5.3 Problem specification
5.3.1 Objective function
Many criteria may be selected for which an optimal temperature profile could be found. The general objective functional (see Eq (5.7)) can handle most of them. The cost of heating the starting material and the reactor was neglected. The reaction time and amount of added enzymes should be minimized while maximizing the yieldΦof the product. The following functional formalizes these criteria:
J=−tf·M.U.h−1+C1·φ
| {z }
ff
−C2·E˜RCS(t=0) +E˜ECS(t=0) 2
| {z }
fi
(5.37)
The weighting factorsC1 in Monetary Units (M.U.) and C2 in M.U. convert yield φ and normalized enzyme concentration into the profitJ expressed inM.U.. If no other value is stated,C1=6 M.U.and C3=96 M.U.. In this study, the yield φ is defined as the amount of monomer units present as free monomers and dimers at the final time divided by the total initial amount of monomers, which is a similar definition to the one used by Ouyang et al. [17]
φ=n1(tf) +2·n2(tf) P˘kmax
i=1 i·ni(t=0)
·100%. (5.38)
E˜RCS(t=0)andE˜ECS(t=0)are the initial concentrations of the Random-chain scission (RCS) and End-chain scission (ECS) performing enzymes divided by the initial concentration of enzyme necessary to cut the percentage of bonds specified in the case definition. The concentration of the enzyme can be varied in some cases to obtain a more economical process. Furthermore, in all cases the temperature can be varied freely.
IfC1 andC2 are set to arbitrary positive values and the initial enzymes concentrations, the reac-tor operation time, and the temperature profile are optimized simultaneously, one obtains the optimal operating conditions. However, for ease of discussion, four simpler objectives are investigated in this work:
Maximizing yield in fixed time with fixed enzyme concentration
If the reaction time of the process is fixed, e.g. due to limitations of the up- or downstream processes, one might be interested in finding a temperature profile that enables the highest possible yield. Thereby, the profit is maximized. By fixing the final timetf as well as the enzyme concentration to be constant (E˜RCS(t=0) =E˜ECS(t=0) =1), the benefit of temperature profile optimization alone can be studied.
Minimizing reaction time with a fixed yield and enzyme concentration
A certain yield might be required but the reaction time tf of the process can be varied. Then it is desirable to find a temperature profile which minimizes the reaction time. In this case, the enzyme concentration is again kept constant (E˜RCS(t=0) =E˜ECS(t=0) =1). By specifying a desired yield Φdes, the time optimal problem (i.e., the shortest possible processing time) is formulated. If the yield is fixed, the functiongis given asΦ(tf)−Φdes.
Trade-off between reaction time and yield with fixed enzyme concentration
A longer reaction time allows higher yield. But in order to handle the same amount of starting material in a given time the reactor needs to be larger or there must be more reactors which increases the capital costs. For this case, the enzyme concentration is kept constant (E˜RCS(t=0) =E˜ECS(t=0) =1). By settingC1 to an arbitrary positive value the goal of maximizing profit for a specified enzyme dosage is realized. C1 indicates how much the product is worth, e.g. a value ofC1=6 M.U.means that a one percent increase in yield justifies an increase of reactor occupation by six hours.
Trade-off between reaction time and enzyme concentration at a fixed yield
Increasing the enzyme concentration allows one to reach a fixed yield in shorter time. However, due to the increased cost of enzymes this might make the process less economically viable. By settingC2to an
arbitrary positive value, and specifying a desired yield, the profit at this yield is maximized.C2indicates the cost of the enzyme, e.g. a value ofC2=96 M.U.means that the initial enzyme concentration needed to cut the desired percentage of bonds costs as much as operating the reactor for four days. This optimization can be performed either by keeping the ratio betweenERCS andEECSconstant (Constant enzyme mixture composition), in which case only the amount of enzyme cocktail is optimized, or by letting the optimizer find the best ratio (Optimizing the enzyme mixture composition), in which case the enzyme mixture is also optimized. Both problems might be encountered in industrial practice depending on whether one uses a mixture of enzymes or two pure enzymes.
5.3.2 Parameters
Ho et al. [48] have experimentally validated a Population Balance Model (PBM) of enzymatically cat-alyzed depolymerization of linear and soluble starch at isothermal conditions. The enzymes were de-scribed by a simplified version of the Subsite Theory. The Michaelis constant was computed by the Subsite Theory, but the catalytic constant was approximated by a ramp function. Furthermore, they assumed that the single enzyme performed ECS. In another paper [49] they performed simulations that suggest that the presence of a RCS performing enzyme is beneficial for the conversion. As this model is able to describe experimental data an analog model was used. Later on a study to investigate the effects of using the product distribution predicted by the Subsite Theory and the effect of branching is performed.
Base case
The model used in this study takes temperature effects into account by using the equations derived with the Equilibrium model. Both the catalytic constant and the Michaelis constant are computed using the Subsite Theory but instead of using the Product Distribution Function (PDF) derived using the Subsite Theory the simplified extreme cases of RCS and ECS are used. The values of the parameters of the base case are provided in Table 5.1 and 5.2. Each initial enzyme concentration is chosen such that at a temperature of50◦Cduring a reaction time of24 hrsthe single enzyme cuts 5% of the bonds.
Parameter study
The amount of bonds cut is not taken from a literature reference. It is therefore desirable to study the effect of the amount of enzymes on the optimal temperature profile and process performance.
Accordingly, the case "Doubled RCS activity" with twice as many bonds cut by RCS and "Doubled ECS activity" with twice as many bonds cut by ECS were considered. As with all other cases, the other parameters are kept constant at the value of the base case.
In order to study the influence of the equilibrium temperature, two additional cases were considered.
In the case "Equal equilibrium temperature" both equilibrium temperature are set to67.8◦C, and in the case "Switched equilibrium temperature"TRCS,eq is set to the base case value ofTECS,eqand vice versa
Table 5.1: Parameters of the base case [3] (With kind permission from Springer).
Name Symbol Value Unit
Mass concentrationa 38.4 g l−1
Mean amount of monomer unitsa 160
Polydispersitya 1.325
Maximal amount of monomer unitsa k˘max 878
Enthalpic change associated with enzyme folding RCSc ∆Heq,RCS 192 kJ mol−1 Activation energy of hydrolysis RCSc ∆G†cat,RCS 6.9 kJ mol−1 Activation energy of irreversible inactivation RCSc ∆G†inact,RCS 105 kJ mol−1
Equilibrium temperature of RCSc Teq,RCS 90 ◦C
Position of the catalytic center of RCS z˘− 6
Enthalpic change associated with enzyme folding ECSe ∆Heq,ECS 225 kJ mol−1 Activation energy of hydrolysis ECSe ∆G†cat,ECS 64 kJ mol−1 Activation energy of irreversible inactivation ECSe ∆G†inact,ECS 96 kJ mol−1
Equilibrium temperature of ECSe Teq,ECS 67.8 ◦C
Position of the catalytic center of ECS z˘− 2
Reference temperature Tre f 30 ◦C
aRef: [48]
bRef: [238]
cEstimated from [207]
dRef: [28]
eRef: [242]
Table 5.2: Subsite binding energies inkJ mol−1from Ho et al. [49].
Enzyme 1 2 3 4 5 6 7 8
Random-chain scission 4.6 10.0 0 2.5 10.0 -13.0 0 5.0 End-chain scission -Inf 0 20.3 6.7 1.8 0.9 0.5 0.4
for ECS.
As starting material quality varies in a real process, the effect of variation is investigated. In the case
"Sweet potato starch" the values (a mean amount of monomer units of 4100, a maximal amount of monomer units of 22496, and a polydispersity of 1.324) provided by Ho et al. [48] for sweet potato starch are used.
Effect of the Product Distribution Function and branching
Ho et al. [48, 49] used a simplified PDF instead of using the PDF derived using the Subsite Theory.
Using the parameters of the base case including the initial enzyme concentration, the effect of using the more realistic but also more complicated PDF computed using the Subsite Theory is investigated.
In order to investigate the effect of branching on optimal control the optimal temperature profile max-imizing the yield in fixed time is computed for a branched polymer. The average branching density is taken to be 5 % and all parameters, including the initial enzyme concentration, of the base case are used. Additionally, a maximal amount of branchings of 88 is used which is sufficiently large to have almost no polymer above this value. In order to include the effect of branchings on the reaction rate, the reaction rate derived by the extension of the Subsite Theory to a branched polymer is used. To stay consistent also the PDF derived using this extension is used.