### Reviewer 1: Mark Little, Imperial College Faculty of Medicine, London, UK

#### General comments

This is an interesting and generally clearly-written paper, although short of methodological detail. Arguably, the model presented is merely schematic, useful nevertheless as illustrating the idea that random+directed motility could aid tumour growth, at least in early stages of tumour development. The detailed mathematical assumptions made (e.g., in relation to the expressions on pp.10-11 describing the chemo-taxis process in the Methods) I suspect have no biological basis, even more so the particular parameter values assumed. What is missing is any biological justification of the gradient-following mechanism. Do tumour cells follow-gradients?

There are a few missing, or poorly explained, things in the model, such as the relation between cancer stem cells and other tumour cells. [Perhaps I missed this somewhere.] I assume that most of the progeny of cancer stem cells are non-stem cells, but clearly there must be some proliferation of stem cells for the model to give the results it does.

#### Specific comments (page/line)

p.4 l.-6 Should this be "source of the gradient, clustering"?

p.10-11 I guess this chemotaxis process is equivalent to the standard Keller-Segel model [16, 17] but this should be referenced.

p.11 l.6 What is the point of the *S*/(*S*+1) term in this expression? It is similar to that in the expression at end of 2nd para on p.10? However, the point in the expression on p.10 was clear (to make the probabilities sum to 1). These *G* do not sum to 1, so I assume they are renormalized to make sure that this is the case. Also, where has the
vector gone?

### Authors' Response

The conclusions are surprisingly robust over substantial variations in parameters, highlighting the power of the arguments. Our point is essentially what the reviewer has already concluded - random plus directed motility is a universal promoter of tumor development for reasons that can be traced to simple cell kinetics. One does not need to invoke higher-order processes e.g. angiogenesis to see these results. That said, the actual values we assumed for the frequency of symmetric division of stem cells (1%), the migration rate of tumor cells (15 cell widths per day), and the generational lifetimes of non-stem cells (*ρ* = 10) are reasonably in line with values determined elsewhere.

That tumor cells follow gradients has been known for some time. The notion has its origin as far back as Paget, who 120 years ago put forward the concept of "seed" and "soil" - that cancer cells metastasize nonrandomly to sites of growth. Since then it has been suspected that tumor cells "home" to specific sites. We now know that "chemokine receptors displayed on tumor cells allow the tumor cell to follow a gradient of chemokine to its target organ milieu that expresses the ligand for this chemokine receptor" [18]. It has even been shown that glioma cells follow chemorepellent gradients produced by the cells themselves, arguably to facilitate the freeing of space for better proliferation [19].

We agree that most progeny of cancer stem cells are non-stem cells. We stated that stem cells undergo "symmetric division" (i.e. reproduce themselves) at frequency p_{s}. The rest of the time, their divisions would be asymmetric, i.e., produce a stem and a non-stem cell. We clarified this further in the manuscript by emphasizing that p_{s} is low, and defining what symmetric and asymmetric division mean. We also define *ρ* as the generational lifetime of non-stem cells, and clarify that the estimate *ρ* = 10 is used. Thank you.

Our model, originally conceived, is quite different. The chemotactic stimuli in our model are not produced by the cells themselves but pre-exist in the milieu (in the spirit of Richmond et al.'s experimental observations [18]). Additionally, the basic premise of the Keller-Segel model is that the cell responds to fluctuations in estimates of the concentration of the critical substrate, rather than to the average concentration. By contrast, we assume the cell reads the local concentration gradient deterministically, but responds stochastically.

There are *S*+1 possible sites into which a cell may move in directed fashion (including the 'null movement'). Of the *S* nontrivial movements, we weight each by the factor *cos(angle*
*θ*
*between the gradient*
*and the movement direction*
*for each possible movement)*, for the sake of calculating the angle-weighted relative opportunity for movement in each available direction. The *cos(θ)* are summed over the *S* available sites to get *AS* (*A* is the average cosine value). Treating the effective "*cos(θ)*" of non-movement to be 1, the total sum of cosines is *AS*+1, where *AS* corresponds to actual movement. The total angle-weighted relative opportunity for movement (to one of the *S* sites) due to directed migration is thus *AS*/(*AS*+1) (we correct a misprint) where *A* is the average of the *cos(θ)* over these sites. This value, times the gradient strength |
|, is now mapped to a probability of mobilization *G* through an arbitrary chemotactic/haptotactic responsiveness function H(z) for the cells, defined such that *H*(0) = 0 and *H* → 1 monotonically as *z* → ∞. The result is *G* = *H*{ |
|*AS*/(*AS*+1)}. *G* is the probability the gradient succeeds in mobilizing the cell, and takes into account the proportion of movement possible and the strength of the gradient. Once mobilized, where the cell then goes is weighted by the respective cosines for movement, so that *cos(θ)G/(AS)* is the probability of movement to the site at angle *θ*. The probabilities of actual movement sum to *G*, and *1-G* is the probability there is no directed movement.

These points have been added to the paper to clarify the origin of *G*. Thank you.