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Mathematical Models of Gene Amplification with Applications To Copyright 0 1990 by the Genetics Society of Alnerica Mathematical Models of Gene Amplification With Applicationsto Cellular Drug Resistance and Tumorigenicity Marek Kimmel"" and David E. Axelrod+ *Investigative Cytology Laboratory, Memorial Sloan-Kettering Cancer Center, New York, New York 10021, and TWaksman Institute, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08855-0759 Manuscript received July 3 1, 1989 Accepted for publication March 12, 1990 ABSTRACT An increased number of copies of specific genes may offer an advantage to cells when they grow in restrictive conditions such as in the presence of toxic drugs, or in a tumor. Three mathematical models of gene amplification and deamplification are proposed to describe the kinetics of unstable phenotypes of cells with amplified genes. The models differ in details but all assume probabilistic mechanisms of increase and decrease in gene copy number per cell (gene amplification/deamplifica- tion). Analysis ofthe models indicates that a stable distribution of numbers of copies of genesper cell, observed experimentally, exists only if the probability of deamplification exceeds the probability of amplification. The models are fitted to published data on the loss of methotrexate resistance in cultured cell lines, due to the loss of amplified dihydrofolate reductase gene. For two mouse cell lines unstably resistant tomethotrexate the probabilities of amplification and deamplification of the dihydrofolate reductase gene on double minute chromosomes are estimated to be approximately 2% and lo%, respectively. These probabilities are much higher than widely presumed. The models explain the gradual disappearance of the resistant phenotype when selective pressure is withdrawn, by postulating that the rate of deamplification exceeds the rate of amplification. Thus it is not necessary to invoke a growth advantage of nonresistant cells which has beenthe standardexplanation. For another analogous process, the loss of double minute chromosomes containing the myc oncogene from SEWA tumor cells, the growth advantage model does seem to be superior to the amplification and deamplification model. In a more theoretical section of the paper, it is demonstrated that gene amplification/deamplification can result in reduction to homozygosity, such as is observed in some tumors. Other applications are discussed. HE amount of DNA per cell remains constant cation of oncogenes has been observed in many hu- T fromone cell generationto another because man tumor cells and may confer a growth advantage during each cell cycle the entire content of DNA is on cells which overproduce the oncogene products. duplicated and then at each mitotic cell division the The inheritance of cellular phenotypes conferred DNA is evenly apportioned to two daughter cells. by gene amplification differs from that confered by However,recent experimental evidence shows that classical genetic mutations in several ways. The rate for a fraction of DNA, the number of gene copies per at whichcells with new phenotypes appear is very cell may not be constant. We are concerned with the rapid (RATH,TLSTY and SCHIMKE1984), the pheno- distribution of those genes which exist in variable types are unstable and rapidly reversible, subpopula- numbers of copies. tions with a broad range of phenotypes can coexist, An increase in gene copy number is referred to as and each subpopulation can regeneratethe entire gene amplification. Gene amplification can been- range of subpopulations (KAUFMANN,BROWN and hanced by conditions that interferewith DNA synthe- SCHIMKE 198 1). sis (BROWN,TLSTY and SCHIMKE1983; TLSTY, In this communication we present mathematical BROWN andSCHIMKE 1984), andis increased in some models of gene amplification that may help to under- mutant (GIULOTTO,KNIGHTS and STARK1987) and stand the inheritance of cellular phenotypes associated tumor cells (ALITALOand SCHWAB 1986). Increased with variation in gene copy number. We focus on the numbers of genes may produce an increased amount kinetics of phenomenarather than on molecular of gene products, and in tumor cells confer resistance mechanisms..All the models assume that at each cell to chemotherapeutic drugs (SCHIMKE1984). Amplifi- generation, there is a probabilistic mechanism of in- creasing or decreasing the number of gene copies per I,1 resent address. Department of Statistics,Rice University. Houston, Texts 7725 1. cell. We analyze the behavior of these models in order Genetics 125 639-644 (July, 1990) 634 M. Kimmel and D. E. Axelrod TABLE 1 Comparison of the essential features of the models of gene amplification Geometric amplification Feature nod el 2 Each double minute repli- Unspecified All copies replicated w.p. /3, cated independently w.p. p; or not replicated w.p. 1 not replicated w.p. 1 - p Division of gene copies be- Each double minute randomly Unspecified Copies split equally between t\veell daughter cell assigned to one of the daughters w.p. 6, or a11 cop- daughter cells ies to a random selected daughter W.p. 1 - 6 Resulting stochastic pIocess Galton-Watson branching Random walk with 0 as Random walk with 0 21s ab- process absorbingboundary sorbing boundary andaddi- tional transition to 0 Probabilities of mmplification, deamplification and gene loss: b b d d C 0 Abbrevi;ttion: w.p., with probability. to determine under what conditions a population of formally by, cells will have a stable distribution of gene copy num- bers, as is observed experimentally.We then apply the models to published experimental data in order to explain the loss of the dihydrofolate reductase gene, Process {x,) assumes all nonnegative integer values. and the myc oncogene of SEWA mouse tumor cells. In the following paragraph we give a detailed de- We also consider more theoretical issues, among them scription of our models. Essentials are collected in the reduction to homozygosity observed in some tu- Table 1. mors. Model 1, independent amplification:It is assumed that all the extrachromosomalelements containing the genes of interest, are independent of each other. During cell’s S-phase, each element is replicated once MODELS OF GENEAMPLIFICATION AND DEAMPLIFICATION with probability (w.P.) equal top, orit is not replicated with probability 1 - p. Then,at the time ofcell In this section we present three probabilistic models division, each element is assigned randomly to one of of amplification and deamplification of genes on ex- the daughter cells. In a lineage of cells a randomly trachromosomalelements. Models 1 and 3 are de- selected daughter will inherit, for each extrachromo- signed to represent two extremely different possibili- somal element present in the mother cell, two ele- ties, one ofwhich is that each of the elementsamplifies ments with probability b = p/4, oneelement with or deamplifies independently of the rest, and the other probability 1 - b - d = 1/2 and noelements with that they amplify or deamplify jointly. Predictions of probability d = (2 - p)/4. Thus, in accordance with these idealized models will be compared to the data, the assumed independence, the stochastic process {X,, in order to determinewhich of them is more likely to n 2 0), is the Galton-Watson process with progeny be true. generating function,f(s) = bs2 + (1 - b - d)s -t d. Model 2 represents the simplest conceivable alter- In a wider sense, model 1 may represent any mech- native of amplification/deamplification, virtually with anism by which extrachromosomal genes are ampli- no structural hypotheses. fied (w.p. b) or deamplified (w.p. d)independently of In all three models we consider the stochastic proc- each other. ess tX,, n 2 01 of the number of gene copies in the n- Model 2, geometric amplification: In this model th generation of the randomly selected cell lineage. no particular mechanism of gene amplificdtion/deam- In models 2 and 3, X, may assume only values 0, 1, 2, plification is assumed. It is only postulated that from 4, 8, 16, . Therefore, we will also employ the one generation to another the numberof gene copies process {xn, n 2 01, with reduced state space, defined onextrachromosomal elements may double with Models of Gene Amplification 635 probability b, stay the same with probability 1 - b - to limiting values. In mathematical terms: d or halve with probability d. Therefore process (x,; n 2 0) is a denumerable Markov chain with the follow- Kan; model 1, 1 - Pix, = 01 (2) ing infinite transition Matrix I: - Kan/n3/';models 2 and 3, 0 0 0 d (1 -b-d) b 0 d (1 - b - d) b 0 d (1 -b-d) b *a* MATRIX 1 It may be classified as a random walk with an absorb- Pix, = ilXn # 0) + c,, i L 1, (3) ing boundary at 0. Assumptions of this model have as n + m), whererates CY,the proper probability been first introduced by PETERSON(1983, 1984), in distributions (ci), and constants K are defined in the an informal way, in reference to chromosomal genes. terms of parameters b, d and c (different for each Model 3, joint amplification: It is assumed that all model; see Table 2). Mathematical details are deferred the extrachromosomal elements are replicated, or not to the APPENDIX. replicated, jointly. The probability of replication is P Thus, in the subcritical case, there exist limit distri- < 1. At division, the elements areeither divided butions of gene copy number, conditional on nonex- equally between daughter cells, with probability 6 < tinction of the gene. The expected value of such 1, or assigned to one randomly chosen daughter cell, distribution is denoted by E,, with probability 1 - 6. In a lineage of cells, a randomly selected daughter will inherit twice her mother's num- E, = 2 ic,. (4) ber of extrachromosomal elements with probability b t = p(1 - 6)/2, exactly this number with probability 1 - b - d - c = (1 - P)(1 - 6)/2 + pa, half of her Convergence to the limit distribution, for model 2, mother's number with probability d = (1 - @)a, and is illustrated by simulations in Figure 1.
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