# Evolutionary suppressors

Evolutionary suppressors are the counterparts of evolutionary amplifiers. For the spatial Moran process these population structures tilt the balance between selection and random drift in favor of random drift such that the fixation probabilities $$\rho$$ of advantageous mutants ($$r>1$$) is smaller than in the original Moran process or on circulation graphs, $$\rho<\rho_1$$. Because selection is suppressed, this also implies that disadvantageous mutants ($$r<1$$) have an increased fixation probability, $$\rho>\rho_1$$. Evolutionary suppressors are also characterized by hierarchical population structures but without feedback loops.

The simplest and most extreme case of an evolutionary suppressor is given by a linear chain where the offspring of each individual consistently replaces e.g. the occupant of the vertex to its right. Thus, the leftmost vertex is a root and is never replaced whereas the offspring of the rightmost, tail vertex is lost. This generates a flux through the population from left to right such that no mutant can reach fixation unless the mutation occurs in the root vertex. This happens with the probability $$1/N$$ in a chain of length $$N$$ but then fixation occurs with certainty. Consequently, the fixation is simply \begin{align} \rho_0 = \frac1N, \end{align} irrespective of the mutant’s fitness. Selection is eliminated and random drift rules. If a graph contains multiple root nodes then a single mutation can never reach fixation, $$\rho=0$$.

Evolutionary suppressors have a very simple, almost trivial structure but at the same time they turn out to be highly relevant in biological systems. While mutations enable populations to adapt to changing environments, they are generally pathogenic when they occur within an organism. Especially dangerous are mutations that increase the net reproductive rate of a cell because this may later develop into cancer (Vogelstein and Kinzler, 1998). In order to prevent accumulation and spreading of detrimental mutations, organisms take advantage of evolutionary suppressors (Nowak et al., 2003). Epithelial tissue, such as our skin or the colon, is organized into small compartments (crypts in the colon) and each compartment is arranged in multiple layers of cells of increasing degrees of differentiation - ranging from few undifferentiated stem cells to terminally differentiated epithelial cells. With the exception of the stem cells, all cells are regularly renewed by new cells from precursor layers. This exactly matches the setup of the linear chain and thus cancerous mutations will be eventually washed out, unless they happen to occur in one of the stem cells. In case this occurs, then the compartmentalization confines the mutants and prevents further spreading. Another impressive example of a complex hierarchical arrangement is given by our blood system where the stem cells reside in the bone marrow and divide only about once a week and give rise to a series of precursor cells that generate the terminally differentiated red blood cells with a production of the order of $$10^{12}$$ cells every day. The architecture of all these systems is shaped to prevent malignant mutations from spreading.

## Evolutionary dynamics on supressor graphs

#### Evolutionary dynamics on the linear chain graph

The linear chain graph is an extreme example of an evolutionary suppressor, which completely eliminates selection. Even if a mutant in the vicinity of the root vertex succeeds in giving rise to a lineage, it will always be a transient state and all mutants are eventually displaced by the offspring of the resident in the root vertex. A highly beneficial mutant is likely to rapidly take over all downstream vertices but nevertheless cannot prevent the spreading of the slower reproducing upstream residents.

## Fixation probabilities on supressor graphs

#### Fixation probabilities on the linear chain graph

The fixation probability on the linear chain graph is zero for all vertices except the root vertex, which fixates with certainty.

## Fixation times on supressor graphs

#### Fixation times on the linear chain graph

The fixation times are zero for all vertices except for the root vertex for which the fixation time is proportional to the length of the chain $$N$$. In contrast, the absorption times linearly decrease from the root vertex to the tail vertex because the transient sojourn times of the mutant lineage depends on the location of the initial mutant and decreases towards the tail vertex.

## Publications

1. Lieberman, E., Hauert, C. & Nowak, M. (2005) Evolutionary dynamics on graphs Nature 433 312-316 doi: 10.1038/nature03204.
2. Hauert, C. (2008) Evolutionary Dynamics p. 11-44 in Evolution from cellular to social scales eds. Skjeltorp, A. T. & Belushkin, A. V., Springer Dordrecht NL.

### References

1. Nowak, M. A., Michor, F., & Iwasa, Y. (2003) The linear process of somatic evolution, Proc. Natl. Acad. Sci. USA 100 14966–14969.
2. Vogelstein, B. & Kinzler, K. W. (1998) The Genetic Basis of Human Cancer, Toronto, McGraw-Hill.