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. Author manuscript; available in PMC: 2022 Dec 1.
Published in final edited form as: IEEE Control Syst Lett. 2020 Dec 23;5(6):1952–1957. doi: 10.1109/lcsys.2020.3046612

A Predictive Reaction-diffusion Based Model of E.coli Colony Growth Control

Changhan He 1,†, Samat Bayakhmetov 2,†, Duane Harris 1, Yang Kuang 1,‡, Xiao Wang 2,‡
PMCID: PMC8021091  NIHMSID: NIHMS1683703  PMID: 33829120

Abstract

Bacterial colony formations exhibit diverse morphologies and dynamics. A mechanistic understanding of this process has broad implications to ecology and medicine. However, many control factors and their impacts on colony formation remain underexplored. Here we propose a reaction-diffusion based dynamic model to quantitatively describe cell division and colony expansion, where control factors of colony spreading take the form of nonlinear density-dependent function and the intercellular impacts take the form of density-dependent hill function. We validate the model using experimental E. coli colony growth data and our results show that the model is capable of predicting the whole colony expansion process in both time and space under different conditions. Furthermore, the nonlinear control factors can predict colony morphology at both center and edge of the colony.

Index Terms—: Synthetic biology, bacterial colony expansion, partial differential equations, diffusion

I. Introduction

THE process of bacterial colony formation provides an excellent experimental platform to study a broad range of biological phenomena such as biofilm [1], multicellular interactions [2], and pattern formation [3]. All of these processes are intimately intertwined with bacterial growth, so its control is an essential tool in regulating other processes within the colony. Formation of bacterial colony cannot be simply regarded as the outcome of cell reproduction. The mechanisms behind it can be rather complicated and several recent works have provided us with valuable insights. For instance, Hwa et al. [4] thoroughly explained all of the forces and first to provide quantitative data on key elements that affect spatiotemporal establishment of bacterial colonies on hard agar. They also were the first to emphasize the role of surface tension in radial cell expansion. In another study, Syu et al. [5] points out that the balance in interplays of the forces have a profound impact on the overall morphology of colony.

Although the morphology of bacterial colony can be clearly observed in the experiments and certain control factors, including temperature and nutrient concentrations have been proposed [2],[3],[4],[5], determining the key factors and study the mechanisms behind them remain challenging. During the past two decades, many mathematical models were developed to describe bacterial colony growth as well as pattern formation, and most of them were established based on Fisher’s equation [6],[25],[27]. In 2000, Mimura et al. [6] modeled different bacterial colony expansion with reaction-diffusion equations. And later in 2013, Leyva et al. [27] proposed a model with non-linear degenerate cross diffusion to estimate the growth velocity and generation of colony pattern. However, although these models were able to describe certain aspects of bacterial colony growth on solid media, they still have limitations and could not fully capture key temporal characteristics of colony growth. For example, a bacterial colony does not grow at a constant speed and may contradict with the assumption in [6]. on the other hand, cell reproduction may not always obtain the highest rate at the beginning when the colony has the lowest cell density. This implies that modeling cell reproduction rate as a monotonic function of cell density may not always be true [27]. The cause of a colony’s physical profile is rather complicated. There are a lot of important factors acting during colony growth such as nutrient concentration, agar dehydration and temperatures. Therefore, establishing a comprehensive and experimentally validated predictive mathematical model with multiple control factors taken into account becomes necessary. Some recent works can bring us valuable insights on developing control models, which describes the dynamics of bacterial colony growth under different control factors. For example, a Hill-function-based model was introduced by Vecchio et al. [7] describing the interactions between resource competition and gene circuits expression. Recently, Murray et al. [8] presented a comprehensive review of control designing and modeling.

In this paper, we proposed a novel reaction-diffusion based mathematical model with nonlinear density-dependent diffusion rate and growth rate to describe the colony growth under different control strategies. We also designed corresponding experiments of E.coli colony growth on semi-solid agar under different conditions. The model is capable of describing non-constant diffusion of the cells as well as the non-monotonic relationship between cell reproduction rate and cell density. Using this dynamic model, we not only compared the quantitative results with experimental data under different control strategies, but also examined how the model performs on capturing the colony side spatial profile. We concluded that the non-constant growth of E.coli colony results from multiple control factors can be well-described by nonlinear density dependent functions which are significant on simulating colony spatial profiles.

II. Results

A. General Results

To describe the morphology and dynamics of a growing bacterial colony, we focus on its radial and vertical growth as illustrated by schematics in Fig. 1A. After seeding, the colony immediately begins to grow, eventually taking on a circular shape when viewed from above and resembling a semielliptical arch when viewed from the side. Microscopy images (Fig. 1B) show that a clear circular pattern can usually be observed 24 hours after seeding. From here, a colony continues to grow in size while maintaining its circular shape, which is consistent with previous findings [4]. We observe that a colony usually has a grey edge and a dark core in the earliest images, which are taken at the 24th hour. As the colony grows, the grey edge becomes barely visible as the size of the dark core increases. The strong contrast between the colonies and the agar surfaces indicates that the colonies have sharp edges. A MATLAB program was created to detect the colony edge (Fig. 1B) using these phase contrast images. The visible threshold was setup based on the background noise and then the colony center was found by fitting the detected colony edge with a circle. This provides an efficient and accurate method to measure its size. The detected edge is used to find the best fitting circle which provides the radius value and center location of the colony. Furthermore, to determine the influence of the two control strategies, we performed the following quantitative analysis. Our control strategies are tuning of culture temperature (T) and amino acid concentration (AA) in the growth media, respectively. Based on earlier findings [16],[17],[18], both factors are modeled to be directly proportional to the growth rate and thus can be represented by a1 and μ in (4) in our model (see section III for model derivation details).

Fig. 1.

Fig. 1.

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Schematics and experimental observations of the growth and morphology of a bacterial colony. (A) Schematics of a bacterial colony growth and its cross-sectional profile. Cells are indicated by small green bars and black circles indicate the edges of an agar plate. Left column: cross-sectional profiles of abacterial colony at various time. (B) Phase contrast images of an E.coli colony growing on a semi-solid agar plate taken every 24 hours after seeding (t=0). Darker color implies higher cell density and the pink trace at 96h colony indicates the edge detected by computer program.

B. Growth at Different Temperatures

The first set of experiments were performed at two different T, 30°C and 37°C. Each colony was inoculated as a single cell growing on a semi-solid agar plate for up to 5 days. The clear circular patterns could be observed after 24 hours, so the first data point of this experiment is t=24h and experimental data was collected every 24 hours. Four colonies were recorded at each T, and the average radii of all colonies were computed at every time point for the purposes of data fitting. We set the initial radius for each colony smaller than its first data point and fit the experimental data with the new model by varying the parameters a1 and μ, in (4) via MATLAB. For comparison, we also fit the experimental data with Fisher’s equation by varying the parameters D and μ, in (1) via MATLAB. Despite the simplicity of Fisher’s equation, it still maintains the linear growth profile and is an appropriate example for comparison. These results are shown in Fig. 2. At 37°C, the colony radius shares a non-constant increasing speed while the colony radius increases at an approximately constant speed at 30°C. At each phase, the colonies at 37°C always have higher colony expansion rates than colonies at 30°C. Here, our model provides comparable fitting results and captures both the non-constant increase of the colony radius at 37°C and constant increase at 30°C. This is because under lower T, bacterial colony will grow slower and have more linear growth profile [26]. Note that the Fisher’s equation can only capture the constant increase profile in low T case, but has much higher fitting errors in high T cases. The insert box of Figure 2 presents a comparison of the simulations and experimental data for the 37°C case at time t=48h, showing that the experimental results are stable and the simulation results are reasonably accurate.

Fig. 2.

Fig. 2.

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Fitting results of two models under different T. Y-axis represents colony radius in micrometer, while X-axis is time points at which colony was imaged, starting from 24h to 96h. Red and blue colors represent low (30°C), and normal (37°C) T, respectively. Blue and red curves represent simulated results of the new model proposed in this paper, dashed lines show the results of Fisher’s equation, the dots are averaged experimental data while error bars indicate the standard error between exact data and their averages. In the box, blue circle is the fitted simulation, while purple circles are the circumference of the colonies taken from the raw images (Fig. 1B, 96h). Fitting errors for the new model are: 86.28 (30°C), 105.68 (37°C). Fitting errors for Fisher’s equation are: 53.42 (30°C), 288.76 (37°C).

C. Growth at Different Amino Acid Concentrations

The second set of experiments was performed with two different AA, 0.01% and 0.1%. Each colony was inoculated as a single cell growing on semi-solid agar plate for up to 4 days. In both cases, the first data point was at t=24h and the experimental data was collected every 24 hours. There were six colonies for each case and the average radius of all colonies for each case was computed at each time point for the purposes of data fitting. similar simulations to section II B were performed via MATLAB and are shown in Fig. 3.

Fig. 3.

Fig. 3.

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Fitting results of two models with different AA. Y-axis is colony radius in micrometer, while X-axis is time points at which colony was imaged, starting from 24h to 96h. Red and blue colors represent normal (0.01%), and high (0.1%) AA, respectively. Blue and red curves represent simulated results of the new model proposed in this paper, dashed lines show the results of Fisher’s equation, the dots are averaged experimental data while error bars indicate the standard error between exact data and their averages. Fitting errors for the new model are: 24.23 (0.01%), 88.22 (0.1%). Fitting errors for Fisher’s equation are: 223.92 (0.01%), 301.83 (0.1%).

Surprisingly, colonies with higher AA always have a lower colony expansion rate when compared at the same time period to colonies with lower AA. The key reason for this could be that colonies with higher AA grow much faster in the first 24 hours, which leads to a larger colony size and a higher cell density, which subsequently slows down the growth. Based on the experimental data, AA may only change the speed at which an E.coli colony grows. Thus, both cases show non-constant increasing speed of the colony radius. Our model captures this profile and presents comparable fitting results with a much smaller fitting error than Fisher’s equation, which only presents a linear increase of the colony radius.

D. Analysis of E. coli Colony Spatial Profile

In addition to radial colony growth, we also quantitatively analyzed the spatial profile of the colonies. We estimated the cell density by using the grey scale data of a phase contrast image. since cell densities can be viewed as the colony height (from colony surface to agar plate), the grey scale data can be used as a reporter of experimental density, so plotting the grey scale data ranging from the colony center to the edge can be regarded as an approximation of the spatial profile. To minimize the background noise, we exported the grey scale data ranging from the center to the leftward edge and flip the image along the vertical axis. In this paper, we have chosen one of the colonies growing at 30°C and have plotted its profiles at t=72h and t=96h. Note that due to different noise levels, the threshold for the colony edge may vary. The corresponding simulation results of (4) are also presented. Note that rather than aiming at predicting the exact shape of the colony edge, we are focusing on developing a framework to capture the general shape of the colony boundary. since the exact profile of a bacterial colony may vary in different experiments, the exact fitting of the boundary would likely require much more detailed modeling of the internal mechanics of colony formation and is left for future explorations.

III. Materials and Methods

A. Experimental Set-up

For the experiments, we took a K12 MG1655 E.coli strain – with lac −/−(Invitrogen). The cells were first cultured overnight in a 5mL liquid Luria-Bertani (LB) broth medium with 100μg/mL ampicillin in 15 mL tubes in a shaker-incubator at 220 rotations per minute (rpm), at 37°C. Then, in order to observe a clear image, we prepared 0.3% semi-solid agarose (G-Bio sciences) supplemented with minimal salt medium (M9), 100%g/mL ampicillin, 0.45% glucose, 1mM MgSO4, 100μM CaCl2, and 0.01%g/mL amino acid mixture, which was then transferred into 5cm plates. The original culture was diluted to 5×105 ~5×106 fold with premix of plates contents, but without agarose and 5μL dilution were evenly plated on to 5cm plate. The cells were grown in the incubator at 37°C, with an extra tray filled with water to decrease the dehydration from the plates. Then, in order to slow down the growth rate, T was decreased to 30°C, whereas to accelerate it AA was increased to 0.1g/ml, respectively.

B. Imaging and Microscopy

Images were taken starting from 24 hours, if visible, and then every 24 hours, using Nikon Eclipse Ti inverted microscope (Nikon, Japan) at. The objective lenses were set at 2×/0.10, OFN25, WD8.5, and eye piece lenses at CFI 10×/22. For taking the image Nikon’s NIS-software was used. Exposure times were kept the same during the time course experiments. Bright field exposure was set up for 4ms as default with DIA brightness 2.7. Exposure time were normalized to analyze the time course results. Brightness and contrast were slightly tuned for presentation of images in the research. About 4~5 individual colonies were imaged from each plate and repeated 3 times on different days.

C. Model Derivation

The growth of E.coli colonies on homogenous semi-solid agar consists of reproduction and expansion, which always presents circular patterns and can be systematically modeled by one dimensional reaction-diffusion equations. Many mathematical models were developed upon Fisher’s equation during the past two decades, which takes the following form:

∂N∂t=∂∂x(D∂N∂x)+μ(1−NK)N (1)

where N=N(x, t) is the cell density at time t and spatial position x (distance to colony center), which can also be regarded as the colony height. D is diffusion coefficient which describe the speed of colony expansion, μ is the maximum reproduction rate and K is the maximum cell density.

However, most of previous works assumed that cells diffused at a constant rate D or cell reproduction rate was a monotone function of cell density [6],[25],[27], which are not consistent with experimental observations. In fact, the bacteria colony growth is controlled by heterogeneous factors coming from a cell’s metabolism and external environment. Recent research shows that cell reproduction can be affected by nutrient concentration [9] and acetate accumulation [10], while colony expansion on the agar surface can be slowed down by physical friction [4]. Based on experimental observations, an E.coli colony grows at a non-constant speed, which is consistent with previous findings that colony expansion of mobile cells is density dependent [11],[12]. Although E.coli on agar are not mobile, they still share similar growth characteristics. In general, a colony grows fastest in the first 24 hours after it forms a circular pattern, which implies that treating cell diffusion rates as constant may be an oversimplification and misleading, since constant diffusion rate will lead to linear colony growth profile (Fig. 5A dashed lines). Therefore, presenting the diffusion rate as a function of cell density N becomes reasonable, noticing that density-dependent diffusion has been introduced and validated in relevant fields such as modeling Glioblastoma growth [13] and modeling E. coli growth [11]. Here we describe the diffusion rate using a nonlinear density-dependent function:

D(N)=a1NNm+1 (2)

with the assumption m>1 which ensures the diffusion is slow when the cell density is either too low or too high. With the non-linear density dependent diffusion, the model can capture the growth profile of bacteria colony (Fig. 5A solid lines).

Fig. 5.

Fig. 5.

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(A) Comparison of Fisher’s equation and our new model via simulations. Y-axis represents colony radius in micrometer, while X-axis is time points starting from 0h to 96h. The dashed curves show the change of a colony radius at various times with different diffusion rate D. The solid curves represent the change of a colony radius at various times with different values of a1. (B) Normalized colony volumes under different experimental conditions at various times. Y-axis represents normalized colony volumes computed from experimental data, while X-axis is time points at which colony was imaged, starting from 0h to 96h. One colony copy was randomly picked in each set of experiment, and the normalized colony volume was computed based on the grey scale.

Meanwhile, the grey scale data of a phase contrast image can be used to estimate the cell density (Fig. 4). With assumptions that the whole colony surface as a calotte and the distribution of the cell inside colony is uniform [4], we can compare the total population of the cell by calculating the area of colony’s half-side profile. Furthermore, we can estimate the relative reproduction rate of a bacteria colony at different time points. Based on the experimental data, we found out that the total population of the cell shared similar profile of the colony size, as it grew slowly during the first 24 hours after seeding, then reached a much higher growth rate in the next 24 hours (Fig. 5B). Such profile indicates that the reproduction rate may not be a monotone function of cell density N.

Fig. 4.

Fig. 4.

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Comparison of quantified temporal and spatial grey scale of experimental observations and simulated cross-sectional colony profiles. Dashed curves represent the grey scale of a bacterial colony (T: 30° C) starting from colony center (detected by computer program) at different times. The simulated colony profiles starting from colony center of corresponding time are shown in solid curves. Black dashed lines indicate the location of colony edges and grey arrows indicated the colony spreading direction.

Based on above analysis, we modified the Logistic growth by considering a density-dependent inhibition described by Michaelis-Menten kinetics, which takes the following form:

F(N)=NK1+N (3)

where K1 is the Michaelis constant. Note that this inhibition effect is strong when cell density is low and weak when cell density is high, this profile shows that such inhibition mainly affects at early stage of a bacterial colony growth. Combining the two regulations, our final model takes the following form:

∂N∂t=∂∂x(D(N)∂N∂x)+μF(N)(1−NK2)N (4)

where the first term on the right side of (4) describes cell diffusion as well as colony expansion, with a density-dependent diffusion rate; the second term modeled cell growth with a non-monotonic growth rate about the density N.

D. Parameter Estimations

The two control factors we focus are T and AA. It is clear that the E. coli cell will obtain both higher diffusion rate and higher reproduction rate if growing under higher T or AA, which will result in higher values of a1 and μ in our model. Therefore a1 and μ will be the fitting parameters.

To find the values of a1 and μ that best fit the observed radii data, we use the simplex method from Lagarias et al. [14], which is implemented in MATLAB (version 2018b) function fminsearch. The objective function begins by performing a simulation assuming radial symmetry on an infinite domain. The initial condition is created by looking at the initial observed colony radius. Any spatial nodes that are closer to the colony center than this initial radius receive a cell density value equal to the carrying capacity, while all other spatial nodes are set to zero. Next, the colony radius generated by the model is found by taking the radius of the farthest spatial node from the center that has a cell density greater than the visibility threshold, equal to 0.03 in this case. Finally, the error is calculated using the least squares formula given by:

Error=12∑i=1M(ri−r^i)2 (5)

where M is the total number of data points, ti is the time of the ith data point. Here ri represent the radii values at time ti produced from the model, and r^i correspond to the experimental observations at time ti.

Based on the experimental observations, the cell density at colony center is stable at the end of experiment, even under different conditions. This implies that the maximum cell density K2 can be regarded as a constant. Moreover, since K1 is the Michaelis constant and m determines the profile of function D(N), they are all fixed in the simulations. Parameters m=2.0, K1=63.0, and K2=189.0 are fixed.

IV. Discussion

In this paper, we proposed a partial differential equation model describing the growth of E.coli colonies on semi-solid agar starting from a single cell with multiple control factors. For each set of conditions, we designed a corresponding experiment and compared its outcome with the model simulations. E.coli are common bacteria that can reproduce on semi-solid agar surfaces and after a certain period of time, form a circular colony. They can also propel themselves by means of long hair-like appendages known as flagella. Our experiments show that as E.coli colonies grow, their profiles evolve in a qualitatively similar way despite being subjected to differing T or AA. in the first 24 hours, the radial growth is slow in comparison to the vertical growth. However, over the course of the next 48 to 72 hours, the radial growth speeds up before it slows down. The key factors that control colony growth include physical friction among cells and between the cells and the agar surface, and inhibition of the cell reproduction potentially caused by agar dehydration. Also, cell colonies present similar growth profiles throughout the experiments, but there are notable differences in colony radius. However, when changing the AA, the final sizes of all colonies were very close. This phenomenon might imply that T affects both the growth speed and final size of a colony, while nutrient concentration mainly affects the growth speed. Meanwhile, it is necessary to point out that although T and AA are independent variables, the mechanisms of how they will affect cell’s metabolism and movement are still unclear. Fortunately, the outcomes of tuning these variables are predictable and can be described via cell density. The density-dependent functions presented in our reaction-diffusion based model can be a possible approach. Furthermore, our model can not only capture these characteristics but also produce simulation results comparable to the experimental data, which shows that the interactions of these control factors can be well-described by nonlinear density dependent functions. Moreover, the model can provide comparable simulation results on a colony’s spatial (cross-sectional) profile at various times.

To focus only on radial expansion, it is worth noting that here we have omitted vertical growth because recent studies have shown that these two processes are independent [15]. Moreover, the forces that govern each propagation factor are distinct, and can be incorporated separately. Key parameters that were tested are AA and T [16],[17]. These factors are tunable and have dramatic effects on colony expansion [18]. The default settings for T and AA were 37°C and 0.01%g/mL respectively. By varying these key parameters, we were able to either accelerate or decelerate the growth rate. specifically, we found that decreasing the T to 30°C doubles the time it takes for the colony to proliferate. On the other hand, increasing AA by a factor of 10 leads to an escalated initial growth rate (Fig. 2D). After 24 hours, we see that the colony with a high AA has a diameter that is twice as large as the control. However, this difference diminishes as both colonies look similar in size by the end of 4th day. There are many mechanisms contributing to these visible changes, including nutrient availability, cell-agar friction, surface tension and adhesion, as well as dehydration [18],[19],[20],[21]. Most of these points were taken into consideration while designing our experiments. For instance, to decrease mechanical forces between agar and cells, we prepared a semi-solid agar with 0.3% agarose by weight, so that the colony could grow with less friction and adhesion, but still maintains a proper shape. Also, water tanks were placed inside the incubators to minimize evaporation from the surfaces of the plates.

The model we propose here, which produces quantitative results that are comparable to the experimental data, is not only a reasonable example of integrating E.coli colony expansion with multiple control factors, but also a potentially powerful tool that can be used to help develop strategies for controlling colony growth under different conditions and for interpreting the mechanisms of other biological phenomena such as spatial pattern formation. In future studies, parameters for the experimental variables (T and AA) in the model could be included explicitly, and impacts to other density-dependent factors besides the maximum growth rate and diffusion rate could be considered, both would require additional experimental validation. With consideration of more complex effects of cell growth and metabolism as well as environmental factors, the model can be used to depict both the radial and vertical growth of other microbe colonies. This will provide valuable mathematical insight and help improve experimental designs. Furthermore, this model can be a framework to systematically present spatial pattern formation. The growth of a bacterial colony will not only interact with circuit expression [22],[23] and pattern formation [3],[24], but will also control the pattern boundary. Developing a mechanistic model that builds on this colony growth model can be an interesting way to study the variable boundary problem seen in pattern formation.

Acknowledgments

Research supported by NIH grant 5R01GM131405.

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