Spatial Moran Models I. Stochastic Tunneling in the Neutral Case By

We consider a multistage cancer model in which cells are arranged in a d-dimensional integer lattice. Starting with all wild-type cells, we prove results about the distribution of the first time when two neutral mutations have accumulated in some cell in dimensions d ≥ 2, extending work done by 1. Introduction. The accumulation of mutations is important not only for cancer initiation, progression, and metastasis, but also for the emergence of acquired resistance against chemotherapeutics, radiation therapy, or targeted drugs. For this reason there is a large and growing literature on the waiting time τ k until some cell has acquired k prespecified mutations. In all the models we consider, type i individuals mutate to type (i +1) at rate u i+1. The dynamics considered have most often been studied in multi-type Moran models with a homogeneously mixing population of constant size. Here we will concentrate on how results change when one considers a spatial Moran models, and as is the case for much earlier work we will concentrate on the behavior of τ 2. We suppose that cells of type 0 and type 1 have relative fitness 1 and λ. Since we will only consider the waiting time for the first type 2, the relative fitness of type 2's is not important. In this work we will consider situation in which λ is so close to 1 that the mutations are essentially neutral. For cancer applications, this is a restrictive assumption, and it will be removed in the companion paper (part II) by Durrett, Foo and Leder [6]. However, the current result applies to the important case of tumor suppressor genes. In that case, when both copies of the gene are inactivated trouble develops, but while there is one working copy the cell can function normally. We begin by recalling results for the Moran model in a homogeneously mixing population of size N. Here and in what follows the mutation rates u i and selection coefficient λ depend on N , even though this is not indicated in the notation, and we write a N b N if a N /b N → 0 as N → ∞. The next result made its first appearance on page 16,230 of Nowak et al. [17]. Since then it has appeared in