an important role in body weight changes. Because molecular diffusion is accompanied by energy transference, we then describe the molecular diffusion based process with energy diffusion.*

*Our purpose is not to do data fitting exercise, but to use the data from the Minnesota human starvation study to verify the validity of our molecular diffusion based model*.

Furthermore, this data fitting exercise leaves a lot to be desired: e.g., only the mean body weight over time were analyzed, as presented in Figures 1–3; no body weight changes from individual’s baseline was analyzed; and no statistical analysis, such as confidence intervals, for predicted body weight changes were provided.

Author reply: *Please see Appendix A and Appendix B*.

Editorial issues:

Pages 7–8. Something must be wrong: it is unlikely that parameters are estimated to be identical when different data sets from S1-S24 and S1–S12 are used.

Author reply: *Corrected*.

First line on top of p9: “are” should be deleted.

Author reply: *Corrected*.