
MBW:Big BacteriaFrom MathBioThis article is a summary and analysis of Big Bacteria. ^{[1]} All information is taken from this source unless otherwise noted. ContentsOverviewExecutive SummaryTypically when considering bacteria, they are often assumed to have nominal size of about 1 micron. The truth is that bacteria have a large range of sizes, from 0.2 micron nanobacteria, to the largest bacteria, Thiomargarita namibiensis, which have a diameter of 750 microns. Consequently, the biomass of different bacteria can vary by over 10 orders of magnitude. The fact that these microorganisms rely on diffusion to receive food and nutrient molecules can pose a problem. As the concentration of the substrate in the surrounding fluid increases, larger bacteria can be sustained. In natural environments, there is an advantage for micron sized bacteria; however, as previously mentioned, certain bacteria can grow to be hundreds of times larger. The mathematical reasoning for the limit of bacteria does not take into account strategies to overcome the diffusion limited growth. In the natural world we find several strategies that bacteria have developed to allow them to grow beyond their diffusion limited size. Two general mechanisms that have been found are chemotaxis, where a motile microorganism moves toward higher concentrations of substrate, reducing the effective volume of the microorganism with vesicles for internal energy storage. History and ContextPrior to the early 1990s, the sizescales of eukaryotes was thought to be roughly 100 times that of typical prokaryotes. This size discretion had been the most common way to visually differentiate between eukaryotes and prokaryotes. In the gut of the surgeonfish, bacteria have been discovered on the order of 100 microns in size, two to three orders of magnitude larger than typical bacteria. ^{[2]} The largest of these, which have been measured at 80 by 600 microns, are roughly the same size as a typical eukaryote, making their classification as prokaryotes much more difficult. Additionally, previous theories on prokaryotic structure and development, most specifically regarding the expected maximum size of bacteria, must be reconsidered. Further disproving prior theories, Heide Schulz discovered an even larger bacteria, the Thiomargarita namibiensis, off the coast of Namibia in 1997. ^{[3]} These bacteria grow to around 750 microns in diameter, but instead of living within the gut of a fish, they feed off of sulfurrich sedimentary rock. Several other gigantobacteria were discovered to also live in these sulfurrich environments, leading Schulz and others to further investigate how they apparently violate the laws of diffusion, as they do in "Big Bacteria." Mathematics, Model Type & Biological System StudiedThis article uses the mathematical relationship between the integrals representing particles in a space with those moving in and out of the space. From here it finds the corresponding differential equation representing the change in particle distribution in a 1, 2 or 3 dimensional space, based on particle movements. It is solved to get the distribution of particles in space, and the diffusion time of particles. It also looks at formulas for the concentration of substrate molecules surrounding a bacteria, based on distance from the bacteria. This is used to find the maximum metabolic rate (the rate at which a bacteria cell metabolizes substrate molecules), related to the inverse square of the radius of the bacteria cell. It outlines further mathematics for predicting bacteria size from diffusion times and the substrate environment (densities). The article also considers the extra effects from chemotaxis, the movement of bacteria that is caused by a chemical gradient. These supply another method for bacteria to get the substrates they need.
Derivation of the AdvectionDiffusion EquationThe following derivation is taken from Britton's Essential Mathematical Biology. ^{[4]} The derivation is based on the conservation of mass, but has a source/sink term to account for possible reactions that often are happening in biological applications. The number of particles inside a volume at a time t+dt is equal to the number of particles in the volume at time t, plus the net particles entering, plus the net creation of particles in the volume. Mathematically this can be written as: Where u is the particle concentration, J is the flux, and f(x,t) is the sink/source density at (x,t). Now by using the divergence theorem, and subtracting the first term on the right side of our equation from both sides, dividing by and taking the limit as , we get: This is true for arbitrary volumes V and the integrand must be zero, This is the conservation of matter with a source term. Because , the advection equation with a source term is: The net flow of particles down a concentration gradient is proportional to its magnitude, This is of course Flick's Law. The diffusion equation becomes: If both advection and diffusion are present, then it can be written as the advection diffusion equation: It can be solved in whichever coordinate system you prefer, in 1, 2, or 3D. In spherical coordinates (without source/sink term) this becomes To use this equation we assume that the system is in steady state i.e. . Next by plugging in the appropriate boundary conditions: In 1D typically: In 2D and 3D: where These then give the following results: In 1D: In 2D radially symmetric flow: In 3D spherically symmetric flow:
to find the diffusion time given here as In 1D: In 2D: In 3D: This is the time it takes for diffusion to move a particle a distance L (or ba) and can be used to predict the size of bacteria. Mathematical ModelSchulz and Jorgensen take the diffusion equation derived above, and applying it in three dimensions over the boundary conditions of a bacteria of set size, determine the result discussed below. In the paper it is not clear where their extra factor of comes from exactly, it has been left in to for consistancy with the original publication. The amount of time, t, required for a molecule to diffuse through an average distance of L through liquid is described by
Specific Metabolic RateThe above model fails to consider the typically very low concentrations of substrate molecules surrounding a bacteria. In general, the concentration of substrate increases with the distance away from the bacteria:
ResultsBacteria Size EstimationThe model described by Schulz and Jorgensen successfully explains the general mechanisms by which bacteria absorb substrate. It predicts the time for particles to diffuse and can be used to predict the size of bacteria. where One can use the above equations to calculate the expected size of a bacterium of a particular mass density living in a certain environment. The following numbers are taken from Gikas and Livingston.^{[5]}: The diffusion constant of oxygen in water is:
Bacteria have been found to be tens of times larger than the limited size set by diffusion alone. Although overarching principles governing the maximum size of bacteria remain elusive, there are several examples of big bacteria whose specific mechanisms for increasing their metabolic rate have been observed. The following table, taken from Schulz and Jorgensen, shows the wide range of possible sizes of known bacteria.
Overcoming the Diffusion Size Limit: ChemotaxisTaxis just means a movement toward or away from an external stimulus. Chemotaxis refers to movement in response to a chemical gradient. Depending on the type of bacterium, this movement can come through a variety of mechanical systems. For example Escherichia coli has a "runandtumble" mechanism where it does a sudden movement in a given direction, then tumbles in place and then moves in a new direction. It tumbles less when moving up an attractive gradient and more when moving down one. This effectively keeps it generally moving in its desired direction. The nitratestoring sulfur bacterium Thioploca lives in sediment layers in the ocean. It forms pathways in the sediment so that it can move from a nitrate rich top layer to a hydrogen sulfate rich lower layer. It moves in order to take up nitrate and then reduce the sulfate to store it as sulfur. Many bacteria anchor themselves to a solid surface with water, such as a sedimentary rock. In this region we find what is called a diffusive boundary layer. The bacteria try to keep within an optimal range of this layer. In some cases the bacteria will form a membrane of many bacteria that all work together to effectively pump water through the membrane, maintaining the chemical gradient. The main consistency among all of these giant bacteria is that by using chemotaxis, they are no longer solely reliant on molecular diffusion to get the substrates they need. The added movement changes the mathematics; a simple way to adapt the diffusion equation is to change the flux. Keeping in mind that the average velocity that the bacteria responds to the gradient is itself proportional to the gradient, the flux should be proportional to the gradient and the density of bacteria.
where is the chemotactic coefficient is the density of bacteria and is the chemical concentration.
ConclusionIn general, prokaryotes such as bacteria are significantly smaller than eukaryotes because of limited substrate concentration for use as a diffusion nutrient. Some bacteria overcome this limitation in different ways, such as using chemotaxis to orient properly and get as many nutrients as possible to the entire surface of the cell, as the gigantic bacteria in the gut of the surgeonfish do, or growing at the interface of sulfurrich sediment, like the bacteria discovered off the coast of Namibia. However, the advantages to such extreme sizes remain mostly unclear, and as an overwhelming majority of bacteria are of the expected size, studying the long term development of gigantic bacteria can be very difficult. The oxidation of hydrogen sulfide carried out by the bacteria living at the sulfurrich sediment interfaces have obvious energy efficient benefits, but most other types of bacteria have not yet seemingly adapted to have larger size be advantageous. This, in addition to the issue of classifying bacteria when they are of the same size scale as eukaryotes, should make the future studies of big bacteria both difficult and enlightening. Citations to The Paper ("Big Bacteria")This paper has been cited in over 100 different papers. One such paper is The Selective Value of Bacterial Shape, which considers why certain bacteria have certain shapes (and sizes). It claims that bacterial shapes are important biologically, since specific morphologies are consistently chosen from many possibilities, since some cells can change shape when necessary, and since morphology can be tracked through evolutionary lineages. It aims to explore the conditions that favor specific morphologies. It considers 8 different environmental or behavioral conditions that can affect bacteria. They believe that, in a way, bacteria "integrate" over (some of) these factors to produce an optimal shape given their circumstances. "Big Bacteria" is cited when discussing the first of the 8 conditions that could affect bacteria shape, nutrient access. It explains that diffusion (to areas of greater nutrient concentration) is the main factor that determines how well bacteria can "eat", and that diffusion is affected greatly by cell size, and possibly by shape. It refers to the large bacteria that appear in "Big Bacteria", as well as the analysis of the effect of diffusion on cell size. It also mentions the results found for the relationship between nutrient transportation rate and surface area of the cell, or how smaller/less spherical cells with specific shapes will have better nutrient transfer. It mentions the "diffusion sphere" and the time for molecules to "meet", which are also introduced in "Big Bacteria". References
