The optimal rate of chromosome loss for the inactivation of tumor suppressor genes in cancer

Abstract

Many cancers are characterized by chromosomal instability (CIN). This phenotype involves the deletion and duplication of chromosomes or chromosome parts and results in a high degree of aneuploidy. The role of CIN for cancer progression is a very important, yet unresolved question. It has been argued that CIN contributes to cancer initiation because chromosome loss can unmask a mutated tumor suppressor (TSP) gene. At the same time, CIN is costly for the cell because it destroys the genome and therefore compromises clonal expansion. Here, we use mathematical models to determine whether CIN can accelerate the generation and expansion of TSP(-/-) cells in the context of this tradeoff. Comparing cells with different degrees of CIN, we find that the emergence and growth of TSP(-/-) cells is optimized if the rate of chromosome loss is of the order of 10(-3) to 10(-2). This result is very robust, is independent of parameter values, and coincides with experimental measures using colon cancer cell lines. However, if we consider all of the steps in the pathway, including the generation of the CIN phenotype from stable cells, then it turns out that the emergence and growth of TSP(-/-) cells is never accelerated by CIN. Therefore, CIN does not arise because it accelerates the accumulation of adaptive mutations. Instead, it arises for other reasons, such as environmental factors, and is subsequently fine-tuned by selection to minimize the time to further cancer progression by means of the inactivation of TSP genes.

Fig. 1.

Schematic diagram illustrating the principles of the mathematical model. The model considers the chromosome pair that bears the TSP gene. We start with a cell that has two wild-type copies of the TSP gene. Relevant genetic alterations comprise two processes: point mutations occur with a rate u = 10^-7 and can inactivate a copy of the TSP gene; chromosome loss can occur with a rate p, which can unmask an already mutated TSP gene. Therefore, a TSP gene can be inactivated in two basic ways: by two successive point mutations (A → B → C), or by a combination of a point mutation and a chromosome loss. The latter can occur through various pathways. The most likely pathway is indicated in red (A → B → E). That is, first a point mutation inactivates one copy of the TSP gene, and the other copy is eliminated by chromosome loss. Dashed arrows indicated possible but irrelevant steps because they do not have the potential to speed up the inactivation of the TSP gene. Cells with an inactivated TSP gene are marked in purple. They are assumed to undergo clonal expansion.

Fig. 2.

The effect of chromosome loss (CIN) on the generation and growth of TSP^-/- cells. (a) This graph plots the number of TSP^-/- cells against time, assuming that all cells have the CIN phenotype and are characterized by different rates of chromosome loss, p. An intermediate chromosome loss rate (of the order of magnitude of ≈10^-2 to 10^-3) results in the fastest growth of the TSP^-/- cell population. The simulations are based on the mathematics presented in Supporting Methods. Parameters were chosen as follows: a = 1, u = 10^-7, k = 1. (b) This graph plots the expansion of TSP^-/- cells assuming that, before chromosome loss can occur, the cell has to generate the CIN phenotype at a rate u_c. Dashed lines represent simulations that assume that CIN is generated. The solid line assumes that CIN is not generated and cancer initiation occurs by two successive point mutations. The figure shows that the generation and growth of TSP^-/- cells in the context of CIN is significantly slower than in the context of two successive point mutations unless the CIN phenotype is acquired at rates that are orders of magnitude higher than the physiological mutation rate (10^-7). Simulations are based on the mathematics presented in Supporting Methods. Parameters were chosen as follows: a = 1, u = 10^-7, k = 23. Genetically unstable cells are characterized by the optimal rate of chromosome loss whereas, in the stable cell population, chromosome loss is assumed not to occur.

Fig. 3.

Schematic summary of the results about the effect of chromosome loss on the generation and growth of TSP^-/- cells. (a) We compare the growth rate of TSP^-/- cells in two scenarios: The TSP^-/- phenotype is generated by two successive point mutations in cells that do not have CIN; or the TSP^-/- phenotype is generated in CIN cells by the combination of a point mutation and a chromosome loss event. In this case, we observe an optimal rate of chromosome loss, for which the generation and growth of TSP^-/- cells occur fastest and which is better for the cancer than being stable. (b) On the other hand, if we assume that CIN first has to be generated before chromosome loss can occur, the generation and growth of TSP^-/- cells is always slower with the CIN pathway compared with the stable pathway (unless stable cells acquire CIN at a very high rate that is orders of magnitude larger than the basic mutation rate).