Lower growth rates evaluated by the optical turbidity
The wild-type E. coli strain W3110 was cultured in minimal medium M63 and was temporally sampled in three different assays to measure the optical turbidity (OD600), colony formation (colony forming unit, CFU) and ATP abundance, which are based on the physical, biological and chemical properties of the growing cell population, respectively. Although these three methods are well established and widely used in experimental and theoretical studies on bacterial growth, it is unclear whether they result in a common conclusion regarding bacterial growth. The three methods of measuring the OD600, CFU and ATP represent the density of the biomass, number of living cells and cellular activity, respectively. The growth dynamics represented by the changes in these different parameters must be varied in consequence. To verify this assumption, we performed three assays in parallel on identical cell cultures. According to the growth curves generated from the OD600, CFU and ATP, the growth rates were calculated by regression toward the time sampling records during the exponential growth phase and as described in the Materials and Methods. Repeated cell cultures (N = 3~ 5) showed significant differentiation of the growth rates estimated by measuring the optical turbidity (OD), colony formation (CFU) and ATP abundance (Fig. 1). The growth rates estimated by measuring the CFU and ATP were comparable, whereas those from OD600 were lower (p < 0.05), regardless of identical population growth. The results demonstrated that the bacterial growth dynamics, which are represented by changes in optical turbidity, were different from those evaluated by changes in the number of active cells, consistent to the previous report [32]. How and what caused the differences in growth rates and whether the differences were common under other conditions were further investigated by comparing two representative methods, OD600 and CFU.
Decay effect in optical turbidity triggered the lower estimated growth rate
A decay effect was observed when the optical turbidity results were compared to the records of colony counts and OD600 values. Both the CFU assay (Fig. 2a, upper) and OD600 reads (Fig. 2a, middle) were performed in parallel to monitor the same population growth. The temporal changes in both records were supposed to be similar. Nevertheless, a temporal decrease in the ratio of the OD600 reads and CFU counts was observed (Fig. 2a, bottom). Since the decrease was initiated from the early exponential phase, the growth rates evaluated by optical turbidity (μOD) were lower than those evaluated by colony formation (μCFU) (Fig. 1b); this decrease was considered to be due to the decay effect of the OD600 values. It was carefully tested whether the decay effect was caused by limitations of the optical measurement. No significant differentiation in the optical reads were detected during the exponential growth phase (OD600 < 0.5) compared with the significant decrease in the reads of OD600 during the stationary phase (OD600 > 1) (Additional file 1: Figure S1). It was verified that the decay of the OD600/CFU ratio during the exponential phase was not attributed to mechanical errors of the optical reads but was mainly due to biological issues, such as a difference in cellular properties.
The lower estimated growth rate according to optical turbidity was also detected either under other growth conditions or in other genotypes. Wild-type E. coli cells W3110 grown under various conditions (D3, D4 and Cm described in the figure legend) also showed lower growth rates (N = 3, P < 0.05) estimated by OD600 (Fig. 2b). In addition, the growth dynamics of four different E. coli strains with reduced genomes (Strain Nos. 7, 14, 20 and 28), which were previously constructed [33] and characterized [34], were assayed by the two methods (N = 3~ 4). Similarly, the values of μOD tended to be smaller than those of μCFU (P < 0.05) regardless of the multiple deletions of genomic sequences (Fig. 2c). The results indicated that the lower growth rates estimated by optical turbidity, which is the most commonly used method in bacterial growth studies, frequently occurred in different genotypes and under various growth conditions. Collectively, a lower μOD than μCFU was found to be common, which indicated the growth associated cellular changes might occur.
A decay rate introduced into the logistic model offset the decay effect
Considering the cellular changes that accompanied growth, a modified logistic model is proposed in the present study by introducing a novel parameter, the decay rate. The theoretical revision differed from the previous study that applying a numeral calibration of the measurements [32]. The logistic equation is the most popular model that is used to represent bacterial growth dynamics. This equation is commonly applied to fit growth curves based on optical turbidity, especially in systems and computational biology to understand the dynamics of living systems. Considering the decay effect in OD600 during population growth, the logistic model was revised. A new parameter, the decay rate (d), was defined, which has the same unit of h− 1 as the growth rate.
Logistic model:
$$ N(t)=\frac{K}{1+\frac{K-N0}{N0}\times {e}^{- rt}} $$
Modified logistic model:
$$ N(t)=\frac{K}{1+\frac{K-N0}{N0}\times {e}^{-r\left(1- dt\right)t}} $$
where N0, K, r, and d represent the initial cell concentration, saturated cell concentration, growth rate, and decay rate, respectively.
To evaluate whether the modified logistic model was applicable, a large number of growth data sets were fitted to the two logistic equations. A total of 710 precisely measured growth curves from 29 different E. coli strains in either rich (Luria-Bertani broth, LB) or poor (M63) medium were subjected to the theoretical fitting. The histograms of parameter K estimated by the two models were completely overlapped (Fig. 3a). However, the histograms of the growth rate, r, were different between the two models (Fig. 3b). The mean growth rates estimated by the modified logistic model were larger and with a larger variation. The results indicated that the addition of the decay rate d in the logistic model did not influence K, the saturated population density or the carrying capacity, but contributed to the growth rate r independent of the nutritional conditions for population growth (Additional file 1: Figure S2).
Intriguingly, the modified logistic model presented improved goodness of fit compared to the original model. The residual errors, which were designated as the sum of the squared errors (SSE), of fitting the total 710 growth curves to the modified logistic model were smaller than those from the original model (Fig. 4a). Thus, the goodness of fit (R2) was better for the modified logistic model (Fig. 4b), independent of either the genotype or medium (Additional file 1: Figure S3). The results showed that the modified logistic model better explained the growth dynamics of bacterial cells, which strongly indicated that the revision of the logistic model was theoretically practical. Considering the fact that the increase in OD600 was decided not only by exponential changes in the number of cells but also by unclear changes in the cellular contents, such as the cell size and macromolecular abundance, revision of the logistic model by introducing an additional parameter of the decay rate was also biologically reasonable.
A correlation between the decay rate and the genome reduction in a nutritional dependent manner
Interestingly, a correlation between genome reduction and the decay rate was observed when applying the modified logistic model to the experimental bacterial growth data. Fitting the growth data of the reduced genomes to the modified logistic equation, as described above, resulted in the estimated constants of r and d, which are the growth and decay rates, respectively. Both the mean growth rates and mean decay rates of the repeated growth assays (N = 12~ 28) were calculated for a total of 29 strains (Nos. 0~ 28). The results showed that a decrease along with a genome reduction was detected not only in the growth rates but also in the decay rates (Fig. 5a). The larger the genome reduction, the lower the detection of decay. This finding was supported by the fact that the changes in growth rates evaluated by OD600 and CFU were smaller with the genome reduction (Fig. 5b). Fewer changes between μOD and μCFU resulted in a smaller decay effect in OD600. Reduction of the genome size might change the cellular properties that are linked to population growth.
A correlation between the growth rate and genome reduction has been previously reported according to the manual calculation [34]. The same conclusion was drawn from the theoretical fitting suggested here, and the modified logistic model was applicable. Notably, an additional conclusion of the correlation between the decay rate and genome reduction was drawn based on the modified logistic model. The Pearson correlation coefficients of the genome size to the decay rate were highly significant (cor = 0.684, p = 4e-5) and were comparable to the growth rate (cor = 0.772, p = 6e-7). This result indicated that the changing cellular features that accompanied the population growth disturbed the optical properties. Note that the decline in the decay rates of the reduced genomes was significantly detected in the cell growth in the M63 medium but not LB medium (Additional file 1: Figure S5), which indicated that the nutritional richness affected the significance of the cellular changes that contributed to the decay rate. This finding agreed with the result that the difference of the fitting goodness between the two models was slight in LB but significant in M63 (Additional file 1: Figure S3).
The decay rate was partially attributed to the changes in cell size
As the decay rate of the reduced genome was much smaller than that of the wild-type genome (Fig. 5), we investigated whether any biological features could be identified to explain the decrease in the decay rate. As the growth rate is often discussed with cell size [35] and the genome reduction was found to somehow contribute to the change in cell size [34], growth accompanying changes in cell size might be linked to the decay rate. To verify the assumption, cells carrying either the wild-type (No. 0) or the reduced (No. 28) genomes were cultured in M63 medium, and the size distributions at a varied population density were analyzed. Since the cell size of an identical population often showed a large variation (Fig. 6a), the median of the cell size of the population was used as a representative parameter [36]. The results showed that the median cell size was smaller compared to the increase in population density (Fig. 6b, triangle). The feature of a higher population density linked to a smaller cell size was clearly identified in the wild-type genome, but not in the reduced genome (Fig. 6b, circle). This result suggests that the population density dependent size effect became insignificant due to genome reduction, consistent with the findings of the decay rate, and significantly declined in the reduced genome (Fig. 5). Thus, cell size changes of the growing population might be one of the reasons that cause the decay effect in OD600. Population density dependent changes in cell size were supposed to happen during the stationary phase due to nutritional depletion [35, 37]; however, the results showed that the changes in cell size also occurred even in the exponential phase. This result agrees well with the temporal decrease in the ratio of OD600/CFU from the exponential phase (Fig. 2a).