Peer review of: Feedback-controlled active brownian colloids with space-dependent rotational dynamics (2020)

Reviewers' comments: 
Reviewer #1 (Remarks to the Author): 


This manuscript presents an active matter system with a tunable rational diffusion coefficient, an 
avenue exploited by several biological systems. The authors implement this with a clever magnetic 
control system, and show, with the aid of simulations, that this system has controllable diffusive 
behavior from sub-diffusive to ballistic. They also argue for localization appearing in the system, and 
present an interesting analogy to the Nyquist theorem to explain their results. 

The authors clearly demonstrate fine control over rotational diffusion, and show that spatial 
modulation can lead to a variety of behavior, but I do not believe the results are sufficiently novel to 
warrant publication in this journal. Both aspects of the work (a swimmer with tunable D_R and spatial 
modulation of activity) have been explored recently (Karani et al., PRL 123 208002 (2019), Koumakis 
et al., Soft Matter 15 7026-7032 (2019), Arlt et al., Nature Communications 10 2321 (2019)), albeit 
separately. Additionally, the most compelling aspect of the work, the localization presented in Figure 4, 
is shown only in simulations, with no explanation for why this phenomena was (or was not) observed 
in the experiments. To add further confusion, in the section of the manuscript discussing figure 4 in is 
not at all clear that the authors have switched to discussing simulations. 



Reviewer #2 (Remarks to the Author): 

This manuscript investigates the dynamics of active Brownian particles whose rotational diffusivity 
depends on their position in the 2D-plane. The authors first present an experimental realisation of 
such a system, consisting of Janus particles that propel in the 2D plane when an AC electric field is 
applied. In addition, the coated hemisphere of the particles is made to possess a magnetic moment in 
the direction of propulsion, that aligns with an applied magnetic field. By imposing random variations 
of the latter combined to a feedback loop, one can alter the dynamics of rotational diffusion, above or 
below the equilibrium value imposed by the thermal bath. Their feedback loop allows to define a 
spatially dependent rotational diffusivity, here a chessboard of low and high values, that the particles 
sense with a given sampling time. They experimentally show that a rich non-Gaussian translational 
dynamic emerges, which markedly differs from that of active particles with a constant rotational 
diffusivity, but resembles that of supercooled liquids of interacting passive particles. Using simulations, 
they show an accumulation (resp. depletion) of particles in regions of high (resp. low) rotational 
diffusivity, and the sharpness of this pattern can be optimised by varying the sampling time. 
I found the manuscript interesting and well organised. The authors demonstrate a nice control of 
rotational diffusivity above and below its equilibrium value. The experiments and simulations address 
an important paradigm in collective and transport phenomena, namely how the variation of motility 
and diffusivity of active particles induce pattern formation. The authors convincingly illustrate that 
spatial variations of the rotational diffusivity induce pattern formation only if individuals adapt to their 
environment with some finite delay. They show how non-Gaussian dynamics can emerge as a 
combination of space-dependent rotational diffusivity and constant self-propulsion, which is not 
obvious at first sight. 
However, I find that the discussions related to Fig. 3 and 4 are unclear and need revision before 
publication (see my points below). I remain on the fence as for the relevance to a journal like Nature 
Communications due to the impact of the experimental system. Obtaining good statistics seems 
prohibitively long in the experiment so many aspects are studied via simulations. Because multiple 
particles in the field of view would have the same orientations and noise history, it seems to me that 
the experimental system does not allow for the study of collective effects, which is a drawback. That 
said, this experimental system may have potential as a probe to drive out of equilibrium various 
complex materials in a controlled manner. 

1. The experimental system is well characterised, and in particular, the control of the rotational 
diffusivity is well established: the measured distribution of angles corresponds to the imposed one, 
which testifies that no lag is present in the response of the particle to the field actuation. Could the 
authors include a short comment on the influence of the field’s intensity to achieve such control 
though? 

2. Fig. 2(e) it takes a very trained eye to evaluate the gaussian/non-gaussian character of the 
distributions. Would it be possible to plot these in y-log scale instead (gaussians will appear as 
parabolas and the non-gaussian tails will be clearly visible)? Also, I was wondering what those 
distributions look like for the rotational degree of freedom. 

3. Fig. 2(f) the non-gaussian parameter does not fully decay to zero here but it does in the simulated 
data. Is it physical or simply due to a lack of statistic in the experiment? Or to the fact that the time 
lag does not go beyond 100s, i.e. does not exceed 1/D_R^L? 

4. The first important result shown here is that non-interacting active Brownian particles with constant 
propulsion speed exhibit non-gaussian translational dynamics when their rotational diffusivity varies as 
a function of position. The dynamics resemble passive interacting particles in supercooled liquids but is 
markedly different from pure active Brownian particles. Nevertheless, I found this section very 
descriptive of the plots and not always clear. What I am missing is a statement that would summarise 
and conclude on the influence of the two parameters L/v and \tau to help the reader. I would also like 
the following points to be addressed: 

- Page 7, line 135: “For small L/v, the length scale of the spatial variation in D_R is smaller than any 
persistence length”. I guess this regime is achievable in simulations, but not in experiments since the 
smallest square size is 16mum, while the smallest persistence length is v/D_R^H=3.5/10=0.35mum. 
I think I need clarification on why the behaviour is that of a homogeneous rotational diffusivity here. 

- In experiments, is L/v varied through L only or v as well? 

- Page 8, line 153: “\alpha reveals a super-diffusive scaling …” The different dynamic regimes are 
easier to see from the MSD than the non-gaussian parameter. The mentioned cut-offs for the sub-
diffusive plateau L/v and 3L/v seem loosely defined: the yellow band in Fig. 3(d) is clearly much wider 
than the actual plateau. 

- Page 9, line 159: the peaks at odd multiples of L/v in Fig. 3(c) were not obvious to me on an x-log 
scale. In all honesty, I thought it was just a lack of statistics at first and would have missed this 
feature. Could they be magnified in an inset? What would the corresponding MSD look like? 

- Page 9, line 160: “Consequently, the MSDs on timescales L/v decrease and the sub-diffusive scaling 
gradually vanishes (see Figure 3e). For \tau >L/v, \alpha decreases again with no detectable peaks. In 
this regime, the MSDs increase across all timescales (Fig. 3f).” I am afraid I cannot see a decreasing 
MSD in Fig. 3(e) and the last statement is a bit empty, since all visible MSDs do increase. 

5. The second important result is that pattern formation can be achieved with a spatially varying 
rotational diffusivity combined to a non-zero sampling time. Spatial variation of the rotational 
diffusivity alone does not lead to pattern formation, mainly because it is absent from the flux balance 
condition. As shown by the authors using simulations, the particle assembly is homogeneous at low 
and high sampling time. At intermediate values though, particles are localized mainly in regions of 
high rotational density. The authors discuss the behaviour of the optimum localisation in terms of the 
sampling times and the crossing time L/v. I like that the authors propose a simple model to estimate 
the maximum particle fraction that can be localized in high rotational diffusivity regions. But I find the 
argument unclear and unnecessary long: 

- Page 10, line 178: it should be specified that D_R here is D_R^H 
- From line 178 still, the penetration time into a ballistic zone is \tau but then another time, \delta t*, 
is introduced in line 185. Aren’t they the same? 
- The explanation of the model is a bit disorganised and gets heavy in the SI while getting Equation 3 
only takes a few steps I find (see additional information in the attached document). 

Other minor points: 
- Page 2, line 50: “a change of tumbling rate can be straightforwardly mapped into an effective 
increase of rotational diffusivity”. From Ref. 14 it is true that the mapping itself is straightforward but 
the conditions of its validity seem less so. 
- Page 2, line 51: “nonetheless, translational and rotational fluctuations are coupled via the thermal 
bath and their lower bound is dictated by thermal fluctuations”. This is only true for active Brownian 
particles but for run and tumble particles, the tumbling rate is an internal degree of freedom which is 
not coupled to the bath. 
- Page 3: the magnetic moment is mentioned in line 65 but the external control of rotational diffusivity 
is only described much later in line 73. I think this is a bit confusing for the reader. 
- Page 5, line 101: there is a typo in the word “instantaneous”. 
- Page 7, line 127: it may be worth specifying that this non-gaussian parameter is defined for 1D 
displacements. Why is there a factor 1/5 in its definition? 
- Page 7, line 131: I think I would rephrase and say “zero on both short and long timescales”. 
- Page 7, line 139: \gamma_R is not introduced in the main text. 
- Page 10, line 192: <\eta_t> is not introduced in the main text. 



Reviewer #3 (Remarks to the Author): 

In their paper the authors demonstrate a quite remarkable control of the rotational diffusion of an 
active Janus swimmer using a rotating magnetic field. This has never been achieved before and opens 
the road towards a full control of active swimmers. 

The paper is well written, the data and methods are clearly presented and discussed. 

And I only have a few minor points to adress (see below) 

I therefore recommend publication as soon as these minor points are solved. 

Minor comments 

Fig 2 (e-f) : indicate that the black points are experimental data and that the red lines are numerics 

Fig 2 End of caption : one of the D_R^L must be replaced by D_R^H 

Just before Formula 1 : replace At equilibrium by In the steady state 

The description of the diffusive regimes is a bit lengthy and lacks some explanations, which only come 
in the Conclusion and Discussion part. Some reorganization of the writing could easily fix that. 

It would be nice to synthesize the overall findings in a diagram as a function of \tau L^{-1} v. 
A comment about the role of v/(D_R^H L^2) would also be welcome.