Rate p, fast step

Slow growth

Fig. 6.2 TSP inactivation and clonal expansion, (a) In the case where chromosome losses occur rarely (p = 0), we have the following sequence of events: first, a mutant appears which has one copy of the TSP inactivated. After a while, a second mutation may occur producing a cell with two inactivated copies of the TSP. This leads to clonal expansion, (b) If losses of chromosomes are possible, then one of the progeny of the first TSP+/- cell may lose the chromosome containing the functional copy of the TSP, thus giving rise to cells with one inactivated TSP copy and one "missing" TSP copy. Such cells enter a phase of clonal expansion, but this happens at a slower rate compared to (a) because of frequent chromosome loss events resulting in dead or non-reproductive cells.

Let us denote the rate at which small-scale genetic events happen by u (per cell division per gene), and the rate of chromosome loss by p (per cell division per chromosome). The basic rate at which such mutation events occur in stable cells has been estimated to be approximately u — 10"7 per cell division per gene. The inactivation of the first allele of the TSP will happen with the rate 2u, because there are two alleles. The inactivation of the second allele can happen with the rate u by a mutation, and with the rate p by loss of chromosome, see Figure 6.2. Let us first suppose that the rate of chromosome loss is zero, p = 0, Figure 6.2a; a TSP gene can only be inactivated by two consecutive, independent (small scale) genetic events. This is possible, but the probability of such a double mutation is very low. Next, let us consider the opposite extreme, where the rate of LOH is very high, such that p u, Figure 6.2b. Now, the second inactivation event happens with probability p, that is, it is greatly accelerated compared to the case p = 0. However, the price that the cell lineage has to pay is a very high rate at which non-viable mutants are produced. This will considerably slow down the expansion of the TSP-negative phenotype.

Therefore, there must be an intermediate, optimal (for cancer!) value of the rate of chromosome loss, for which wild-type cells have a high chance of inactivating the TSP gene, without having to pay too high a price in non-viable or non-reproductive mutants.

6.2 Calculating the optimal rate of chromosome loss

The model set-up. We model epithelial tissue organized into compartments. In the simplest case, there is one stem-cell per compartment. For example, in colon this would correspond to crypts with a stem-cell situated at the base of each crypt. Stem cells divide asymmetrically producing one (immortal) stem-cell and one differentiated cell. Here we concentrate on the dynamics of the stem cells. Each division event is equivalent to a replacement of the old stem cell with a copy of itself. Upon division of a stem cell, the immortal daughter cell might (i) acquire a silencing mutation in one of its alleles of the APC gene with probability u per cell division, or (ii) lose one of its chromosomes, with probability p per cell per cell division per chromosome. Once both copies of the TSP gene have been inactivated, the cell will be able to escape homeostatic control and create a growing clone. We will describe the clonal expansion by a deterministic model.

Uncertainties still exist about the exact cellular origins of cancer, see Chapter 5. According to the stem-cell theory, it is the stem cells which are at risk. The de-differentiation theory suggests that partially differentiated cells could be targets for cancerous mutations. If we do not want to restrict ourselves to one or the other theory, we can solve the problem of optimization for a number of different assumptions. According to one scenario, cancer is initiated in adult stem cells. There are or several stem cells per compartment, such as the crypt of the colon. Alternatively, we can assume that a healthy compartment contains a population of partially differentiated cells, which are subject to a constant turnover, but still maintain a constant size of the compartment. Depending on the number of cells, this can be described either with a stochastic or a deterministic model. Again, inactivation of a TSP gene results in clonal expansion. It turns out that the results remain very similar in the context of the different assumptions.

Optimal rate of chromosome loss. Suppose that a stem cell has a probability to lose a chromosome p per chromosome per cell division. First we calculate the probability to inactivate the TSP gene by time t. The sequence of events can be expressed by the following simple diagram, y o

lose 2k chromosomes di(fc)

lose 2k — 1 chromosomes

Here yi is the probability for the stem cell to have i inactivated copies of the TSP gene. The first event of inactivation happens by a fine-scale genetic event (probability u times two for two alleles), and the second event is a loss of the chromosome with the remaining copy of the TSP gene (probability p). The parameter k is related to the cost of chromosome loss, as explained below.

A very important issue here is the exact cost of LOH events for the cell and its reproductive potential. In the most optimistic (for cancer) scenario, (a), there is no reduction in fitness due to the loss of any other chromosomes: the only chromosome that "counts" is the one containing the TSP gene. At stage yo, a loss of either copy damages the cell, and at stage yi, a loss of the chromosome with the mutated copy of the TSP gene is harmful (and a loss of the other copy leads to a clonal expansion). An alternative interpretation of this extreme case is that while loss of a single chromosome copy would reduce fitness, this is buffered by duplication events. In the most pessimistic scenario (b), a loss of any chromosome results in cell death, unless it leads to a TSP inactivation. It is safe to say that the reality is somewhere between these extreme scenarios.

For scenario (b), we set d0(k) = 1-(1 -p)2k and di{k) = l-(l-p)2fc-1, where k = 23 is the number of chromosomes. For scenario (a), the death rates can be expressed by the same formulas with k = 1.

We can write down the Kolmogorov forward equations for all the probabilities (skipping the argument k of do and di), jfe = [(1 - dx)(l - 2u) - 1] j/o, (6-1)

yi = (1 - d0)2uy0 + [(1 - di)(l - p) - 1] »i, (6.2)

with the initial condition j/o(0) = 1. We need to calculate the probability distribution of creating a TSP-/- mutant as a function of time, which is given by y2. We have,

p + d\ — u where we assumed that ut -C 1. Note that the argument given here holds without change for (constant) populations of more than one cell, as long as the number N of cells satisfies N < 1/u and N < 1 /^/p. Otherwise, the calculations can be easily adapted to include the effect of tunneling, see Chapter 3.

Once a TSP-/- cell has been produced, it starts dividing according to some law which is (at least, initially) close to exponential. Starting from one cell at time t = 0, by time t we will have Zy(t) cells, with

The parameter a is the growth rate of the initiated cells, and 0 < (3 < 1 is the cost due to the fact that a chromosome is missing from all CIN cells because of the inactivation of the TSP by a loss of chromosome. The factor [1 - di(k)] comes from the probability for a CIN cell to produce a nonviable mutant, which for scenario (a) only happens if only one particular chromosome is lost, and for scenario (b) - if any chromosome is lost.

If we now include the mutation stage, we will need to evaluate the convolution,

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