Biofilm Growth and Fossil Form

Stromatolites can grow under the influence of microbial processes, but it is often unclear whether and how the macroscopic morphology of these rocks records biological processes. Conical stromatolites, which formed in the absence of sedimentation, provide a comparatively simple record of the interplay between microbial growth and lithification. Here, we show that the dynamics shaping conical stromatolites result from diffusive gradients within the overlying microbial mat. These gradients cause minerals to precipitate faster in regions of high curvature, resulting in measurable properties of the shapes of stromatolite laminas. This model allows us to estimate the thickness of ancient stromatolite-forming mats to be approximately $1\mathrm{mm}$, consistent with modern systems. Proceeding from the assumption that the ubiquitous process of diffusion is recorded in the translating form of a stromatolite, we derive the shape of a diffusion-driven stromatolite. The conical morphology—a distinctive feature of stromatolites growing in the absence of sedimentation—arises from these dynamics. This form is quantitatively consistent with the shape of conical stromatolites that grew for more than $2.9\times {10}^9$ yrs of Earth history.

Popular Summary

Stromatolites are laminated, lithified rocks that are commonly regarded as the fossilized remains of ancient microbial mats. A particular class of stromatolites with conical shapes can be as old as $3.4\times {10}^9\mathrm{yrs}$ and remain in hot springs today. How did such a conical form come about? Did this persistence of form over billions of years imply some fundamental generality in the physical, chemical, and biological processes underlying the formation? In this paper, we show that diffusion of calcium ions and molecules of inorganic carbon species is one of the keys to addressing these questions.

Understanding the growth dynamics of rocks that stopped growing billions of years ago seems a daunting task. Fortunately, the color and texture of the precipitated minerals in stromatolites, in the shape of periodic mineral bands, provide a very good record of their growth and indications of the physical and chemical conditions for the growth as well as biological makeup of the microbial mats. This information not only gives us clues about how to make our theoretical hypotheses about the ancient growth process but also allows us to verify our predictions against it.

Our theory views the ancient growth process of a conical stromatolite as follows: Each growing stromatolite had a thin microbial mat as its surface layer. Mineral precipitation and their diffusion through the mat were the two primary processes. However, as mineral precipitation was limited by diffusion through the microbial mat, the rate of precipitation was faster in regions of high curvature. Mathematical formulation of this conceptual picture in terms of a differential equation about the form of the stromatolite leads to the prediction of conical shapes that compare quantitatively well with those observed in fossil records. Equally compelling is our estimation, based on the model and its comparison with the fossil records, of the 1-mm thickness of ancient microbial mats. This number is quite consistent with the thickness of modern-day microbial mats.

Our work highlights the microscopic diffusion process as a dynamical basis for the ubiquity of conical stromatolites throughout Earth’s history.

Figure 1

Conical stromatolites from the Archean and Proterozoic, as seen in the field and in thin sections. (a) A side view of a rock formed by many small conical stromatolites from the Pongola Supergroup [38]. The scale bar is 1 cm. (b) Thin section of a stromatolite from (a). The scale bar is 0.5 cm. (c) Large conical stromatolite from the Bakal Formation [39]. The hammer is approximately 30 cm long. (d) Thin section of a large cone from the Bakal Formation showing thin bands called laminas. These changes in color and texture record the shape and position of the stromatolite surface at different points in time. The scale bar is 1 cm.

Figure 2

A schematic of a simplified stromatolite. (a) The stromatolite grows as chemicals diffuse through the microbial mat and precipitate at the base of the mat. The thickness $d$ of the mat varies over the surface of the stromatolite. The shape of the interface between the microbial mat and the stromatolite is described by the local curvature $H$. At the tip of an axisymmetric stromatolite, $H^{-1}$ is the radius of curvature. (b) The transformation from Cartesian coordinates to the local coordinates of the surface $f(x,y)$. The unit vectors ${\widehat{s}}_1$ and ${\widehat{s}}_2$ are tangent to the surface. The vector $\widehat{n}$ is the local normal. The position of the red circle on the stromatolite surface can be expressed in Cartesian coordinates as $[x_0,y_0,f(x_0,y_0)]$ or in the local coordinates as $(s_1,s_2,0)$, where $s_1$ and $s_2$ are measured along the stromatolite surface from some known point. For the sake of illustration, we have shown $f$ as a paraboloid.

Figure 3

The shape of stromatolite laminas records the stromatolite’s evolution in time. (a) A $2.9\times {10}^9$-yr-old conical stromatolite from the Pongola Supergroup in South Africa cut vertically through the center. The transitions between light and dark laminas record the shape of the stromatolite at different points in time. The two solid blue curves represent the surface of the stromatolite at different times. This curve was measured by visually following the transition between a light lamina and a dark lamina. (b) The width of a lamina increases ($R^2=0.57$, $p<{10}^{-8}$) with curvature. Thus, the rate at which this stromatolite grew is an increasing function of curvature, consistent with Eq. (15). The fit of the measured lamina width and curvature to a straight line gives an estimate of the thickness of the overlying microbial mat $\Delta =1.2\pm 0.6\sim 1\mathrm{mm}$. Two additional laminas from this sample are shown in the Supplemental Material. Additional curvature-thickness plots are shown in Fig. 5 and the Supplemental Material [41].

Figure 4

A balance between curvature-driven growth and translational growth sets the shape of a conical stromatolite. (a) When a curve $f(r)$ (red parabola) evolves due to curvature-driven growth, the normal velocity $c$ is inversely proportional to the mean of the two radii of curvature at that point. When a curve translates forward, there is a geometric relationship between the speed at which a point translates $c_t$ and the speed at which it grows in the normal direction $c$. (b) The shape of the stromatolite depends on two parameters. ${\chi }^{-1}$ is the dimensionless speed of upward growth. The dimensional mat thickness $\Delta$ gives the importance of curvature-driven growth. Values of $\Delta$ are given relative to the thickness of the mat $d_0$. The scale bar is $5d_0$.

Figure 5

Comparison of curvature-driven growth to the shapes of large and small conical stromatolites sectioned vertically through the center. (a) Thin section through a $2.9\times {10}^9$-yr-old conical stromatolite (photograph) [38]. Solid blue lines represent the upper and lower boundaries of two laminas. The red points lay along a curve equidistant from the traced laminas. (b) Thin section of a $1.6\times {10}^9$-yr-old stromatolite from the Bakal Formation [39, 40] with traced laminas. (c) The thickness of the upper laminas from (a) increases with lamina curvature ($R^2=0.81$, $p<{10}^{-17}$, $\Delta =1.9\pm 0.7\mathrm{mm}$), consistent with Eq. (15). (d) Similar agreement is found for the lower lamina ($R^2=0.64$, $p<{10}^{-8}$, $\Delta =3.5\pm 1.7\mathrm{mm}$) and in the (e) Proterozoic sample shown in (b) ($R^2=0.88$, $p<{10}^{-44}$, $\Delta =9.7\pm 1.6\mathrm{mm}$). (f),(g) Fitting two parameters, the measured shapes of the (f) upper and (g) lower laminas (points) are consistent with the shape (solid curve) of a lamina growing by the curvature-driven growth equation (19). (f) Fit of the sample from (b) to Eq. (19). All samples are cut through the central axis of the cone.

Figure 6

Two estimates of the thicknesses of ancient stromatolite-forming microbial mats produce consistent values. ${\Delta }_t$ is estimated from the relation between the curvature of a lamina and the distance to the neighboring laminas. ${\Delta }_s$ is estimated from the shape of a single lamina. The black line shows equality. The measured slope is $m=2.3\pm 1.6$, consistent with the predicted value $m=1$. The intercept $b<0$ reflects that ${\Delta }_t$ is systematically larger than the estimate of ${\Delta }_t$. Solid blue points represent stromatolites from the $2.9\times {10}^9$-yr-old Pongola Formation [38]. Open red points are from Proterozoic samples.

Figure 7

A toy model of mat growth. Nutrients (arrow) diffuse through a microbial mat and are consumed. Near the nutrient-rich mat surface, the mat grows. At the nutrient-poor mat bottom, the mat decays. As described in Eq. (a15), mat thickness is determined by the balance of growth and decay. The equilibrium thickness varies with the curvature of the underlying stromatolite.