Signalling architectures can prevent cancer evolution

Cooperation between cells in multicellular organisms is preserved by an active regulation of growth through the control of cell division. Molecular signals used by cells for tissue growth are usually present during developmental stages, angiogenesis, wound healing and other processes. In this context, the use of molecular signals triggering cell division is a puzzle, because any molecule inducing and aiding growth can be exploited by a cancer cell, disrupting cellular cooperation. A significant difference is that normal cells in a multicellular organism have evolved in competition between high-level organisms to be altruistic, being able to send signals even if it is to their detriment. Conversely, cancer cells evolve their abuse over the cancer’s lifespan by out-competing their neighbours. A successful mutation leading to cancer must evolve to be adaptive, enabling a cancer cell to send a signal that results in higher chances to be selected. Using a mathematical model of such molecular signalling mechanism, this paper argues that a signal mechanism would be effective against abuse by cancer if it affects the cell that generates the signal as well as neighbouring cells that would receive a benefit without any cost, resulting in a selective disadvantage for a cancer signalling cell. We find that such molecular signalling mechanisms normally operate in cells as exemplified by growth factors. In scenarios of global and local competition between cells, we calculate how this process affects the fixation probability of a mutant cell generating such a signal, and find that this process can play a key role in limiting the emergence of cancer.

1 Introduction 19 In all models presented here, a single mutant cell emerges, capable of gener-20 ating a molecular signal. The signal affects the cell producing the signal and 21 cells within a radius R. Cells producing the signal get a benefit b (constant 22 in Sections 2-4, cumulative in Section 5) and pay a cost c for producing the 23 signal, whereas cells in the vicinity receive the benefit b but pay no cost. The is implemented so that mutants are always adjacent to each other. We will 28 consider two versions of the model: one with global competition, and one 29 with local competition. We start by analysing local competition, since the 30 analysis is a bit simpler, and then turn to global competition. One key measurement for a stochastic process is the transition probability. 34 Starting with N S mutant cells what is the probability of increasing in number 35 to N S + 1 or decreasing to N S − 1. We will start our analysis with one 36 signalling cell (N S = 1). We calculate the transition probabilities in this case 37 and compare them to neutrality. We will then generalise our results for an 38 arbitrary number of signalling cells.  Figure S1: In the simplified model growth and death occur only within a distance L of the boundaries between the cell types.
The chance to decrease is: The ratio between these two is: We see that the ratio is < 1 when c > 0. We see that the dynamics of 79 the system is like a system with a genotype with a fitness detriment of s = 80 −c/(1 + b).

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When N S < L, we get: The chance to decrease is giving again, the same ratio. The chance to fix is, therefore, the same chance

Case L > R
Now there will be three types of cells. We again start with N S > L.
And their ratio is Again we have a dynamics that is equivalent to constant selection, with a 94 selection pressure that also depends on R. Selection will be negative when 95 b c < L L−R , which can give us a lower bound both on R or c.

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When N S < L we get Giving again, the same ratio. Again the system will behave exactly like a 100 system with a mutant with a constant s.
Giving a ratio of: We see that with global competition the selection pressure is dependent on 109 N S , so the system is not equivalent to a system with constant selection 110 pressure. We can, however, look at the direction of selection. If we rewrite 111 the above expression as 1 + s, we get If the range of benefit given, i.e., R, is very small, 114 the cost has to be larger than the benefit to have negative selection. On the 115 other hand, when α is close to 1, a positive cost results in negative selection. 116 We see that in non-local competition the benefit has to be given to a large 117 fraction of cells to prevent the invasion of the mutant. In two dimensions a cell will compete with cells within a radius L. An 120 assumption equivalent to the analysis done above with competition within a 121 range L of the boundary is competition within distance L perpendicular to 122 the boundary, and L along the boundary (see Figure S2). The analysis is 123 then identical to the one-dimensional case.

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L L L L Figure S2: Simplified 2D model. Only cells within a distance L of the boundary are taken into account for reproduction. Other could reproduce, but that will not affect the frequency of the cell types. When the indicated cell reproduces it can replace any cell in the competition region -a distance L from the boundary, and L along the boundary.

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We also simulated 2D models. signalling cells have a fitness of (1 + b − c).  we need an analytical expression for the total payoff of each cell type. extreme. Then, the cluster will be present when N S ≥ (R + 1).

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Before the cluster is formed (N S < (R + 1)), each N S cell will receive N S After the cluster is formed (N S ≥ (R + 1)), the first N C cell closer to the N S 153 cells will receive R unities of benefit, the second N C cell will receive (R − 1) 154 unities of benefit and so on. As this pattern is present at both sides of the N S cluster, we obtain that the total payoff for N C cells is R i=1 i = R (R + 1).

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In a similar way one can show that, after the cluster is formed, N S cell will 157 receive a total payoff equal to R (2N S − R − 1) + N S .

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Summarising, for the signalling cells the amount of benefit they will re-159 ceive: For the non-signalling cells, the benefit they will receive: The interplay between the number of signalling cells N S , the radius R and 166 the level of locality of the competition L generates scenarios that are not easy 167 to tackle analytically. We performed numerical simulations to study the evo-168 lutionary output of the interaction between these variables and parameters. The main result of the present study is that long-range diffusion (large secretion, and release of the Wnt signal depends on Wntless (wls) [3]