Alexander P. Petroff

Morphological record of oxygenic photosynthesis in conical stromatolites

Conical stromatolites are thought to be robust indicators of the presence of photosynthetic and phototactic microbes in aquatic environments as early as 3.5 billion years ago. However, phototaxis alone cannot explain the ubiquity of disrupted, curled, and contorted laminae in the crests of many Mesoproterozoic, Paleoproterozoic, and some Archean conical stromatolites. Here, we demonstrate that cyanobacterial production of oxygen in the tips of modern conical aggregates creates contorted laminae and submillimeter-to-millimeter-scale enmeshed bubbles. Similarly sized fossil bubbles and contorted laminae may be present only in the crestal zones of some conical stromatolites 2.7 billion years old or younger. This implies not only that cyanobacteria built Proterozoic conical stromatolites but also that fossil bubbles may constrain the timing of the evolution of oxygenic photosynthesis.

Formation and stability of oxygen-rich bubbles that shape photosynthetic mats

Gas release in photic-zone microbialites can lead to preservable morphological biosignatures. Here, we investigate the formation and stability of oxygen-rich bubbles enmeshed by filamentous cyanobacteria. Sub-millimetric and millimetric bubbles can be stable for weeks and even months. During this time, lithifying organic-rich laminae surrounding the bubbles can preserve the shape of bubbles. Cm-scale unstable bubbles support the growth of centimetric tubular towers with distinctly laminated mineralized walls. In environments that enable high photosynthetic rates, only small stable bubbles will be enclosed by a dense microbial mesh, while in deep waters extensive microbial mesh will cover even larger photosynthetic bubbles, increasing their preservation potential. Stable photosynthetic bubbles may be preserved as sub-millimeter and millimeter-diameter features with nearly circular cross-sections in the crests of some Proterozoic conical stromatolites, while centrimetric tubes formed around unstable bubbles provide a model for the formation of tubular carbonate microbialites that are not markedly depleted in (13)C.

Biophysical basis for the geometry of conical stromatolites

Stromatolites may be Earth’s oldest macroscopic fossils; however, it remains controversial what, if any, biological processes are recorded in their morphology. Although the biological interpretation of many stromatolite morphologies is confounded by the influence of sedimentation, conical stromatolites form in the absence of sedimentation and are, therefore, considered to be the most robust records of biophysical processes. A qualitative similarity between conical stromatolites and some modern microbial mats suggests a photosynthetic origin for ancient stromatolites. To better understand and interpret ancient fossils, we seek a quantitative relationship between the geometry of conical stromatolites and the biophysical processes that control their growth. We note that all modern conical stromatolites and many that formed in the last 2.8 billion years display a characteristic centimeter-scale spacing between neighboring structures. To understand this prominent—but hitherto uninterpreted—organization, we consider the role of diffusion in mediating competition between stromatolites. Having confirmed this model through laboratory experiments and field observation, we find that organization of a field of stromatolites is set by a diffusive time scale over which individual structures compete for nutrients, thus linking form to physiology. The centimeter-scale spacing between modern and ancient stromatolites corresponds to a rhythmically fluctuating metabolism with a period of approximately 20 hr. The correspondence between the observed spacing and the day length provides quantitative support for the photosynthetic origin of conical stromatolites throughout geologic time.

Ramification of stream networks

The geometric complexity of stream networks has been a source of fascination for centuries. However, a comprehensive understanding of ramification—the mechanism of branching by which such networks grow—remains elusive. Here we show that streams incised by groundwater seepage branch at a characteristic angle of 2π/5 = 72°. Our theory represents streams as a collection of paths growing and bifurcating in a diffusing field. Our observations of nearly 5,000 bifurcated streams growing in a 100 km2 groundwater field on the Florida Panhandle yield a mean bifurcation angle of 71.9° ± 0.8°. This good accord between theory and observation suggests that the network geometry is determined by the external flow field but not, as classical theories imply, by the flow within the streams themselves.

Geometry of valley growth

Although amphitheatre-shaped valley heads can be cut by groundwater flows emerging from springs, recent geological evidence suggests that other processes may also produce similar features, thus confounding the interpretations of such valley heads on Earth and Mars. To better understand the origin of this topographic form, we combine field observations, laboratory experiments, analysis of a high-resolution topographic map and mathematical theory to quantitatively characterize a class of physical phenomena that produce amphitheatre-shaped heads. The resulting geometric growth equation accurately predicts the shape of decimetre-wide channels in laboratory experiments, 100 m-wide valleys in Florida and Idaho, and kilometre-wide valleys on Mars. We find that, whenever the processes shaping a landscape favour the growth of sharply protruding features, channels develop amphitheatre-shaped heads with an aspect ratio of n.

Bifurcation dynamics of natural drainage networks.

As water erodes a landscape, streams form and channellize the surficial flow. In time, streams become highly ramified networks that can extend over a continent. Here, we combine physical reasoning, mathematical analysis and field observations to understand a basic feature of network growth: the bifurcation of a growing stream. We suggest a deterministic bifurcation rule arising from a relationship between the position of the tip in the network and the local shape of the water table. Next, we show that, when a stream bifurcates, competition between the stream and branches selects a special bifurcation angle α=2π/5. We confirm this prediction by measuring several thousand bifurcation angles in a kilometre-scale network fed by groundwater. In addition to providing insight into the growth of river networks, this result presents river networks as a physical manifestation of a classical mathematical problem: interface growth in a harmonic field. In the final sections, we combine these results to develop and explore a one-parameter model of network growth. The model predicts the development of logarithmic spirals. We find similar features in the kilometre-scale network.

Longitudinal profile of channels cut by springs

We propose a simple theory for the longitudinal profile of channels incised by groundwater flow. The aquifer surrounding the stream is represented in two dimensions through Darcys law and the Dupuit approximation. The model is based on the assumption that, everywhere in the stream, the shear stress exerted on the sediment by the flow is close to the minimal intensity required to displace a sand grain. Because of the coupling of the stream discharge with the water table elevation in the neighbourhood of the channel head, the stream elevation decreases as the distance from the streams tip with an exponent of 2/3. Field measurements of steephead ravines in the Florida Panhandle conform well to this prediction.

Reaction-diffusion model of nutrient uptake in a biofilm: theory and experiment.

Microbes in natural settings typically live attached to surfaces in complex communities called biofilms. Despite the many advantages of biofilm formation, communal living forces microbes to compete with one another for resources. Here we combine mathematical models with stable isotope techniques to test a reaction-diffusion model of competition in a photosynthetic biofilm. In this model, a nutrient is transported through the mat by diffusion and is consumed at a rate proportional to its local concentration. When the nutrient is supplied from the surface of the biofilm, the balance between diffusion and consumption gives rise to gradients of nutrient availability, resulting in gradients of nutrient uptake. To test this model, a biofilm was incubated for a fixed amount of time with an isotopically labeled nutrient that was incorporated into cellular biomass. Thus, the concentration of labeled nutrient in a cell is a measure of the mean rate of nutrient incorporation over the course of the experiment. Comparison of this measurement to the solution of the reaction-diffusion model in the biofilm confirms the presence of gradients in nutrient uptake with the predicted shape. The excellent agreement between theory and experiment lends strong support to this one-parameter model of reaction and diffusion of nutrients in a biofilm. Having validated this model empirically, we discuss how these dynamics may arise from diffusion through a reactive heterogeneous medium. More generally, this result identifies stable isotope techniques as a powerful tool to test quantitative models of chemical transport through biofilms.

Hydrodynamics and collective behavior of the tethered bacterium Thiovulum majus

Significance In this paper we examine the dynamics underlying a form of collective behavior exhibited by bacteria. In a nutrient gradient, Thiovulum majus cells aggregate into a community in which cells attach to one another using mucus tethers. As tethered cells beat their flagella, they pull nutrient-laden water through the community. The flow of water created by many cells gives rise to variations in the nutrient concentration. As cells reorganize in response to these nutrient gradients, they change the flow of water. We show that the coupling between the motion of water, nutrients, and cells generates large-scale flows that dramatically increase the rate at which nutrients are pulled to the cells. The ecology and dynamics of many microbial systems, particularly in mats and soils, are shaped by how bacteria respond to evolving nutrient gradients and microenvironments. Here we show how the response of the sulfur-oxidizing bacterium Thiovulum majus to changing oxygen gradients causes cells to organize into large-scale fronts. To study this phenomenon, we develop a technique to isolate and enrich these bacteria from the environment. Using this enrichment culture, we observe the formation and dynamics of T. majus fronts in oxygen gradients. We show that these dynamics can be understood as occurring in two steps. First, chemotactic cells moving up the oxygen gradient form a front that propagates with constant velocity. We then show, through observation and mathematical analysis, that this front becomes unstable to changes in cell density. Random perturbations in cell density create oxygen gradients. The response of cells magnifies these gradients and leads to the formation of millimeter-scale fluid flows that actively pull oxygenated water through the front. We argue that this flow results from a nonlinear instability excited by stochastic fluctuations in the density of cells. Finally, we show that the dynamics by which these modes interact can be understood from the chemotactic response of cells. These results provide a mathematically tractable example of how collective phenomena in ecological systems can arise from the individual response of cells to a shared resource.

Shape and dynamics of seepage erosion in a horizontal granular bed

We investigate erosion patterns observed in a horizontal granular bed resulting from seepage of water motivated by observation of beach rills and channel growth in larger scale land forms. Our experimental apparatus consists of a wide rectangular box filled with glass beads with a narrow opening in one of the side walls from which eroded grains can exit. Quantitative data on the shape of the pattern and erosion dynamics are obtained with a laser-aided topography technique. We show that the spatial distribution of the source of groundwater can significantly impact the shape of observed patterns. An elongated channel is observed to grow upstream when groundwater is injected at a boundary adjacent to a reservoir held at constant height. An amphitheater (semicircular) shape is observed when uniform rainfall infiltrates the granular bed to maintain a water table. Bifurcations are observed as the channels grow in response to the groundwater. We further find that the channels grow by discrete avalanches as the height of the granular bed is increased above the capillary rise, causing the deeper channels to have rougher fronts. The spatiotemporal distribution of avalanches increase with bed height when partial saturation of the bed leads to cohesion between grains. However, the overall shape of the channels is observed to remain unaffected indicating that seepage erosion is robust to perturbation of the erosion front.