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A stochastic model for gene induction
J. theor. Biol. (1991) 153, 181-194
A Stochastic Model for Gene Induction MINORU S. H. K o
Furusawa MorphoGene Project, Exploratory Research for Advanced Technology ( ERA TO ), Research Development Corporation of Japan ( JRDC ), 5-9-6 Tohkohdai, Tsukuba 300-26, Japan (Received on 12 October 1990, Accepted in revised form on 22 March 1991) Expression levels of individual copies of an inducible gene have been presumed to be identical to the averaged level of many copies and to change in a smooth and predictable way according to the concentration of an inducing molecule. However, our recent experiments using a steroid-inducible system showed that the expression levels of individual copies are very heterogeneous and do not necessarily coincide with the averaged expression level of many copies (Ko et aL, 1990, EMBO J. 9, 2835-2842). To explain this result, I present a stochastic model for gene induction here and its analysis using computer simulation. Stochasticity in the model is derived from the randomness corresponding to the random timing of molecular collisions and dissociations between transcription factors and a gene copy, since at any instant each copy is thought to be either "switched on" by having a transcription complex bound to it, or "switched off' by not having a transcription complex bound. This model can produce two types of gene induction that depend on the stability of the transcription complex on the regulatory region of the gene. An unstable transcription complex causes a homogeneous level of gene induction among individual copies, while a stable transcription complex causes a heterogeneous level. Since the recent consensus formed by in vitro transcription experiments is that the transcription complex is generally very stable, the latter case (the non-deterministic one) is highly possible. Since typical eukaryotic cells have just two copies for any gene in a single cell, this possibility of heterogeneous gene induction indicates that the phenotypes of individual cells cannot be precisely determined by just environmental signals, such as inducers. This may prompt us to reconsider many problems related to gene induction, including morphogenesis.
1. Introduction Induction or repression of a battery of genes by environmental signals play a central role in determining the fate of cell differentiation. For a typical eukaryote inducible gene, the expression level of an individual copy is thought to change in a smooth and predictable way according to the concentration of an inducing molecule (say a steroid h o r m o n e or a retinoid). This is a general basis for theories related to gene induction and for experiments using inducible gene expression systems. Since this notion is extrapolated from results obtained by measuring the averaged activity o f many gene copies, it is essential to know whether the expression levels o f the individual gene copies are really determined in the concentration-dependent manner of inducers. Accordingly, development of both theoretical and experimental strategies to analyze the behavior of individual gene copies becomes essential. 181
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Experimentally, we previously reported such attempts using a glucocorticoid-inducible gene expression system (Ko et al., 1990). In this paper, a theoretical approach to this problem is presented. Recent studies of gene expression regulation have elucidated many intriguing mechanisms and serve to illustrate how genes are switched on and off (for reviews see Johnson & McKnight, 1989; Saltzman & Weinmann, 1989; Ptashne & Gann, 1990). In brief, the cis-acting transcriptional regulatory region, or the enhancer and promoter regions, are located upstream of the transcribed DNA sequences and are specific binding domains for the trans-acting factors or the transcription factors. These proteins bind to the enhancer/promoter region and form a transcription complex, presumably by protein-protein interactions. This complex is then recognized by RNA polymerase and transcription begins. Thus, at any instant each gene copy is either "switched on" by having a transcription complex bound to it, or "switched off' by not having a transcription complex bound. If we were to measure directly the transcription rate of each gene copy, we should find it fluctuating randomly between two quantized levels---the randomness corresponding to the random timing of molecular collisions and dissociations and the quantized levels corresponding to the fact that we are in effect observing the association of a single molecule with its ligand-activated transcription factor. Based on this concept, I have built a stochastic model and analyzed it in a quantitative fashion by computing the expression levels of individual gene copies. I have found two types of gene induction that depend on the half-life of the " o n " state of the gene and, therefore, on the extent of transcription complex stability. During the gene induction, one generally observes not the instantaneous rate of gene transcription, but the quantity of gene product accumulated over some characteristic time interval. When the accumulation time is long compared with the half-life of the " o n " state of the gene, the quantity of gene product represents a time-average of the state of the gene, and varies in a more or less smooth and predictable way with the concentration of the inducer. I call this a homogeneous or deterministic gene induction type. In contrast, when the accumulation time is relatively short, or equivalently when the half-life of the " o n " state is long, we should observe random fluctuations. I call this a heterogeneous or non-deterministic gene induction type. To the best o f my knowledge, there is no clear example for the former case, in which a homogeneous and concentration-dependent induction occurs at the level of a single gene copy. However, the existence of heterogeneous, or non-deterministic, gene induction was supported by our recent results in which the induction levels of individual gene copies were shown to be heterogeneous in a steroid hormoneinducible gene expression system (Ko et al., 1990). In the last part of this paper, the problems ofgene induction prompted by previous and current results, especially as they relate to morphogenesis are discussed. 2. The Model
For simplicity, the model is limited to consideration of genes transcribed by RNA polymerase II, although essentially the same argument can be applied to the
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transcription systems of RNA polymerase I and III. Recent in vitro transcription experiments show that the assembly of some trans-acting factors containing TATAbinding proteins on the enhancer/promoter region of the target gene switches on transcription by RNA polymerase II (for reviews, see Lillie & Green, 1989; Johnson & McKnight, 1989; Saltzman & Weinmann, 1989; Ptashne & Gann, 1990). At least part of this complex probably remains bound to the promoter region throughout multiple rounds of transcription (Hawley & Roeder, 1987; Van Dyke et aL, 1989). Figure 1 shows a simplified illustration of this process. Although usually a single transcribed region has several cis-elements including a TATA-box, a CAAT-box and enhancer sequences, the simplified model gene of Fig. 1 has only two such regulatory'sequences (RSI, RS2). The system begins when a defined concentration of inducer is added. By binding an inducer molecule, an inactive trans-acting factor (designated as trans-acting factor 1) molecule is transformed into an active transacting factor 1 molecule (step a). The concentration of the active trans-acting factor 1 is proportional to the concentration of the inducer. In this model, activation of the trans-acting factor 1 is brought about by the inducer, although other mechanisms are also possible such as phosphorylation or de novo synthesis. In the next step, the active trans-acting factor 1 binds to RSI (step b). This is the rate-limiting step for the overall process. A second trans-acting factor (designated as trans-acting !
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factor 2) then binds to RS2 (step c). This process may be facilitated by pre-existing bound trans-acting factor 1 through protein-protein interactions. RNA polymerase II recognizes this transcription complex comprised of the trans-acting factors bound to the enhancer/promoter (step d), and RNA synthesis begins (step e). While the transcription complex remains bound to the enhancer/promoter region (step f), RNA polymerase II repeatedly recognizes this transcription complex (step d) and RNA is synthesized (step e). Once the transcription complex dissociates from the enhancer/promoter sequences, the whole process is initialized (step g). There are several essential features required for the success of this simplified model: (i) the concentration of the active trans-acting factor 1 is proportionally determined by the concentration of the inducer; (ii) binding of one trans-acting factor activated by the inducer, whether the first one or the second one, to the enhancer/promoter region is a rate-limiting step for whole process; (iii) when the transcription complex remains on the enhancer/promoter region, RNA polymerase II can recognize it and initiate multiple rounds of transcription; and (iv) during the existence of the transcript ion complex, the transcription rate by RNA polymerase II is constant. The first point (i) is valid under ordinary conditions where many molecules of inducer and trans-acting factor 1 are present. The second point (ii) seems to be a consensus. In inducible gene expression systems, trans-acting factors that already exist in a cell, yet are somehow masked in their activating function, are known to be rate-limiting for formation of the transcription complex on the enhancer/promoter region (for review see Johnson & McKnight, 1989). The third point (iii) is established by in vitro transcription experiments (Hawley & Roeder, 1987; Van Dyke et al., 1989). The fourth point (iv) is a reasonable assumption, if the concentration of RNA polymerase II and other accessory factors for transcription is sufficient. To simulate the behavior of individual gene copies, we have to trace every step with temporal sequences of'individuals. Figure 2 shows the transition diagram for a single gene molecule based on the above model. There are two states (A and B) of the gene in this system, because it is supposed that the binding of trans-acting Stort
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factor 2 (step c) and the recognition by R N A polymerase II (step d) occur immediately after the binding of active trans-acting factor 1 (step b). Thus, the time for step c and d is negligible, since the step b is the rate-limiting step for the overall process. The system starts at time 0 immediately after the addition of an inducer. The concentration o f active trans-acting factor 1 is determined proportionally according to the concentration of the inducer. At every unit time, the state of the gene proceeds through an arrow o f the transition diagram. A gene copy in state A produces no transcribed R N A molecules, while a gene copy in state B produces a defined number of RNA molecules. Since collisions and dissociations o f molecules are a stochastic, these processes are properly represented by probabilities. The probability o f step b is referred to here as PI and is a function o f the number o f available active trans-acting factor 1 molecules and the affinity between active trans-acting factor 1 and RS1. This affinity is assumed to be constant through the whole process, and PI becomes a function of the number of molecules of active trans-acting factor 1. The probability of initialization (step g) is represented by P2 and is a function of the stability of the transcription complex. Since many proteins interact with each other in the transcription complex and stabilize the complex, P2 is not simply a function of the affinity between active trans-acting factor 1 and RS1. It is worth noting that the rate-limiting trans-acting factor (here the active transacting factor 1) is not necessarily involved in the transcription complex, but might just be a trigger for the formation of the transcription complex. In such a case, a trans-acting factor 3 should be included in the model for stabilizing the transcription complex by protein-protein interactions. When P1 and P2 are fixed, we can trace every transition o f g e n e copies individually. When the system starts, the computer produces a random positive number (-< 1). If that number is smaller than P~, the state of the gene proceeds to state B. In this state, R N A polymerase II produces a constant amount of transcripts per unit time (arbitrarily defined here as five molecules of R N A per unit time). For simplification, degradation of transcripts are not considered, and products accumulate. In the next step, the computer again produces a random positive number (-<1). If that n u m b e r is smaller than P2, the state of the gene proceeds to the state A. At every transition step, the gene copy takes one of two states. When the gene is in the A state, there is no transcription. When the gene is in the B state, there is a constant a m o u n t of transcription. After 51 steps of these processes, the accumulated transcripts o f an individual copy of the gene are calculated. Thus, the maximum amount o f R N A is 255 molecules and the minimum amount of R N A is 0 molecules. 3. Results
Simulation analyses were made at various values o f P~ and P2 for the individual copies. Figure 3(a) shows the gene expression levels (amounts o f accumulated R N A molecules after 51 steps of processes) of 1000 gene copies at various concentrations of a rate-limiting active trans-acting factor 1(P1), when the transcription complex is very stable (P2 = 0.01). The induction level of an individual gene copy was very heterogeneous among copies even under identical conditions. When P1 took the
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FIG. 3. The distribution of the induction level at various trans-acting factor concentrations. Different possible levels of induction of a single gene copy (accumulated quantities of transcripts) are plotted along the X-axis. The probability for a gene to show a given level of induction is plotted on the Y-axis. T h e numerical value attached to each curve indicates the averaged amount of R N A molecules produced. (a) A stable transcription complex ( ~ = 0.01). (b) A n unstable transcription complex (P2 = 0.9).
values ranging from 0.01-0.1, the induction level was distributed across a wide range. For example, in the presence of very low concentrations of trans-acting factor 1 (P~ =0-01), some copies show a maximum expression level (255 molecules). In contrast, when the transcription complex is unstable (/>2=0.9) [Fig. 3(b)], the induction level of an individual copy is more homogeneous among the gene copies and the peak level of induction increased as the concentration of trans-acting factor 1 (P1) increased. In both cases, the averaged level of gene induction is proportional to the concentration of the active trans-acting factor 1 (P~) [Fig. 3(a) and (b)]. This means that when we consider the averaged behavior of gene induction, the gene expression mechanism of both stable and unstable transcription complex cannot be distinguished. However, when we consider the expression level of individual gene copies, these two cases are totally different. To evaluate the heterogeneity of gene induction levels, the standard deviation was used as a reference. In the simulation, where P~ = 0-01 and P2 = 0 . 0 1 (mean: 52-2), the standard deviation was 76.9. In contrast, where P1 = 0.5 and P2 = 0.9 (mean: 91.4), the standard deviation (S.D.) was 11.3. According to Chebyshev's
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inequality, more than 90% of the data should exist within the region of M + 3 x S.D. (M: mean). Thus, in the former case, the value of the standard deviation means that the most data is distributed over the full spectrum. Thus, the larger the standard deviations, the greater the heterogeneity. Although it is difficult to determine the border between heterogeneity and homogeneity, this index is useful to view the overall behavior of gene copies. Figure 4 shows cumulative data presented with the standard deviations for four levels of stability of the transcription complex. The dose-response curves of the averaged activity of gene expression show the wellknown gradual increase as P~ increases. Conventionally, the data for such curves have been thought to represent the induction levels of individual gene copies. However, as the standard deviation of individual copies shows, the behavior of individuals is very heterogeneous when the transcription complex is relatively stable, such as when P2 = 0.0012 o r P2 = 0-01. These data indicate that the heterogeneity of gene induction level is mainly determined by the stability of the transcription complex (P2). The simulation data is summarized in a contour map of the standard deviations as depicted in Fig. 5. This map illustrates that as the transcription complex stabilizes, the induction level of individual gene copies becomes heterogeneous and individuals
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behave in a totally different manner. In contrast, when the transcription complex is unstable, homogeneous gene induction levels are observed. By monitoring the behavior of individual gene copies through the induction time, it was found that the different frequency of formation of the transcription complex on the enhancer/promoter region caused the different behaviors of transcriptional activation (Fig. 6). Once a stable transcription complex is formed on the enhancer/ promoter region of the gene, it remains there for a substantial number of steps [Fig. 6(a)]. The amount of transcripts finally accumulated from the downstream gene is determined mainly by when the transcription complex is formed on the individual gene copy. With the condition that not all the copies are induced, those from a non-induced to a fully-induced state exist at the same time and heterogeneity is observed [Fig. 6(a)]. In contrast, when the transcription complexes formed on the enhancer/promoter region are unstable, homogeneous gene induction is obtained [Fig. 6(b)]. Even as the transcription complexes are formed, they immediately disassemble. In this case, the total expression level of individual copies during induction is determined by the assembly frequency of transcription complexes on the gene molecules. When the number of active trans.acting factor 1 molecules increases, the frequency of forming a transcription complex increases. By taking
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Total 225 255 185 180 175
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P~=0.5 010011001000101000101001010100010000000001010001001 100101001011100100101000100101000101000000000101011 110100101001010101001010111011001001100100001000101 010010101001010100110001010000101010001000010101110 001011010110000101010000100000101000100000010100001
Total 80 95 115 100 80
FtG. 6. A "life history" of individual gene copies. The numbers from left to right show the state of the gene for each time unit (0, state A in Fig. 2; 1, state B in Fig. 2). Data for five representative gene copies are shown (a)-(e).
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the appropriate induction time, the expression o f an individual gene copy can be averaged. This reduces variations in gene expression and leads to homogeneous gene expression of individual gene copies [Fig. 6(b)].
4. Discussion
4.l. AGAINST THE DETERMINISTIC VIEWPOINT OF GENE INDUCTION In this paper, the importance o f individual gene copy analyses for elucidation of gene induction mechanisms is proposed. At the level o f an individual gene molecule, the association and dissociation of the trans-acting factors and the enhancer/ promoter region on that gene is no doubt determined in a stochastic manner. On this basis, I have built a model and carried out an analysis o f that model by computer simulation. When the transcription complex on the e n h a n c e r / p r o m o t e r region was relatively unstable, the transcription complex formed and dissociated many times during the induction time. Thus, the total induction level of individual gene copies was integrated by the time factor. In this case, differences in the induction level among the gene copies was small and homogeneous gene induction was obtained. In contrast, when the transcription complex on the e n h a n c e r / p r o m o t e r region was relatively stable, the transcription complex existed continuously on the region during the induction time. Thus, the total induction level o f individual copies was directly influenced by the stochastic association and dissociation of the trans-acting factors and the e n h a n c e r / p r o m o t e r region. In this case, differences in the induction level among the gene copies were very large and heterogeneous gene induction was obtained. The most c o m m o n ideas about the mechanism of the gene induction seems to be based on a deterministic viewpoint. When the concentration of the inducer is determined, the expression levels of all the genes are thought to be nearly identical after a substantial induction time. The p h e n o m e n o n that the induction level measured biochemically using many gene copies as a whole is proportional to the concentration of the inducer in a dose-response relationship is interpreted as every gene copy showing a concentration-dependent induction level. However, the analysis of the model in this paper shows that the induction levels of individual gene copies are totally different when the transcription complex is stable, even if the concentration of inducer is determined. Thus, a non-deterministic viewpoint of gene induction mechanisms should be considered as well as the deterministic viewpoint. In this relation, the concept o f non-deterministic gene induction should be clearly distinguished from that of stochastic cell commitment or differentiation. Although a model of stochastic cell proliferation is presented (Till et ai., 1964) and it is well-established that the commitment decision for each cell is made in a stochastic manner in murine erythroleukemia cell differentiation (Gusella et al., 1976), there has been no evidence that these stochastic cell differentiation processes occur at the gene induction level. A key parameter for the difference between heterogeneous and homogeneous gene induction is the stability of the transcription complex on the e n h a n c e r / p r o m o t e r region. It is known that the stability o f the transcription complex can be augmented
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by protein-protein interactions among trans-acting factors (Ptashne, 1986). A direct inference from this observation is that even in a system using the same genes, the same regulatory regions, and the same rate-limiting trans-acting factor, the stability of the transcription complex can be altered by exchange of other trans-acting factors. Recent findings show that the same regulatory sequences can be bound by several distinct trans-acting factors (Schaffner, 1989; Johnson & McKnight, 1989; Ptashne & Gann, 1990), and this transformation from heterogeneous to homogeneous induction and vice versa, might work in an in vivo system. In different cell types or tissues, the same genes may use eithe:" types of induction mechanism, even if the same inducer is used. Although only the heterologous protein-protein interaction is modeled and discussed so far, homologous protein-protein interactions can also be considered in the same way. One of the best known examples of this is the co-operative binding of the repressor molecules on the operator sequences in the lambda phage system (for review see Ptashne, 1986). Although this kind of system shows a steeper sigmoidal curve in the dose-response relationship (Ptashne, 1986), essentially the same argument as the heterologous protein-protein interaction system can be made. 4.2. V A L I D I T Y O F T H E M O D E L
Since the time unit, total induction time, and the stability of transcription complex is relatively defined in the current model, I would like to discuss the validity of the model by comparison with a known system of gene induction. The averaged time for the existence of the transcription complex on the enhancer/promoter was about 38 time units in the stable case (P2 = 0.01) and about 1.2 time units in the unstable case (P2=0.9). Suppose that the total induction time is 12hr. This supposition seems to be reasonable, since the ordinary induction time for biologically significant phenomena, such as mesoderm induction in animal development, is from a few hours to a day. Since the total unit times for induction is 51 in the current model, one unit time calculates to about 14 min. In the representative case of heterogeneous gene induction (P2=0.01), the averaged time for maintaining the transcription complex on the enhancer/promoter region is calculated as about 9 hr. In the representative case of homogeneous gene induction (P2 = 0.9), that is calculated as about 17 min. Since in vitro analysis of gene expression regulation shows that the transcription complex stays on the promoter region for more than several hours, heterogeneous gene induction is possible. The choice of five molecules of RNA per 1 unit time (14min) for the transcription frequency by RNA polymerase II is reasonable, considering that on a gene of high transcription efficiency, such as the ovalbumin gene in tubular gland cells, the RNA polymerase II runs on the gene at 6 sec intervals (Palmiter, 1975). 4.3. E X P E R I M E N T A L
CONSIDERATION
The glucocorticoid-inducible gene expression system of the mouse mammary tumor virus (MTV) is an appropriate experimental model. The nature of its activation is well-documented and similar to the simulated model system in that changes in
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the extracellular glucocorticoid hormone concentration lead to changes in the concentration of intracellular active receptor-hormone complexes. These then bind and activate the MTV enhancer/promoter (Yamamoto, 1985; Beato, 1989). Recent molecular cloning of glucocorticoid receptor genes has revealed the major pathway for activation of glucocorticoid-inducible gene expression systems (for reviews, see Green & Chambon, 1988; Evans, 1988; Beato, 1989). In brief, glucocorticoids diffuse passively into the cells, then bind and activate specific receptor proteins. The active hormone-receptor complexes bind to glucocorticoid response elements by their zinc-finger domain and activate expression of the downstream gene. It is wellestablished that the amount of glucocorticoid receptor protein is a rate-limiting substance for the level of induction (Vanderbilt et al., 1987). To investigate the mechanism of gene induction experimentally, we introduced the fl-galactosidase gene under regulation of the MTV enhancer/promoter at a single or a few copies per cell (Ko et al., 1990). By measuring the/3-galactosidase expression level in an individual cell after 48 h of induction, we were able to measure the expression level of individual gene copies. We showed that the expression level of 3-galactosidase is heterogeneous and the dose-response relationship induced by the glucocorticoid hormone is caused by changes in the number of transcriptionally active genes (Ko et al., 1990). These data indicate the existence of heterogeneous gene induction that contrasts with the commonly accepted homogeneous gene induction. The current model provides an explanation for this unexpected behavior of a steroid-inducible gene expression system. If this model truly underlies the experimental results, the transcription complex on the steroid-inducible enhancer/promoter should be very stable. This seems to be supported by recent in vitro transcription results (Freedman et al., 1989; Klein-Hitpass et al., 1990). 4.4. M O R P H O G E N I C
FIELD
In the development of many organisms, the gradient of a certain substance has been expected to play an important role for pattern formation (Wolpert, 1969, 1989). The expression of a specific gene which gives a differentiated phenotype to the cells in the morphogenic field is supposed to be induced by a morphogen in a concentration-dependent manner. To produce a clear boundary, defining a group of cells expressing that gene in a monotonic concentration gradient of a morphogen, a positive feedback mechanism was presented (Lewis et al., 1977; Meinhardt, 1982). The fundamental basis of these hypotheses is that the behavior of all gene copies must be uniformly identical. That is to say, the induction level of an individual gene can be precisely determined by an inducer (a morphogen) in a concentrationdependent manner. However, according to the current analysis, this kind of deterministic gene induction is limited to the special case of genes with a very unstable transcription complex relative to the total induction time. Recently, it was reported that the gradient of retinoic acid may determine the specificity of the anteroposterior axis of X e n o p u s laevis neural development (Durston et al., 1989) and the vertebrate limb pattern formation (for review, see Eichele, 1989). Since the retinoic acid receptor is a member of the steroid/thyroid hormone
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superfamily (Evans, 1988; Green & Chambon, 1988), the type of gene induction mechanism may be the same as the glucocorticoid-inducible gene expression system, which is shown to be a heterogeneous gene induction type (Ko et al., 1990). In fact, essentially the same results were obtained using a retinoid-inducible/3-galactosidase gene transfected into F9 embryonic carcinoma cells (M. S. H. Ko & Naomi Takahashi, unpublished). These data indicate that while the retinoic acid-inducible gene expression system is operating in morphogenesis and thus should be the deterministic gene induction type, the system behaves experimentally as the nondeterministic type. What kind of mechanism can explain this apparent discrepancy ? First, it is possible that the arrangement of the cells is altered after or during the induction by sorting out cells according to their expression level of the gene. Second, cell-to-cell communication before or during the induction may modify the inducibility of the gene in individual cells. Since these two possibilities require cell contact, the validity of these cases can be tested experimentally under conditions in which cells dissociated from the limb bud or the neural tissue are exposed to various concentrations of retinoic acids. Third, as I pointed out in the previous section, transformation of the gene induction type from heterogeneous to homogeneous is possible by replacing the species of trans-acting factor. Since we have tested the gene induction type in F9 embryonic carcinoma cells, it is possible that different species of trans-acting factors are involved in the transcription complex in the developing neuronal tissues or the limb bud. Fourth, the concentration gradient may be steeper than just a monotonic one. In the retinoic acid-inducible gene expression system, changes in the distribution of the retinoic acid receptor molecule can magnify differences in the effective concentration of the hormone-receptor complex. Fifth, the gene copy number per cell may be changed. If a single cell contains many copies, the heterogeneity of the gene induction level of an individual copy can be averaged. For example, it is known that the puffs of the Drosophila polytene chromosomes show a dose-dependent enlargement of ¢cdysone-inducible puffs (Ashburner et al., 1974). In a single genetic loci of polytene chromosomes, there are about 1000 copies of the genes. This amplification of genetic material may be an evolutionary strategy to avoid a stochastic gene induction mechanism. I thank Ms Naomi Takahashi for encouragement, Dr Mitsuru Furusawa for support and Dr Deborah J. Stearns-Kurosawa for critical reading of this manuscript. I also thank Dr Julian Lewis for useful comments on this manuscript. REFERENCES ASHBURNER, M., CHIHARA, C., MELTZER, P. & RICHARDS, G. (1974). Temporal control of puffing activity in polytene chromosomes. Cold Spring Harbor Syrnp. quant. Biol. 38, 655-662. BEATO, M. (1989). Gene regulation by steroid hormones. Cell 56, 335-344. DURSTON, A. J., TI MMERMANS, J. P. M., HAGE, W. J., HENDRI KS, J. F. J., DE VRIES, N. J., HEIDEVELD, M. & NIEUWKOOP, P. D. (1989). Retinoic acid causes an anteroposterior transformation in the developing central nervous system. Nature, Lond. 340, 140-144. EICHELE, G. (1989). Retinoids and vertebrate limb pattern formation. Trends Genet. 5, 246-251. EVANS, R. M. (1988). The steroid and thyroid hormone receptor superfamily. Science 240, 889-895.
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FREEDMAN, L. P., YOSHINAGA, S. K., VANDERBILT, J. N. & YAMAMOTO, K. R. (1989). In vitro transcription enhancement by purified derivatives of the glucocorticoid receptor. Science 245, 298-301. GREEN, S. & CHAMBON, P. (1988). Nuclear receptors enhance our understanding of transcription regulation. Trends Genet. 4, 309-314. GUSELLA, J., GELLER, R., CLARKE, B., WEEKS, V. & HOUSMAN, D. (1976). Commitment to erythroid differentiation by Friend erythroleukemia cells: a stochastic analysis. Cell 9, 221-229. HAWLEY, D. K. & ROEDER, R. G. (1987). Functional steps in transcription initiation and reinitiation from the major late promoter in a HeLa nuclear extract. 3". biol. Chem. 262, 3452-3461. JOHNSON, P. F. & McKNIGHT, S. L. (1989). Eukaryotic transcriptional regulatory proteins. Annu. Rev. Biochem. 58, 799-839. KLEIN-HITPASS, L., TSAI, S. Y., WEIGEL, N. L., ALLAN, G. F., RILEY, D., RODRIGUEZ, R., SCHARADER, W. T., TSAI, M. -J. & O'MALLEY, B. W. (1990). The progesterone receptor stimulates cell-free transcription by enhancing the formation of a stable preinitiation complex. Cell 60, 247-257. Ko, M. S. H., NAKAUCHI, H. & TAKAHASH l, N. (1990). The dose dependence of glucocorticoid-inducible gene expression results from changes in the number of transcriptionally active templates. E M B O J. 9, 2835-2842. LEWIS, J. H., SLACK, J. M. W. & WOLPERT, L. (1977). Thresholds in development. Z theor. Biol. 65, 579-590. LILLIE, J. W. & GREEN, M. R. (1989). Gene transcription: activator's target in sight. Nature, Lond. 341, 279-280. MEINHARDT, H. (1982). Models for Biological Pattern Formation. London: Academic Press. PALPvIITER,R. D. (1975). Quantitation of parameters that determine the rate of ovalbumin synthesis. Cell 4, 189-197. PTASHNE, M. (1986). A Genetic Switch. Palo Alto, CA: Cell Press and Blackwell Scientific Publication. PTASHNE, M. & GANN, A. A. F. (1990). Activators and targets. Nature, Lond. 346, 329-331. SALTZMAN,A. G. & WEINMANN,R. (1989). Promoter specificity and modulation of RNA polymerase II transcription. F A S E B J. 3, 1723-1733. SCHAFFNER, W. (1989). How do different transcription factors binding the same DNA sequence sort out their jobs? Trends Genet. 5, 37-39. TILL, J. E., MCCULLOCH, E. A. & SIMI NOVITCH, L. (1964). A stochastic mode/of stem cell proliferation, based on the growth of spleen colony-forming cells. Proc. hath. Acad. Sci. U.S.A. 51, 29-36. VANDERBILT, J. N., MIESFELD, R., MALER, B. A. & YAMAMOTO, K. R. (1987). Intracellular receptor concentration limits glucocorticoid-dependent enhancer activity. Molec. Endocrinol. 1, 68-74. VAN DYKE, M. W., SAWADOGO, M. & ROEDER, R. G. (1989). Stability of transcription complexes on class II genes. Molec. cell. Biol. 9, 342-344. WOLPERT, L. (1969). Positional information and the spatial pattern of cellular differentiation. J. theor. Biol. 25, 1-47. WOLPERT, L. (1989). Positional information revisited. Development 107, (Suppl.) 3-12. YAMAMOTO, K. R. (1985). Steroid receptor regulated transcription of specific genes and gene networks. Annu. Rev. Genet. 19, 209-252.
A Stochastic Model for Gene Induction MINORU S. H. K o
Furusawa MorphoGene Project, Exploratory Research for Advanced Technology ( ERA TO ), Research Development Corporation of Japan ( JRDC ), 5-9-6 Tohkohdai, Tsukuba 300-26, Japan (Received on 12 October 1990, Accepted in revised form on 22 March 1991) Expression levels of individual copies of an inducible gene have been presumed to be identical to the averaged level of many copies and to change in a smooth and predictable way according to the concentration of an inducing molecule. However, our recent experiments using a steroid-inducible system showed that the expression levels of individual copies are very heterogeneous and do not necessarily coincide with the averaged expression level of many copies (Ko et aL, 1990, EMBO J. 9, 2835-2842). To explain this result, I present a stochastic model for gene induction here and its analysis using computer simulation. Stochasticity in the model is derived from the randomness corresponding to the random timing of molecular collisions and dissociations between transcription factors and a gene copy, since at any instant each copy is thought to be either "switched on" by having a transcription complex bound to it, or "switched off' by not having a transcription complex bound. This model can produce two types of gene induction that depend on the stability of the transcription complex on the regulatory region of the gene. An unstable transcription complex causes a homogeneous level of gene induction among individual copies, while a stable transcription complex causes a heterogeneous level. Since the recent consensus formed by in vitro transcription experiments is that the transcription complex is generally very stable, the latter case (the non-deterministic one) is highly possible. Since typical eukaryotic cells have just two copies for any gene in a single cell, this possibility of heterogeneous gene induction indicates that the phenotypes of individual cells cannot be precisely determined by just environmental signals, such as inducers. This may prompt us to reconsider many problems related to gene induction, including morphogenesis.
1. Introduction Induction or repression of a battery of genes by environmental signals play a central role in determining the fate of cell differentiation. For a typical eukaryote inducible gene, the expression level of an individual copy is thought to change in a smooth and predictable way according to the concentration of an inducing molecule (say a steroid h o r m o n e or a retinoid). This is a general basis for theories related to gene induction and for experiments using inducible gene expression systems. Since this notion is extrapolated from results obtained by measuring the averaged activity o f many gene copies, it is essential to know whether the expression levels o f the individual gene copies are really determined in the concentration-dependent manner of inducers. Accordingly, development of both theoretical and experimental strategies to analyze the behavior of individual gene copies becomes essential. 181
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Experimentally, we previously reported such attempts using a glucocorticoid-inducible gene expression system (Ko et al., 1990). In this paper, a theoretical approach to this problem is presented. Recent studies of gene expression regulation have elucidated many intriguing mechanisms and serve to illustrate how genes are switched on and off (for reviews see Johnson & McKnight, 1989; Saltzman & Weinmann, 1989; Ptashne & Gann, 1990). In brief, the cis-acting transcriptional regulatory region, or the enhancer and promoter regions, are located upstream of the transcribed DNA sequences and are specific binding domains for the trans-acting factors or the transcription factors. These proteins bind to the enhancer/promoter region and form a transcription complex, presumably by protein-protein interactions. This complex is then recognized by RNA polymerase and transcription begins. Thus, at any instant each gene copy is either "switched on" by having a transcription complex bound to it, or "switched off' by not having a transcription complex bound. If we were to measure directly the transcription rate of each gene copy, we should find it fluctuating randomly between two quantized levels---the randomness corresponding to the random timing of molecular collisions and dissociations and the quantized levels corresponding to the fact that we are in effect observing the association of a single molecule with its ligand-activated transcription factor. Based on this concept, I have built a stochastic model and analyzed it in a quantitative fashion by computing the expression levels of individual gene copies. I have found two types of gene induction that depend on the half-life of the " o n " state of the gene and, therefore, on the extent of transcription complex stability. During the gene induction, one generally observes not the instantaneous rate of gene transcription, but the quantity of gene product accumulated over some characteristic time interval. When the accumulation time is long compared with the half-life of the " o n " state of the gene, the quantity of gene product represents a time-average of the state of the gene, and varies in a more or less smooth and predictable way with the concentration of the inducer. I call this a homogeneous or deterministic gene induction type. In contrast, when the accumulation time is relatively short, or equivalently when the half-life of the " o n " state is long, we should observe random fluctuations. I call this a heterogeneous or non-deterministic gene induction type. To the best o f my knowledge, there is no clear example for the former case, in which a homogeneous and concentration-dependent induction occurs at the level of a single gene copy. However, the existence of heterogeneous, or non-deterministic, gene induction was supported by our recent results in which the induction levels of individual gene copies were shown to be heterogeneous in a steroid hormoneinducible gene expression system (Ko et al., 1990). In the last part of this paper, the problems ofgene induction prompted by previous and current results, especially as they relate to morphogenesis are discussed. 2. The Model
For simplicity, the model is limited to consideration of genes transcribed by RNA polymerase II, although essentially the same argument can be applied to the
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transcription systems of RNA polymerase I and III. Recent in vitro transcription experiments show that the assembly of some trans-acting factors containing TATAbinding proteins on the enhancer/promoter region of the target gene switches on transcription by RNA polymerase II (for reviews, see Lillie & Green, 1989; Johnson & McKnight, 1989; Saltzman & Weinmann, 1989; Ptashne & Gann, 1990). At least part of this complex probably remains bound to the promoter region throughout multiple rounds of transcription (Hawley & Roeder, 1987; Van Dyke et aL, 1989). Figure 1 shows a simplified illustration of this process. Although usually a single transcribed region has several cis-elements including a TATA-box, a CAAT-box and enhancer sequences, the simplified model gene of Fig. 1 has only two such regulatory'sequences (RSI, RS2). The system begins when a defined concentration of inducer is added. By binding an inducer molecule, an inactive trans-acting factor (designated as trans-acting factor 1) molecule is transformed into an active transacting factor 1 molecule (step a). The concentration of the active trans-acting factor 1 is proportional to the concentration of the inducer. In this model, activation of the trans-acting factor 1 is brought about by the inducer, although other mechanisms are also possible such as phosphorylation or de novo synthesis. In the next step, the active trans-acting factor 1 binds to RSI (step b). This is the rate-limiting step for the overall process. A second trans-acting factor (designated as trans-acting !
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factor 2) then binds to RS2 (step c). This process may be facilitated by pre-existing bound trans-acting factor 1 through protein-protein interactions. RNA polymerase II recognizes this transcription complex comprised of the trans-acting factors bound to the enhancer/promoter (step d), and RNA synthesis begins (step e). While the transcription complex remains bound to the enhancer/promoter region (step f), RNA polymerase II repeatedly recognizes this transcription complex (step d) and RNA is synthesized (step e). Once the transcription complex dissociates from the enhancer/promoter sequences, the whole process is initialized (step g). There are several essential features required for the success of this simplified model: (i) the concentration of the active trans-acting factor 1 is proportionally determined by the concentration of the inducer; (ii) binding of one trans-acting factor activated by the inducer, whether the first one or the second one, to the enhancer/promoter region is a rate-limiting step for whole process; (iii) when the transcription complex remains on the enhancer/promoter region, RNA polymerase II can recognize it and initiate multiple rounds of transcription; and (iv) during the existence of the transcript ion complex, the transcription rate by RNA polymerase II is constant. The first point (i) is valid under ordinary conditions where many molecules of inducer and trans-acting factor 1 are present. The second point (ii) seems to be a consensus. In inducible gene expression systems, trans-acting factors that already exist in a cell, yet are somehow masked in their activating function, are known to be rate-limiting for formation of the transcription complex on the enhancer/promoter region (for review see Johnson & McKnight, 1989). The third point (iii) is established by in vitro transcription experiments (Hawley & Roeder, 1987; Van Dyke et al., 1989). The fourth point (iv) is a reasonable assumption, if the concentration of RNA polymerase II and other accessory factors for transcription is sufficient. To simulate the behavior of individual gene copies, we have to trace every step with temporal sequences of'individuals. Figure 2 shows the transition diagram for a single gene molecule based on the above model. There are two states (A and B) of the gene in this system, because it is supposed that the binding of trans-acting Stort
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FIG. 2. Transitiondiagram of a single gene for computersimulation.
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factor 2 (step c) and the recognition by R N A polymerase II (step d) occur immediately after the binding of active trans-acting factor 1 (step b). Thus, the time for step c and d is negligible, since the step b is the rate-limiting step for the overall process. The system starts at time 0 immediately after the addition of an inducer. The concentration o f active trans-acting factor 1 is determined proportionally according to the concentration of the inducer. At every unit time, the state of the gene proceeds through an arrow o f the transition diagram. A gene copy in state A produces no transcribed R N A molecules, while a gene copy in state B produces a defined number of RNA molecules. Since collisions and dissociations o f molecules are a stochastic, these processes are properly represented by probabilities. The probability o f step b is referred to here as PI and is a function o f the number o f available active trans-acting factor 1 molecules and the affinity between active trans-acting factor 1 and RS1. This affinity is assumed to be constant through the whole process, and PI becomes a function of the number of molecules of active trans-acting factor 1. The probability of initialization (step g) is represented by P2 and is a function of the stability of the transcription complex. Since many proteins interact with each other in the transcription complex and stabilize the complex, P2 is not simply a function of the affinity between active trans-acting factor 1 and RS1. It is worth noting that the rate-limiting trans-acting factor (here the active transacting factor 1) is not necessarily involved in the transcription complex, but might just be a trigger for the formation of the transcription complex. In such a case, a trans-acting factor 3 should be included in the model for stabilizing the transcription complex by protein-protein interactions. When P1 and P2 are fixed, we can trace every transition o f g e n e copies individually. When the system starts, the computer produces a random positive number (-< 1). If that number is smaller than P~, the state of the gene proceeds to state B. In this state, R N A polymerase II produces a constant amount of transcripts per unit time (arbitrarily defined here as five molecules of R N A per unit time). For simplification, degradation of transcripts are not considered, and products accumulate. In the next step, the computer again produces a random positive number (-<1). If that n u m b e r is smaller than P2, the state of the gene proceeds to the state A. At every transition step, the gene copy takes one of two states. When the gene is in the A state, there is no transcription. When the gene is in the B state, there is a constant a m o u n t of transcription. After 51 steps of these processes, the accumulated transcripts o f an individual copy of the gene are calculated. Thus, the maximum amount o f R N A is 255 molecules and the minimum amount of R N A is 0 molecules. 3. Results
Simulation analyses were made at various values o f P~ and P2 for the individual copies. Figure 3(a) shows the gene expression levels (amounts o f accumulated R N A molecules after 51 steps of processes) of 1000 gene copies at various concentrations of a rate-limiting active trans-acting factor 1(P1), when the transcription complex is very stable (P2 = 0.01). The induction level of an individual gene copy was very heterogeneous among copies even under identical conditions. When P1 took the
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FIG. 3. The distribution of the induction level at various trans-acting factor concentrations. Different possible levels of induction of a single gene copy (accumulated quantities of transcripts) are plotted along the X-axis. The probability for a gene to show a given level of induction is plotted on the Y-axis. T h e numerical value attached to each curve indicates the averaged amount of R N A molecules produced. (a) A stable transcription complex ( ~ = 0.01). (b) A n unstable transcription complex (P2 = 0.9).
values ranging from 0.01-0.1, the induction level was distributed across a wide range. For example, in the presence of very low concentrations of trans-acting factor 1 (P~ =0-01), some copies show a maximum expression level (255 molecules). In contrast, when the transcription complex is unstable (/>2=0.9) [Fig. 3(b)], the induction level of an individual copy is more homogeneous among the gene copies and the peak level of induction increased as the concentration of trans-acting factor 1 (P1) increased. In both cases, the averaged level of gene induction is proportional to the concentration of the active trans-acting factor 1 (P~) [Fig. 3(a) and (b)]. This means that when we consider the averaged behavior of gene induction, the gene expression mechanism of both stable and unstable transcription complex cannot be distinguished. However, when we consider the expression level of individual gene copies, these two cases are totally different. To evaluate the heterogeneity of gene induction levels, the standard deviation was used as a reference. In the simulation, where P~ = 0-01 and P2 = 0 . 0 1 (mean: 52-2), the standard deviation was 76.9. In contrast, where P1 = 0.5 and P2 = 0.9 (mean: 91.4), the standard deviation (S.D.) was 11.3. According to Chebyshev's
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inequality, more than 90% of the data should exist within the region of M + 3 x S.D. (M: mean). Thus, in the former case, the value of the standard deviation means that the most data is distributed over the full spectrum. Thus, the larger the standard deviations, the greater the heterogeneity. Although it is difficult to determine the border between heterogeneity and homogeneity, this index is useful to view the overall behavior of gene copies. Figure 4 shows cumulative data presented with the standard deviations for four levels of stability of the transcription complex. The dose-response curves of the averaged activity of gene expression show the wellknown gradual increase as P~ increases. Conventionally, the data for such curves have been thought to represent the induction levels of individual gene copies. However, as the standard deviation of individual copies shows, the behavior of individuals is very heterogeneous when the transcription complex is relatively stable, such as when P2 = 0.0012 o r P2 = 0-01. These data indicate that the heterogeneity of gene induction level is mainly determined by the stability of the transcription complex (P2). The simulation data is summarized in a contour map of the standard deviations as depicted in Fig. 5. This map illustrates that as the transcription complex stabilizes, the induction level of individual gene copies becomes heterogeneous and individuals
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behave in a totally different manner. In contrast, when the transcription complex is unstable, homogeneous gene induction levels are observed. By monitoring the behavior of individual gene copies through the induction time, it was found that the different frequency of formation of the transcription complex on the enhancer/promoter region caused the different behaviors of transcriptional activation (Fig. 6). Once a stable transcription complex is formed on the enhancer/ promoter region of the gene, it remains there for a substantial number of steps [Fig. 6(a)]. The amount of transcripts finally accumulated from the downstream gene is determined mainly by when the transcription complex is formed on the individual gene copy. With the condition that not all the copies are induced, those from a non-induced to a fully-induced state exist at the same time and heterogeneity is observed [Fig. 6(a)]. In contrast, when the transcription complexes formed on the enhancer/promoter region are unstable, homogeneous gene induction is obtained [Fig. 6(b)]. Even as the transcription complexes are formed, they immediately disassemble. In this case, the total expression level of individual copies during induction is determined by the assembly frequency of transcription complexes on the gene molecules. When the number of active trans.acting factor 1 molecules increases, the frequency of forming a transcription complex increases. By taking
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Total 80 95 115 100 80
FtG. 6. A "life history" of individual gene copies. The numbers from left to right show the state of the gene for each time unit (0, state A in Fig. 2; 1, state B in Fig. 2). Data for five representative gene copies are shown (a)-(e).
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the appropriate induction time, the expression o f an individual gene copy can be averaged. This reduces variations in gene expression and leads to homogeneous gene expression of individual gene copies [Fig. 6(b)].
4. Discussion
4.l. AGAINST THE DETERMINISTIC VIEWPOINT OF GENE INDUCTION In this paper, the importance o f individual gene copy analyses for elucidation of gene induction mechanisms is proposed. At the level o f an individual gene molecule, the association and dissociation of the trans-acting factors and the enhancer/ promoter region on that gene is no doubt determined in a stochastic manner. On this basis, I have built a model and carried out an analysis o f that model by computer simulation. When the transcription complex on the e n h a n c e r / p r o m o t e r region was relatively unstable, the transcription complex formed and dissociated many times during the induction time. Thus, the total induction level of individual gene copies was integrated by the time factor. In this case, differences in the induction level among the gene copies was small and homogeneous gene induction was obtained. In contrast, when the transcription complex on the e n h a n c e r / p r o m o t e r region was relatively stable, the transcription complex existed continuously on the region during the induction time. Thus, the total induction level o f individual copies was directly influenced by the stochastic association and dissociation of the trans-acting factors and the e n h a n c e r / p r o m o t e r region. In this case, differences in the induction level among the gene copies were very large and heterogeneous gene induction was obtained. The most c o m m o n ideas about the mechanism of the gene induction seems to be based on a deterministic viewpoint. When the concentration of the inducer is determined, the expression levels of all the genes are thought to be nearly identical after a substantial induction time. The p h e n o m e n o n that the induction level measured biochemically using many gene copies as a whole is proportional to the concentration of the inducer in a dose-response relationship is interpreted as every gene copy showing a concentration-dependent induction level. However, the analysis of the model in this paper shows that the induction levels of individual gene copies are totally different when the transcription complex is stable, even if the concentration of inducer is determined. Thus, a non-deterministic viewpoint of gene induction mechanisms should be considered as well as the deterministic viewpoint. In this relation, the concept o f non-deterministic gene induction should be clearly distinguished from that of stochastic cell commitment or differentiation. Although a model of stochastic cell proliferation is presented (Till et ai., 1964) and it is well-established that the commitment decision for each cell is made in a stochastic manner in murine erythroleukemia cell differentiation (Gusella et al., 1976), there has been no evidence that these stochastic cell differentiation processes occur at the gene induction level. A key parameter for the difference between heterogeneous and homogeneous gene induction is the stability of the transcription complex on the e n h a n c e r / p r o m o t e r region. It is known that the stability o f the transcription complex can be augmented
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by protein-protein interactions among trans-acting factors (Ptashne, 1986). A direct inference from this observation is that even in a system using the same genes, the same regulatory regions, and the same rate-limiting trans-acting factor, the stability of the transcription complex can be altered by exchange of other trans-acting factors. Recent findings show that the same regulatory sequences can be bound by several distinct trans-acting factors (Schaffner, 1989; Johnson & McKnight, 1989; Ptashne & Gann, 1990), and this transformation from heterogeneous to homogeneous induction and vice versa, might work in an in vivo system. In different cell types or tissues, the same genes may use eithe:" types of induction mechanism, even if the same inducer is used. Although only the heterologous protein-protein interaction is modeled and discussed so far, homologous protein-protein interactions can also be considered in the same way. One of the best known examples of this is the co-operative binding of the repressor molecules on the operator sequences in the lambda phage system (for review see Ptashne, 1986). Although this kind of system shows a steeper sigmoidal curve in the dose-response relationship (Ptashne, 1986), essentially the same argument as the heterologous protein-protein interaction system can be made. 4.2. V A L I D I T Y O F T H E M O D E L
Since the time unit, total induction time, and the stability of transcription complex is relatively defined in the current model, I would like to discuss the validity of the model by comparison with a known system of gene induction. The averaged time for the existence of the transcription complex on the enhancer/promoter was about 38 time units in the stable case (P2 = 0.01) and about 1.2 time units in the unstable case (P2=0.9). Suppose that the total induction time is 12hr. This supposition seems to be reasonable, since the ordinary induction time for biologically significant phenomena, such as mesoderm induction in animal development, is from a few hours to a day. Since the total unit times for induction is 51 in the current model, one unit time calculates to about 14 min. In the representative case of heterogeneous gene induction (P2=0.01), the averaged time for maintaining the transcription complex on the enhancer/promoter region is calculated as about 9 hr. In the representative case of homogeneous gene induction (P2 = 0.9), that is calculated as about 17 min. Since in vitro analysis of gene expression regulation shows that the transcription complex stays on the promoter region for more than several hours, heterogeneous gene induction is possible. The choice of five molecules of RNA per 1 unit time (14min) for the transcription frequency by RNA polymerase II is reasonable, considering that on a gene of high transcription efficiency, such as the ovalbumin gene in tubular gland cells, the RNA polymerase II runs on the gene at 6 sec intervals (Palmiter, 1975). 4.3. E X P E R I M E N T A L
CONSIDERATION
The glucocorticoid-inducible gene expression system of the mouse mammary tumor virus (MTV) is an appropriate experimental model. The nature of its activation is well-documented and similar to the simulated model system in that changes in
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the extracellular glucocorticoid hormone concentration lead to changes in the concentration of intracellular active receptor-hormone complexes. These then bind and activate the MTV enhancer/promoter (Yamamoto, 1985; Beato, 1989). Recent molecular cloning of glucocorticoid receptor genes has revealed the major pathway for activation of glucocorticoid-inducible gene expression systems (for reviews, see Green & Chambon, 1988; Evans, 1988; Beato, 1989). In brief, glucocorticoids diffuse passively into the cells, then bind and activate specific receptor proteins. The active hormone-receptor complexes bind to glucocorticoid response elements by their zinc-finger domain and activate expression of the downstream gene. It is wellestablished that the amount of glucocorticoid receptor protein is a rate-limiting substance for the level of induction (Vanderbilt et al., 1987). To investigate the mechanism of gene induction experimentally, we introduced the fl-galactosidase gene under regulation of the MTV enhancer/promoter at a single or a few copies per cell (Ko et al., 1990). By measuring the/3-galactosidase expression level in an individual cell after 48 h of induction, we were able to measure the expression level of individual gene copies. We showed that the expression level of 3-galactosidase is heterogeneous and the dose-response relationship induced by the glucocorticoid hormone is caused by changes in the number of transcriptionally active genes (Ko et al., 1990). These data indicate the existence of heterogeneous gene induction that contrasts with the commonly accepted homogeneous gene induction. The current model provides an explanation for this unexpected behavior of a steroid-inducible gene expression system. If this model truly underlies the experimental results, the transcription complex on the steroid-inducible enhancer/promoter should be very stable. This seems to be supported by recent in vitro transcription results (Freedman et al., 1989; Klein-Hitpass et al., 1990). 4.4. M O R P H O G E N I C
FIELD
In the development of many organisms, the gradient of a certain substance has been expected to play an important role for pattern formation (Wolpert, 1969, 1989). The expression of a specific gene which gives a differentiated phenotype to the cells in the morphogenic field is supposed to be induced by a morphogen in a concentration-dependent manner. To produce a clear boundary, defining a group of cells expressing that gene in a monotonic concentration gradient of a morphogen, a positive feedback mechanism was presented (Lewis et al., 1977; Meinhardt, 1982). The fundamental basis of these hypotheses is that the behavior of all gene copies must be uniformly identical. That is to say, the induction level of an individual gene can be precisely determined by an inducer (a morphogen) in a concentrationdependent manner. However, according to the current analysis, this kind of deterministic gene induction is limited to the special case of genes with a very unstable transcription complex relative to the total induction time. Recently, it was reported that the gradient of retinoic acid may determine the specificity of the anteroposterior axis of X e n o p u s laevis neural development (Durston et al., 1989) and the vertebrate limb pattern formation (for review, see Eichele, 1989). Since the retinoic acid receptor is a member of the steroid/thyroid hormone
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superfamily (Evans, 1988; Green & Chambon, 1988), the type of gene induction mechanism may be the same as the glucocorticoid-inducible gene expression system, which is shown to be a heterogeneous gene induction type (Ko et al., 1990). In fact, essentially the same results were obtained using a retinoid-inducible/3-galactosidase gene transfected into F9 embryonic carcinoma cells (M. S. H. Ko & Naomi Takahashi, unpublished). These data indicate that while the retinoic acid-inducible gene expression system is operating in morphogenesis and thus should be the deterministic gene induction type, the system behaves experimentally as the nondeterministic type. What kind of mechanism can explain this apparent discrepancy ? First, it is possible that the arrangement of the cells is altered after or during the induction by sorting out cells according to their expression level of the gene. Second, cell-to-cell communication before or during the induction may modify the inducibility of the gene in individual cells. Since these two possibilities require cell contact, the validity of these cases can be tested experimentally under conditions in which cells dissociated from the limb bud or the neural tissue are exposed to various concentrations of retinoic acids. Third, as I pointed out in the previous section, transformation of the gene induction type from heterogeneous to homogeneous is possible by replacing the species of trans-acting factor. Since we have tested the gene induction type in F9 embryonic carcinoma cells, it is possible that different species of trans-acting factors are involved in the transcription complex in the developing neuronal tissues or the limb bud. Fourth, the concentration gradient may be steeper than just a monotonic one. In the retinoic acid-inducible gene expression system, changes in the distribution of the retinoic acid receptor molecule can magnify differences in the effective concentration of the hormone-receptor complex. Fifth, the gene copy number per cell may be changed. If a single cell contains many copies, the heterogeneity of the gene induction level of an individual copy can be averaged. For example, it is known that the puffs of the Drosophila polytene chromosomes show a dose-dependent enlargement of ¢cdysone-inducible puffs (Ashburner et al., 1974). In a single genetic loci of polytene chromosomes, there are about 1000 copies of the genes. This amplification of genetic material may be an evolutionary strategy to avoid a stochastic gene induction mechanism. I thank Ms Naomi Takahashi for encouragement, Dr Mitsuru Furusawa for support and Dr Deborah J. Stearns-Kurosawa for critical reading of this manuscript. I also thank Dr Julian Lewis for useful comments on this manuscript. REFERENCES ASHBURNER, M., CHIHARA, C., MELTZER, P. & RICHARDS, G. (1974). Temporal control of puffing activity in polytene chromosomes. Cold Spring Harbor Syrnp. quant. Biol. 38, 655-662. BEATO, M. (1989). Gene regulation by steroid hormones. Cell 56, 335-344. DURSTON, A. J., TI MMERMANS, J. P. M., HAGE, W. J., HENDRI KS, J. F. J., DE VRIES, N. J., HEIDEVELD, M. & NIEUWKOOP, P. D. (1989). Retinoic acid causes an anteroposterior transformation in the developing central nervous system. Nature, Lond. 340, 140-144. EICHELE, G. (1989). Retinoids and vertebrate limb pattern formation. Trends Genet. 5, 246-251. EVANS, R. M. (1988). The steroid and thyroid hormone receptor superfamily. Science 240, 889-895.
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