The French flag problem revisited: Creating robust and tunable axial patterns without global signaling

Reviewer #1 (Public Review):

Summary:

In this article, Kremser et al set off to explore how local interactions between cells can drive pattern formation by focusing on the French flag problem whereby an initially homogeneous system breaks axial symmetry to form three distinct regions of different cell fates. The authors use a cellular automata model together with evolution searches on possible rules that determine cell state and tissue level patterning. It is assumed that three cell states are possible and that at each time iteration each cell updates its fate according to the current state of itself and its neighbours. The authors use a computational procedure based on evolution algorithms to identify "fit" update rules that can successfully drive patterning into three distinct domains and go on to provide insights with regards to the function of these rules as well as their properties such as robustness and patterning dynamics. The article is generally well-written, the results seem solid, and the analysis and methods are thorough and generally well-explained. A main concern is the lack of connection between the biology that motivated the analysis and the results, this could be improved in the discussion by making the methods somewhat more concise to allow space to make links back to potential biological mechanisms when the results are presented. We raise some general points and some more specific questions and suggestions for clarification below that we hope will help improve the MS and make it more accessible to a wider audience.

General points:

• Although the authors motivate their work on the premise that biological patterns at the tissue level often are driven by local cell-cell interactions, by the end of the analysis any possible connection to the underlying biology is lost. For example, it would have been useful to discuss how the rules that evolved to dominate the patterning process in the results section could be implemented by cells. Is there a connection that could be made back to Notch signalling and its multiple ligands or to morphogens that diffuse only locally? Would the large number of rules possible in the cellular automata context reflect transcriptional feedback? This is an important point to bring the work "home". At the moment, it feels like a nice computational analysis of cellular automata but the links to the systems that motivate the work are lost in the process.

• When growth is considered (p.14-15) a discussion of timescales seems pertinent. Often patterning takes place at a timescale faster than cell division so the system could be allowed to reach a steady state before a new division event takes place. What are the time scales of updating the phenotype compared with the time scales of division in the model and in relevant biological systems? How would different limiting cases impact conclusions, e.g. new cells added and pattern allowed to reach steady state before more growth versus cells added while patterning dynamics are still updating?

• An interesting question is whether certain elements of rules (out of the 27 possible elements for the system with 3 states) are more or less likely to appear together in an evolved final rule. This may give a mechanistic understanding of what combinations of elements are likely to drive the optimal pattern and which combinations are avoided altogether.

Reviewer #2 (Public Review):

Summary:

In this paper, the authors seek to identify strategies that can be used to generate robust one-dimensional large-scale patterns through the sequential application of only local, unchanging, space-independent rules. This is an important general question in developmental biology.

Strengths:

The authors do a nice job of laying out the problem, which they explore through cellular automaton (CA) modeling. The modeling framework is well described, as are the methods used for computational identification of effective (most "fit") strategies. As many biologists are unfamiliar with CA models, the clarity of description offered by these authors is especially important, as is the attention that was paid to useful visualization of results.

Ultimately, the authors use their approach to converge on certain generic strategies for achieving robust patterns. In the case when there are only three states (no hidden or transient states) available to cells, they rationalize the consensus strategy that emerges to involve a combination of "sorting" and "bulldozer" modules, which are relatively easy to rationalize. In cases involving a fourth state, a more complicated set of strategies arise and are considered.

As a pure modeling paper, I find the work to be very well done, and the conclusions are well supported by the data and analyses. In terms of the long-term importance of this approach to biologists studying pattern formation, I see this paper as primarily laying a foundation for taking the next step, which is moving into two (or three dimensions). Clearly, the complexity of rules becomes much greater, but one may expect some big qualitative differences to show up in higher dimensions, where simple strategies like sorting and bulldozing cannot work quite as simply. It will be interesting to see where this leads.

Weaknesses:

Ultimately, the relevance of this work to biology rests with its ability to provide insight into important biological problems. In terms of explaining the challenging nature of generating long-range patterns using short-range rules, I think the authors do a good job. However, they could do a better job of relating the results of the work back to biology. For example, are there examples of "sorting module" and "bulldozer module" behavior in biology? Could they be involved in explaining actual biological patterns?

It also would have been helpful for the authors to generalize more about the way in which their CA rules achieve global patterns with other patterning mechanisms. For example, in a Wolpert positional information model, patterning information is distributed over space in a steady-state gradient. In the CA model, no information spreads more than one cell at any one time point, but over time information still spreads, so in a sense a stationary spatial gradient has been traded for a moving spatial discontinuity. Because the discontinuity moves without decrement, any stationary state ends up being determined by the boundaries of the system, which goes a long way to explaining the robustness they observe, as well as why the result is quite sensitive to growth (which keeps changing the boundary).