This is an important and well-written paper that deals with the efflux P-glycoproteins (P-gp), exchange of P-gp between cells including a mathematical model of this process, and the role of the P-gp transfer in bringing about extragenetic resistance to chemotherapeutic treatment of breast cancer.

### 1. General Comments

1. I think more biological discussion of P-gp activity and the nature of P-gp transfer is warranted. In particular, what is the time scale for transfer processes and is the level of cellular P-gp activity heritable?

*Quantitative values, relatively to arbitrary fluorescent units given by flow cytometry, are now added in the results in term of heritable cellular protein and activity, as well. This question and the point concerning the time scale for transfers are also discussed more extensively*.

2. To put formation of chemotherapeutic drug resistance into a more practical perspective, it would be helpful to mention whether normal breast tissue cells display P-gp based efflux activity and whether P-gp exchange may occur between normal and malignant cells.

*The one and only article showing a possibility of P-gp transfers between normal and cancer cells is the report of Rafii et al. (2008) detailed at the beginning of the discussion. The possible basal P-gp activites in normal breast and its consequences in terms of MDR are discussed p18*.

3. Pp. 7-8. The mathematical model is introduced too abruptly. I believe the authors should write a paragraph or two about underlying general assumptions, the "big" stochastic picture behind their model, explain the formula for the transfer operator including the role of normalization in more detail, and give stochastic interpretation of coefficient τ.

*A part concerning data conditioning and normalization has been added p7 in Methods section. We have also modified the presentation of the model, recalling the precise assumptions for the transfer process*.

4. P. 8. The paper deals with the model of P-gp transfer with ε = h = 0. Given the predominantly biological readership of the journal, I think it would be better to omit these terms in the model and refer the reader to paper [20] for a more general model accounting for diffusion and drift.

*We agree with the referee, and we modified the system accordingly to the comment*.

5. Paper [20] establishes the existence, under certain simplifying assumptions, of the limiting equilibrium distribution of P-gp activity. What is it in a slightly more general case of two thresholds considered in the present work (in the absence of cell proliferation)? How fast is this equilibrium reached? Is this relaxation process slower or faster than cell cycling?

*The two threshold case was in fact also considered in the paper by Hinow et al. Up to now the authors did not study the speed of convergences to the equilibrium distributions (which is indeed a difficult question). In the present model the cell cycle is simplified but includes the growth rate of cells (neglecting the oscillation induced by the cell cycle). This problem will be considered in future works*.

6. What are the relative effects of P-gp transfer and cell proliferation on chemotherapeutic activity of DOXO? One can compare survival of two-compartment tumor cell population exposed to DOXO under two scenarios: (1) no cell proliferation, just P-gp exchange and (2) cell proliferation alone assuming equilibrium distribution of P-gp activity. This would give some idea about the net effect of P-gp transfer.

*Indeed we did not consider the point (1) since in cancer the case with no cell proliferation does not arise. Point (2) corresponds to Figure*8A&8C, *where cell growth of a two-compartment tumor cell population was plotted without transfer*.

7. It is not enough to say that model parameters were "estimated by using intensive parameter estimation computations". Which experimental data were used for model fitting? How was the method of least squares implemented? What weights were used, if any? Also, what are the confidence limits or standard deviations of model parameters? If the confidence intervals are too wide the reported parameter estimates may be misleading.

*This point has been detailed*.

8. References [22–44]-46] were never mentioned in the text. Am I missing a supplement to the paper?

*Corrected*.

### 2. Technical Comments

1. P. 7, last line. "... fluorescence p is assumed to be a function of P-gp expression on the cell surface." It is important to emphasize that this function is strictly increasing.

*OK, this has been added*.

2. P. 9, lines 6-7 from below. Shouldn't the sign of the transfer terms be plus in both cases? Otherwise, if p1 < p2 then the cell with smaller activity loses it even further while the cell with larger activity gains it.

*The signs are correct in the paper, and the part has been carefully explained in the companion paper of Hinow et al*.

3. P. 9. What is the biological basis of the two thresholds, δmin and δmax?

*The mentioned thresholds are a conjecture introduced to investigate whether biological data contained constraints in the transferred quantity when fitted with the proposed model. See answer to general comment 3*.

4. P. 14, lines 6-7 and p. 18, lines 2-3. Please explain in more detail the interpretation of coefficient τ. Is P-gp transfer thought of as some kind of Poisson process? See also general comment 3.

*Yes, the time between transfers is assumed to follow an exponential law. This part is now contained in the paper; see assumption (A2) page 8*.

5. P. 15, lines 5-7, Figure 8B and p. 26, legend to Figure 8B. Where does the function ρ(p) displayed in Figure 8B come from? Is it determined experimentally?

*This function has been established empirically given that it is known that resistance increases as a function of efflux activity which depends on the expression of P-gp (see ref 15, 39 and 40). However, there is no precise study linking the number of P-gp copies at the membrane and the efflux activity. It has to be considered that P-gp pumping is regulated in the minute range, at least by the phosphorylation state of the protein. As a consequence for a given cell at a particular time, a heterogeneous population of P-gps, with pumping activities varying from 0 to a maximal rate, can coexist in the membrane*.

6. P. 24, legend to Figure 3, lines 1-2 from below. Where are these "vertical solid lines"?

*It was corresponding to an older version of the figure. This sentence has been removed*.

7. P. 25, legend to Figure 5. Please explain the role and selection of quadrants in more detail.

*This whole legend has been re-written for clarification*.

8. P. 25, legend to Figure 6. What are the cell volumes matched to? Give the matching volumes or a reference.

*This whole legend has been re-written for clarification. This particular issue is developed in the result section p12 lines 1-5 from below*

9. P. 27. Check the estimates for δmin and δmax. Shouldn't they be 1.4 and 1.9? What are the units?

*It is correct in the paper. The confusion comes from the fact that we used for numerical simulation log(p) in Figure*6*(as well as for the numerical simulations) while in Table*1*the result is expressed only using p. Figure*6*has been corrected*.

*We thank the referee to point out this problem!*

10. P.35, Figure 7. Fluorescence counts are reported on the log scale. Were parameters estimated on the same scale? Changing scale may affect the estimates of model parameters.

*See the answer of point 9 above*.

### 3. Stylistic Comments

1. P. 5, line 10. The paper denoted [ref] is not specified.

*This work is submitted. The given reference is related to an oral communication based on the study*.

2. Use a uniform name for the cell line MCF-7/DOXO. In some places in the text it is called MCF-7/doxo, see e.g. pp. 6 and 7.

*We harmonized the name of cell line to MCF-7/Doxo throughout the text*

3. P. 6, line 2 from below. What is S.E.M.?

*It is the standard error of the mean, now specified in the text*.

4. P. 8. The role and nature of coefficient (or function) h was never explained.

All coefficients, including coefficient h, are exposed in table 1.

*The function h has been removed*.

5. P. 8, line 7 from below. "describe" should be "describes".

*Ok, this has been changed*

6. P. 8, last line. Delete "of" in "transfer of operator T".

*Ok, this has been deleted*

7. P. 14, line 6 from below. What is "volumes of distribution"?

*The volume of distribution is a pharmacological concept used to quantify the distribution of a drug between plasma, extracellular fluid, intracellular fluid and other compartments of the body after administration. This concept is useful for studies carried out in vivo*.

8. P. 18, lines 4-5 from below. The phrase "biological data as well as modelized transfer parameters suggest an organized quantal trogocytosis of P-gp" is very obscure. What is the meaning of the words "modelized" and "quantal"?

*The sentence as been modified as following: In this respect, biological data as well as transfer parameter f suggest an organized quantal transfer of P-gp, in which the minimum unit of exchange would probably consist in a cluster of several copies of the transporter*.

9. P. 19, line 4. Shouldn't "administrated" be changed to "administered"?

*Ok, this has been changed*.

10. P. 19, lines 7-8. What does it mean "separating the injection to the measures"?

*In some studies, the effective doxorubicin concentration in the tumor environment is followed after intravenous administration of a dose of the drug. Given that doxorubicin is distributed in several physiological compartments, metabolized and excreted (pharmacokinetics), its concentration in the tumor decreases over time and is all the more low than the time separating the i.v. infusion to the measure is high. The sentence has been slightly modified*.

11. P. 20, ref 5. Volume number and pages are missing.

*Ok, Volume number and pages have been added*.

12. P. 25, line 9. D-E should be D-F.

*Ok, this has been changed*.

13. P. 25, line 10. Change "analysis" to "analyses".

*Ok, this has been changed*.

*The recommendations of the referees were most helpful to us in the improvement of this manuscript*.

**Reviewer 2:** Anna Marciniak-Czochra, University of Heidelberg, Institute of Applied Mathematics, Heidelberg, Germany

The paper is devoted to an interdisciplinary study of the dynamics of cancer cell drug resistance induced by the expression of the drug efflux protein, P-gp. A mathematical model is developed to investigate and validate the hypothesis of cell-to-cell transfer of resistance. The model describes the dynamics of cancer cell population structured by the level of P-gp activity corresponding to the resistance level. It is expressed in the form of a diffusion - drift equation with a nonlocal reaction term describing the transfer between cell subpopulations having different levels of P-gp activity. The model is calibrated and validated based on in vitro experiments with co-cultures of drug-resistant and drug-sensitive human breast carcinoma cells. The model is interesting and shows the evolution of the resistance structure in cancer cell population. Model predictions are compared to experimental observations and biologically relevant conclusions are drawn.

I recommend the manuscript for publication in Biology Direct after minor revisions.

I think that for completeness of paper it would be good to clarify:

1. What is the function h(p)?

**This question is related to a comment of Reviewer 1:** "The paper deals with the model of P-gp transfer with ε = h = 0. Given the predominantly biological readership of the journal, I think it would be better to omit these terms in the model and refer the reader to paper [20] for a more general model accounting for diffusion and drift".

*We agree with both referees, and we modified the system accordingly to the comment*.

2.Why is spatially homogeneous approach correct? Are there any mixing conditions imposed in the experimental setting? Or it is just a first approach ? Do the authors expect that including spatial effects will influence the results? It would be good to comment on this issue, when model is introduced.

*Of course, this is a first approach. From a biological point of view, there is a lot to do in order to give an underlying mechanism for P-gp transfers and several directions have to be explored. Another manuscript will be submitted soon concerning mechanistic. A comment as been added p10 on this issue*.

3.How important is the "diffusion" process included in the model? How large is

ε? Figure 7 showing that there exist no cells with some small levels of P-gp

activity suggests that such diffusion process is very small, if any.

*The term ε has been removed*

#### Minor remarks

1. The quality of Figures 4, 6, 8, 9 is not satisfactory, in particular fonts are too small.

*Quality of mentioned figures has been enhanced*.

Figure 4
*is now* Figure 3
*CD*

Figure 6
*is now* Figure 5

Figure 8
*is now* Figure 7

Figure 9 *is now* Figure 8

2. The description of Figure 3 is confusing; what is (C) and (D)?

*It was corresponding to an older version of the figure. This whole legend has been re-written for clarification*.

3. On p.5 the reference is missing

*This work is submitted. The given reference is related to an oral communication based on the study*.

4. P. 11, line 8 from below: it is written that the peak of activity is shifted to the left towards the region of higher activity. Is it correct? It seems that "left" means "lower activity".

*The description is correct. In fact, in these experiments, the cells were loaded with a fluorescent dye that is a substrate of the efflux pump protein. As a consequence, cells expressing a low activity level accumulate intracellular fluorescence and are located in the right part of the graph after analysis, in region of high fluorescence light (FL). On the contrary, cells exhibiting higher activity levels actually efflux the dye and are shifted to the left, a region corresponding to less intracellular fluorescence and thus to higher pumping activity. This is given p12, line 5 from below in the revised version of the manuscript*.

**Reviewer 3:** Marek Kimmel, Houston, United States of America

The paper concerns the important mechanism of horizontal transfer of agents of multiple drug resistance among cancer cells. Based on their own experiments and published data, the authors designed a mathematical model kinetics of which are consistent with the observations. The mathematical model, in the form of a Boltzman-type equation, will serve as a realistic component for models of chemotherapy. The current version of the paper has been improved based on referees' remarks.