6.3 Microbial growth in a chemostat The chemostat is a wiely-use apparatus use in the stuy of microbial physiology an ecology. In such a chemostat also known as continuous-flow culture), microbes such as bacteria an yeast cells) can be grown uner precisely controlle conitions. Fig. 6.9 shows a iagram. The microbes grow in the main culture vessel which contains a well-stirre culture meium. This meium is replenishe from a reservoir shown on the left. As a constant volumetric flow F of fresh meium enters the culture vessel propelle by a pump, an overflow 21 allows an equal flow to leave the culture vessel, so that the vessel retains a constant volume of culture at all times. The experimenter can control the pump an thus the flow F ; the reservoir nutrient concentration C R can of course also be varie at will by the experimenter. However, in the classic set-up, the experimenter keeps both F an C R at fixe values an waits for the system to settle on a steay state. 21 Such a evice is alternatively known as a porcelator.
98 Growth of populations an of iniviuals Fig. 6.9 The continuous-flow culture vessel or chemostat. nutrient reservoir pump cell culture instruments for analysis waste 22 Derive eqn 6.22) from eqn 6.21) an the efinitions given. 23 Check this; apply the efinition D = F/ first. 6.3.1 Basic chemostat equations Let W enote the microbial biomass in the culture vessel an N the molar amount of nutrient in the vessel. We have two conservation equations: t W = µw F W 6.20) t N = F C R F N SµW. 6.21) The term µw represents the increase in biomass ue to growth; the factor µ is the per capita reprouction rate, which we calle relative growth rate before but which is known in these systems more commonly as the specific growth rate. Biomass leaves the system at a rate F W/ which makes intuitive sense if W/ is viewe as the biomass concentration in the culture vessel. The secon equation escribes the amount of nutrient. The first two terms on the right-han sie again escribe the influx an efflux of nutrient, whereas the thir term escribes the conversion of nutrient into biomass. The factor S expresses the stoichiometry of this conversion. The ratio F/ plays an important role in the operation of the chemostat an therefore has a name of its own: it is calle the ilution rate, traitionally enote D. For the nutrient concentration C = N/ we have the following ynamics: 22 t C = D C R C) Sµ W. 6.22) The steay state is, as per usual, characterize by the conitions t W = 0 an tc = 0, whence23 µ = D an C = C R SW/ at steay state. The experimenter controls the ilution rate D, which is just a setting on the pump in the apparatus shown in Fig. 6.9. Since µ = D at
6.3 Microbial growth in a chemostat 99 steay state, the experimenter effectively forces the cells to grow at the rate she esires. In fact, this phenomenon is the great avantage of continuous culture: after the chemostat settles on steay state, the cells can be sample an their physiological state at various stationary relative growth rates can be stuie. Waiting for the system to attain steay state is stanar practice. However, as we shall presently see, the transient behaviour actually contains important information about the unerlying biology. The specific growth rate µ will generally vary with time, as a function of the internal state of the organism an perhaps also the ambient conitions such as the nutrient concentration C). However, we can alreay say something about the ynamical behaviour of the chemostat even when, for now, we leave µ as an unspecifie function of time. Consier the following quantity: X = W + N S. 6.23) This represents the actual biomass in the culture vessel W ) plus the biomass equivalent of the nutrient in the vessel N/S). Its time course is given by: 24 Xt) = C ) R S + CR S X 0 exp{ Dt} 6.24) 24 Derive the ifferential equation for X: t X = D C R S DX an solve it to fin eqn 6.24). where X 0 is the initial conition, the value of X at time t = 0. From eqn 6.24) we can infer that after an initial transient with relaxation time D 1 which occurs if we suenly change the reservoir concentration C R ), the quantity X settles own on its equilibrium value C R /S. From then onwars the following useful conservation law for the culture vessel applies: C R = Ct) + S W t). 6.25) This equation states that biomass ensity an nutrient concentration are complementary. 25 25 Can you think of a quick way to see why this result is not so surprising? 6.3.2 Logistic growth in the chemostat To procee further, we nee to postulate a moel for the relative growth rate µ. One simple moel is as a simple proportionality: 26 µ = ϑc. Using eqns 6.25) an 6.20) we obtain: t W = W ϑc R D ϑs ) W. 6.26) But this is just the logistic growth equation in isguise 27. Of course, we must not forget about the initial transient while Xt) approaches C R /S, so the growth curve may not be logistic uring some initial perio, but if we pick the initial nutrient an biomass ensity in the culture vessel so that they satisfy eqn 6.25), this equation will be vali for all t. 26 Although simple, this moel is reasonable since the cells cannot grow when there is no nutrient, so we woul expect µ = 0 when C = 0, an we woul also suppose that a linear approximation is reasonably accurate for some range of sufficiently low values of C. 27 Cast eqn 6.26) in the traitional form of the erhulst equation by efining r an K in terms of the present parameters
100 Growth of populations an of iniviuals Fig. 6.10 Glycogen poly-glucose) granules in Escherichia coli cells: an example of nutrient storage in microorganisms. 6.3.3 Towars a moel for the relative growth rate In the last section the chemostat equations were introuce. The crucial task for the mathematical moeller is to relate the relative growth rate µ to the ambient conitions an/or the internal state of the cells. In this section we follow this moelling process in etail, to highlight the backan-forth between observations an theory cf. Fig. 1.6). As we saw before, the microbial culture in the vessel settles, after a while, on a steay state when the ilution rate D of the chemostat an the reservoir nutrient concentration C R are kept at a fixe value. In the steay state, we can observe the biomass W, the nutrient concentration C, an the relative growth rate µ which at steay state equals the ilution rate D the bar over the symbols signifies stationarity in time). Obtaining these ata for a range of ilution rate settings, the experimenter accumulates a ata set consisting of triples {µ, W, C} i.e. a table with three columns). The next step is to look for systematic relationships in this ata set. Plotting µ against C, the experimenter fins that the ata points closely ahere to the following simple mathematical expression: µ = α C β + C 6.27) where α an β are two positive parameters which the experimenter is able to estimate using the non-linear least squares criterion see Section 4.2). This hyperbolic relationship immeiately remins the experimenter of the Michaelis-Menten relationship for the nutrient uptake system see Section 2.3), which suggests that β represents the K m of this uptake system. Moreover, this ientification suggests a more general principle: The specific growth rate is irectly proportional to the saturation fraction of the uptake system which is given by C/K m + C)). I) This principle is propose to apply also to the transient situation, where C is time-varying. Next, the experimenter acquires an instrument which allows her to analyse the chemical contents of the cells. It is foun that the cells have internal stores of the nutrient. The total amount of this nutrient in the culture at steay state is enote Q Q for quota). The experimenter calls the ratio Q/W the relative nutrient quota. Plotting the steaystate growth rate against the relative nutrient quota, again a very goo agreement is foun with a simple mathematical relationship: µ = γ Q/W δ + Q/W 6.28) 28 By the general state we mean not only the steay state, but transients as well. where γ an δ are two new positive parameters. This suggests another principle for the general 28 state: The specific growth rate is a hyperbolic function of the relative nutrient quota. II) We now have two equations, an two principles, an the experimenter
6.3 Microbial growth in a chemostat 101 wants to etermine which is vali. Let us begin with the easy question: can equations 6.27) an 6.28) both be vali escriptions of the steaystate ata? The answer is: certainly. Moreover, we can inform the experimenter 29 that she will fin another hyperbolic relationship if she 29 Derive eqn 6.29) from plots the relative nutrient quota against the concentration: eqns 6.27) an 6.28). Q W = αδ γ α C βγ/γ α) + C. 6.29) To her elight, this comes out as preicte. The choice between principles I) an II) is more complicate. These are putative laws which exten beyon the regime uner which the ata were obtaine the steay state). Fortunately, moelling can help here, by preicting how the culture will respon ynamically to perturbations, if one or the other principle is assume. The perturbations might be step changes in ilution rate or fee reservoir nutrient concentration. For each perturbation, we can evaluate the chemostat equations to work out what shoul happen. Principle I) leas to the following system of ifferential equations: 30 30 Derive eqns6.30) an 6.31) by combining the chemostat equations of Section 6.3.1 with eqn 6.27). t W = W α C ) β + C D 6.30) t C = D C R C) αsw C β + C. 6.31) servation law as our starting point: 32 The experimenter carries out the perturbations an fins a poor agreement with the preictions: in general, the cells seem to respon more sluggishly to changes in nutrient concentration C. For instance, in a wash-out experiment were C R is set to zero at time t = 0 this is one by connecting the supply tube to a reservoir containing meium without nutrient), the preicte curve for the nutrient concentration in the culture vessel agrees very well with the ata, whereas the biomass shows a istinct lag before it starts to ecrease, as shown in Fig. 6.11. Also, the experimenter enlists the help of a biochemist who etermines the K m of the nutrient uptake system, an fins that the latter is much larger than β, whereas principle I) asserts that they shoul be about equal, 31 Consier a microbiologist who allowing for slight variations ue to the fact that ifferent experimental firmly believes in principle I). As far approaches were use. 31 as he is concerne, the chemostat curve is simply a way of etermining K m. The experimentalist now asks us, the moellers, for another ynamic He asserts a iscrepancy between two preiction, this time assuming principle II). Since any ecent moel methos of measuring K m. What must satisfy basic conservation principles, we might as well take a con- coul be one to change this microbiologist s min? 32 t Q = mw C SµW DQ 6.32) K m + C where m W expresses the maximum nutrient uptake rate, m being a proportionality constant that expresses how much of the uptake machinery is present per unit biomass. The constant S represents the stoichiometric conversion of nutrient into biomass, an D is the ilution rate of the chemostat. The steay state associate with eqn 6.32) is compatible Justify eqn 6.32).
102 Growth of populations an of iniviuals 10 8 Fig. 6.11 Comparison of biomass top curve) an nutrient concentration bottom curve) preicte from principle I) with ata in a wash-out experiment. Whereas the ata o not accor well with principle I), they are well escribe by principle II) curve not shown, as the ata correspon closely to the curve obtaine uner principle II). 6 4 2 0 0 2 4 6 8 10 time t 33 Demonstrate this. Start from steay state assume C constant, put Q = 0 an set µ = D) an compare the equation you obtain with equa- t tions 6.27) 6.29). 35 Derive eqn 6.34); use the quotient rule. with equations 6.27) 6.29) only if the parameter values satisfy certain ientities. Specifically, compatibility requires the following parameter ientifications: 33 m µs α = µ ; β = K m ; γ = µ ; δ = S 6.33) µs + m µs + m where we have use the symbol µ to inicate the maximum growth rate. 34 On these ientifications we obtain a very simple equation for the ynamics of the relative nutrient quota: 35 Q t W = µ m µ C K m + C Q W ). 6.34) This equation states that the relative nutrient quota relaxes exponentially towar the saturation fraction of the uptake machinery. This equation forms a ynamical system together with the chemostat equations. which for principle II) take the following form: t W = W µ Q/W ) S + Q/W D 6.35) t C = D C R C) mw C K m + C. 6.36) This moel explains the iscrepancy foun in the wash-out experiment. Since the relative growth rate epens irectly on the relative nutrient quota Q/W which lags behin the external nutrient concentration, microbial growth tens to continue at almost the steay state rate for some time after C R has been set to zero. This may seem to violate basic conservation principles, since the external nutrient concentration behaves 34 There is a conceptual problem with the ientification δ = S if we view principle II) an eqn 6.28) as a escription of how the microbe s internal regulatory system ajusts the growth rate as a function of the relative nutrient quota. On this interpretation, δ is a property of the molecular machinery involve in regulation, an hence some complex compoun involving the rate at which certain molecules bin DNA et cetera. It is not a priori clear why this compoun shoul equal the macrochemical conversion constant S. This oes not really present an insurmountable problem for the experimental finings: if δ an S are roughly of the same orer, the actual equations are somewhat more complicate than hyperbolas but they will still have the same general shape an conform to the ata. Similarly, instea of eqn 6.34) we obtain slightly more complex ynamics.
much the same uner principles I) an II). However, the aitional growth in the latter case is effectively pai for by epleting the nutrient store in the cells. Does the story en there? Shoul the experimenter conclue that principle I) is false an that principle II) is true? No. The situation is more subtle. A careful stuy of the two systems of ynamical equations reveals that the lag effect will be very small if the relative nutrient quota stays well below S. This will be the case if the parameter m is relatively small. This parameter expresses how many builing blocks glucose, amino acis) the microbial cell allocates towar the molecular machinery which processes the nutrient. Thus, the organism can in fact switch between principle I) moe an principle II) moe by ajusting the amount of machinery it synthesises. 36 Moreover, microbes generally require more than one chemical species from their environment to grow an subsists e. g. a carbon nutrient, an energy nutrient, an various minerals) an thus the growth rate will epen on the ambient concentrations of various substances as well as the internal stores of these species). This aitional complexity is taken into account in multiple nutrient limitation theory, 37 which relies on further iterations between experimental observations an theoretical preictions. 36 To be precise, the cell also has the option of partially) isabling the catalytic machinery that is alreay in place, for instance by means of attaching chemical moieties to this machinery. 37 See Exercise 6.9 for a brief introuction.