TEMPERATURE DEPENDENCE OF GROWTH RATE CONSTANT OF ESCHERICHIA COLI BACTERIA
Group R1
4/26/1998
Mike Dolan
Bethany Gallagher
Robert T Jenkins
Emily McCourt
Abstract
The rate of growth of Escherichia Coli bacteria was determined using a Milton – Roy Spectronic 20D spectrophotometer to measure the absorbance of light through a sample every ten minutes. The experimenters sought to determine the dependence of the growth rate of the bacteria on temperature. The growth rate constant of the bacteria at 37 oC during week one was determined to be .01572 .0056 min-1. The growth rate constant for week two at 37 oC was found to be .01629 .00161 min-1 and at 42 oC was found to be .01885 .00309 min-1. The growth rate constant for week three at 37 oC was found to be .01540 .00194 min-1 and at 32 oC was found to be .01101 .00063 min-1. From this data, the experimenters concluded that growth rate constant of the Escherichia Coli bacteria was dependant on temperature. Furthermore, the growth rate of said bacteria was observed to follow a different temperature relationship from 32 – 37 oC than from 37 – 42 oC.
Background
The fact that a bacterial population follows a characteristic growth curve throughout its lifecycle is important in understanding this experiment. The life span of the E. Coli strain used in this experiment spans approximately six hours (1). The curve can be divided into four distinct regions.
- Lag Phase: the time during which the cells do not increase in number, but prepare for reproduction by synthesizing DNA and various inducible enzymes needed for cell division.
- Growth Phase: the logarithm of the biomass increases linearly with time. During the growth phase, the bacteria obey first order chemical kinetics illustrated by the following equation:
= (1/T) * Ln(t/o)Eq. 1
In this equation, is the growth rate constant (1/min), T is time (min), Xt is concentration at time T (cells/ml), and Xo is concentration at T = 0.
- Stationary Phase: the number of bacteria have reached a maximum and the growth rate equals the death rate. This often occurs when a necessary nutrient in the growth medium has been exhausted, when inhibitory end products accumulates, or when conditions are no longer suitable for growth.
- Death Phase: number of viable cells decline and no further divisions occur. The death rate often follows the reverse kinetics as the exponential growth phase.
The spectrophotometer is the tool used to measure this relative growth of the bacteria. The machine sends ultraviolet light waves through the solution and measures the amount light absorbed by the sample. This number is directly related to the concentration of cells present in the solution according to the following equation:
C=BA Eq. 2
This equation relates the concentration of cells to the absorbance reading on the spectrophotometer by a proportionality constant of B=1.9*10^8 cells/mls per unit absorbance for E. Coli (2).
Two characteristics of bacterial cells, which can be determined during the active cellular division of the log phase, are the exponential growth rate constant and the generation, or doubling, time. The slope of the logarithmic phase of the growth curve represents the exponential growth rate constant. The generation time is calculated using the following equation. In this equation, t equals the elapsed time during which growth is measured and Xo and Xt are the number of bacteria at times zero and t:
Tgen= t(ln 2)/(ln Xt-ln Xo) Eq. 3
When temperature is varied, a new relationship between the temperature and the growth rate constant is revealed. Arrhenius developed a relationship between the velocity of a chemical reaction and its temperature which can be applied to this experiment when it is put into the following form,
k=Ae^(B/T)Eq. 4
Where k, A, B, and T represent the growth rate constant, a collision factor, activation energy, and absolute temperature, respectively (3,4). When this equation is used to plot the natural log of growth rate versus the reciprocal of the temperature, there is a region where there is a straight line; this region is called the normal of Arrhenius range. In the ranges above and below the Arrhenius range, the growth rate is lower. The graph below is an example of an Arrhenius plot (3,4).
Figure 1
Literature Arrhenius Plot for E. Coli Growth Rate (k) vs. 1000/T
Note in the above graph that k is on a log scale, and the units of k are hrs-1
In this graph, the normal of the Arrhenius range is 21 to 37 degrees Celsius. However, it is extremely important to note that this range is not common to all types of bacteria, or to all strains of the E.Coli bacteria. Each strain of E.Coli is different in its nutritional needs and overall metabolism (4). Several factors influence the range in which the Arrhenius plot is linear besides the temperature at which it is growing. The most influential in this experiment are the pH of the solution and the nutrients of the medium (4). When the temperature of the E.Coli culture is changed, the amount of different proteins in the culture also change. One enzyme in the Methionine pathway, homoserine trans-succinylase, has been proven to be extremely temperature sensitive and is an important enzyme to the growth of the cells. When there is a change in the concentration of this enzyme, the metabolisim of the cells changes accordingly. Thus, with changes in the temperature, there are changes in the growth rate (5).
Another reason that there is change in the E.Coli at different temperatures is due to the changes in the membrane structure. When the temperature is decreased, the membrane becomes less fluid. The cell, in order to compensate for this, adds more unsaturated fatty acids to the membrane. Unsaturated fatty acids, because of their bent hydrocarbon tails, restore this fluidity to the membrane. Therefore, the change in the membrane structure influences the metabolism of the cells and thus their growth rate (4).
Procedure and Materials
The same growth medium was prepared each week, which consisted of 10g of Bacto-Tryptone, 5g yeast extract, and 10g of Sodium chloride added to 1000ml of deionized water. The solution was shaken to insure homogeneity. The pH of this solution was taken and, if needed, adjusted to 7 by adding additional 5 M Sodium Chloride solution. pH readings were also taken every hour to make sure that the pH did not deviate from 7. The sterile (via the autoclave) PennCell culture apparatus was used to grow the E.Coli. The PennCell apparatus enables the experimenter to maintain a relatively sterile environment while taking samples and measuring temperature and pH levels. Air flow was set at .75 and agitation was kept constant at 200 RPM.
The E.Coli were added to the solution and an absorbance reading was taken every ten minutes in the Milton-Roy Spectrotonic 20D Spectrophotometer. In the first week, the cells were kept on a hot plate at 37 degrees Celsius until the stationary phase. However, because the hot plate broke, a temperature regulator was used the next two weeks. The temperature regulator sends water at the desired temperature through a tube and a coil inserted into the growth solution. In week two, the temperature was changed at t = 130 to 42 degrees and the growth rate continued to me measured. This temperature change was executed by changing the dial on the temperature regulator and the temperature changed within 9 minutes. In week three, the growth rate was changed to 32 degrees Celsius at t = 130 by adding ice to the bath in which the Penn-cell was placed. Within 16 minutes, the experimenters were able to reach the new temperature.
Results
The E. Coli bacteria followed the growth patterns expected (lag phase followed by logarithmic growth) for all three weeks in which experiments were run. The following is an example of this type of growth taken from the first week of experimentation.
Figure 2
Ln (Concentration) vs. Time (min)
This graph is representative of the logarithmic growth phase of the E Coli. Bacteria studied in this experiment. The temperature was maintained at a constant 37oC for the length of the experiment for this particular trial.
In weeks two and three, the graph of natural log concentration (ln X) versus time is different as the temperature was varied in the middle of logarithmic growth. The following is a representative graph taken from the third trial of the experiment in which the temperature was changed from 37oC to 32oC in the middle of logarithmic growth.
Figure 3
Ln(Concentration) vs. Time (min)
Note the decrease in growth rate(slope of line) at the lower temperature.
The growth rate constant () - which is the slope of the best fit line on the graph of natural log (concentration) versus time - was calculated using the excel regression function. The 95% confidence limits were used in said calculations and then tabulated. This information can be found in the following table.
Table 1
Mean and 95% Confidence Ranges for Growth Rate Constants
Trial / Actual / Lower 95% / Upper 95%Week 1 (37oC) / 0.015723279 / 0.015162997 / 0.016283561
Week 2 (37oC) / 0.01628551 / 0.014666534 / 0.017904485
Week 3 (37oC) / 0.015395161 / 0.013459208 / 0.017331115
Average (37oC) / 0.015801317 / 0.01442958 / 0.017173054
Week 2 (42oC) / 0.018853493 / 0.015756613 / 0.021950373
Week 3 (32oC) / 0.011010576 / 0.010384227 / 0.011636925
This table shows the growth rates obtained for each trial and at each temperature along with the upper and lower confidence limits of the trials.
A paired two sample t-test for means was also performed on the growth rates of the data obtained for all three trials as well as their 95% confidence limits. Variable one consisted of all the slopes between the temperatures 42oC and 37oC. Variable two consisted of all the slopes between the temperatures 37oC and 32oC. The t-test showed that tcrit was greater than tstat, meaning that these two set of data are statistically different. The following is the said t-test.
Table 2
T-Stat Paired Mean Test For Growth Rate Constants at Different Temperatures
t Stat / 1.844034525P(T<=t) one-tail / 0.05120486
t Critical one-tail / 1.85954832
P(T<=t) two-tail / 0.102409719
t Critical two-tail / 2.306005626
Note that tcritis greater than tstat
Using methods similar to those used to determine growth rates of bacteria, the activation energy of their metabolic reactions was found. Instead of using a linear fit an exponential fit was used of the form Ae(B/T). A representative graph is shown.
Figure 4
Arrhenius Plot Relating Growth Rate to Temperature Between 32 and 37 oC
In this graph, the growth rates as well as their respective upper and lower 95% boundaries are plotted for the temperatures 37oC and 32oC. the activation energy of the reaction if proportional to the exponent of the line on the graph.
The activation energy – which is directly proportional to the exponents of all graphs - (actual data and 95% confidence limits) were calculated and tabulated. These exponents are visible in the following table.
Table 3
Activation Energy of Bacteria Growth Range for Each Trial
37 - 42 Trial / 37 – 32 Trial65470.91136 / 80011.96552
24031.94176 / 52902.03288
-20263.85568 / 23849.02496
Activation energy (J/mol E Coli.)
A paired two sample t-test for means was also performed on this set of data, where variable one was all activation energies from the second trial and variable two was all activation energies from the third week of experimentation. For this t-test, tcrit was greater than tstat. This means that these two sets of data are significantly different. The following is said t-test.
Table 4
T –Test Between Activation Energy Mean for 32 – 37 oC and 37 – 42 oC
t Stat / -3.417034843P(T<=t) one-tail / 0.038004392
t Critical one-tail / 2.91998731
P(T<=t) two-tail / 0.076008783
t Critical two-tail / 4.302655725
Note that the tcrit is great than the tstat
In the interest of more thorough data analysis, the Arrhenius plot (and 95% confidence limits) for the 32-37 C trial was extrapolated out to 42 C. This data was plotted on the same graph as the Arrhenius plot (and 95% confidence limits) for the 37-42 C trial. The Arrhenius plots for each week were then compared to see if they were significantly different.
Figure 5
Note that the Arrhenius plot for 37-42 C average falls below the extrapolated 32-37 C average, although the 37-42 C plot does remain within the 95% confidence limits for 32-37 C.
In an effort to determine if the graphs for 32 – 37 oC and 37 – 42 oC were different, a t-stat test was conducted for the coefficients of the exponential fits of each range and its 95% confidence limit. These t-stat tests are shown in Tables 5 and 6.
Table 5
T-Stat Test Between Coefficient (A)
A (32 - 37) / A (37 - 42)t Stat / 1
P(T<=t) one-tail / 0.2113249
t Critical one-tail / 2.9199873
P(T<=t) two-tail / 0.4226497
t Critical two-tail / 4.3026557
Table 6
T-Stat Test Between Coefficient (B)
B (37 - 42) / B (32 - 37)t Stat / -3.417034843
P(T<=t) one-tail / 0.038004392
t Critical one-tail / 2.91998731
P(T<=t) two-tail / 0.076008783
t Critical two-tail / 4.302655725
Note that in the tables the T critical is greater than the Tstat, indicating that the means for each range are significantly different. This, in turn, indicates that the equations themselves are significantly different.
Discussion
The two goals of this experiment were to prove the temperature dependence of the growth rates and to prove that those growth rates did not follow an Arrhenius plot for all temperatures.
In order to prove that the growth rate was temperature dependent, the following methodology was used. First, the cell kinetics lab was run under conditions identical to those found in the bioengineering manual (37oC, .75 air flow). The growth rate was calculated using the absorbencies recorded during the day. The next two trials were identical to the first except for the fact that temperature was changed. In the middle of logarithmic growth, the temperature was changed from 37oC to 42 oC in the second week and 37oC to 32 oC in the third week. There was a corresponding change in growth rate. A paired two sample t-test for means was done on this data and their respective upper and lower 95% points. The t-test concluded that these sets data were statistically different. As the only variable that was changed was temperature, the experimenters conclude that the growth rate of the E. Coli bacteria is temperature dependent.
A graphical analysis was used to prove that the growth rate of the E Coli. does not exhibit the same temperature dependence for all temperature ranges. The data that were analyzed consisted of the growth rates at all temperatures (42oC, 37 oC, and 32 oC) as well as their 95% confidence limits (see Figure 5). In this graph, the growth rates for both temperature ranges (32 – 37 oC and 37 – 42 oC ) are plotted as if they follow an Arrhenius equation (this is because the Arrhenius equation is frequently used to relate the speed of chemical reactions, or growth rate constant, and the temperature). The relationship found between growth rate constant and 1/Temperature for the 32 – 37 oC range was extrapolated to 42 oC and plotted on the same graph. If the bacteria do follow the Arrhenius equation for all temperatures, the experimenters would expect the extrapolated data for the 32 –37 oC trial to coincide with the measured data for the 37 – 42 oC trial. However, the extrapolated average value for the 32 – 37 oC trial did not coincide with the measured 37 – 42 oC trial, indicating that the bacteria do not follow the same relationship at all temperatures. In fact, the graph for the 37 – 42 oC trial was less than the extrapolated 32 – 37 oC graph, which was expected (the experimenters postulated that the growth rate constant does not rise as sharply with 1/Temperature in the 37 – 42 oC range because of an inhibition of an enzyme involved in the growth of the bacteria). However, the graph also shows that the average growth rates for the 37 – 42 oC trial fall well within the 95% confidence limits of the extrapolated 32 – 37 oC trial, indicating that no exact conclusion can be determined from the data. More trials are needed to confirm our hypothesis about growth rate vs. 1/Temperature relationship of the E. Coli bacteria.
In an effort to add mathematics to the graphical analysis of the data, the experimenters conducted a t-stat test on the coefficients of the exponential fit for the data in the 32 –37 oC range as well as the fit for the data in the 37 – 42 oC range (and their 95% confidence limits). The theory was that if the coefficients for these exponential fits were significantly different, then the graphs themselves must be significantly different as well (because an exponential equation is defined only by its coefficients). The t-stat experiments showed that the coefficients for these exponential fits were significantly different between the 32 – 37 oC and the 37 – 42 oC trials. This lends support to our contention from the graphical analysis that the bacteria do not follow the same equation in the 37 – 42 oC range as in the 32 – 37 oC range.
There are several sources of the error that should be mentioned. The two quantifiable sources of error were the measurement of time and absorbance. It was estimated that the percent error in time was 6%, and that Absorbance was .005. The error in the absorbance corresponded to a .095e7 error in the concentration measurements, as per Equation 5. This equation equates the percent error in absorbance with the percent error in the proportionality constant (B) plus the percent error in the concentration ().
(Absorbance/Absorbance) = (B/B) + (/)Eq. 5
However, the error in concentration was insignificant in comparison to the error in time when determining the error in growth rate constant, as illustrated in Equation 6, which compares the percent error in with the percent error in time and concentration.
(/) = (T/T) + (t/t)* (Ln(t/o))-1Eq. 6
Because the error in concentration is multiplied by such a small fraction, it is insignificant in determining error in . Thus the percent error in was considered to be 6%. The percent error in the doubling time was also 6%. The random error in the system was accounted for in Excel’s regression function. The result of this error is the 95% confidence limit.
The experimenters noticed that after the “zero” cuvette (the cuvette used to zero the spectrometer) was filled with water, gas bubbles would form in the water. It is unknown how long this occurred, but we believe that this phenomenon was limited to the last week of the experiment (37oC- 32oC). These gas bubbles would have changed the amount of light that was absorbed by the cuvette, and would have therefore affected the absorbance that was recorded. The experimenters attempted to correct for this phenomenon by changing the water in the zero cuvette after every two trials. Because the water was all from the same source, it was estimated that the error involved in changing said water was less than the error caused by the bubbling.