Dear Dr. Mofrad,

Thank you for forwarding us the reports on our manuscript "Excitable actin dynamics and amoeboid cell migration". We thank the reviewers for their assessment of our work. We have followed the reviewers' suggestions, which have helped us to improve the quality of the presentation of our results. A detailed response to the reviewers may be found below.

In the meantime, we have become aware of very recent experimental work that provides direct evidence for the mechanism underlying actin polymerization wave generation that we have implemented in our work (Kamps et al, Cell Rep 2020). We now refer to this work in our revised manuscript.

We have also now made available our code for solving the dynamic equations numerically. See https://github.com/NEc-7/Cell-Motility-caused-by-Polymerization-Waves, to which we reference in the revised manuscript (Ref.[41]). 

Finally, we have made some small edits to increase clarity and to correct typos in the original manuscript.

With kind regards,
Nicolas Ecker
Karsten Kruse.

Response to Reviewer #1:

This is a decent theor-phys study of previously derived mathematical
model of wavy actin dynamics. The authors use one of the soft-matter
models where actin network is characterized by density and direction;
there are simple polymerization and disassembly; the polymerization
though is induced by 'nucleators'; there are a couple of nonlinear
feedbacks that make the system excitable. The authors first analyze the
time-dependent system only and notice its similarity to the well-known
F-N model. Then, they add diffusion and do the linear stability analysis,
numerical solutions and variational ansatz, demonstrating emergence of
stationary and traveling-wave patterns. Finally, they couple the wavy
actin dynamics to the established phase field model of motile cell,
and see what the numerical solutions predict. The result is an interesting
diagram showing how the character of cell movements and shapes depends on
model parameters. Interesting diffusive and chaotic trajectories are revealed,
as well as cell splitting.

The study is solid; for biophysical or bio journal the authors would likely
face a number of very hard questions, but for PLoS One this is a very good
fit.

> We thank the reviewer for his/her assessment of our work.

Some minor critique:

As this study is heavily based on [9,15,20], the authors should describe
more explicitly what exactly is new.

> As we clarify now in the first paragraph of Section II, the basic mechanism of the coupled actin-nucleator dynamics was introduced in Ref. [15], but not coupled to a deformable domain represented by a phase field. This was done in Ref. [20], but there the coupling of the phase-field to the nucleator current had a form that led to strong leaking of nucleators from the cell interior. This was remedied in Ref. [9], which focuses on experiments and lacked a detailed study of the dynamic equations, which is the purpose of the present article.

Eqs 1 and 2 - it'd be good to have an intuitive explanation of the terms
for readers who are not working in the field of soft matter. For example,
why grad c term is the source for p? I can guess, but don't know for sure.

> In this work, we introduce the dynamic equations as a phenomenological description that respects the symmetries of the system. This approach has the advantage of not relying on specific mechanisms, but we agree with the reviewer that it often fails to provide an intuition for the various terms that appear. In the paragraph containing Eqs. 1 and 2, we thus now make a connection with a more microscopic approach that was introduced in Ref.[20]. From this we see that the term v_a\nabla\mathbf{p} in Eq. 1 describes changes of the actin density resulting from the addition of actin monomers at the ends of actin filaments (removal of actin monomers is captured by the degradation term -k_d c). The term v_a\nabla c in Eq. 2 indicates that changes in the polarization are linked to the actin polymerization current v_ac.

The authors have to explain how did they solve PDE and phase field equations
numerically.

> We have added a new Appendix A that explains how we solved the dynamic equations numerically and deposited our code plus further information about our algorithm on GitHub.

The authors have to explain better how the actin dynamics is coupled to
the phase field model of the free boundary of the cell.

> In the paragraph containing Eqs. 30-33, we now explain that the coupling of the actin density c and the polarization field \mathbf{p} to the phase field is obtained by restricting the corresponding source terms to the cell interior by multiplying them by \Psi.

Some proofreading would help - there are numerous typos in the text.

> We have carefully read the manuscript again and corrected typos. 


Response to Reviewer #2:

The authors extend their earlier work (Stankevicins et al. PNAS. Doi: 10.1073/pnas.1907845117) and develop a computational actin network model featuring simple nucleation, polymerization and disassembly. They also incorporate a (non-linear) feedback feature to mimic excitable systems. Additionally, their model features diffusion enabling wave patterns and dynamics.

I believe this is an interesting study and I am happy to recommend it for publication subject to the following suggestions:

> We thank the reviewer for his/her assessment of our work.

1. I wonder how hydrodynamics interactions if they were to be incorporated, could affect the results, in particular the wave patterns and dynamics. (please see Chandran et al. Averaged implicit hydrodynamic model of semiflexible filaments. Phys. Rev. E 2010. DOI:10.1103/PhysRevE.81.031920)

> The reviewer raises an interesting point as hydrodynamic flows are indeed present in cells. However addressing this question would go far beyond the scope of the present work and we would like to leave it for future work. We have added a remark on this question in the third paragraph of the discussion, where we also reference the work suggested by the reviewer.

2. Please comment the link between the phase field model and the actin dynamics patterns and how they are implemented.

> In the paragraph containing Eqs. 30-33, we now explain that the coupling of the actin density c and the polarization field \mathbf{p} to the phase field is obtained by restricting the corresponding source terms to the cell interior by multiplying them by \Psi. The numerical method for solving the dynamic equations is explained in the new Appendix A.

3. The literature survey could benefit from other computational and continuum models for actin networks (e.g. see Chandran et al. Band-like Stress Fiber Propagation in a Continuum and Implications for Myosin Contractile Stresses. CMB.
DOI:10.1007/s12195-009-0044-z)

> In the third paragraph of the discussion, we now discuss possible ways to implement molecular motors into our description. In this context we refer to other continuum models for actin networks.

4. There are several grammatical and typographical errors in the text. An English check through will benefit the paper.

> We have carefully read the manuscript again and corrected typos.