Abstract
Induction of gene expression can be accomplished either by removing a restraining element (negative mode of control) or by providing a stimulatory element (positive mode of control). According to the demand theory of gene regulation, which was first presented in qualitative form in the 1970s, the negative mode will be selected for the control of a gene whose function is in low demand in the organism's natural environment, whereas the positive mode will be selected for the control of a gene whose function is in high demand. This theory has now been further developed in a quantitative form that reveals the importance of two key parameters: cycle time C, which is the average time for a gene to complete an ON/OFF cycle, and demand D, which is the fraction of the cycle time that the gene is ON. Here we estimate nominal values for the relevant mutation rates and growth rates and apply the quantitative demand theory to the lactose and maltose operons of Escherichia coli. The results define regions of the C vs. D plot within which selection for the wildtype regulatory mechanisms is realizable, and these in turn provide the first estimates for the minimum and maximum values of demand that are required for selection of the positive and negative modes of gene control found in these systems. The ratio of mutation rate to selection coefficient is the most relevant determinant of the realizable region for selection, and the most influential parameter is the selection coefficient that reflects the reduction in growth rate when there is superfluous expression of a gene. The quantitative theory predicts the rate and extent of selection for each mode of control. It also predicts three critical values for the cycle time. The predicted maximum value for the cycle time C is consistent with the lifetime of the host. The predicted minimum value for C is consistent with the time for transit through the intestinal tract without colonization. Finally, the theory predicts an optimum value of C that is in agreement with the observed frequency for E. coli colonizing the human intestinal tract.
THE life cycle of a microbe, in the simplest case, consists of alternative phases. The demand for expression of some effector genes will be high in one phase and low in the other, and adapting the level of expression to this varying demand requires a functional regulatory mechanism. It has long been known that the same regulatory function, for example, induction of gene expression, can be accomplished in one of two different modes: the negative mode involves the removal of a restraining element, which permits expression from a highlevel promoter, whereas the positive mode involves the provision of a stimulatory element, which facilitates expression from a lowlevel promoter. The demand theory of gene regulation provides a selectionist explanation for this fundamental duality (e.g., see Savageau 1977, 1989).
The components of a minimal regulatory mechanism consist of a promoter site, a modulator site, and a regulator gene encoding the protein that binds the modulator site in response to environmental cues. Each of these components is subject to a rate of mutation that is determined by the number of critical bases in its nucleotide sequence and the mutation rate per base per round of DNA replication. A mutant altered in one of these components may exhibit two different phenotypes depending upon the phase of the life cycle in which it is expressed. The growth rate of the organism serves as the relevant phenotype, and selection is based upon differences in growth rate among wildtype and mutant organisms.
The quantitative development of demand theory (Savageau 1998) combines these elements of life cycle, ecology, physiology, and molecular genetics to predict regions of the C vs. D plot within which selection for the wildtype regulatory mechanisms is realizable. These regions define minimum and maximum values for demand. This theory ties together a number of important variables, including growth rates, mutation rates, and minimum and maximum demands for gene expression. We apply demand theory here to the lactose (lac) and maltose (mal) catabolic systems of Escherichia coli, and we show that this theory also yields predictions for the rate and extent of selection and for minimum, maximum, and optimal cycle times of E. coli that are in reasonable agreement with independent experimental data.
Life cycle of E. coli: The normal life cycle of an organism defines the demand for expression of its effector genes. The life cycle of E. coli as it passes from one host to another will be considered here in terms of two different environments (Figure 1). The first will be identified with the proximal portions of the digestive tract for a lactosetolerant host that ingests both lactose and starch (which consists largely of maltose). This is the environment in which rapid growth occurs during the transition between stable association with one host and then another. The second environment will be identified with the distal end of the small intestine and the colon of the host in which colonization and slow growth take place. Also included in the second environment will be the host's surroundings through which the bacteria pass to enter a subsequent host. This is admittedly a simplification of a more complex ecology (Cooke 1974; Freter 1976; Savageau 1983), but nevertheless it captures the essential features for our purposes here. Additional environments and more complex linkages among them in principle can be handled by the same methods.
Ecology and gene expression: We shall consider the lac operon as representative of a lowdemand function governed by the negative mode of control (Miller and Reznikoff 1980) and the mal operon as representative of a highdemand function governed by the positive mode of control (Schwartz 1987). These systems are well studied at the molecular level, and the evidence regarding their mode of control is clear.
Evidence regarding demand for expression of the lac and mal operons of E. coli comes from studies of intestinal ecology. Lactose is a relatively rare sugar in nature (Shallenberger 1974). The host's lactase enzymes, which hydrolyze this disaccharide and thereby permit its utilization by the host, are located in the proximal small intestine and are subject to developmental regulation (Dahlqvist 1961; Koldovsky and Chytil 1965; Walker 1968). In contrast, maltose, the breakdown product of all dietary starch, is among the most abundant sugars (Widdas 1971). The host's maltase enzymes, which hydrolyze this disaccharide and thereby permit its utilization by the host, are located at the distal end of the small intestine and in the colon (Dahlqvist 1961; Rosensweig and Herman 1968). This information suggests that the lac operon of E. coli is likely to be expressed at high levels in the first environment and at low levels in the second, whereas the mal operon is likely to be expressed at high levels in the second environment and at low levels in the first.
The time required for E. coli to pass through the highdemand environment for lactose utilization is about 3 hr. This is onehalf the average time required to reach the colon (Madsen 1992); the 3hr figure is also based on measured patterns of lactose utilization (Bond and Levitt 1976; Malageladaet al. 1984). Much of the ingested lactose is hydrolyzed to constituent sugars that are absorbed by the host in the proximal small intestine; the remainder is catabolized by the bacteria so that very little lactose normally reaches the colon (Bond and Levitt 1976). From this 3hr figure for time in the highdemand environment, one can predict that the cycle time of E. coli will be inversely related to the demand for expression of its lactose operon C = 3/D.
We estimate the time for passage through the lowdemand environment for maltose utilization to be ~6 hr. This is the average time required for a bolus of ingested food to reach the distal portions of the small intestine and colon (Madsen 1992). We assume that free maltose is sparse in the proximal portions of the small intestine and that it becomes abundant only in the distal portion of the intestinal tract where the host's maltase enzymes are localized (Dahlqvist 1961; Rosensweig and Herman 1968). From this 6hr figure for time in the lowdemand environment, one can predict that the cycle time of E. coli will be inversely related to 1 minus the demand for expression of its maltose operon C = 6/(1 − D).
ESTIMATION OF PARAMETER VALUES
The demand theory of gene regulation involves three levels of parameters (Savageau 1998): constituent parameters, individual growth rates and mutation rates, and macroscopic rate constants. Estimates for the values of the constituent parameters in our model are given below. In a subsequent section we will determine the consequences of these choices by examining other values for each of the parameters. As we shall see, the exact values are not critical for about half of the parameters, whereas the values for two of them are extremely influential.
Reference mutation rate, μ: For E. coli, the spontaneous mutation rate is estimated to have a nominal value of μ_{0} = 6E–10 per base per DNA replication (Drake 1991). The spontaneous mutation rate for loss of function in the modulator or promoter sites of our model can be determined from estimates of the spontaneous mutation rate per base and the number of critical bases that define these sites.
Relative mutation rate for loss of a highlevel promoter site, π: Promoters encompass a region of ~75 nucleotides upstream of the RNA start site (Harley and Reynolds 1987; Lisser and Margalit 1993). Although most of the information in promoter sequences is localized within two 6base blocks with variable spacing between them (the “consensus” elements at positions −10 and −35 relative to the start site), there is only limited base conservation at most positions. If we assume that each of 10 nucleotides is critical for the definition of a highlevel promoter (Schneideret al. 1986), then the relative mutation rate is π_{0} = 10.
Relative mutation rate for gain of a highlevel promoter site, ν: Spontaneous uppromoter mutations with the positive mode occur at about onetenth the frequency of spontaneous downpromoter mutations with the negative mode (G. Gussin, personal communication). If we assume that a lowlevel promoter can be converted to a highlevel promoter by a single mutation in a critical base, then the relative mutation rate for this gain of a highlevel promoter site is ν_{0} = 1.
Relative mutation rate for loss of a regulator's functional target site, τ: Targets for the binding of regulator proteins are the modulator sites—operator sites in the case of the negative mode and initiator sites in the case of the positive mode. Operator sites span a region of ~100 nucleotides upstream of the RNA start site (Realla and ColladoVides 1996). If we assume that each of 20 nucleotides is critical for the definition of the operator (Schneideret al. 1986), then the relative mutation rate in this case is τ_{0} = 20. Although the sizes of initiator sites are about onehalf those of operator sites, 50 nucleotides upstream of the RNA start site (Realla and ColladoVides 1996), the information needed to locate these sites within the genome is similar to that for operators (Schneideret al. 1986). If we assume that each of 20 nucleotides also is critical for the definition of the initiator, then the relative mutation rate in this case is τ_{0} = 20.
Relative mutation rate for loss of a functional regulator protein, ρ: We shall assume that a typical regulator protein has 30 amino acid residues that are critical for binding to its modulator site (operator in the case of the negative mode or initiator in the case of the positive mode) and for properly affecting transcription initiation. This implies that the regulator gene has ~60 bases that are critical because the identity of the base in the third codon position is largely irrelevant. Thus, we obtain a relative mutation rate of ρ_{0} = 60 for loss of a functional regulator protein.
Relative mutation rate as a function of gene expression, ε: There is evidence to suggest that the rate of spontaneous mutation increases with the rate at which the DNA is being transcribed (Datta and JinksRobertson 1995). There also is evidence to suggest that the rate of spontaneous mutation decreases because of transcriptioncoupled repair mechanisms (Francinoet al. 1996). This is one of the questions we wish to examine further in our quantitative analysis. We initially assume that there is no significant effect on the mutation rate either way; therefore, we assign a value of ε_{0} = 1 for this relative mutation rate. If one were to assume a change in mutation rate that is proportional (or inversely proportional) to the rate of transcription, then the mutation rate relative to the reference would be given by ε = k × 100 (or k/100), where k is the proportionality constant. (Recall that the capacity for regulation is assumed to be 100 and that expression is assumed to be fully ON or fully OFF.)
Reference growth rate, γ: We shall assume that E. coli grows with a doubling time of 1 hr in the nutritionally richer of the two environments; thus, the nominal value for the reference growth rate is γ_{0} = 1.0. This is not an unreasonable value because it is known that bacteria like E. coli can double in a period as short as 20 min (Maaløe and Kjeldgaard 1965). In any case, the simple value of unity provides a convenient reference; should the actual value be different, one can simply rescale the time accordingly, and none of our results would change.
Relative growth rate with loss of normal expression, λ: Because expression is either fully ON or fully OFF and the capacity for regulation is 100, which supports the nominal growth rate, a failure of expression is assumed to result in a basal level of expression, which would support only a 100fold reduction in growth rate if there were no other carbon source in the environment. However, in the complex environment of the intestinal tract there are multiple carbon sources, and the reduction in growth rate will therefore be less. We shall assume a 3% reduction in growth rate. Thus, the nominal value for this parameter is set at λ_{0} = 0.97.
Relative growth rate with superfluous expression, σ: When the demand is such that a function is normally turned OFF and a regulatory mutation causes the function to be fully expressed under inappropriate circumstances, the cell unnecessarily expends resources for material and energy. Experimental evidence in the case of βgalactosidase expression in E. coli (Novick and Weiner 1957; Koch 1983) suggests that such inappropriate expression decreases the growth rate by <1%; we shall assume a 0.1% reduction. The growth rate, relative to the reference growth rate, is thus assigned a nominal value of σ_{0} = 0.999.
Relative growth rate in the more nutritionally deficient of the two environments, δ: From measurements of the mean transit time through the human intestinal tract (Cummings and Wiggins 1976; Gearet al. 1980), and the assumption that it is a wellstirred chemostat, one can calculate that the average doubling time for net growth of E. coli in the intestinal tract is about 40 hr (Savageau 1983). Because the intestinal tract is not a wellstirred chemostat, but rather a very heterogeneous environment in which the growth is undoubtedly faster in the proximal regions and slower in the distal, the doubling time of E. coli in the more deficient distal environment will be longer than the average. There are no good measurements to go by, so we will arbitrarily set the doubling time for growth in the more deficient environment to be two times the average value given above. Thus, the growth rate in the more deficient environment, relative to the reference growth rate in the richer environment, is given by a nominal value of δ_{0} = 0.0125.
Criterion for selection, θ: Our criterion for selection is that each mutant population shall be reduced to no more than 0.05% of the wildtype population or, alternatively, that the sum of the two mutant populations shall be reduced to no more than 0.1% of the wildtype population. This is similar to values that are found in the literature (Leclercet al. 1996). Thus, the criterion for selection is assigned a nominal value of θ_{0} = 0.0005.
Estimation of macroscopic parameters: The values of the macroscopic parameters in each environment and for each mode of control are determined as follows. First, the constituent parameters given above are combined to represent the relevant growth rates (g_{w}, g_{p}, g_{m}, g_{d}). The growth rate of the wildtype organism g_{w} in the first environment is γ (the reference), and in the second it is γ multiplied by δ, the relative growth rate in the nutritionally deficient environment. The growth rates of the promoter mutants g_{p}, modulator mutants g_{m}, and promoter/modulator (double) mutants g_{d} in the two environments are the same as those of the wild type, but multiplied when appropriate by relative growth rates that reflect either loss of expression that is normally ON (λ) or superfluous expression that is normally OFF (σ). For example, the growth rate of a lac modulator mutant (g_{m}) is γ in the first environment, where its pattern of gene expression mimics that of the wild type, and γδσ in the second, where expression of the lac operon is superfluous. The growth rate of a mal modulator mutant is γ in the first environment, where its pattern of gene expression mimics the wild type, and γδλ in the second, where there is a failure to express the mal operon.
Second, the constituent parameters are combined to represent the mutation rates between populations (m_{pw}, m_{mw}, m_{dp}, m_{dm}). Each mutation rate is given by the product of the number of critical bases that define the structure in question (π, ν, τ, or ρ), the spontaneous mutation rate per base per DNA replication (μ), and a factor reflecting transcriptionrelated mutation or repair (ε) when appropriate. For example, the rate of production of lac promoter mutants from wildtype organisms (m_{pw}) is πμε in the first environment, where the lac operon is being actively transcribed, and πμ in the second, where it is not. The rate of production of mal promoter mutants from wildtype organisms is νμ in the first environment, where the mal operon is not being transcribed, and νμε in the second, where it is. Finally, the growth rates and mutation rates are combined to represent the macroscopic rateconstant parameters that characterize the population dynamics (α_{ww}, α_{pp}, α_{pw}, α_{mm}, α_{mw}, α_{dd}, α_{dp}, α_{dm}). For example, the rate constant for net growth of the promoter mutant α_{pp} is given by its intrinsic growth rate g_{p} minus the rate of loss due to the production of double mutants, which is given by the mutation rate per DNA replication m_{dp} times the intrinsic growth rate of the promoter mutant g_{p}. The rate constant for production of promoter mutants from the wildtype population α_{pw} is given by the mutation rate per DNA replication m_{pw} times the intrinsic growth rate of the wildtype organism g_{w}. The other rateconstant parameters are determined in a similar fashion. Thus, α_{ww} = [1 − (m_{pw} + m_{mw})]g_{w}, α_{pp} = (1 − m_{dp})g_{p}, α_{pw} =m_{pw}g_{w}, α_{mm} = (1 −m_{dm})g_{m}, α_{mw} = m_{mw}g_{w}, α_{dd} = g_{d}, α_{dp} = m_{dp}g_{p}, α_{dm} = m_{dm}g_{m}.
SELECTION OF WILDTYPE REGULATORY MECHANISMS
Determination of the thresholds for selection: The threshold for selection of a wildtype promoter or modulator is determined as follows. As described above, the constituent parameters are combined to represent the relevant growth rates and mutation rates that enter into the macroscopic parameters that characterize the population dynamics of mutant and wildtype organisms. The population dynamic equations are solved in the two environments to yield an equation for the ratio of mutant to wildtype population numbers. This ratio is set equal to θ, the criterion for selection, and the resulting equation expresses the relationship between cycle time C and the demand for gene expression D that constitutes the threshold for selection.
The threshold for selection of a wildtype promoter or modulator (regulator) can be obtained by solving the threshold equation for C as a function D using the method of bisection (Presset al. 1988). These numerically calculated thresholds and the analytically determined asymptotes (Savageau 1998) are nearly indistinguishable, except in the region of transition between low and highC asymptotes. Only those values of C and D that lie in the region of overlap below both the promoter and modulator thresholds will allow selection of the wildtype regulatory mechanism.
Regions in which selection for negative and positive modes is realizable: When nominal values are assumed for the parameters of the model (see Table 1), one finds the thresholds plottedin Figures 2 and 3. The thresholds shown in Figure 2 for the negative mode of control (3% selection coefficient against the promoter mutant and 0.1% against the modulator mutant) exhibit a narrow region of overlap. In contrast, the thresholds shown in Figure 3 for the positive mode of control (0.1% selection coefficient against the promoter mutant and 3% against the modulator mutant) exhibit a wide region of overlap. As predicted (Savageau 1998), the positive and negative modes are associated with asymmetric regions of the C vs. D plot in which selection is realizable. Note that the horizontal axis in the case of the positive mode is plotted as values of 1 − D, instead of D; this allows us to distinguish more clearly the threshold values near unity when plotting demand in logarithmic coordinates. The extent of the asymmetry is perhaps more evident when the thresholds for selection against the modulator mutants are plotted on the same scale for both the negative and positive modes (Figure 4). The regions for which selection is realizable are nonoverlapping, which indicates discriminate selection for the positive and negative modes.
Influence of parameters on minimum and maximum values for demand: Selection requires the demand for gene expression to be greater than the asymptote for the minimum threshold and less than the asymptote for the maximum threshold, that is, D_{min} < D < D_{max}. The parameters in our model influence these asymptotic values to various degrees. It is important to examine a range of values for each of the parameters because there is some uncertainty in the nominal values for many of them. We have systematically varied each parameter about its nominal value given in Table 1 and observed the resulting changes in the minimum and maximum values for demand. The results are shown in Figure 5.
Five classes of influence can be discerned in Figures 5A, 5B, 5C and 5D. First, in many cases there is no discernible influence [Figure 5A (π and λ), Figure 5B (ρ, τ, ε, and σ), Figure 5C (ν, τ, ε, and σ), and Figure 5D (ρ, τ, and λ)]. Second, in several cases there is a nearly linear variation with the change in parameter value [Figure 5A (μ, ρ, ε, δ, and θ), Figure 5B (μ, π, δ, and θ), and Figure 5D (μ, ν, ε, δ, and θ)]. Third, in five cases there is a nearly cuberoot influence [Figure 5A (τ) and Figure 5C (μ, ρ, δ, and θ)]. Fourth, in two cases there is a moderate (order of magnitude) amplification of the response to a change in parameter value [Figure 5B (λ) and Figure 5C (λ)]. Finally, in two cases there is an extreme (1000fold) amplification of the response to a change [Figure 5A (σ) and Figure 5D (σ)]. The results obtained for the negative and positive modes exhibit different patterns. The influences in the local region about the nominal values can be summarized numerically by the parameter sensitivities (Shiraishi and Savageau 1992), as shown in Table 2.
From these results one can see that the most influential parameter is the selection coefficient that reflects the diminished growth rate of the organism when there is superfluous gene expression. For the negative mode of gene control, this corresponds to the diminished growth rate of the modulator (regulator) mutants that express the effector function constitutively when it should be OFF. For the positive mode, this corresponds to the diminished growth rate of the promoter mutants that express the effector function at a high level when it should be OFF.
Time course of selection: The numbers of wildtype, modulatormutant, and promotermutant organisms are represented by the variables X_{w}, X_{m}, and X_{p}. The ratio X_{w}/(X_{m} + X_{p}) is equal to the reciprocal of the mutant fraction, which we define as f_{m}. If we start with equal numbers for the two types of mutants and a ratio X_{w}/(X_{m} + X_{p}) that is onetenth of its steadystate value, then the enrichment of the wildtype regulatory mechanism with time is obtained from the solution of the population dynamic equations (Equation 9 in Savageau 1998). Given the nominal values for the parameters in Table 1, the time course of selection for various values of demand D is as shown in Figure 6. The temporal behavior of the populations is independent of the cycle time C. The time scale is actually discrete, given by values of nC, where n is the number of cycles. Thus, within a fixed time period, the same degree of enrichment can be achieved with either a large value for C and a small number n, or a small value for C and a larger number n. Note that the negative mode of gene control emerges more rapidly as demand for gene expression increases, whereas the positive mode of gene control emerges more rapidly as demand for gene expression decreases (Figure 6). The extent and rate of selection are examined in greater detail below.
Extent of selection: We define the extent of selection as the steadystate value of the ratio X_{w}/(X_{m} + X_{p}), which is the inverse of the mutant fraction in the population (1/f_{m}). Although there is selection for the wildtype regulatory mechanism throughout the region of overlap beneath the thresholds (e.g., Figures 2A and 3A), the extent of the selection varies as a function of cycle time C and demand D. For a given value of C, the extent of selection reaches its maximum at a value of D that is roughly the geometric mean of its threshold values. With the nominal values for the parameters (Table 1), the results for the negative mode of gene control are as shown in Figure 7A; the results for the positive mode are similar to those for the negative mode, except that the allowable values for demand now occur in the highdemand region of the plot (Figure 7B). The maximum extent of selection for the positive mode of gene control is ~10fold greater than that for the negative mode.
Rate of selection: The rate at which selection occurs is independent of cycle time. We define response time as the time required for the ratio X_{w}/(X_{m} + X_{p}) to reach 99% of its steadystate value, starting from an initial state in which the numbers of the two types of mutants are equal and the ratio is equal to onetenth of its steadystate value. Recall that the time points are given in units of nC, where C is the cycle time and n is the number of cycles. The same temporal behavior is obtained regardless of whether C is large (n small) or small (n large). However, the resolution is poorer for large values of C because the minimum value of n is one.
Like the extent of selection, the rate of selection is strongly dependent on the demand for gene expression. Although selection in the case of the negative mode can occur near the lower limit of allowable values for D, the response time is very long. Response time decreases in an inverse fashion as D increases, until a lower plateau is reached (Figure 8A). The break in the curve occurs at approximately the value of D that yields the maximum extent of selection (see Figure 7A). The minimum response time with the nominal values for the parameters is ~294,000 hr (~36 yr).
Similar results are found with the positive mode of gene control (Figure 8B), except that the long response times occur near the upper limit of allowable values for D. The response time decreases as D decreases until a minimum is reached, and then it increases. For the same extent of selection as the negative mode (18,400), the positive mode exhibits a faster response time (~17,000 vs. 294,000 hr); alternatively, for the same response time as the negative mode (294,000 hr), the positive mode exhibits a greater extent of selection (~214,000 vs. 18,400). Thus, it appears that the positive mode of gene control is capable of achieving greater extents of selection with faster response times than is the case for the negative mode of control.
Minimum cycle time: Estimates for the minimum cycle time of E. coli passing from one host to another can be obtained by combining the information in Figure 2A with the inverse relationship between C and D for the lactose operon of E. coli. (Recall from the ecology and gene expression section that C = 3/D.) The intersection of this inverse relationship with the threshold for selection of the wildtype modulator (regulator) gives a value of C_{min} = 26 hr (Figure 9A). Another estimate of minimum cycle time can be obtained by combining the information in Figure 3A with the inverse relationship between C and (1 − D) for the maltose operon of E. coli. [Recall from the ecology and gene expression section that C = 6/(1 − D).] The intersection of this inverse relationship with the threshold for selection of the wildtype modulator (regulator) gives a value of C_{min} = 10 hr (Figure 9B).
Maximum cycle time: Estimates for the maximum cycle time of E. coli passing from one host to another also can be obtained by combining the information in Figure 2A with the inverse relationship between C and D for the lactose operon of E. coli. The intersection of the inverse relationship C = 3/D with the threshold for selection of the wildtype promoter (Figure 9A) gives a value of C_{max} = 580,000 hr (~66 yr). Again, the data for the maltose operon in Figure 3A provide an alternative estimate. The intersection of the inverse relationship C = 6/(1 − D) with the threshold for selection of the wildtype promoter (Figure 9B) gives a value of C_{max} = 502,000 hr (~57 yr).
Optimal cycle time: Although the estimates for minimum and maximum cycle time in the preceding sections are of some interest, perhaps the more relevant issue is the nominal value of the cycle time for E. coli in its natural environment. We argue that the most probable values for the cycle time will be those corresponding to the values for demand that lead to the optimal extent and rate of selection. For the negative mode, the optimum extent and rate of selection occur with a value of demand D_{op} ≈ 0.001 (Figures 7A and 8A). Combining this optimum value for D with the inverse relationship C = 3/D in Figure 9A for the lactose operon yields an estimate for the nominal value of the cycle time, namely, C_{op} = 3000 hr (~4 mon). For the positive mode, the optimum extent and rate of selection occur with a value of demand 1 − D_{op} ≈ 0.01 (Figures 7B and 8B). Combining this optimum value for 1 − D with the inverse relationship C = 6/(1 − D) in Figure 9B for the maltose operon yields an estimate of C_{op} = 800 hr (~33 days).
DISCUSSION
The application of demand theory presented in this article provides an opportunity to test a number of the theory's quantitative implications. The results in Savageau (1998) led to the prediction of welldefined regions within the C vs. D plot where selection for the positive and negative modes of gene control is realizable. These regions allow one for the first time to specify precisely what is meant by high and low demand. With the nominal values for the parameters of the lactose and maltose operons in E. coli, selection of the negative mode of control requires a demand between 0.000005 and 0.1 (Figure 2A), whereas selection of the positive mode requires a demand between 0.2 and 0.999985 (Figure 3A). Furthermore, these regions were predicted to exhibit an inherent asymmetry with the positive mode having the larger region within which selection is realizable. This is clearly seen in the case of the lactose and maltose examples analyzed here (Figure 4).
Although the minimum and maximum values of demand are influenced by a number of parameters, by far the most influential parameter is σ, which reflects the reduction in growth rate when there is superfluous expression of a gene (Figure 5A and 5D). The nominal value for this parameter was set at 0.1%, on the basis of data for the lactose operon that suggest a value <1% for the reduction in growth rate of operatorconstitutive mutants in a lowdemand environment. In the case of the positive mode, the same value was used to characterize the reduction in growth rate of an uppromoter mutant in a lowdemand environment. A 0.1% variation in σ yields a twofold change in the value of D_{max} for both the negative and positive mode (Table 2). The remaining parameters have much less influence on the limits of D; approximately onehalf exhibit a nearly linear influence, whereas the other half have a negligible influence.
The ratio of mutation rate to selection coefficient is the most relevant determinant of the realizable region for selection. Indeed, if the target sizes for the various types of mutations are increased by an order of magnitude (e.g., to match the footprint for binding a regulator protein to its modulator site on the DNA) at the same time the selection coefficients are increased by an order of magnitude, then the results are essentially unchanged (data not shown).
The results in Figure 5 suggest that the effect of transcription on mutation rate may be significant only if it reduces the mutation rate. The parameter ε, which represents this effect, has no influence on the selection of the wildtype promoter when there is a negative mode of control (Figure 5B). This is counter to the intuitive expectation that suggests a lower mutation rate would aid the selection of the wildtype promoter when it is not in use. The results in Figure 5A show that the parameter ε can represent an increased selection for the wildtype repressormodulator interaction (increased D_{max}) if there is an increase in transcriptioncoupled repair (decrease in ε). In the case of the positive mode, ε has negligible influence on the selection of the wildtype activatormodulator interaction (Figure 5C), and its influence on the selection of the wildtype promoter (Figure 5D) would appear to be of little consequence because the threshold value of D_{max} in this case is already so high. Given the nominal values we have used for the parameters, ε does not seem to be highly significant, and similar effects can be achieved by varying other parameters; nevertheless, ε might still be important for selection under other conditions.
The equations that characterize the population dynamics of mutant and wildtype organisms (Equations 7–11 in Savageau 1998) led to the prediction that the extent of selection is a function of cycle time C and maximal at intermediate values of demand D, whereas the rate of selection is independent of cycle time C. Indeed, with the parameter values in Table 1, the extent of selection increases, reaches a maximum, and then declines as demand increases (Figure 7). As seen in Figure 8, the time required to reach full selection decreases until a minimum is reached with increasing demand (negative mode) or decreasing demand (positive mode). The combination of these results suggests that the optimum extent and rate of selection occurs at around D = 0.001 for the negative mode and 1 − D = 0.01 for the positive mode. In the case of the positive mode, this represents a choice of 1 − D that yields a rate of selection that is nearly equivalent to the optimum for the negative mode.
The quantitative theory reveals a number of new relationships involving cycle time that can be tested against experimental data in the case of the lactose and maltose operons of E. coli. The first such relationship provides an estimate for the minimum value of the cycle time C_{min}. We obtained values of 26 hr (Figure 9A) and 10 hr (Figure 9B), which is on the same order of magnitude as the 40 hr required on average for transit through the entire intestinal tract (Cummings and Wiggins 1976; Gearet al. 1980; Savageau 1983). Under these circumstances, E. coli is simply passing through the intestinal tract without colonizing the colon. Clearly, the cycle time can be no shorter than this period.
The second relationship provides an estimate for the maximum value of the cycle time C_{max}. We have estimated this value to be ~580,000 hr (~66 yr) in the case of the lactose operon (Figure 9A) and 502,000 hr (~57 yr) in the case of the maltose operon (Figure 9B). These values for C_{max} are on the same order of magnitude as the 120yr maximum for the life span of humans (Hayflick 1977). Clearly, the cycle time for E. coli can be no longer than the life time of the host because the bacteria will die with the host if they do not colonize a new host.
The final relationship provides an estimate for the optimum value of the cycle time C_{op}. The optimum extent and rate of selection determined for the lactose operon suggest a demand in the neighborhood of D_{op} = 0.001. This value of D, taken together with the relationship D = 3/C, predicts an optimum cycle time of C_{op} = 3000 hr (~4 mon). The corresponding estimate based on the maltose operon is C_{op} = 800 hr (~33 days). These predicted values for the cycle time of E. coli are comparable with the cycle times (recolonization rates) of months to years that have been observed in humans for resident strains of E. coli (Searset al. 1950; Sears and Brownlee 1952; Caugantet al. 1981).
In summary, the quantitative development of demand theory presented in Savageau (1998) and applied here provides the first estimates for the minimum and maximum values of demand that are required for selection of the positive and negative modes of gene control. The specific application to the maltose and lactose operons of E. coli suggests that the positive and negative modes of control for these genes are subject to selection throughout the full range of cycle times that are possible for this microbe. Moreover, the cycle times predicted on the basis of the optimal extent and rate of selection are in agreement with the typical cycle times that have been observed experimentally.
Acknowledgments
I thank Drs. S. Cooper, R. G. Freter, D. E. Kirschner, J. V. Neel, and M. S. Swanson for critically reading the manuscript and two anonymous reviewers who made valuable suggestions for improving the parameter estimates. This work was supported in part by U.S. Public Health Service grant RO1GM30054 from the National Institutes of Health and U.S. Department of Defense grant N000149710364 from the Office of Naval Research.
Footnotes

Communicating editor: R. H. Davis
 Received December 15, 1997.
 Accepted May 6, 1998.
 Copyright © 1998 by the Genetics Society of America