The authors have constructed a linear array of coupled, microscale patches of …

• Abstract
• Results
• Discussion
• Conclusions
• Methods
• Acknowledgements
• References
• Figures

Biology Articles » Biophysics » Bacterial metapopulations in nanofabricated landscapes » Results

Results

- Bacterial metapopulations in nanofabricated landscapes

Fig. 3A shows the dynamics of an E. coli population in a single MHP [zero-dimensional (0D) device described in Fig. 2D] with all nanoslits open (_max). Although initially the density of bacteria follows a pattern of exponential growth, the weak coupling to an external resource (via ) results in some unusual behavior: oscillations occur at least at two distinctive frequencies (high and low).

A simple analysis of diffusion through the nanoslit to the MHP gives us an expression for the contribution of a single nanoslit to the exchange rate between the MHP and the feeding channels:

where A is the nanoslit's total cross-sectional area, V is the MHP's volume, l is the length of the nanoslit, and D_w is the average diffusion coefficient of resources and waste. From the volume of a MHP (V = 3 x 10⁵ µm³), the approximate diffusion constant of small molecules such as amino acids (D_w 10^–5 cm²·s^–1), the area of a 200-nm-deep and 20-µm-long nanoslit (4 µm²), and the width of the nanoslit (l = 15 µm), we find that * 10^–3·s^–1.

The population density (t) of an MHP can be modeled by the logistic equation (16),

Here, the per capita growth rate is determined by two factors: space and resources. Space limitation is represented as the logistic (1 – /K) environmental resistance (17), where the parameter K represents the carrying capacity of the MHP. Resource-based growth rate r(w) is a function of habitat quality 0 w 1 inside the MHP but relative to the concentration of resources in the feeding channels. Without these resources, cells cannot grow. Thus, following resource competition theory (18), we use

as our resource utilization function. Here, 1/_m represents the rate of cell death and 1/_r represents the birth rate achieved when the medium inside the MHP is fresh LB (w = 1). After the biomass of the cells starts growing and transforming the medium, w decreases. Waste-saturated medium means w = 0.

The feeder channels supply the MHP with fresh LB by -limited diffusion into (and waste out of) it through its nanoslits. The rate dw/dt at which resource quality changes inside the MHP is then the difference between inward diffusion and consumption by E. coli, normalized by the efficiency by which resources are converted into bacteria. Thus,

From the known volume of a MHP (V) and the approximate volume of a single E. coli (0.5 µm³) we can estimate (an upper limit) that a close-packed MHP can hold about 10⁶E. coli cells. In practice, however, a MHP typically saturates at about K* 10⁴E. coli cells.

The chief concern of our 0D theory is to understand the effects of space and resource limitation upon bacterial growth (Fig. 3A). There are three (see Supporting Appendix) scenarios: (i) an extinction solution (Fig. 3B Left), with the equilibrium values = 0 and = w(0); (ii) a resource-limited solution (Fig. 3B Center), where = x ([1/w*] – 1)/ and = w*; and (iii) a space-limited solution (Fig. 3B Right), where = 1 and = 1/(1 + [/]). Here, life-history parameter w* = _r/_m (dimensionless) corresponds to the total number of cell divisions in a cell's lifespan. Thus, a given strategy [, w*] can have a totally different fate, depending on the value of habitat parameter nanostructured onto a given MHP. On the other hand, for a fixed environment , organisms can adapt by changing their strategy [, w*].

When our microecosystem is in a regime of resource limitation, the picture goes like this; as food resources are depleted, Eqs. 1 and 2 predict that the growth rate w/_r will become less than the death rate 1/_m, and the population in the MHP will start going extinct. However, resources can diffuse in from the feeder channels and growth can reinitiate; this can give rise to oscillations in the population density due to the diffusional lag between consumption and supply. Although high-frequency spikes and lower-frequency bumps exist, our model cannot accommodate both at the same time. Only a single frequency basic oscillation for population density vs. time is expected from our model. For a fixed environment , the frequency of the oscillation is determined by an organisms life-history strategy [, _r, _m]. A consortium of phenotypes would be expected to exhibit more frequencies.

Metapopulation Dynamics in a Flat Landscape. The next step up in complexity is a 1D array of equivalent MHPs, which we call a flat landscape. Fig. 4 shows the spatial dynamics of E. coli growing on a flat landscape where all MHPs have all their nanoslits open (_i = _max 10*, i). Fig. 4A Right consists of a time-ordered stack of epifluorescence images of all 85 MHPs. Here, we scanned the array every 10 min (t) 300 times, sampling a total of 3,000 min (t = 2.1 days). Each row of images _t(x, y) of the 85 MHP array represents the configuration of the array at time t (Fig. 4A and Movie 1, which is published as supporting information on the PNAS web site). Local (MHP) population density average _i at any given time t can then be calculated by integrating epifluorescence intensity _t for all pixels (x, y) within the ith MHP (see Fig. 7A in Supporting Appendix, which is published as supporting information on the PNAS web site).

The dynamics of the landscape average (t) = _ii/85 (Figs. 4B and 7B), resemble what is seen in batch cultures; after a lag period of 400 min during which little growth occurs in the array, a period of growth (exponential phase) followed by landscape saturation (stationary phase) at 10⁴ cells per MHP (K* 3 x 10¹⁰ per ml) is observed. Because the landscape is flat, we would expect that over time the bacteria would inhabit all of the MHPs. However, because coupling is weak (small J_i,i+1), a metapopulation emerges. Thus, whereas the population density of an individual MHP (local scale) shows sharp rises and falls in density, occupancy of the entire array (landscape scale) shows a much slower growth rate and smoother dynamics than the single MHP (Fig. 4B). The fit (dotted red) of the logistic map (Eq. 1) to the globally averaged occupancy/MHP (solid blue) shown in Fig. 4B yields a _T 250 min (4 h). The reason for this slow growth is clear from Fig. 4: there are localized E. coli populations distributed over the landscape, interacting through local extinction and colonization processes operating at multiple spatial and temporal scales (Figs. 4 and 7). Notice that although the density seems constant (stationary phase) at the global scale, at the local MHP scale there are clear dynamics (Fig. 4B). On the other hand, although the global averages are at exponential phase because of continuous range expansion, individual populations can be in stationary as well as in death phases.

In Fig. 4A Right, we show a (mesoscale) 5-MHP-wide, "parent" population giving "birth" to a new population spreading to the right and settling six MHPs away. The message: in a flat habitat landscape, E. coli aggregates its biomass at multiple scales satisfying a careful balance between vacancy and occupancy. These multiscale aggregates correspond to spatial versions of the classical phases of growth: lag, log, stationary, and death. Zooming into a particular MHP (Fig. 4B, black solid line), we notice pulses of exponential growth with a 10- to 20-min time constant (local colonization events), stationary-phase, and death-phase oscillations (local extinction events) occurring at multiple spatial and temporal scales. Unlike the zero dimensions case, in one dimension, the bacteria can migrate into near by MHPs, so growth can continue in a delayed fashion throughout the MHP array.

Adaptation in a Black and White Landscape. To induce a fitness pressure on the E. coli metapopulation, we built a "black and white" (B&W) adaptive landscape by patterning two different ecotopes on each side of the landscape. Fig. 5 presents the basic idea of such adaptive landscape; the bar code (at the top) indicates the number of nanoslits that are opened in each MHP (indexed by i); in this case the nanoslits are fully closed on the left side (a stress-based black ecotope) and fully open (white ecotope) on the right side. The response of the bacteria to this landscape can be broken down into three basic epochs based on episodic expansions of its range of landscape occupancy (see Fig. 5 and Fig. 8A in Supporting Appendix).

Epoch I.  At first, there is a high increase in landscape average density (from 10¹ to 10² cells per MHP) due to invasion and growth of bacteria in the white ecotope. Initially, this growth is confined only to this ecotope, and it saturates at 10² cells per MHP. However, at around t 700 min (Fig. 5), the bacteria do probe into the black ecotope, thus expanding their range to the whole landscape. After this range expansion, the global average density achieved is larger but still 10² cells per MHP. Unfortunately for the bacteria, they quickly die out (death phase) in most of their range, going down to a very low spatially homogeneous density of 0.5 x 10¹ cells per MHP at the end of epoch 1, at about t 1,000 min after inoculation. Notice the large-scale correlation of the extinction event (Fig. 5); almost the whole metapopulation disappears from the landscape leaving very few colonizing cells.

Epoch II.  Average density is down to 0.5 x 10¹ cells x MHP. However, it is localized into a discrete set of few surviving populations (3 or 4). These populations compete for the landscape, increasing the average density to 0.4 x 10² cells per MHP. This growth is distributed across the whole landscape, unlike the localized growth observed in the first third of epoch I. Bacterial history repeats itself: another large-scale extinction event (Fig. 5; t 1,800 min) wipes out most of the population. This time, however, a larger number of local populations survive (8 or 10). Some of these populations are indeed inhabiting the low-nutrient, "stressful" region. Indeed, at least two of these stress-tolerant populations persist and ultimately merge with a new landscape-wide one (at t 3,000 min). This time, the landscape-wide expansion in range is sustained for a longer period (1,000 min) than the one observed during epoch I (100 min); yet, it is also followed by massive extinction. However, this time more populations (10–15) survive. At the end of the epoch, a massive regrowth into the stress region is triggered from the right part of the landscape. Notice, however, that the speed of the dispersal front into the left of the array is much slower than the incursion of the second third of epoch I. As these populations expand to the left, they merge with other local ones. So far, densities nowhere in the landscape have >10² cells per MHP.

Epoch III.  At the beginning of epoch III (t 1,800 min), populations are expanding consistently to the left, into the stress domain of the landscape (densities are still low, at 2 x 10¹ cells per MHP). During the first third of the epoch (t most growth is widespread in the full supply region (_max) while highly localized in the stress region (_min) of the landscape. At around t = 2,500 min, populations from the _max ecotope (right side) start expanding into stress territory (left side); until at t = 3,000 min they absorb the stress-tolerant population located around the 19th MHP of the landscape. After this event, two things seem to happen: (i) the territorial expansion into the stress region accelerates (Fig. 5), and (ii) the landscape average growth also accelerates (Figs. 5 and 8A). The following period (3,000 min t 4,000 min) consists of vigorous growth in the _max region and sustained growth in the _min region. However, the bacteria in the stress and full-supply ecotopes have growth rates that differ by a factor of 2 (Fig. 8A Inset). At a late period of this epoch (t 4,000 min), a sharp density boundary has emerged between the two ecotopes, although the range of the metapopulation includes the whole landscape. This boundary divides the population in two different classes of abundance. On the left, populations reach densities of 10² cells per MHP, whereas on the right part of the landscape, populations reach densities of 10³ to 10⁴ cells per MHP. This boundary is similar to the boundary that separated populations at the beginning of epoch I. Then, the average densities were 10¹ cells per MHP on the left and 10² cells per MHP on the right. Notice the extinction events (around t 4,500 min), eliminating two (competing) nearby high-density (10⁴ cells per MHP) populations located around the 50th and 60th MHPs. Landscape saturation is hard to reach with this (B&W) adaptive topology. Even after 5,000 min, the metapopulation is not fully adapted to life in the stress region.

Adaptation to a Rugged Landscape. Evolutionary dynamics on adaptive landscapes are a function not only of the local fitness value but also a function of higher spatial derivatives of the landscape (19). Thus, the sharp fitness barrier (like the one in the B&W case) separating stress and full-supply ecotopes can be mixed with more "rugged" parts of the landscape where only fractions of nanoslits are closed or open, thus introducing more "intermediate" niches. To investigate the role of higher correlations in landscape parameters on adaptation, we developed a "rugged" landscape composed of three basic regions described by the black and white bar at the top of Fig. 6.

Notice that in this landscape only 10% of the MHPs are not stress MHPs; the MHP array as a whole then has less supply exchange-area than in the previous case (B&W landscape). However, the total amount of resources in storage in the feeder channels remains the same. The niche differences between this adaptive landscape and the previous (B&W) one are in the spatial localization and the rates of delivery, not in the total amount of resources. The adaptive topology of these two landscapes is therefore different: the number of local niches is higher in the rugged landscape because the B&W one only had two. Fig. 6 shows the response of bacteria to such a landscape. Here, as before, we once again break the bacterial response into epochs based on the range expansions of landscape occupancy.

Epoch I.  The first half (t by the high localization of rapid growth around the central cluster (ecotope) of _max niches. Bacteria grow to saturation at around K* 10⁴ cells per MHP but remain localized to the center (around the 40th MHP), exploiting the better-supplied part of the landscape. Landscape average densities, however, are still below 3 x 10² cells per MHP because of the small scale of the range of occupancy. As these patches are exhausted at about t 500 min, the bacteria expand their range to the right side of the landscape where more clusters of opportunity (_i 0) are located (Fig. 6). When the bacteria start exploring these ecotopes, landscape average growth increases (Fig. 8B; t 600 min), a daughter population emerges around the 55th MHP, and global densities climb to 10³ cells per MHP rapidly before a wide-range extinction takes place at about t 700 min (Fig. 6).

Epoch II.  After the population crash described above, we enter Epoch II. Landscape-wide average densities now are 0.5 x 10² cells per MHP. The few survivors are once again fully localized to the MHP cluster at the center of the landscape. Slow growth of bacteria is observed, with little further range expansion, increasing local density from post-crush levels back to 0.5 x 10³ cells per MHP. At around t 1,500 min, this central population starts expanding its range once again to the rugged (right) side of the landscape. By t 1,800 min, the landscape begins to saturate (Fig. 8B) with most of the bacteria inhabiting the rugged (right) side of the landscape and reaching very high cell densities (Fig. 6). A huge-density barrier separates the left (10² cells per MHP) from the right (10⁴ cells per MHP) demographic states of the metapopulation. As with the B&W landscape, bacteria in the different (stress, partial, and full supply) ecotopes, have growth rates that now differ by a factor of 3 (see Fig. 8B Inset). Even the supply to the MHPs is apparently less efficient. It indeed allows the metapopulation to adapt faster to the habitat landscape. After 1,800 min, populations saturate at 10⁴ cells per MHP. Interestingly, even though the landscape is fully saturated on the right side, it holds plenty of capacity to the left. After a spatial lag, a large range expansion takes place at t 2,000 min.

Epoch III.  This new range expansion is slow. Vigorous growth of bacteria in the right region of the array during epoch II (while experiencing different, nearby variable ecotopes for at least five generations; Fig. 6) sets the conditions for the dynamics to come. Now, these populations start rapidly expanding into the left side (Fig. 8B) where the stress ecotope is located. Notice that the landscape-wide density remains constant at its saturation level (Fig. 6) The colonization of the stress (_min) region of the landscape is completed by t = 2,500 min. Interestingly, at the same time the adaptive radiation is completed, there is a widely correlated decrease in density from 10⁴ to 0.5 x 10⁴ cells per MHP. This decrease in density coincides with the appearing of a less dense pattern of occupancy, propagating from the central region into both sides of the landscape. By t 3,000 min, a new pattern of occupancy spans the whole landscape. Metapopulation dynamics as in the case of the flat landscape are observed. In this state, MHPs fluctuate locally between having 10³ cells per MHP or 0.5 x 10⁴ cells per MHP. The landscape average density is stationary as expected. Now the bacteria are fully adapted to live on the entire landscape.