Strongly Deterministic Population Dynamics in Closed Microbial Communities

Biological systems are influenced by random processes at all scales, including molecular, demographic, and behavioral fluctuations, as well as by their interactions with a fluctuating environment. We previously established microbial closed ecosystems (CES) as model systems for studying the role of random events and the emergent statistical laws governing population dynamics. Here, we present long-term measurements of population dynamics using replicate digital holographic microscopes that maintain CES under precisely controlled external conditions while automatically measuring abundances of three microbial species via single-cell imaging. With this system, we measure spatiotemporal population dynamics in more than 60 replicate CES over periods of months. In contrast to previous studies, we observe strongly deterministic population dynamics in replicate systems. Furthermore, we show that previously discovered statistical structure in abundance fluctuations across replicate CES is driven by variation in external conditions, such as illumination. In particular, we confirm the existence of stable ecomodes governing the correlations in population abundances of three species. The observation of strongly deterministic dynamics, together with stable structure of correlations in response to external perturbations, points towards a possibility of simple macroscopic laws governing microbial systems despite numerous stochastic events present on microscopic levels.

Popular Summary

If it were possible to seed microbial life on several identical exoplanets and return to those same planets later, would one find the same outcomes? Or should one expect that life on the planets would be very different because of the influence of random, historical processes? We perform such an experiment in the laboratory with ten identical communities, each composed of three microbial species and kept in identical external conditions for several months. We find that each replicate exhibits nearly identical dynamics in the species abundances over space and time. We also find that despite very complex interactions occurring between thousands of microbes living in each container, all systems respond to variation in external conditions in a simple way.

We place three microbial species that are not typically found together in nature—a green alga, a bacterium, and a ciliate—together in ten small containers. We carefully control the external conditions (e.g., illumination and temperature) and use noninvasive digital holography to resolve many individuals of each species every few minutes for several months. The resulting data set represents nearly 10 microscope-years of millions of single cells, including their sizes and spatial locations in three dimensions. Our analysis of this rich data set shows that despite many sources of randomness, including genetic and morphological changes, birth, and death, the dynamics of population abundance and the spatial distribution of each species are nearly identical from one replicate to the next. Furthermore, when we change the external conditions, the systems respond identically, with a strong pattern in the response of the three species that we call an “ecomode.”

We propose that the regularities that we observe in these systems arise in spite of randomness as a result of basic macroscopic laws. We anticipate that our work will stimulate further study to uncover these laws.

Figure 1

Closed ecosystem population dynamics by digital holography. (a) Imaging schematic: A red laser beam diverges from a focus at the surface of a gradient-index (GRIN) lens (cylinder on lower right) and illuminates a region within the cuvette containing the microbial ecosystem, forming an interference pattern or hologram, which is recorded by a camera sensor (rectangle, left). Arrows above and below the cuvette indicate the direction of homogeneous white illumination provided by LEDs. (b) A single plane, near the wall of the cuvette facing the camera sensor, is reconstructed computationally from the hologram in (a). Hundreds of such planes form a 3D image of the observed volume. Gray scale represents intensity. The scale bar is $200\mu \mathrm{m}$. Many T. thermophila and some C. reinhardtii are visible. E. coli are not readily distinguished by eye in this image (see Appendix pp2). Inset: Magnified images of, from left to right, C. reinhardtii, E. coli, and T. thermophila at the same magnification and contrast. The scale bar is $10\mu \mathrm{m}$. (c) Population dynamics from previous work [13], measured by fluorescence microscopy, in 24 replicate ecosystems. The curve indicates the geometric mean abundance across replicates, and the shaded region includes 1 geometric standard deviation above and below the geometric mean. Vertical black dashes along the bottom indicate measurement times. Green ($A$—algae, right axis): C. reinhardtii; red ($B$—bacteria): E. coli; blue ($C$—ciliates): T. thermophila. (d) Population dynamics measured by digital holography, in 10 replicate ecosystems with $I=I_0$ (1200 lx) constant illumination. Species abundances were measured every 400 s. A gap occurred between 1.1 and 2.3 days, when abundances were too high to measure. The horizontal dashes on the right indicate the offset applied to zero counts to avoid divergence of log transforms. The individual time series used to calculate these statistics are shown in Fig. 2. (e) Geometric mean and standard deviation for 10 replicates in $I=0.25I_0$ illumination [see individual time series in Fig. 2]. Note that for (c–e) the vertical axis on the left of panel (c) applies to $N_B$ and $N_C$ for all three panels, while the vertical axis on the right of panel (e) applies to $N_A$ in all three panels.

Figure 2

Population time series and phase portraits at low and high illuminations. (a) Time series of population dynamics for 10 replicate ecosystems at $I=0.25I_0$ illuminance. Green ($A$—algae, right axis): C. reinhardtii; red ($B$—bacteria): E. coli; blue ($C$—ciliate): T. thermophila. (b) Time series for 10 replicates at $I=I_0$, as in (a). (c) Population dynamics are plotted as trajectories in the space of one species abundance versus another, for two illumination conditions: left column, $I=0.25I_0$ [low illumination, 10 replicates, Figs. 1 and 2]; right column, $I=I_0$ [high illumination, 10 replicates, Figs. 1 and 2]; top row, E. coli vs C. reinhardtii; center row, T. thermophila vs C. reinhardtii; bottom row, E. coli vs T. thermophila. Gray curve, geometric mean trajectory over replicates; white region, 68% confidence interval of logarithmic abundance over replicates; blue curve, trajectory for a single replicate; red ellipse, 95% confidence interval for initial inoculation abundance; numbered points, timing along trajectory in weeks, starting at the beginning of week 1. The typical time between inoculation and the first labeled point was 20 h.

Figure 3

Morphological dynamics at low illumination. (a) An example of morphological dynamics from a single replicate at low illumination ($I=0.25I_0$). Upper panel: $N_C(t)$. Lower panel: The fraction of cells at each time point that are classified as large (${\pi }_2$, see Appendix pp2-s6). The times ${\tau }_p$ and ${\tau }_m$ are defined as shown by the arrows (see text for details). (b) Scatter plot of ${\tau }_p$ versus ${\tau }_m$ as defined in (a) for 7 CES in the $I=0.25I_0$ condition. ${\tau }_p$ can only be defined for seven of ten replicates in which the peak in $N_C(t)$ was observed. For the remaining three replicates, the large values of ${\tau }_m$ (not shown) suggest that the peak in $N_C(t)$ would have occurred at times after data acquisition terminated. Error bars in ${\tau }_m$ were computed by varying the threshold applied to ${\pi }_2$ by $\pm 25\%$. The correlation coefficient between ${\tau }_m$ and ${\tau }_p$ is 0.90 ($p=0.006$).

Figure 4

Spatial distributions of microbes in the imaging volume. Using data from the ten replicates in the $I=I_0$ illumination condition [Figs. 1 and 2], the time-averaged spatial density $g_s^{(i)}(x,y,z)$ of each species $s$ and replicate $i$ was calculated as described in Appendix pp2-s4. The mean $⟨g_s(x,y,z)⟩$ and coefficient of variation, $\sigma [g_s(x,y,z)]/⟨g_s(x,y,z)⟩$, of the time-averaged spatial density were calculated over replicates. These three-dimensional statistics of spatial density were sliced along five planes of constant $y$, as shown by the transparent red slices in the central schematic view of the monitored volume [see Fig. 1], and plotted as heat maps within each slice (left panel, normalized mean; right panel, coefficient of variation). Within each panel, species are sorted by column [left column, C. reinhardtii ($A$); center column, E. coli ($B$); right column, T. thermophila ($C$)]. The normalized mean ${\gamma }_s⟨g_s(x,y,z)⟩$ is shown, where ${\gamma }_s^{-1}={\max }_{x,y,z}⟨g_s(x,y,z)⟩$.

Figure 5

Structure of fluctuations across ecosystems in a range of conditions. (a) Population dynamics, as in Fig. 1, for 63 replicate ecosystems maintained at 7 illumination conditions between $I=0.2I_0$ and $I=I_0$. The geometric mean (bold curves) and standard deviation (shaded regions) are computed across all 63 replicates and 7 illumination conditions. Data before 9 days are missing in some conditions because of overly high cell abundance. (b) Dynamics of the correlation matrix: The correlation matrix $C(t)$ of logarithmic abundances of the three species was calculated for the data shown in (a). The eigenvalues of $C(t)$ were sorted by size and are plotted as time series in the top panel: $L$ (large, black line); $M$ (medium, cyan line); $S$ (small, magenta line). The composition of the corresponding eigenvectors is plotted in the bottom three panels. Green: C. reinhardtii ($A$), red: E. coli ($B$), blue: T. thermophila ($C$).

Figure 6

Effect of pH buffer. Time series for nine CES measured by holography with the Tris-acetate pH buffer omitted from the medium. Geometric mean and standard deviation are shown as in Figs. 1.

Figure 7

Final viability and total algal density. Top four panels: Viability of each species, measured as described above, as a function of illumination. Green, red, and blue symbols show the fraction of systems with $A$, $B$, and $C$ viable, respectively. Black squares show the fraction of replicates where all three species are viable. The two experiments in each illumination condition are represented by circle and square symbols. Bottom panel: Density of C. reinhardtii, as measured by flow cytometry, from a well-mixed sample of each replicate, at the end of each experiment as a function of illumination condition. A single point is plotted for each replicate with symbols denoting different experiments within each illumination condition.

Figure 8

Signal subtraction detail. (a) Focal plane of reconstructed T. thermophila, to the upper left. (b) Same T. thermophila from (a), $290\mu \mathrm{m}$ out of focus. An E. coli, indicated by a red arrow, is barely visible, to the right—see (f) and (h) for better visibility at high contrast. (c) Focal plane from (a), after the signal from the T. thermophila has been subtracted. (d) Focal plane from (b), after signal subtraction, showing unperturbed E. coli. (e)–(h) Duplicate panels (a)–(d) at high contrast. The scale bar is $20\mu \mathrm{m}$. (i) Intensity profile across the horizontal coordinate in (b) and (f), through the E. coli cell, before subtraction. The large high-intensity region around $70\mu \mathrm{m}$ is the out-of-focus T. thermophila, while the peak at $140\mu \mathrm{m}$ is the in-focus E. coli cell. (j) Intensity profile through the same row of pixels as in (i), after signal subtraction—see (d) and (h).

Figure 9

Autocorrelation of count-smoothing residuals. The autocorrelation function of the residual from the count-smoothing procedure, ${\delta }_i=\log a_i-w_i$, was calculated for one of the replicates and each species, with lag measured in time points (400 s). The blue line shows the 95% confidence interval. The rapid decay observed here was typical for all replicates.

Figure 10

$I=0.25I_0$ spatial distributions of microbes in imaging volume. $\gamma ⟨g⟩$ and $\sigma [g]/⟨g⟩$ are defined in the caption to Fig. 4 for the $I=I_0$ condition.

Figure 11

Gaussian mixture model fit for one replicate in the $I=0.25I_0$ condition. The upper panel shows the means of the two modes (${\mu }_1$ and ${\mu }_2$), the second panel shows the weights of these modes in the mixture distribution (${\pi }_1$ and ${\pi }_2$, respectively), the third panel shows a histogram of the raw data, and the fourth panel shows the inferred probability density at each time point. In both the third and fourth panels, each column (time point) is normalized independently. The bottom panel plots the classification with large cells in red and small cells in black; note that there is no overlap between these classes.

Figure 12

Ecomodes in individual environmental conditions. Eigensystems of the correlation matrix (as in Fig. 4) for 10 CES in the low-illumination condition ($I=0.25I_0$, left column) and the high-illumination condition ($I=I_0$, right column). Eigenvalues are plotted in the upper panels, and eigenvector composition in the lower panels [as in Figs. 5 and 5, respectively].