Gene Amplification and Drug Resistance

Amplification of a gene is an increase of the number of copies of that gene in a cell. Amplification of genes coding for the enzyme dihydrofolate reductase (DHFR) has been associated with cellular resistance to the anticancer drug methotrexate (MTX). A resistant population with an increased number of DHFR gene copies per cell can be obtained after a sensitive population is grown in increasing concentrations of the drug. Increased resistance is correlated with increased numbers of gene copies on small extrachromosomal DNA elements. These elements are visible in the microscope and resemble pairs of small chromosomes; they are called double minute chromosomes or double minutes. The number of DHFR genes on double minutes in a cell may increase or decrease at each cell division. This is because double minutes are acentric, i.e., they do not have centromeres like real chromosomes. Centromeres are required for the mitotic apparatus to faithfully segregate chromosomes into progeny cells.

In populations of cells with the double minutes, both the increased drug resistance and the increase in number of gene copies are reversible. The classical experiment confirming this includes transferring the resistant cell population into a drug-free medium. When these populations are grown in the absence of the drug, they gradually lose resistance to the drug, by losing extra gene copies.

The population distribution of numbers of copies per cell can be estimated by the experimental technique called flow cytometry. In the experiments described, two features of these distributions are notable. First, as expected, the proportion of cells with amplified genes decreases with time. Second, less obvious, the shape of the distribution of gene copy number within the subpopulation of cells with amplified genes appears stable as resistance is being lost. This stable distribution is depicted in Fig.1 taken from Brown et al. (1981), Molecular and Cellular Biology 1: 1077-1083. The distribution of cells with amplified genes retains its shape, only the area under the distribution gradually decreases while the peak corresponding to sensitive cells increases.

Figure 1 Loss of the amplified copies of the DHFR gene during cell growth in MTX-free media. The 3T6 cells resistant to the MTX were grown for different times in MTX-free medium. The fluorescence level is proportional to the number of gene copies per cell. The values in parentheses are the percentages of cells with gene copy numbers greater than those for sensitive cells. a. Dotted line, 3T6 sensitive cells; solid line, resistant cells. b. Cells grown for 17 generations without MTX. c. Cells grown for 34 generations without MTX. d. Cells grown for 47 generations without MTX. (Source: Brown et al. 1981)

Galton–Watson Process Model of Gene Amplification and Deamplification

Recall the following limit theorem from Lecture 12. It is known as Yaglom’s theorem.

Theorem 12.2 If \(\mu<1\), then \[ \lim_{n\to \infty} P(X_n=k\ |\ X_n>0)=p^*_k, \qquad \sum_{k=1}^\infty p^*_k=1. \] The probabiility generating function of the limiting distribution \[ f^*(s)=\sum_{k=1}^\infty p^*_ks^k \] is the solution of the equation \[ 1-f^*(\Pi_{X_0}(s))=\mu (1-f^*(s)), \] where recall that \(\Pi_{X_0}(s)\) is the offspring PGF.

Material of this section is based on the paper by Kimmel and Axelrod (1990), Genetics 125: 633-644. It is an example of application of the Yaglom’s theorem to the analysis of the asymptotic behavior of a subcritical Galton-Watson process

We consider a cell, one of its progeny (randomly selected), one of the progeny of that progeny (randomly selected), and so forth. The cell of the \(n\)th generation contains \(X_n\) double minutes carrying the DHFR genes. During cell’s life, each double minute is either replicated with probability \(a\), or not replicated, with probability \(1 − a\), independently of the other double minutes. Then, at the time of cell division, the double minutes are segregated to progeny cells. If the double minute has not been replicated, then it is assigned to one of the progeny cells with probability 1/2. If it has been replicated, then either both copies are assigned to progeny 1 (with probability \(\alpha/2\)), or to progeny 2 (with probability \(\alpha/2\)), or they are divided evenly between both progeny (with probability \(1-\alpha\)). Let us note that the two double minutes segregate independently to progeny cells only when \(\alpha =1/2\). Otherwise, they either preferentially go to the same cell (\(\alpha>1/2\)), or to different cells (\(\alpha<1/2\)). The randomly selected progeny in our line of descent contains (Fig.2),

• No replicas of the original double minute (with probability \((1 − a)/2 + a\alpha/2)\), or

• One replica of the original double minute (with probability \((1 − a)/2 + a(1 − \alpha))\), or

• Both replicas of the original double minute (with probability \(a\alpha/2)\).

Figure 2 Schematic representation of the mathematical model of amplification and deamplification of genes located on double minute chromosomes. The sequence of events is presented for one of the possibly many double minutes present in the cell. During cell’s life, the double minute is either replicated, or not replicated. At the time of cell division, the double minute is assigned to one of the daughter cells (segregation). If it has not been replicated, it is assigned to one of the daughter cells. If it has been replicated, then either both copies are assigned to daughter 1, or to daughter 2, or they are divided evenly between both daughters. Probabilities of the events involved are presented in the graph. (Source: Kimmel and Axelrod 1990)

Therefore, the number of double minutes in the \(n\)th generation of the cell lineage is a Galton-Watson process with the offspring PGF \[ \Pi_{X_0} (s) = d + (1 − b − d)s + bs^2, \] where \(b = a\alpha/2\) and \(d = (1−a)/2+a\alpha/2\) are the probabilities of gene amplification and deamplification, respectively. Since in the absence of selection double minutes gradually disappear from the cell population, it is assumed that deamplification (loss of gene copies) exceeds amplification, so that the process is subcritical. In mathematical terms, \(b<d\) and \[ \mu = \frac{d}{ds}\Pi_{X_0}(s)\big|_{s=1} = 1 + b − d < 1. \] ## Mathematical Model of the Loss of Resistance

We will call a cell resistant if it carries at least one double minute chromosome with the DHFR gene. Otherwise it is called sensitive. In the experiments described above, a population of cells resistant to MTX, previously cultured for \(N\) generations in medium containing MTX, consists only of cells with at least one DHFR gene copy, i.e., \(X_N\) > 0. Therefore, the number of gene copies per cell is distributed as \(\{X_N\ |\ X_N > 0\}\). If \(N\) is large then, since the process is subcritical, by the Yaglom’s Theorem, this distribution has PGF \(f^*(s)\) satisfying the functional equation given in the theorem. Also, based on the estimates of \(1−\Pi_{X_n}(0)\sim \mu^n/(f^*(1))'\) as \(n\to \infty\), the resistant clone grows, in each generation, by the factor \(2\mu\) on the average. After the \(N\) initial generations, the resistant clone has been transferred to the MTX-free medium. The overall number of cells now grows by factor two in each generation, while the average number of resistant cells continues to grow by the factor \(2\mu\). Let us denote by \(R(n)\) and \(S(n)\), the number of resistant and sensitive cells in the population, n generations after transferring the cells to the MTX-free medium; \(r(n) = R(n)/[R(n) + S(n)]\) is the fraction of resistant cells. We obtain, \[ R(n) = (2\mu)^nR(0),\qquad S(n) + R(n) = 2n[S(0) + R(0)], \] hence \[ r(n)/r(0) = \mu n. \] This means that the proportion of resistant cells decreases geometrically, while the distribution of gene copy number among the resistant cells remains close to the limit distribution of the Yaglom theorem. This behavior is consistent with the experimental data of Fig.1.

Probabilities of Gene Amplification and Deamplification from MTX Data

Probabilities \(b\) and \(d\) can be estimated from the loss of resistance experiments similarly as in Kimmel and Axelrod (1990), using data on the S-180(R1A) cells in Kaufman et al. (1981), Molecular and Cellular Biology 1:1084-1093. The resulting estimates are \(b = 0.47\), \(d = 0.50\), yielding \(a = 1 − 2(d − b) = 0.94\) and \(\alpha = 2b/a = 1\). The interpretation is that while the frequency of replication of the double minute chromosomes is quite high, both copies are assigned almost always to the same progeny cell.