The chemostat is a basic piece of laboratory apparatus, yet it occupies a central place in mathematical ecology. Its importance stems from the many roles it plays. It is a model of a simple lake, the ideal place to study competition in its most primitive form – exploitative competition. It is also used as a model of the wastewater treatment process. In its commercial form the chemostat plays a central role in certain fermentation processes, particularly in the commercial production of products by genetically altered organisms (e.g., in the production of insulin)… In addition to being known as a chemostat, other common names are "continuous culture" and, in the engineering literature, CSTR (continuously stirred tank reactor)… In biology there are relatively few accepted mathematical models. The chemostat is one – and, in microbial ecology, perhaps the only one – such model that does seem to have wide acceptance. This is true, at least in part, because the parameters are measurable and the experiments confirm that the mathematics and the biology are in agreement. – Hal L. Smith and Paul Waltman (1995)
A chemostat is a continuous culture devise that consists of broth-based growth media (that is, liquid media). The media is well mixed within a growth chamber and there is a continuous flow of fresh broth into the growth chambers along with cultured broth out, with both of the latter occurring at the same rate. As such, a chemostat resembles a pond, where input is a stream or a spring and output is over a spillway.
Chemostats are useful to experimental evolution studies for two reasons. The first is that they can provide constant conditions for microbial growth (including in terms of nutrient density and temperature) and the second is that these conditions may be maintained without great input of either labor or technology. On the other hand, continuous culturing, as typically practiced, is limited in the kinds of microbial growth that can be sustained. In particular this consists of a stationary phase type continuous growth at a nutrient density that in part is under organism control. These conditions nonetheless may be representative of planktonic microbial growth as it occurs in the wild given constant nutrient input. In addition, conditions can be more complex than desired due to the development of wall populations of the organisms being cultured, essentially unintended biofilms. Still, chemostats, due to their simplicity, are accessible to simple mathematical ecological theory. This theory, in turn can provide insight into evolutionary processes.
To truly appreciate chemostats and continuous culture, from a population biology perspective, one must understand them quantitatively. That is, it is necessary to wade, at least a bit, into chemostat theory and modeling. This I present conceptually, mostly showing how concepts fit together qualitatively before providing a few mathematical terms and expressions. The important point is to understand chemostats in terms of births, deaths, washout, and resource consumption. From there we can consider competition between different organism types, and even consider a means of predicting winners over losers.
For bacteria, birth rates can occur as a function of resource concentration and consequently can be modeled using a simple relationship known, in microbiology, as the Monod equation, which I describe below. More complex situations can occur in which consumers not only consume but also have predators, which I briefly dwell upon but without considering the mathematics. I end the section with an actual model, and then describe the relationship of chemostat growth to the more familiar and more basic ecological concept of logistic growth. An outline of the process as it applies to bacteria consists of the following:
The last step can be viewed as being nutrient "death" as well as bacterial "death", and so too are wastes lost via outflow, though that is not something that we will consider. In a chemostat, properly operating, all of these births and deaths completely balance such that both nutrient and bacterial densities remain constant over time.
"Births" refers to the reproductive gains of organisms. This idea is applicable not just to those organisms that in fact are "born" in the sense that, for example, placental mammals are born. Thus, births for bacteria occur as a consequence of binary fission – for most bacteria, at least – whereas for viruses births occur as a consequence of virion release from virus-infected cells. Even more broadly and as outlined immediately above, the concept of births can also be used to describe the gain of any material within a system, such as the "birth" of nutrients within chemostat growth chambers via the inflow of fresh media. Deaths, by analogy, can be viewed as any means of loss of materials such as loss of resources to consumption by bacteria, loss of resources (or organisms) to washout from chemostats, or loss of bacteria to predation.
In chemostats, washout occurs as a consequence of outflow, which, as you will recall, is directly balanced by a corresponding inflow, that is, of fresh media. The washout of culture medium from chemostats provides a number of results. First, since inflow and outflow are balanced, it means that culture volume is kept constant. Second, it means that organism wastes are both diluted (by inflow) and removed (by outflow). Third, the organisms themselves are lost to outflow. Together these properties mean that limits exist on the degree to which components of chemostats can build up in number or concentration, i.e., such as in terms of cell and waste densities. Furthermore, the loss of cells to washout can be viewed as an equivalent to "non-age specific death, predation or emigration" (Smith and Waltman, 1995) quoting Williams (1971).
Consumption is the utilization of resources by organisms such as those located within a chemostat growth vessel. In simple chemostat systems these resources are equivalent to the nutrients that flow into the chemostat growth chamber. Consumption can be accounted for on a per-organism basis as well as on a per-population basis. Rates of consumption can vary depending on an organism's growth rate, which in turn can vary as a function of nutrient densities, or in terms of population densities. Nutrient densities consequently can vary as functions of organism densities, per-organism nutrient-utilization rates, and flow rates through the chemostat. As noted above, in terms of the mathematical modeling of chemostats it can be helpful to conceptualize the consumption process as resource "death" as mediated by the microorganisms contained within the chemostat growth vessel as well as by washout or outflow from the growth chamber. Resource "births" by contrast are associated with the inflow of fresh media.
The Monod equation is a means of modeling birth rates as a function of limiting-resource densities. It is an example of saturation kinetics where rates are controlled by resource densities, which also is an example of bottom-up control. At lower resource densities, birth rates are slower but increase approximately as a linear function of those densities (that is, a doubling in resource densities results in approximately a doubling in birth rates). At higher resource densities, however, this continuing increase in rates is limited by intrinsic properties of the entity in question, which are indicated by the parameter, r, that is, an organism's intrinsic growth rate.
The idea of saturation kinetics was originally formulated in terms of enzyme kinetics (i.e., Michaelis-Menten kinetics) and has been extended to describe virus population growth rates as functions of target-bacteria density (Abedon, 2009b). A single equation may be employed to describe all of these saturation kinetics, i.e., the Monod equation (a.k.a., the Michaelis-Menten equation). The equation, in words, is:
7.1: Birth Rate = Maximal Rate × Resource-Density Function,
where "Maximal Rate" is the intrinsic growth rate of a population within an environment as seen under low-organism density conditions (again, it is the variable known as r, which contrasts with the steady-state variable known as K). The "Resource-Density Function", in turn, approaches one (that is, 1.0) when resources are not limiting, i.e., when otherwise limiting resources are present in excess amounts. Alternatively, it approaches zero when resource densities are very low (and therefore truly limiting). In the Monod equation the "Resource-Density Function" (RDF), is defined as:
7.2: RDF = [Resource] / (Constant + [Resource]),
where the brackets indicate concentration (e.g., [Resource] is density of resource as found within the chemostat growth chamber). In words, it is the concentration of a limiting resource divided by the sum of a constant, as yet undefined, and the same resource concentration. Note, in looking at equation 6.2, that as [Resource] becomes larger then RDF increases towards 1, that is, lim[Resource] → ∞ (RTF) = 1.
The "Constant" in equation 6.2 can be described as a half-maximal constant since it is the resource density that supports
7.3: Birth Rate = Maximal Rate × ½,
and which otherwise is an empirically determined value. That is, "Constant" is in units of resource concentrations and is that concentration that supports organism growth at its half-maximal rate as one measures experimentally. This concentration is indicated as "Km" which corresponds to "½Vmax" in the "Saturation kinetics" figure (above).
In the Michaelis-Menten equation this "Constant" is known as the Michaelis-Menten constant and serves as a description of the affinity that an enzyme has for a substrate. A lower value – meaning less substrate is needed to achieve half-maximal enzyme velocity – implies a higher substrate affinity whether by an enzyme or, in the case of the Monod equation, a growing organism. Organisms possessing lower values for this constant can replicate faster at lower resource densities than can organisms with higher values for this constant. In other words, in terms of maximizing RDF, the lower the value of this constant, the better. I provide a numerical example of what this statement means in the following paragraph.
If "Constant" equals an arbitrary value of 1, then RDF = ½ when [Resource] = 1 (that is, in this instance RDF = 1 / (1 + 1) = ½). At a resource density of 1 the organism thus grows half as fast as it possibly can given a "Constant" that is also equal to 1 (i.e., one-half maximal). If instead [Resource] = 10 while "Constant" equals 1, then there is ten times as much resource as is needed for the organisms to display its half-maximal growth rate. As a result, RDF = 10/11, which is a value that approaches 1 (10/11 = 0.91, or 91%). This means that the organism at this resource density is replicating almost as fast as it possible can. Keep in mind as you consider these numbers that "Constant", again, is some empirically determined value while [Resource] = 10 is ten times as much resource as [Resource] = 1.
Alternatively, if [Resource] = 0.1, then RDF = 0.1/1.1, which is approximately 10% of the maximal growth rate, while for [Resource] = 0.01, then RDF = 0.01/1.01, which is approximately 1% of the maximal growth rate. That is, at lower resource densities, Birth Rate declines more or less linearly downward towards zero. This decline can be seen in the above "Saturation kinetics" figure as the left-hand portion of the curve. The more linear aspect of the curve is also seen on the far left. Thus, at low limiting resource densities organism growth rates increase almost as a linear function of resource densities, but at high densities of limiting resources organism growth rates increase much more slowly than as resource densities increase.
Saturation kinetics also may be viewed in terms of logistic growth. Resources are abundant at lower population densities and therefore population growth rates are mostly a function of intrinsic growth rates of the organism in question. At higher population densities, by contrast, resources are no longer abundant and therefore population growth rates are less a function of intrinsic organism qualities and more a function of resource availability.
The key take home message is that as organism densities increase in chemostats, as depicted as logistic growth in the above figure, resources necessarily decline in density, resulting in a slowing of organism population growth rates. The chemostat's carrying capacity (K) occurs when resource densities have been reduced sufficiently, by organism utilization, that organism birth rates come to balance organism death rates. Logistic growth considerations are discussed again further, below.
Given the various concepts just considered, it is possible to model, using simply words, the dynamics of a simple chemostat:
7.4: Rate of change of nutrients = Inflow – Washout – Consumption
7.5: Rate of change of organisms = Births – Washout – Deaths
For nutrients, "Inflow" could be replaced with "Births", though it would have to be understood that these births occur as a direct consequence of inflow. "Washout" too could be replaced with "Deaths", except that for prey deaths occur also as a consequence of predation and it is best, therefore, to separate the two forms into "Washout" versus "Deaths" (note that "Washout" too could be replaced with "Outflow"). "Deaths" thus can take on a variety of meanings including spontaneous losses as well as due to the action of parasites or predators.
If organism deaths are left out, then the equations reduce to
7.6: Rate of change of nutrients = Inflow – Washout – Consumption
7.7: Rate of change of organisms = Births – Washout
Here organisms are increasing in number at the expense of nutrients and are lost, along with nutrients, via outflow, with the system maintained via the ongoing inflow of nutrients (that is, it represents a steady state). Unless inflow and outflow are too rapid, or organism birth rates are too slow, then birth rates will come to be defined by inflow rates, i.e., will be equal to washout rates such that both organism and nutrient densities remain constant over time at the chemostat's steady state densities (where Inflow = Washout + Consumption and Births = Washout). This steady state organism density corresponds to the chemostat's carrying capacity for the organism in question.
In logistic growth, populations at first, that is, while at low densities, experience exponential increases in numbers at rates that are approximately as fast as the environment they are in is capable of allowing. As population densities increase, growth rates decline, however. Eventually growth can reach a long-term sustainable rate that coincides with the maximum population density that the environment is capable of sustaining indefinitely. This maximum, sustainable population density is known as the carrying capacity of the environment for the population in question (K). It can be helpful to realize that while carrying capacities are not necessarily constant in real-world ecosystems, an assumption that they are constant greatly simplifies the mathematics of logistic growth. Nonetheless, environment carrying capacities can vary as a function of factors that are either intrinsic or extrinsic to the focus population. For example, either the focus organisms or its resources (e.g., prey) may evolve.
Cellular microorganism, such as bacteria, that are growing in batch culture will initially display exponential increases of logistic growth. Usually they are unable to stabilize their population density at carrying capacity, however, but instead will overshoot that density. In doing so they will reduce the capacity of the environment to support further population growth. This effect comes about because during batch growth organism loss or death can be inefficient, at least in the short term. Consequently, there is little potential to establish a stationary phase that represents a balance between ongoing births and deaths. During batch growth the population instead reaches a density at which stationary phase occurs that represents an absence of either births or deaths.
In chemostats a different trajectory occurs, one which is essentially logistic growth. That is, at low population densities growth rates approximate the intrinsic rate of increase for the broth medium employed. As population densities increase, nutrient densities decline, resulting in lower rates of population growth. Eventually population growth rates, in association with limiting resource densities, decline to the point where they equal rates of organism loss due to washout (assuming a two trophic level chemostat, i.e., no predators). This growth rate is determined by nutrient densities which in turn are a function of a combination of organism densities and organism metabolic rates. Organism metabolic rates, as captured indirectly by the Monod equation, decline per capita as nutrient densities decline and replication rates as a result also decline.
With logistic growth, greater population densities result in slower population growth rates, while with saturation kinetics reduced resource densities similarly result in slower population growth rates. The connection between the two is that it is higher population densities that give rise to lower nutrient densities – more generally, lower resource densities – and population growth rates thereby are lowered. This slowing of population growth continues until resulting nutrient densities support a rate of organism births that equals the rate of organism deaths (since if deaths exceeded birth rates then population densities would decline, resulting in increases in per capita available nutrient densities). Deaths effectively can be a consequence solely of washout in chemostats so therefore it is the rate of washout, along with the saturation kinetics resource utilization, that determines the steady state population density (i.e., K) attained by organisms. Organisms replicate, in other words, as fast as they are lost to washout and the population density at which they attain this balance is a function of a combination of nutrient availability and efficiency of nutrient utilization. Thus, it is the rate of inflow of new resource (nutrients) along with organism properties such as half-maximal growth rates, that define the carrying capacity, that is, the steady-state organism density of the chemostat.
It is possible to introduce additional trophic levels into chemostats. A standard chemostat contains two trophic levels: the nutrients and the consumers of the nutrients. If one introduces consumers of the consumers, then these additional organisms can be described as a third trophic level. Via such introductions of additional trophic levels, chemostats can support predator-prey-type interactions between organisms. Here a second mode of death is introduced for the prey organisms, i.e., other than washout, and that mechanism is predation.
The presence of these predator organisms can result in a "top-down control" of community dynamics, which is a way of saying that the predators have substantial impacts on the density of prey species. With bottom-up control, by contrast, it is the availability of resources to consumers that has the larger impact on consumer density. Usually what this means is that when few resources are available to consumers, then consumer populations never reach sufficient densities to support substantial predator densities, hence predators play little role in regulating consumer densities. Resources thus would be controlling consumer prevalence and thereby predator prevalence from the bottom of the food chain on up. When resources are less limiting to consumers, however, then their densities can be sufficiently high to support predator densities that in turn are sufficiently large to have a substantial impact on consumers, in this case impacting consumer density from the "top" of the food chain going down.
Note that with predator-prey interactions it can be unusual for an ecosystem to retain steady densities of organisms, even if conditions are otherwise held constant. Instead, there can be a tendency for communities to display what are known as Lotka-Volterra-type predator-prey dynamics. These dynamics, except at very low resource densities that give rise to the above-noted bottom-up control, do not achieve a steady state but instead involve a cycling up and down, with predator densities following prey densities up in density as well as down (the phenomenon described as top-down control). Technically, these top-down controlled ecosystems cannot be described as static, even in terms of resource densities, and therefore technically should not be referred to as chemostats, though nonetheless typically they are referred to as chemostats (and, regardless, certainly the systems remain continuous cultures). These predator-prey type community dynamics tend to be better studied from ecological rather than evolutionary perspectives.
A simple model of such dynamics is the following:
7.8: Rate of change of nutrients = Inflow – Washout – Consumption
7.9: Rate of change of organisms (prey) = Births – Washout – Deaths
7.10: Rate of change of organisms (predators) = Births – Washout – Deaths
Here "Deaths" for prey is a consequence explicitly of predation. "Deaths" can take on a variety of meanings for predators, or for simple models may not occur at all. In addition, there can exist complicating delays between prey deaths and subsequent predator births, none of which will we consider here.
Despite the apparent simplicity of this model, the dynamics it produces can be impressively complicated, though less so at lower limiting-resource densities where bottom-up rather than top-down control is a more important determinant of community dynamics. On the other hand, at sufficiently high nutrient densities, predators will tend to drive planktonic prey to extinction.