Referee 2's Review:

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N.B. These are the comments made by the referee when reviewing an earlier version of this paper. Prior to publication, the manuscript has been revised in light of these comments and to address other editorial requirements.
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This is an interesting study, presented in a somewhat confusing manner. The data are new and certainly important enough to publish, but the analysis is incomplete, and not strong enough to support conclusions about an efficient search strategy.
I think the ms. could be improved by writing separate Result and Discussion sections.

More specifically:
1) Some of the confusion results from terminology:
I interpret “bias” to imply an asymmetry, not just a deviation from the simplest random model.
Other readers might have a similar problem.
I don’t think it is helpful to call any of these results “directed” motion.
Most readers will probably think of “Brownian motion” as “translational Brownian motion”, producing something similar to Fig. 1A, with no obvious persistence. (A purist would restrict the term to cases where the source of the movement is molecular bombardment, and refer to other cases as “similar to Brownian motion”.) If the authors simply want to DEFINE “Brownian motion” to mean any case where mean square displacement/time interval is constant for large enough times, OK, but they should make this clear to the reader.

There have been past attempts to model cell movements as translational Brownian motion with persistence, but this was not a very successful approach. For a motile cell moving at constant velocity, a simpler and appropriate model is just to add turns resulting from a rotational diffusion process -- this is the model described by Equation (1). In some places in the ms. (as in line 3 of page 11), the authors seem to be using the term “Brownian motion” to describe the Equation (1) model, but on page 5 they seem to be using it differently. At the end of the top paragraph on page 11, it is not clear whether “Brownian searching” is referring to the Equation (1) model, or to the low persistence model seen in Fig. 1A. Clarification is needed. Use of the term “random walk” is also confusing. In order to meaningfully discuss “a variety of different random walks” (abstract) I think it is necesary to define the how the term is used here. The abstract states that the data indicate “a special random walk”, but on page 8 it is stated that the data “cannot be fitted by a random walk alone”. Clarification is needed.

2) Fig. 3 shows results consistent with the Equation (1) model, and yields 2 parameters: a diffusion coefficient and a persistence parameter. Would the results in Fig. 3 be significantly different if the data were filtered to remove the high-frequency (oscillatory?) turning? Alternatively, when the model is expanded to include Equation (2), does that change the interpretation of the diffusion coefficient obtained from Fig. 3 data? It is not stated how the 30 min persistence time was calculated. It should be possible to extract it from Fig. 3 data. I see that it agrees with a translational D of about 400 (from Fig 3B) and a swimming speed of about 7 μm/ min. A reference to the theory behind this calculation would be helpful (I used Brokaw, 1959, J Cell Comp Physiol 54:95-101).

Two of the traces shown in Fig. 3 are different, with unusually high persistence. Why are these not discussed further? What model is required to fit these traces? (Applause for showing them!, but maybe they should be (were ?) excluded from the averages.)

3) The data in Fig. 4, for very short times, do not add anything of importance. Should be deleted.

4) The real novelty of this work is in the fine structure of the movement revealed by the analysis of turn angles, as shown in Figure 6. I think the nicest part of this paper is that the process described by Equation (2) may fit both the movement fine structure data and independent observations on pseudopod behavior. Maybe more could be done with this. A biological audience might be helped more by relating Equation (2) to the elementary textbook case of a noisy galvanometer, rather than the more advanced Ornstein-Uhlenbeck reference. Maybe a more helpful analogy for thinking about the extra energy in the PSD. What might cause “stochastic oscillations”? Are the “circling” events seen in Figure 6 consistent with expectation from random rotational diffusion? Given that the authors have created a way to distinguish turns, I would think it appropriate to look at the probability distribution of turn amplitude, and compare it with the predictions of random rotational diffusion. This might be more informative than the PSD approach.

I would expect that at least a portion of the results in Figure 9A would be predicted by the harmonic restoring force in Equation (2). Is this what Figures 9B and 9C are intended to show? The discussion is too incomplete for me to tell. I would also expect these results to be highly dependent upon the method used to identify “turns”. I’m not convinced that these results provide support for an additional stochastic oscillation, for a “memory”, or for characterization as a Markov process. I may be wrong, but I need to be convinced. The analysis would need to show what portion of the turn angle results cannot be explained by the harmonic restoring force, and show that the results are robust, over a range of turn identification parameters such as smoothing distance, turn “threshold”, etc.

5) I am also not convinced by the arguments on page 11. The simplest interpretation would be that the efficiency of searching is primarily determined by the persistence determined by an Equation (1) model, and that the higher frequency motion determined by Equation (2) is not a major factor. Evidence for any increase in efficiency resulting from either the O-U process or an additional stochastic oscillation (Equation (2)) is not presented. To support their interpretation, they need to do simulations showing that the addition of Equation(2) to the model, with parameters obtained from the data, gives more efficient searching than is obtained with Equation
(1) alone.

If the “two-state” model is worth discussing in the Introduction, it would seem to be appropriate to return to it in the final Discussion. Figure 10 is unnecessary. More information about the search simulations is needed: parameters and results.

6) Perhaps these cells do not use a straight line search because they are simply incapable of “steering” well enough to get their persistence time much greater than about 30 min. On the other hand, perhaps life in non-infinite environments has selected for a different search strategy, or at least eliminated any selective advantage for a further increase in persistence. Free-swimming cells increase their persistence by rotating around the velocity vector, and thus generating a helical path. It would be interesting if cells on a surface achieve something similar by Equation (2). In the case of free-swimming cells, the helical path has been suggested to improve the sampling of the environment; perhaps something like that is involved here also. (You might want to look up work by Hugh Crenshaw on helical motion.)