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A chemostat (abbreviation for CHEMical envirOnment is STATic) is a biologically active environment TASKS (or a bioreactor). The central part of this system is a
A chemostat (abbreviation for "CHEMical envirOnment is STATic") is a biologically active environment TASKS (or a bioreactor). The central part of this system is a vessel (tank) that contains a microorganism, such as a bacterial ot yeast species. The fresh medium (support for the growth of a population) is continuously 1. A scientist sets up a chemostat experiment which is governed by Eq. (4) and (5). The scientist provided to this vessel through an inflow. At the same time, excess nutrients and metabolic end products are experimentally learned the constants and initial conditions as given in Table 1 for each project group. continuously removed through an outflow to maintain the culture volume invariant. Chemostats are used (a) Simulate this system and plot the variables N(t) and C(t) of the experiment versus time t. products. (b) Plot the orbit (N(t),C(t)) in the state space for the given initial condition in Table 1. 2. The scientist aims to fix the population and nutrition in the tank at a steady state (N,C). (a) Analytically find all steady states (N,C) as functions of and . Hint: Suppose that x=f(x). If a point x is a steady state (equilibrium) of this system, then x(t)=x is a solution of the system of ODEs; therefore, f(x)=0. (b) Find parameters and such that each of the equilibria is biologically realistic. Hint: Populations or concentrations cannot take any number from real numbers, they should be Figure 1: A chemostat diagram featuring inflow (nutrient support) and outflow (disposal). selected carefully. Mathematical Description of the Chemostat: The volume of the solution V in the culture vessel (c) An equilibrium point (N,C) for each individual project is provided in Table 2. Find the pair is constant. Inflow and outflow rates, F are equal and F is a constant in volume/time. The bacterial of and analytically for which the given (N,C) is an equilibrium of the system. concentration at time t is given with N(t) in mass/volume. The growth medium concentration is supplied with a constant (C0) and the concentration in the vessel at time t is given with C(t) in mass/volume. 3. In such real experiments, it is expected to observe some (small) noise affecting the applications. Under The concentrations in the culture tank are well-mixed, meaning that all the ingredients in the tank are such small fluctuations, it is important to understand the stability of the steady states. homogeneously distributed. We will use N and C for N(t) and C(t), respectively, for the sake of simplicity. (a) Apply local stability analysis to investigate the stability of all steady states (N,C). The chemostat is modeled with the following coupled differential equations as follows: (b) By simulation, check whether or not the orbits starting close to the equilibrium point (N,C) given in Table 2 converge to it. Provide a phase portrait to show your finding. 4. The scientist adds a valve (or filter) to adjust the amount of nutrition disposal from the tank. Then, the equation has an additional control parameter w, and the equation reads as where is the nutrition consumption and r(C) is the growth rate of the bacteria population depending on the availability of the nutrient C. Michaelis-Menten Kinetics is one of the best-known models of dtdN=1+CCNN,dtdC=1+CCNwC+, enzyme kinetics that describes the rate of enzymatic reactions by r(C)=kn+CkmaxC where w[0,1], meaning that if w=0 no nutrition outflow, if w=1 then there is no mutrition filter. where kmax and kn are constants that can be learned from experimental data. If we assume that MichaelisMenten Kinetics is an appropriate choice for our r(C), then Eqs. (1) and (2) can be written as follows: (a) Suppose that the scientist has parameters and fixed as in Table 1 , and changes the parameter w slowly from w=1 to w=0. Check analytically if any transition occurs for the dynamics of dtdN=1+CCNNdtdC=1+CCNC+ the system near each steady state while w is decreasing from 1 to 0. (b) If there exists a transition parameter value at wc, draw the phase portraits of the system before and after the transition. where =FVkmax and =k0C0 Hint: To have sufficiently visible and clear phase portraits for before and after the transition. choose the parameters wot too close to the critical value wc- Table 2 Table 1 A chemostat (abbreviation for "CHEMical envirOnment is STATic") is a biologically active environment TASKS (or a bioreactor). The central part of this system is a vessel (tank) that contains a microorganism, such as a bacterial ot yeast species. The fresh medium (support for the growth of a population) is continuously 1. A scientist sets up a chemostat experiment which is governed by Eq. (4) and (5). The scientist provided to this vessel through an inflow. At the same time, excess nutrients and metabolic end products are experimentally learned the constants and initial conditions as given in Table 1 for each project group. continuously removed through an outflow to maintain the culture volume invariant. Chemostats are used (a) Simulate this system and plot the variables N(t) and C(t) of the experiment versus time t. products. (b) Plot the orbit (N(t),C(t)) in the state space for the given initial condition in Table 1. 2. The scientist aims to fix the population and nutrition in the tank at a steady state (N,C). (a) Analytically find all steady states (N,C) as functions of and . Hint: Suppose that x=f(x). If a point x is a steady state (equilibrium) of this system, then x(t)=x is a solution of the system of ODEs; therefore, f(x)=0. (b) Find parameters and such that each of the equilibria is biologically realistic. Hint: Populations or concentrations cannot take any number from real numbers, they should be Figure 1: A chemostat diagram featuring inflow (nutrient support) and outflow (disposal). selected carefully. Mathematical Description of the Chemostat: The volume of the solution V in the culture vessel (c) An equilibrium point (N,C) for each individual project is provided in Table 2. Find the pair is constant. Inflow and outflow rates, F are equal and F is a constant in volume/time. The bacterial of and analytically for which the given (N,C) is an equilibrium of the system. concentration at time t is given with N(t) in mass/volume. The growth medium concentration is supplied with a constant (C0) and the concentration in the vessel at time t is given with C(t) in mass/volume. 3. In such real experiments, it is expected to observe some (small) noise affecting the applications. Under The concentrations in the culture tank are well-mixed, meaning that all the ingredients in the tank are such small fluctuations, it is important to understand the stability of the steady states. homogeneously distributed. We will use N and C for N(t) and C(t), respectively, for the sake of simplicity. (a) Apply local stability analysis to investigate the stability of all steady states (N,C). The chemostat is modeled with the following coupled differential equations as follows: (b) By simulation, check whether or not the orbits starting close to the equilibrium point (N,C) given in Table 2 converge to it. Provide a phase portrait to show your finding. 4. The scientist adds a valve (or filter) to adjust the amount of nutrition disposal from the tank. Then, the equation has an additional control parameter w, and the equation reads as where is the nutrition consumption and r(C) is the growth rate of the bacteria population depending on the availability of the nutrient C. Michaelis-Menten Kinetics is one of the best-known models of dtdN=1+CCNN,dtdC=1+CCNwC+, enzyme kinetics that describes the rate of enzymatic reactions by r(C)=kn+CkmaxC where w[0,1], meaning that if w=0 no nutrition outflow, if w=1 then there is no mutrition filter. where kmax and kn are constants that can be learned from experimental data. If we assume that MichaelisMenten Kinetics is an appropriate choice for our r(C), then Eqs. (1) and (2) can be written as follows: (a) Suppose that the scientist has parameters and fixed as in Table 1 , and changes the parameter w slowly from w=1 to w=0. Check analytically if any transition occurs for the dynamics of dtdN=1+CCNNdtdC=1+CCNC+ the system near each steady state while w is decreasing from 1 to 0. (b) If there exists a transition parameter value at wc, draw the phase portraits of the system before and after the transition. where =FVkmax and =k0C0 Hint: To have sufficiently visible and clear phase portraits for before and after the transition. choose the parameters wot too close to the critical value wc- Table 2 Table 1
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