First Order Reaction Sequence for solutes A to E in a stirred tank flowing reactor with constant volume, Vol, and step jumps in flow and in reaction rates. This is the basis for a series of models to account for substrate capacitance in enzymatic networks.
A reaction sequence A-->B-->C-->D-->E can be represented many ways to approximate the biological form of the reactions. This version is the simplest: first order unidirectional reactions, here written as clearances, or a progress curve as in a bioreactor, but with additional factor of a flow through the mixing chamber. Both the flow and the reaction are expressed as first order terms for each solute in the sequence, both are "clearances" or removal processes, and compete, e.g. if for the inflowing initial solute A, the flow, ml/sec, is the same as the clearance by the reaction A-->B, G, then the outflow concentration of A in the steady state goes to 50% of the inflow concnetration. (Do this by setting Flow1 = Flow2 = 0.05 ml/sec and setting GA1 and GA2 also to 0.05 and run the program. The other reaction rates, to form C, D, and E are set identidal to that to form B from A, the result is that steady state concentrations for A, B, etc go to 0.5, 0.25, 0.125, 0.0625, and 0.03125, all at one half of its predecessor. This program illustrates the transient delays between steady state, and the form of the transients. There is no enzyme here, and no binding of the reactants in the process of the chemical reaction. All of the transient delay is due to the combination of flow and the reaction. The initial conditions are zero for all solutes so the time constant for the initial entry would be simply the volume divided by flow, Vol/Flow1, in seconds, if there were no reaction. The reaction, augmenting the disappearance of A, shortens the time constant so that it is Vol/(GA + Flow1). The second transient is due to a step increase (or decrease) in flow at time TFjump (at t=30 s with this parameters set (LinearA2E.FGjump181021). The third transient is a step change in the reaction rates at time TGjump. The solution to the differential equation for solute A is accomplished using the numerical solvers, but it is also solvable analytically. The three transients, at t = 0, t= 30 s and t=60 s, are expressed in three analytical equations. Their sum fits exactly the numerical solutions to the systems equations. VERIFICATION! Try this out using the difference ODE solvers: it turns out the DOPRIS5 (an advanced RungeKutta algorithm) gives a more precise fit to the analytical solution than either of the more powerful stiff solvers, CVODE or Radau. At the steep parts of the transients DOPRIS5 is still good to 7 decimal digits, while the other supposedly superior solvers, CVODE and Radau are good only 5 or 6 decimal digits. In the steady state they all give the same correct answers. For solutes B through E, the analytical solutions are more complex, and have not been developed. For these it is much faster to use the numerical solutions to the ODEs; the analytical solutions for solute E would be, because of their complexity, much slower to compute, and maybe not even as accurate as the numerical solutions, even for this rather simple model system. What we know is that the steady state solution match exactly to the predicted steady state values for A through E. This model provides a reference for subsequent models with different reaction forms, namely Michaelis-Menten reactions (Model #0424 ), then fully developed kinetic reactions accounting for enzyme binding and product release (model # ???), and then a variant on Michaelis-Menten that approximates the effect of solute binding to the enzymes (Model #0425). That is done to correct the inadequate kinetics of M-M expressions. This family of models is designed for use as a set of tools to determine the magnitude of the buffering capacitance and consequent delay in transient responses in enzyme systems.
Figure: Progress curves for a sequence of reactions from substrate A to E in a compartment with flow through the compartment. All substrates have initial concentration of zero with A in (CinA) set to 1 mM. Flow doubles from 0.025 to 0.05 ml/sec at t= 30 sec. Consumption (the flow of fluid cleared of substrate per unit time) doubles from 0.1 to 0.2 ml/sec at t= 60 sec. GA is the consumption of substrate A and Anal3 is the analytical solution for substrate concentration A as a function of time.
The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.
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Cascante M, Melendez-Hevia E, Kholodenko B, Sicilia J, and Kacser H. Control analyis of transit time for free and enzyme-bound metabolites: physiological and evolutionary significance of metabolic response times. Biochem J 308: 895-899, 1995. Bassingthwaighte JB. Capacitance in metabolic netowrks. 2019 (in prep for submission to Biophys J)
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The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.
Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.