The apparent clock-like evolution of Escherichia coli in glucose-limited chemostats is reproducible at large but not at small population sizes and can be explained with Monod kinetics

Lukas M. Wicka,1, Hansueli Weilenmann1 and Thomas Egli1

Swiss Federal Institute for Environmental Science and Technology, PO Box 611, Überlandstrasse 133, CH-8600 Dübendorf, Switzerland1

Author for correspondence: Thomas Egli. Tel: +41 1 823 51 58. Fax: +41 1 823 55 47. e-mail: egli{at}eawag.ch


   ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
 
To follow and model evolution of a microbial population in the chemostat, parameters are needed that give an indication of the absolute extent of evolution at a high resolution of time. In this study the evolution of the maximum specific growth rate max) and the residual glucose concentration was followed for populations of Escherichia coli K-12 under glucose-limited conditions at dilution rates of 0·1 h-1, 0·3 h-1 and 0·53 h-1 during 500–700 h in continuous culture. Whereas µmax improved only during the initial 150 h, the residual glucose concentration decreased constantly during 500 h of cultivation and therefore served as a convenient parameter to monitor the evolution of a population at a high time resolution with respect to its affinity for the growth-limiting substrate. The evolution of residual glucose concentrations was reproducible in independent chemostats with a population size of 1011 cells, whereas no reproducibility was found in chemostats containing 107 cells. A model based on Monod kinetics assuming successive take-overs of mutants with improved kinetic parameters (primarily Ks) was able to simulate the experimentally observed evolution of residual glucose concentrations. Similar values for the increase in glucose affinity of mutant phenotypes (Ks(mutant) 0·6xKs(parent)) and similar mutation rates per cell per generation leading to these mutant phenotypes (1–5x10-7) were estimated in silico for all dilution rates. The model predicts a maximum rate of evolution at a dilution rate slightly below µmax/2. With increasing and decreasing dilution rates the evolution slows down, which also explains why in special cases a selection-driven evolution can exhibit apparent clock-like behaviour. The glucose affinity for WT cells was dependent on the dilution rate with highest values at dilution rates around µmax/2. Below 0·3 h-1 poorer affinity was mainly due to the effects of rpoS.

Keywords: continuous culture, maximum specific growth rate, glucose affinity, molecular clock, rpoS

a Present address: Microbial Evolution Laboratory, National Food Safety and Toxicology Center, Michigan State University, East Lansing, MI 48824, USA.


   INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
 
Bacteria grown in chemostats are evolving constantly. In fact, Novick & Szilard (1950) , soon after having invented the chemostat, recognized that the observed shifts in frequency of neutral mutations, such as T5 resistance, are a manifestation of evolution. Mutants with increased fitness arise periodically in chemostat and batch cultures, which results in a replacement of the original population, a phenomenon referred to as periodic selection (Atwood et al., 1951 ; Dykhuizen, 1990 ). Theoretical considerations suggested that for populations where the product of cell number and mutation rate (referred to here as the number of advantageous mutations per cell and per generation) is much higher than one, the effects of selection can be described by deterministic equations and therefore should be reproducible (Dykhuizen, 1990 ; Kubitschek, 1974 ; Wahl & Krakauer, 2000 ). However, experimental data supporting these assumptions are scarce, especially because a high time resolution is needed to demonstrate the degree of reproducibility of the evolution in parallel populations.

Dykhuizen and Hartl (Dykhuizen & Hartl, 1981 ; Hartl & Dykhuizen, 1979 ) compared the evolution of the competitive ability of Escherichia coli populations in glucose-limited continuous culture at dilution (growth) rates of 0·14 h-1 and 0·28 h-1, corresponding to generation times of 5 and 2·5 h, respectively. Over a period of 500 h they removed samples from the continuous culture every 100 h, determined the µmax (maximum specific growth rate) and competed each population with the population isolated 100 h earlier. Whereas µmax increased during the first 100 h and then remained constant, an increase in glucose affinity was observed over the whole 500 h period. Surprisingly, the increase in glucose affinity did not proceed faster in the population with the shorter generation time. In fact, the evolution of the two populations was dependent on the absolute time rather than on the number of generations. This observation was reminiscent of the molecular clock, i.e. that the substitution rates of amino acids in a given protein are fairly constant over time for different species. However, such a molecular clock was proposed to hold only when the amino acid substitutions are selectively neutral (Ayala, 2000 ; Kimura, 1983 ; Zuckerkandl & Pauling, 1965 ). A rationale or even a model explaining why in the case of the Dykhuizen–Hartl experiment a selection driven evolution exhibited an apparent clock-like behaviour was not given.

Different models have been proposed to describe the growth kinetics of a microbial population growing with a single limiting substrate. Statistical analysis by Senn et al. (1994) has shown that out of several tested models, the three models of Monod (1942) , Shehata & Marr (1971) and Westerhoff et al. (1982) described their experimental data equally well. The Monod model describes the relationship between µ and s (substrate concentration) by a type of saturation kinetics:

(1)

where µ is the specific growth rate, µmax the maximum specific growth rate at saturating substrate concentrations, s the substrate concentration and Ks the substrate affinity, i.e. the substrate concentration at which the cells grow at half maximum specific growth rate. The clear advantage of the Monod model is that the parameters µ, µmax, s and Ks have a biological meaning and are experimentally accessible. A weakness of the original Monod model is that it neglects the concept of maintenance energy, which implies a substrate flux even at a specific growth rate of 0 h-1. Different models have been proposed to solve this problem (Kováová, 1996 ; Pirt, 1965 ; Powell, 1967 ; Westerhoff et al., 1982 ). The extended Monod model of Kováová (Equation 2) includes an smin term, which denotes the minimal substrate concentration needed for growth:

(2)

This model provides a good fit for data obtained for steady-state glucose concentrations in cultures of E. coli cultivated in glucose-limited chemostats at different temperatures and dilution rates (Kováová et al., 1996 ).

To follow and model evolution of a microbial population in the chemostat, we need one or more parameters that give an indication of the absolute extent of evolution. Such parameters should be measurable without much labour to obtain a high time resolution of the evolutionary process. In our case the parameter s of the Monod model exhibits these characteristics. Monod Ks values calculated from residual glucose concentrations have shown that populations of E. coli ML 30 (Kováová-Kovar & Egli, 1998 ; Senn et al., 1994 ) and E. coli K-12 (Wick et al., 2001 ) improve their glucose affinity during growth under glucose-limited conditions. In this work we monitored the Monod parameters µmax and s of E. coli K-12 during growth for 500–700 h in glucose-limited chemostats at dilution rates of 0·1 h-1, 0·3 h-1 and 0·53 h-1. Using the extended Monod model we modelled periodic take-overs of the culture by mutant populations with improved glucose affinity. Such periodic take-overs lead to a steady decrease in residual glucose concentration in a continuous culture. By fitting calculated s values to the measured ones we inferred the improved Ks values of the modelled mutant populations and mutation rates (i.e., advantageous mutations per cell per generation) leading to these improved Ks values. The Monod model showed that in special cases, populations of cells cultivated at longer generation times can evolve as quickly as populations of cells grown at shorter generation times. This is because a stronger selection pressure per generation can compensate for fewer generations per time interval. The model proposed here can explain the apparent clock-like behaviour of the selection-driven evolution described previously (Dykhuizen & Hartl, 1981 ; Hartl & Dykhuizen, 1979 ).


   METHODS
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ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
 
Bacterial strains and growth conditions.
Wild-type Escherichia coli K-12 MG 1655, and cultivation conditions for batch and large bioreactors were the same as described previously (Wick et al., 2001 ). An rpoS negative derivative of the WT was constructed by P1 transduction (Miller, 1972 ) of the rpoS13::Tn10 allele from strain E. coli MV2792 (Volkert et al., 1994 ) to E. coli K-12 MG 1655. Successful transduction resulted in tetracycline resistant cells with decreased catalase activity as seen in reduced bubbling of the cells when exposed to hydrogen peroxide (Sak et al., 1989 ).

The glucose-limited medium supported a µmaxSD) of 0·63±0·01 h-1 (n=14) for the wild-type strain. At 0·65±0·01 h-1 (n=7), the growth rate for the rpoS strain was significantly higher (P=0·001, t-test).The parameters and variables used are shown in Table 1.


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Table 1. Parameters and variables used in this communication

 
Small chemostats were self-assembled from Schott flasks and had a working volume of 75 ml. Stirring was achieved by a standard laboratory magnetic stirrer. Air was not introduced into the culture directly as in large cultures but only into the headspace. The pH was buffered at pH 7 with Na2HPO4/KH2PO4 (14·4 mM/4·4 mM). The chemostats were inoculated with approximately 5x106 cells and the medium flow (containing 1 mg glucose l-1; note that the medium in large chemostats contained 100 mg glucose l-1) was started immediately at 0·3 h-1. The total cell number (±SD) in small chemostats, assessed by plate counts, was (3±2)x107 (n=20); the total cell number in large chemostats at 0·3 h-1, assessed by plate counts, was (9±4)x1010 (n=6).

Determination of maximum specific growth rates.
Cells taken from the chemostat were transferred to shake flasks containing glucose minimal medium, where the specific growth rate increased gradually. Cells were always transferred to a fresh batch culture before they reached the late exponential phase. Maximum specific growth rates were determined after the specific growth rate became constant. This was the case after about 3, 5 and 12 h for cells taken from chemostats operated at 0·53 h-1, 0·3 h-1 and 0·1 h-1, respectively. The OD546 values of the batch cultures were determined every 20 min as long as their OD546 was between 0·08 and 0·5. The natural logarithms of these OD546 values were plotted against time and the slope of the resulting linear regression line was taken as µmax. The values obtained for µmax were highly reproducible (see Table 2)


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Table 2. Parameters of predicted mutant phenotypes in E. coli populations evolving at different dilution rates

 
Glucose analysis.
Glucose concentrations in continuous cultures were determined as described previously by Senn et al. (1994) .

Glucose uptake assays.
Cells (30 ml) from the continuous cultures were collected by centrifugation for 10 min at 1200 g at 4 °C, washed twice with buffer (glucose-free medium) and resuspended to an OD546 of 10. For the assays, cells were diluted to an appropriate cell density (OD546 between 0·1 and 0·005), dependent on the glucose concentration in the assay, so that the uptake was in a linear range during the first 2 to 3 min. The assays were performed at 37 °C and were started by adding 80 µl [14C]glucose to 720 µl bacterial suspension. Final glucose concentrations were 40, 10, 4, 2, 1, 0·5, 0·25, 0·125 and 0·0625 µM. Samples (100 µl) were removed at 20, 40, 60, 90, 120 and 150 s, immediately filtered through 0·45 µm cellulose nitrate membrane filters (Millipore) and washed with 10 ml buffer. The filters were dissolved in scintillation liquid (Filter count, Packard Instruments) and the radioactivity was determined with a BETAmatic I liquid scintillation counter (Kontron Analytical). Ks and Vmax values were determined by the direct linear plot method (Eisenthal & Cornish-Bowden, 1974 ). In contrast to the original method, we did not determine the coordinates of intersections graphically, but we calculated them from their linear equations.

Modelling.
For mathematical calculations and modelling the program AQUASIM was used (Reichert, 1994 ). The Monod equation was the basis for the modelling of the time-dependent substrate and cell concentrations in the reactor (Monod, 1942 ). Instead of the classic Monod model, a modified equation including an smin term was employed ( Kováová et al., 1996 ):

(3)

Based on this equation the following differential equations were used to describe the dynamics of cell growth, washout, mutant generation and substrate concentration for all i phenotypes in the continuous culture where phenotype i is created from phenotype i-1 by mutation (the last term of Equation 4; this term is only considered if Xi-1(t) >1/mr):

(4)


(5)

Values for µmax and s were determined experimentally. With the parameter estimation tool of AQUASIM we inferred Ks,i values of mutant phenotypes, which were supposed to periodically take over the population. Simultaneous with the Ks values, the mutation rates (mr) leading to those mutant phenotypes, and the correlation coefficients of the Ks and mr values were inferred by AQUASIM. A weighted least squares parameter estimation was performed for Ks,i and mr by fitting calculated s values (spred) to measured s values (sobs). This was done by minimizing {chi}2 values using the secant method, where {chi}2={Sigma}[(sobs-spred)/SD]2 and SD denotes the estimated standard deviations of the measured s values. SD values for s values were estimated from deviations of measured s values to be 10%. However, values from the first 100 h of chemostats run at higher dilution rates (>=0·5 h-1) showed higher deviations and SD was estimated to be 20%. For s values measured during the first 75 h in 0·1 h-1 chemostats the SD values were set to 30% instead of 10%. This was done because a chemostat population needs about 5–10 volume changes (50–100 h at 0·1 h-1) to reach a steady state, which probably also leads to variations in residual glucose concentrations. Since the model does not take into account this physiological adaptation, the difference between modelled and measured values was allowed to be higher for values taken during this period (see also ‘Model’ Results section).


   RESULTS
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ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
 
Evolution of kinetic parameters during glucose-limited chemostat growth
Chemostats with E. coli populations were set up and run under glucose-limited conditions for 500–700 h at different dilution rates. The evolution of the residual glucose concentration, µmax values and Ks towards glucose was monitored.

Residual glucose concentrations were measured for four chemostats at 0·3 h-1, three at 0·1 h-1 and one at 0·53 h-1. All these chemostats were inoculated with different single clones from the same glycerol stock culture. The time-course of evolution between independent runs was highly reproducible (Fig. 1).



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Fig. 1. Evolution of the residual glucose concentration in glucose-limited chemostats of E. coli K-12 run at dilution rates of 0·1 h-1 (a), 0·3 h-1 (b) and 0·53 h-1 (c). Experimental data from independent cultures (different symbols) and model predictions (thick lines) are shown. Measured values in (b) are from Wick et al. (2001) . The predicted glucose concentrations were calculated using AQUASIM with inferred Ks values and mutation rates that provided the best fit to the measured glucose concentrations. These parameters are listed in Table 2. The insets show a magnification of the boxed areas. For a culture that would evolve exactly as predicted (thick line) the 95% confidence interval of measured values is given by thinner lines in (b), under the assumption that the standard deviation of glucose measurements is 10%.

 
At all dilution rates, µmax increased in the first 50–150 h; the increase was faster for 0·1 h-1 and 0·53 h-1 than for 0·3 h-1. Later, between 150 and 500 h of cultivation the µmax remained constant (Fig. 2). Improved µmax values were calculated as mean values of the measurements taken at time points after the initial steep increase (>120 h for 0·1 h-1, >150 h for 0·3 h-1 and >120 h for 0·53 h-1). The means and standard deviations of these measurements were 0·67±0·01 h-1 (n=9) for cells evolved at 0·1 h-1, 0·73±0·01 h-1 (n=6) for cells evolved at 0·3 h-1 and 0·78±0·01 h-1 (n=10) for cells evolved at 0·53 h-1. The µmax of cells evolved at 0·1 h-1 was significantly higher than the µmax of the WT cells (P=2x10-7, t-test), the µmax of cells evolved at 0·3 h-1 was significantly higher than that of cells evolved at 0·1 h-1 (P=6x10-7, t-test) and the µmax of cells evolved at 0·53 h-1 was significantly higher than that of cells evolved at 0·3 h-1 (P=9x10-6, t-test).



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Fig. 2. Evolution of µmax in glucose-limited chemostats of E. coli K-12 run at different dilution rates. Experimental data (symbols) and model predictions (lines) are shown for 0·1 h-1 ({circ}, {bullet}, +, dashed line), 0·3 h-1 ({triangleup}, {blacktriangleup}, solid line) and 0·53 h-1 ({blacksquare}, dotted line). The predicted values were calculated using AQUASIM with the kinetic parameters listed in Table 2.

 
The Monod model implies that Ks can be calculated from measured µmax and residual glucose concentrations. Another classical method to determine Ks for a particular substrate is to measure the uptake rate (V) of the substrate in dependence of the substrate concentration (s). Ks and Vmax are then calculated by the direct linear plot method described by Eisenthal & Cornish-Bowden (1974) . At different time points Ks values of an E. coli population growing at 0·3 h-1 were determined by both methods (Fig. 3). The two methods gave similar results, the correlation coefficient being 0·89 (P=0·043).



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Fig. 3. Comparison of Ks values for glucose determined by uptake assays with [14C]glucose ({triangleup}) and by calculation from residual glucose concentration ({bullet}) during evolution of a culture of E. coli K-12 in a glucose-limited chemostat run at D=0·3 h-1.

 
Growth rate dependence of initial glucose affinity in continuous culture
Ks values calculated from the initial residual glucose concentrations from chemostats operated at dilution rates of 0·1 h-1, 0·3 h-1 and 0·53 h-1 (Fig. 1) indicated a growth rate dependence of Ks. Since the level of the general stress response regulator RpoS is inversely related to growth rate (Lange & Hengge-Aronis, 1994 ; Notley & Ferenci, 1996 ) and RpoS-attenuated strains exhibit a growth advantage in stationary phase (Zambrano et al., 1993 ), the influence of rpoS on initial Ks values was assessed using an rpoS negative mutant. Continuous cultures were set up at different dilution rates, each inoculated with cells from batch cultures as described in Methods. Residual glucose concentrations were measured during the first 50 h of cultivation. Ks values were calculated from residual glucose concentrations before they started to decrease. The Ks values as a function of dilution rate for the WT and rpoS strain are shown in Fig. 4.



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Fig. 4. Ks values of E. coli K-12 WT ({square}) and of its rpoS negative derivative ({blacktriangleup}). Ks values at different dilution rates were determined from residual glucose concentrations measured in chemostats inoculated with fresh cells. For calculation of Ks glucose concentrations were taken only from the initial phase before they started to decrease. Error bars represent 95% confidence intervals. Intervals for Ks are given as t intervals. Deviations of µmax and smin were taken into consideration for the calculation of the error bars. The degrees of freedom were 7, 7, 40, 6, 3 and 6 for experiments with WT cells in 0·1, 0·2, 0·3, 0·4, 0·5 and 0·53 h-1 chemostats, respectively, and 12, 9, 14, 10 and 12 for rpoS cells in 0·1, 0·2, 0·3, 0·4 and 0·5 h-1 chemostats, respectively. Intervals for D were calculated from estimated standard deviations of the parameters determining D (D=flow rate/reactor volume).

 
Model
The decrease of the residual glucose concentration suggests that mutant phenotypes with increased glucose affinity periodically take over the chemostat population. Some of the important questions immediately arising are now: how much do the Ks values improve at each take-over and what is the rate at which advantageous mutations arise per cell and generation? We used the kinetic simulation program AQUASIM to give estimations for these values. The following assumptions were made for the simulation. (i) The growth of the bacteria in the chemostat can be described with extended Monod kinetics (Kováová et al., 1996 ). (ii) The chemostat is inoculated with a WT strain with given µmax (0·63 h-1) and Ks that depends on specific growth rate (see Fig. 4). An exception was made for the WT Ks of cells growing at 0·1 h-1. This Ks value was not fixed at the value indicated in Fig. 4 but was allowed to roam freely during parameter estimation (see below). (iii) Mutant phenotypes (i.e. phenotype 1) with an improved µmax and Ks are produced from the WT with a certain mutation rate. The mutation rate is defined as the number of advantageous mutations per cell per generation. The µmax of phenotype 1 was as determined experimentally (0·67 h-1, 0·73 h-1 or 0·78 h-1 for populations cultivated at 0·1 h-1, 0·3 h-1 or 0·53 h-1, respectively), whereas the improvement of Ks was estimated by AQUASIM. This leads to an out-competition of the WT by the phenotype 1. (iv) When phenotype 1 has reached a certain population size (>1/mutation rate), phenotype 2 with a lower Ks is generated from phenotype 1 at each replication, whereas µmax now stays constant. Phenotype 2 will out-compete phenotype 1 because of its improved Ks. (v) Phenotypes 3, 4 and 5 are generated from phenotype 2, 3 and 4, respectively, in the same way. It should be noted that we use the term phenotype here not for a cell line carrying a defined mutation but for a group of different mutants that have the same improved kinetic properties with respect to µmax and Ks on glucose [compare with Notley-McRobb & Ferenci (2000 , 1999a , b )]. This process is illustrated in Fig. 5.



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Fig. 5. Schematic illustration of the succession of mutated phenotypes and their relationship with respect to kinetic parameters. Note that each phenotype can represent a mixture of different mutants with the same phenotype with respect to Ks and µmax.

 
AQUASIM estimated mutation rates and improved Ks values of each phenotype by fitting measured residual glucose concentrations to calculated ones (see Methods). The Ks values and mutation rates for the best fits for each dilution rate are given in Table 2. Table 3 shows the correlation coefficients of the estimated parameters. Estimated Ks values and mutation rates were used to predict the evolution of the residual concentration (lines in Fig. 1) and the population dynamics of postulated mutant phenotypes (Fig. 6). The population dynamics show that some of the phenotypes dominate the culture longer than others (Fig. 6). This is reflected in the course of the residual glucose concentration where sometimes plateaux can be recognized between the changeovers from one population to the next. Mostly, such plateaux are very short-lived and smoothed by rapid successive population changeovers.


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Table 3. Correlation coefficients of the fitted parameters (see also Table 2) as estimated by AQUASIM

 


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Fig. 6. Succession of wild-type and phenotypes with improved kinetic properties (compare with Fig. 5) in glucose-limited chemostats of E. coli K-12 run at dilution rates of 0·1 h-1 (a), 0·3 h-1 (b) and 0·53 h-1 (c) as predicted using AQUASIM with the kinetic parameters shown in Table 2.

 
As stated above, the WT Ks at 0·1 h-1 was not determined as in Fig. 4, but estimated by AQUASIM. Because physiological adaptation of cells to chemostat conditions takes 5–10 volume changes, values determined in the first 20 h as those in Fig. 4 are probably too high, because cells are not yet fully adapted. On the other hand, values determined after 5–10 volume changes (50–100 h at D=0·1 h-1) are probably too low because after this time period mutants with improved Ks are very likely to account for a good part of the culture.

Evolution of Ks at low cell numbers
There are 1011 individuals in the large chemostats and the estimated mutation rate (advantageous mutations per cell per generation) lies at about 10-7. The product of the population size (number of cells) times the mutation rate gives the mean number of cells with advantageous mutations that are produced in the whole population within one generation interval. Since this product is much higher than one (1011x10-7=104) in large chemostats the system does not have to wait for a favourable mutation to occur, but only for the existing favourable mutations to sweep the population (Wahl & Krakauer, 2000 ). This is the reason for the high reproducibility and deterministic behaviour of the evolutionary process observed (Fig. 1). If the population size is around 107 cells, the mean number of cells with advantageous mutations arising at each generation interval is only one. This means that there is a high probability of no advantageous mutations taking place during a generation interval in the whole population. Hence, the evolution in populations with a total cell count in the range of 1/(mutation rate) should start to become stochastic and lose its reproducibility. Therefore, we followed the evolution of the residual glucose concentration in four much smaller populations, consisting of total cell numbers between 1x107 and 5x107. To achieve this, we decreased both the glucose concentration (from 100 mg l-1 to 1 mg l-1) and the reactor size (from 1500 ml to 75 ml). As predicted, the time course of the residual glucose concentration was different for all four populations (Fig. 7). Besides the lack of reproducibility between different runs, the evolution of the residual glucose concentration was slower in all four small bioreactors compared to the large bioreactors. The residual glucose concentration started to decrease later in three of the four small bioreactors. Only in one population, which was cultivated for 750 h, the residual glucose concentration decreased to about 50 µg l-1 after 600–700 h. Populations in large bioreactors typically reached this residual glucose concentration after 400–500 h. Moreover, plateaux, which can only be estimated in the large bioreactors, are much more pronounced in the small bioreactors. These plateaux indicate Ks values of mutants. For small populations one would expect to observe quite extended plateaux in the case where a population has to wait for the next mutation to occur and to take over. Therefore, we performed computer simulations of evolutionary processes, where mutants were not produced as a certain fraction of the growing population but introduced into the system at arbitrarily chosen time points. The expected residual glucose concentrations for such evolving cultures are shown as lines in Fig. 7. In fact, the predicted plateaux are in good agreement with the experimentally observed extended plateaux, thus supporting the estimated Ks values of the four mutants listed in Table 2.



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Fig. 7. Evolution of the residual glucose concentrations of four independent glucose-limited chemostats of E. coli K-12 run at a dilution rate of 0·3 h-1 and containing a total population of 15x107 cells. Experimental data (symbols) and model predictions (lines) are shown. At the time points indicated by arrows, the cultures were transferred to a fresh bioreactor vessel. Model predictions were performed with the parameters from Table 2. In contrast to the procedure followed in Fig. 1, the mutated phenotypes (for kinetic properties see Fig. 5 and Table 2) were not produced with a given frequency from the growing population, but were introduced into the population at arbitrarily chosen time points. One cell with a mutated phenotype was introduced into a population of 107 cells at the following time points. (a) Phenotype 1 at 150 h, phenotype 2 at 300 h. (b) Phenotype 1 at 60 h, phenotype 2 at 190 h. (c) Phenotype 1 at 0 h, phenotype 2 at 110 h, phenotype 3 at 300 h, phenotype 4 at 530 h. (d) Phenotype 1 at 80 h.

 
Interestingly the analytically determined residual glucose concentrations in the small chemostats were only about 75% of those measured in the large bioreactors. Therefore, the lines plotted in Fig. 7 are adjusted by this factor. The reason for this reproducible deviation is not known, but wall populations might account for this difference. Wall populations decrease the residual glucose concentration in reactors and would have a larger effect in the small chemostats, because of the higher surface to volume ratio and lower cell density compared to the large chemostats. Such a wall population would have to build up quickly and reproducibly since initial steady state values in the four independent chemostats were similar and established rapidly. Moreover, when we transferred whole cultures to a fresh bioreactor after different cultivation times (indicated by arrows in Fig. 7), the residual glucose concentration increased just after the transfer. Within a few hours after the transfer, the steady state was reached again and the residual glucose concentration was the same as before the transfer. Thus, it is possible that a wall population led to generally lower glucose levels in the small chemostats, but it seems unlikely that the wall population has a considerable effect on the long-term evolution of the residual glucose concentration, on which this study focused.

Evolution of Ks after a shift to pyruvate
To test how the Ks for glucose of a population evolves when the selection pressure to increase glucose affinity is relieved, a culture grown under glucose limitation for 128 h at D=0·3 h-1 (57 generations) was shifted to pyruvate limitation. The culture was then grown under pyruvate limitation at D=0·1 h-1. (D was reduced because µmax for growth with pyruvate was only between 0·25 and 0·3 h-1.) After 353 h (51 generations) the culture was shifted back to glucose limitation and D=0·3 h-1 (Fig. 8a). Astonishingly, the residual concentration of glucose immediately established again at the value measured before the shift to pyruvate and continued to decrease (Fig. 8a). Hence, the temporal shift from glucose to pyruvate limitation did not affect the evolution of the affinity for glucose, because after the shift back to glucose limitation the residual glucose concentration followed a time-course starting at the exact value before the interruption and exactly paralleling the pattern of cultures that were not submitted to the temporal shift (Fig. 8b). This demonstrates that the selective evolution during growth of an E. coli population with glucose can be followed easily in a continuous culture by monitoring the residual concentration of this sugar.



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Fig. 8. Evolution of the residual glucose concentration in a continuous culture of E. coli K-12. The culture was operated at D=0·3 h-1 under glucose-limited conditions for 128 h, then switched to pyruvate-limited conditions at 0·1 h-1 for 353 h and subsequently switched back to glucose-limited conditions at 0·3 h-1. Uncorrected data (a) and data corrected for growth on glucose only (b) are shown ({diamondsuit}). Data from continuous cultivations grown under glucose limitation at D=0·3 h-1 without interruption are shown for comparison in (b) ({circ}, {square}).

 

   DISCUSSION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
 
Effects of Monod kinetics on population dynamics
This study has provided experimental data for an evolutionary process, which can be described with a deterministic model in large populations and which becomes stochastic when the population size is in the range of 1/(mutation rate) or smaller. Moreover, it has been demonstrated how a selection-driven evolution, like the one studied here and investigated previously (Dykhuizen & Hartl, 1981 ; Hartl & Dykhuizen, 1979 ), can proceed as a function of absolute time passed (i.e., clock-like) rather than as a function of the number of generations. We want to emphasize that the model presented here can explain the apparent clock-like behaviour for this special case of selection-driven chemostat evolution under glucose limitation, but of course, it does not serve as a general explanation for the molecular clock observed in amino acid substitution rates (Zuckerkandl & Pauling, 1965 ). The apparent clock-like behaviour becomes evident when we plot the time-course of the improvement of Ks either as a function of time (Fig. 9a) or as a function of generations (Fig. 9b). At dilution rates of 0·3 h-1 and 0·1 h-1 the evolution of Ks is almost identical and, except for the first 100 h, the Ks exhibited by the evolving cultures is always similar. The evolution at 0·53 h-1 is also quite similar, although Ks values are always higher at given time points.



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Fig. 9. Evolution of Ks in glucose-limited chemostats of E. coli K-12 run at dilution rates of 0·1 h-1, 0·3 h-1 and 0·53 h-1. The evolution is given either as a function of time measured in hours (a), or of generations (b). The comparison demonstrates that the glucose evolution pattern is more related to absolute elapsed time than to the number of generations.

 
The model used here predicted similar improvements for Ks for most of the mutations (Fi{cong}0·6, Table 2). Therefore, first we will discuss how fast a population is expected to evolve at different growth rates, when Ks improvements are the same. Later we will discuss the influence of the first mutational step (from WT to phenotype 1), where the improvements in kinetic properties were different at the various growth rates.

For two competing strains with the same µmax but different Ks (Ks,1>Ks,2), the selection coefficient per hour (Sh), i.e. the specific growth rate difference between the two strains growing together in the same chemostat, can be calculated as follows:

(6)

and the selection coefficient per generation (Sg) is calculated as:

(7)

(Dykhuizen & Hartl, 1983 ), where s is the residual glucose concentration in the chemostat. In the term s/(Ks+s), the same proportional change in Ks has a greater impact on the selection pressure at lower than at higher s values. Substrate concentrations in the chemostat become lower with decreasing dilution rates. This is the reason why Sg steadily decreases with decreasing dilution rates (Fig. 10). However, when we calculate the selection coefficient per hour, Sh, there are two effects counterbalancing each other. On the one hand, there is the selection pressure per generation, Sg, which speeds up the evolution of slower growing populations compared to faster growing populations. But on the other hand, there are more generations per time interval at higher dilution rates, speeding up the evolution of faster growing populations compared to slower growing populations. The combination of these effects leads to a maximum rate of evolution at intermediary growth rates. This is visualized in the selection coefficient per hour, Sh, which shows a u-shaped curve (Fig. 10). Note that more negative values mean higher selection and hence faster evolution.



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Fig. 10. Model prediction of the two selection coefficients, selection per hour, Sh (solid line), and selection per generation, Sg (dotted line), for two competing strains with the following kinetic properties: µmax,1max,2=0·73 h-1 and Ks,2/Ks,1=0·6. Selection coefficients per hour are specific growth rate differences between the two strains when grown together at the given dilution rate under the assumption that the residual glucose concentration is as if only the strain with the higher Ks were present in the chemostat. Note that more negative values mean higher selection and hence faster evolution. The selection coefficients per hour for strains with Ks and µmax values calculated for our experimental data are also given: phenotype 1 against WT ({blacksquare}), phenotype 2 against phenotype 1 ({triangleup}), phenotype 3 against phenotype 2 ({square}), phenotype 4 against phenotype 3 ({circ}) and phenotype 5 against phenotype 4 (+). Note that the Sh values for the competition of phenotype 1 against WT at 0·1 h-1 and 0·53 h-1 differed markedly from the other Sh values.

 
In the following we want to calculate this maximum rate of evolution as a function of the different Ks values of the competing strains. Since the residual glucose concentration is constantly changing during a competition experiment, we first need to know what concentrations we have to base our calculations on. Normally competitions are started with equal numbers of the competing strains. In this case the residual glucose concentration in the chemostat is as though only the better of the two strains (with the lower Ks) was in the chemostat (Dykhuizen & Hartl, 1983 ). In contrast, when evolution of successive mutants is considered, mutants are generated as a strong minority (10-7xthe total cell count) in an environment where s is given by the parent strain (with the higher Ks). When the population size is much higher than 1/(mutation rate), consecutive mutants are generated already before the precedent mutant has completely taken over the population. Therefore, the evolution-relevant selection coefficients have to be calculated based on s values as if only the strain with the higher Ks was in the chemostat:

(8)

Hence Sh can be calculated as:

(9)

The minimum of this function is found when the first derivative is set to 0 and it is positioned at

(10)

which can also be written independently of absolute values of Ks,1 and Ks,2 as

(11)

where Fi=Ks,2/Ks,1 and is referred to as the improvement factor. Therefore, a system in which the proportional improvement in Ks is the same at all dilution rates will evolve at the fastest pace (i.e. the most rapid mutant successions) at a dilution rate of Dmin. For values of Fi between 0·9 and 0·1, Dmin lies between 0·49xµmax and 0·24xµmax. Note that Dmin is always lower than 0·5xµmax, since 0·5 is the boundary value of the term (Fi)/(1-Fi) when Fi approaches 1.

For the mean improvement in Ks found in this study, i.e. Fi=0·6, and µmax values that increase from 0·67 h-1 to 0·78 h-1 between the dilution rates of 0·1 h-1 and 0·53 h-1, the fastest evolution should be observed at about 0·32 h-1. However, the first mutational step predicted by the model (from WT to phenotype 1) is special, because not only Ks but also µmax improved. Moreover, the improvements for Ks and µmax were very different between dilution rates. The effects of these differences on the overall rate of the observed evolution will be discussed next.

In the system studied here, the dependence of the initial WT Ks on the dilution rate (Fig. 4) also exhibits a u-shaped pattern similar to that of the Sh values (Fig. 10). As a result and as estimated by AQUASIM (Table 2), the improvement in Ks in the first 100–200 h of cultivation is much higher at 0·1 h-1 than at 0·3 h-1, accelerating the evolution at 0·1 h-1 compared to 0·3 h-1 and thus cancelling the effect of the higher selection per hour at 0·3 h-1 in the initial phase. Later when the improvement in Ks is in a similar proportional range for 0·1 h-1 and 0·3 h-1, predicted population changeovers will become faster at the dilution rate of 0·3 h-1 (Fig. 6). In comparison to the culture run at 0·3 h-1, the predicted mutant phenotype 1 takes over faster at the beginning of the selection process in the chemostats run at 0·1 h-1 and at 0·53 h-1, respectively. However, despite the same outcome, the phenomenon is based on different kinetic reasons. In the 0·1 h-1 chemostat it is mainly the improvement in Ks which makes phenotype 1 better than in 0·3 h-1 chemostats, whereas in 0·53 h-1 chemostats it is mainly the improvement in µmax that makes phenotype 1 of the 0·53 h-1 chemostat better (compare parameter values in Table 2).

Based on the above considerations, we can conclude that there is an apparent overall clock-like behaviour due to the special combination of Monod kinetics (Fig. 10) and the different initial Ks and µmax improvements. When we only look at the sweeps where Ks improvements are the same for the different dilution rates, a maximum rate of evolution slightly below µmax/2 is observed. With higher and lower growth rates the evolution slows down and a certain rate of evolution occurs exactly twice (Fig. 10, solid line). Therefore, it is possible that populations with highly different generation times (i.e. at very high and very low growth rates) evolve at the same speed, but no constant speed of evolution is expected over the whole range of dilution rates.

The model has shown that such an apparent clock-like evolution can be explained with very similar mutation rates per generation and similar improvements in Ks per mutation at different growth rates. It is therefore not mandatory to assume different mutation rates to explain a phenomenon such as that shown in Fig. 9. However, mutation rates can depend on many experimental factors, and growth rate may be one of them (Smith, 1992 ). For E. coli populations growing under glucose-limited conditions there is contradictory information about the dependence of mutation rate on specific growth rate. Whereas studies on mutations to bacteriophage T5 resistance could not find such a dependence (Kubitschek & Bendigkeit, 1964 ), Savva (1982) reported that mutations from tryptophan auxotrophy to prototrophy increased proportionally with specific growth rate. Note that one has to be careful when comparing mutation rates from different studies. Throughout this report we give mutation rates as the number of advantageous mutations per cell per generation [as in Savva (1982) ], whereas in the literature mutation rate is sometimes also defined as the number of mutations per bacterium per hour (Kubitschek & Bendigkeit, 1964 ).

The mutation rates calculated here are in the same range as those reported commonly in the literature, which vary typically between 10-6 and 10-9 mutations per cell and generation for a range of different detectable phenotypes (Merrel, 1981 ). Notley-McRobb & Ferenci (2000) estimated that in their glucose-limited chemostats (D=0·3 h-1), containing 2–3x1010 cells in total, 104 mutants with an improved glucose affinity were generated at each replication round due to loss-of-function mutations in mgl and mlc. This suggests a mutation rate of 3–5x10-7, which is very similar to the mutation rates estimated in this study, namely 1–5x10-7 at all dilution rates tested.

Our results are in good agreement with the studies of Dykhuizen and Hartl (Dykhuizen & Hartl, 1981 ; Hartl & Dykhuizen, 1979 ). We also observed an increase in µmax during the first 50–150 h. After this phase, µmax became stable in both studies for the following 400 h, whereas Ks improved constantly. Also the apparent clock-like improvement in Ks was observed here (Fig. 9). The two studies differed with respect to the observed increase in µmax and the extent of Ks improvement during the 500 h. Whereas Dykhuizen & Hartl (1981) found that the decrease of Ks was tapering off in later periods; we observed that the stepwise improvements in Ks were quite constant. Furthermore, these authors reported similar improvements in µmax at both dilution rates, we observed that the improvement in µmax was greater at higher dilution rates. A selection for increasing µmax with increasing growth rates can be explained with the concept of µmax- and Ks-type strategists, which postulates that batch cultures select for quickly growing (µmax-type) organisms, whereas in a chemostat at low dilution rates, slow-growing cells with a high affinity for the limiting substrate (Ks-type) are selected for (Powell, 1958 ; Veldkamp & Jannasch, 1972 ). In fact, in the chemostat the growth rate can be varied from very slow growth to rates close to µmax; thus it allows mimicking of a transition state between ‘µmax-type-only’ and ‘Ks-type-only’ selection. The values reported here for the improvement in µmax and Ks demonstrated such a transition: with increasing specific growth rate the improvement of µmax became bigger, whereas the improvement in Ks was smaller (Table 2). The genetic events leading to this improvement in µmax are not yet understood.

Growth rate dependence of Ks
In contrast to one of the assumptions of the Monod model and to measurements performed with evolved strains (Kováová et al., 1996 ; Senn et al., 1994 ), starting clones of the WT strain used here exhibited a dependence of Ks on dilution rate. At dilution rates lower than 0·3 h-1, this dependence is mainly due to rpoS as demonstrated by the comparison of the WT and the otherwise isogenic rpoS mutant (Fig. 4). The negative effect on Ks and, hence, on the competitive ability of a strain with high RpoS levels might be one reason why many isolates from carbon-limited environments such as starved batch cultures or environmental habitats show an attenuated RpoS phenotype (Finkel et al., 1998 ; Zambrano et al., 1993 ). The dependence of Ks on dilution rate determined here from residual glucose concentrations is in good agreement with earlier studies by Ferenci and co-workers, who demonstrated that the expression of the high-affinity glucose-uptake pathway is maximal at intermediate growth rates, getting lower towards both higher and lower growth rates (summarized by Ferenci, 1996 , 1999 ). The lower glucose affinity at enhanced dilution rates results from reduced cAMP and endoinducer levels, which are necessary for the induction of the high affinity glucose transport systems (Death & Ferenci, 1994 ; Ferenci, 1996 ).

Limitations in monitoring evolution
Monitoring evolution by measuring residual substrate concentrations has its limits because the growth-limiting substrate(s) must be known. In this study this growth-limiting substrate was glucose. However, as the evolution proceeds, not all cells of the population may show the same values of these parameters.

For example, Adams and co-workers have found that E. coli populations cultivated under glucose-limited conditions evolve polymorphisms maintained by acetate crossfeeding (Rosenzweig et al., 1994 ). Such polymorphisms were found only in populations cultivated for 1615 h (at D=0·2 h-1, corresponding to 468 generations) or more (Treves et al., 1998 ).

However, in the first 500–700 h, evolution of such E. coli populations triggers the improvement of growth kinetics on glucose. Notley-McRobb and Ferenci have analysed mutations occurring in populations during the first 644 h of cultivation (280 generations at D=0·3 h-1). All of 10–20 randomly picked isolates had acquired mutations in the mlc, mgl and mal loci, all of which confer an increased competitive ability in glucose scavenging (Notley-McRobb & Ferenci, 1999a , b ). In a later study the appearance of these mlc and mgl mutations was directly correlated to periodic population shifts (Notley-McRobb & Ferenci, 2000 ).

Proteomic analysis of cells from 500 h old cultures (217 generations at D=0·3 h-1) has shown massive up-regulation of proteins belonging to the mgl and mal operons (Wick et al., 2001 ). No indications for changes other than these triggering increased glucose uptake were found, whereas proteomic studies of clones isolated from an older glucose-limited culture (0·2 h-1, 773 generations, 2667 h) suggested that mutations with pleiotropic effects had taken place (Kurlandzka et al., 1991 ). Comparison of the above-mentioned studies and the data presented here suggest that during the first 500 h in glucose-limited chemostats the growth of E. coli cells is single-nutrient limited and the evolution is indeed driven by improving the Monod parameters. Therefore, experimental determination of these parameters is a good tool to monitor the evolution of the population during this initial time period. After about 500 h, when the potential for optimizing Monod parameters has been exploited, other mutational events such as those reported by Adams and co-workers may take place (Kurlandzka et al., 1991 ; Rosenzweig et al., 1994 ; Treves et al., 1998 ), complicating further monitoring of the evolutionary process.


   ACKNOWLEDGEMENTS
 
We thank Paolo Landini for helpful discussions and for supply of the strain E. coli MV2792, Thomas Ferenci and Uwe Sauer for constructive comments on an earlier version of this manuscript, Peter Reichert for help in the statistical analysis of the data, and A. J. B. Zehnder for support throughout this study. We acknowledge the financial support of this research by a grant from the Swiss National Science Foundation (Grant number 31-50885.97).


   REFERENCES
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
 
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Received 21 March 2002; revised 27 May 2002; accepted 30 May 2002.