The above is a schematic of a four-unit protein purification process aimed at extracting the P and B isomers of protein A. Assume the depicted process has already achieved equilibrium and steady state. The external feed into the first unit consists of a pure stream of protein A with volumetric flowrate QA,in and mass concentration CA,in.. The first unit, the isomerization chamber, is a well-mixed vessel with volume 10L. In this chamber, the solution conditions allow the following conformational transitions happen: Ak1,k1B;Bk2,k2P;Pk3,k3A where ki and ki are the forward and backward rate constants, respectively. You can assume first order rate kinetics, e.g.: rAB=k1CAVrBA=k1CBV which are the rates at which A is converted to B and vice versa. Outside of this chamber, no isomerization is observed. All known rate constants are given below: In the second unit, an external feed of pure monoclonal antibody M is introduced with volumetric flowrate QM,in and mass concentration CM,in.. The second unit is a well-mixed vessel of volume 25L in which M selectively binds to isomer P with high affinity. We can assume the binding is irreversible and follows the "reaction": P+nMk4PMnrPMn=k4(CMV)n(CPV)1 However, the rate constant k4(kgnmin1) and reaction order n are presently unknown. Instead, using the provided experimental data in "mab_kinetics_data.dat", these model constants can be found. Make sure you consider the units and remember that mass is always conserved. The third unit is a protein affinity column used to separate, concentrate, and extract proteins. The column is setup such that one exit stream (with flowrate Qp,out ) contains all of the incoming PMn and M; in other words, this is a concentrated stream of purely PMn and M. The other exit stream (with flowrate QAB,out ) is a concentrated stream of purely A and B, but does not extract all incoming A and B. The third stream, i.e. the recycle stream with flowrate Qcyc, only consists of A,B, and P, but maintains the same A and B mass concentrations of the column input. The fourth unit is a conversion reactor of volume 3L containing heterogeneous enzymes that irreversibly isomerize A into B. The change in mass concentration of A (and B ) is governed by: CA=CA00k5(t)dtCB=CA where is the residence time (=V/Q),CA0 is the incoming mass concentration of A(kg/L) into the reactor, and k5 is an effective conversion rate (1/min). While an analytical expression for k5 is unknown, a calibrated profile for k5 as a function of t has been provided in "AB_conversion_PFR.dat". Apply an appropriate iterative method (e.g. Gauss-Newton or Steepest Descent) and other required numerical techniques to solve and report all unknown mass concentrations (kg/L) in the streams depicted above; achieve a tolerance of 0.000001 (or 1e-6) kg/L. Also report the percent yield of PMn, which is given by: Yield%=100massflowofAin+massflowofMinmassflowofPMnout After doing all of the above, generalize your model into a function such that the percent yield of Pn is returned given any value of CA,in,CM,in, and V of the AB conversion reactor. Include checks for unphysical behavior and report error messages accordingly. Note that any real number for V in the range 1V20L should be acceptable for your function. Using your new function, compute the percent yield of Pn as both CM,in(=0.1,0.15,0.2,0.25, 0.3,0.35,0.4,0.45, and 0.5kg/L) and V(=2,4,6,8,10,15, and 20L) are varied (a total of 63 different operating conditions). Report these values in the form of a plot ( x-axis =CM,in,y-axis = yield, V represented as different lines) with proper labels, legends, etc. Also report the operating conditions (within the ones explored above) which result in the maximum PMn yield. The above is a schematic of a four-unit protein purification process aimed at extracting the P and B isomers of protein A. Assume the depicted process has already achieved equilibrium and steady state. The external feed into the first unit consists of a pure stream of protein A with volumetric flowrate QA,in and mass concentration CA,in.. The first unit, the isomerization chamber, is a well-mixed vessel with volume 10L. In this chamber, the solution conditions allow the following conformational transitions happen: Ak1,k1B;Bk2,k2P;Pk3,k3A where ki and ki are the forward and backward rate constants, respectively. You can assume first order rate kinetics, e.g.: rAB=k1CAVrBA=k1CBV which are the rates at which A is converted to B and vice versa. Outside of this chamber, no isomerization is observed. All known rate constants are given below: In the second unit, an external feed of pure monoclonal antibody M is introduced with volumetric flowrate QM,in and mass concentration CM,in.. The second unit is a well-mixed vessel of volume 25L in which M selectively binds to isomer P with high affinity. We can assume the binding is irreversible and follows the "reaction": P+nMk4PMnrPMn=k4(CMV)n(CPV)1 However, the rate constant k4(kgnmin1) and reaction order n are presently unknown. Instead, using the provided experimental data in "mab_kinetics_data.dat", these model constants can be found. Make sure you consider the units and remember that mass is always conserved. The third unit is a protein affinity column used to separate, concentrate, and extract proteins. The column is setup such that one exit stream (with flowrate Qp,out ) contains all of the incoming PMn and M; in other words, this is a concentrated stream of purely PMn and M. The other exit stream (with flowrate QAB,out ) is a concentrated stream of purely A and B, but does not extract all incoming A and B. The third stream, i.e. the recycle stream with flowrate Qcyc, only consists of A,B, and P, but maintains the same A and B mass concentrations of the column input. The fourth unit is a conversion reactor of volume 3L containing heterogeneous enzymes that irreversibly isomerize A into B. The change in mass concentration of A (and B ) is governed by: CA=CA00k5(t)dtCB=CA where is the residence time (=V/Q),CA0 is the incoming mass concentration of A(kg/L) into the reactor, and k5 is an effective conversion rate (1/min). While an analytical expression for k5 is unknown, a calibrated profile for k5 as a function of t has been provided in "AB_conversion_PFR.dat". Apply an appropriate iterative method (e.g. Gauss-Newton or Steepest Descent) and other required numerical techniques to solve and report all unknown mass concentrations (kg/L) in the streams depicted above; achieve a tolerance of 0.000001 (or 1e-6) kg/L. Also report the percent yield of PMn, which is given by: Yield%=100massflowofAin+massflowofMinmassflowofPMnout After doing all of the above, generalize your model into a function such that the percent yield of Pn is returned given any value of CA,in,CM,in, and V of the AB conversion reactor. Include checks for unphysical behavior and report error messages accordingly. Note that any real number for V in the range 1V20L should be acceptable for your function. Using your new function, compute the percent yield of Pn as both CM,in(=0.1,0.15,0.2,0.25, 0.3,0.35,0.4,0.45, and 0.5kg/L) and V(=2,4,6,8,10,15, and 20L) are varied (a total of 63 different operating conditions). Report these values in the form of a plot ( x-axis =CM,in,y-axis = yield, V represented as different lines) with proper labels, legends, etc. Also report the operating conditions (within the ones explored above) which result in the maximum PMn yield