# Dynamic Modelling and Optimisation of the Batch Enzymatic Synthesis of Amoxicillin

^{*}

## Abstract

**:**

## 1. Introduction

- To introduce temperature dependency into the published kinetic model of batch enzymatic amoxicillin synthesis;
- To understand the attainable performances and inherent trade-offs (via isothermal operation) of varying batch times and operating temperatures;
- To optimise dynamic temperature profiles toward optimal process performance for varying product quality constraints.

## 2. Dynamic Modelling and Optimisation

#### 2.1. Amoxicillin Synthesis Pathway and Kinetic Model

_{E}, equals that considered in the literature [20].

_{i}= the species rate of formation; v

_{h1}= the rate of PHPGME hydrolysis; v

_{h2}= the rate of AMOX hydrolysis; vS = the rate of AMOX synthesis; k = the species inhibition constant; k

_{EN}= the 6-APA adsorption constant; k

_{cat}= the reaction rate constant; K

_{M}= the reaction empirical rate constant; and X

_{max}is the maximum conversion ratio of the enzyme reagent complex into AMOX. Subscripts i and j denote species and reactions, respectively. The system of dynamic ODEs is solved simultaneously using the built-in MATLAB ODE solver ode15s. Equations (1)–(8) are

#### 2.2. Kinetic Parameter Estimation

_{PHPG}(t

_{0}= 0), C

_{AMOX}(t

_{0}= 0) = 0.

_{cat,1}and k

_{cat,2}were regressed from published amoxicillin concentration data at 5, 25, and 35 °C [22], from which Arrhenius parameters could then be estimated. The remaining kinetic parameters were assumed to be temperature-independent, as a much wider kinetic dataset is required for further multiparametric regression. Values of k

_{cat,1}and k

_{cat,2}at different temperatures were regressed by minimising the residual error between the kinetic model (Equations (1)–(8)) and the experimental data using the bound constrained solver “fminsearchbnd” in MATLAB. Table 3 shows the regressed values of k

_{cat,1}and k

_{cat,2}at the given temperatures.

_{cat,1}and k

_{cat,2}, the Arrhenius parameters, k

_{0}(the pre-exponential factor) and E

_{a}(the energy barrier), were then regressed. An Arrhenius-type temperature dependence of k

_{cat,1}and k

_{cat,2}was assumed, according to Equation (9):

_{0}and E

_{a}are the Arrhenius pre-exponential factor and energy barrier, respectively; R is the universal gas constant; and T is the reaction temperature. The fitting methodology for Arrhenius constant regression also used “fminsearchbnd” in MATLAB, as described previously. Figure 3 shows the lines of best fit for both Arrhenius plots, showing a good fit in both cases. The regressed Arrhenius parameter values (listed in Table 4) allowed for good replication of the experimental data (Figure 4). A corroboration of the regressed parameters with a wider dataset will further validate the values used in this work. The experimental data in the literature that were used to regress the kinetic parameters at different operating temperatures (T = 5, 25, 35 °C) provided no error values for the calculated concentrations (estimated via HPLC) or temperatures, and thus error bars are not shown [22]. An investigation of the effect of errors on regressed parameter values (isothermal k

_{cat,1}and k

_{cat,2}and non-isothermal Arrhenius parameters) could be implemented by perturbing their values and observing the effect on the optimisation results as a form of sensitivity analysis.

#### 2.3. Dynamic Simulation and Optimisation

#### 2.3.1. Design Space Investigation and Simulation

#### 2.3.2. Dynamic Optimisation

^{−1}) is minimised. Minimising feedstock consumption in different cases can lead to possible process configurations where reagents and solvent are recycled following desired product (amoxicillin) crystallisation [23,24]. Although recycling options are not explicitly considered in this work, they have been demonstrated in the literature for β-lactam antibiotic synthesis with nanofiltration for by-product (PHPG) removal [25]. This can allow for processes with greater material efficiencies and overall sustainability. Economic analyses of specific process designs can further elucidate the benefits of such designs [5], but are beyond the scope of this work.

_{1}and J

_{2}present competing objectives, the utility of the presented optimisation results is in exploring how to manipulate the optimal temperature control profile over the batch duration to yield the desired amount of amoxicillin while minimising feedstock consumption. Equations (12)–(14) are

_{1}= C

_{6-APA}(t

_{0}) − C

_{6-APA}(t

_{f})

_{2}= –C

_{AMOX}(t

_{f})

_{f}= 500 min, which was observed as the maximum time beyond which amoxicillin degradation via hydrolysis to 6-APA and PHPG dominated (Figure 3 and Figure 4). Equations (12)–(14) become Equations (15)–(18):

_{AMOX}(t

_{f}) ≥ ε

_{f}= 500 min

## 3. Results and Discussion

#### 3.1. Dynamic Simulation

_{h2}and v

_{AMOX}to v

_{PHPG}(as shown in Figure 6), both of which decreased significantly with batch time. The colours in Figure 6 represent the reaction rate ratios shown on the z axis and are shown to aid in the interpretation of the surface plots.

#### 3.2. Non-Isothermal Simulation

_{MAX}), and also selectivity (Equation (10)) and productivity (Equation (11)). The banding observed was due to stepwise temperature profile simulations. Different cases are highlighted on each of these plots: (1) maximum selectivity, (2) maximum amoxicillin concentration and (3) a compromise between selectivity and productivity. Maximum final amoxicillin concentration was achieved by operating at the maximum temperature (35 °C), in agreement with previously observed results [22]. There are inherent trade-offs between different process performance metrics. Doing so facilitates a visualisation of the attainable performance of the process, subject to the rules imposed in generating the set of profiles. Dynamic optimisation can be implemented for temperature profile manipulation in order to investigate the possibility of the process benefitting from non-isothermal operation for specific production objectives.

#### 3.3. Non-Isothermal Dynamic Optimisation

_{AMOX}(t

_{f}) = 5 and 6 mM, the temperature profile decreased until t = 375 min and then increased again, with a higher final operating temperature required for a higher concentration constraint. When the constraint was set to C

_{AMOX}(t

_{f}) = 7 mM, a gradual decrease from 296 to 290 K over the batch duration was implemented. For C

_{AMOX}(t

_{f}) ≥ 8 mM, temperature profiles increased over the batch duration. At C

_{AMOX}(t

_{f}) = 9 mM, a steady increase from 295 to 305 K was observed. For C

_{AMOX}(t

_{f}) = 10 mM, the temperature increased sharply from 292 to 308 K (upper temperature bound) at t = 100 min. For C

_{AMOX}(t

_{f}) = 11 mM, the temperature increase rate was more drastic, increasing to the upper bound at t = 25 min, with subsequent isothermal operation for the remainder of the batch duration.

_{AMOX}(t

_{f}) = 5–7 mM), the target amoxicillin concentration was met before the concentration of 6-APA (i.e., the objective function) was maximised. This indicated that there was scope to also optimise the batch runtime in the dynamic optimisation problem as well, which can be considered in future work. The final species concentrations at the end of the batch runtime (t

_{f}= 500 min) are shown in Figure 10. As the product amoxicillin constraint was increased, the maximum objective function value decreased.

_{AMOX}(t

_{f}) = 11 mM was specified, the constraint was not met. Consideration of the maximum attainable objective function values for a broad range of C

_{AMOX}(t

_{f}) (Figure 11) explains this result. This was a result of there being a maximum product concentration of amoxicillin attainable from the given feed substrate concentrations. Figure 12 plots the attained maximised objective function for different imposed amoxicillin product concentration constraints. Until a certain amoxicillin product concentration (Point A), the maximum attainable objective function value was the same: This was the maximum attainable value from the given initial conditions (feed concentrations). As the product amoxicillin concentration increased, the maximum attainable objective function decreased until the maximum constraint, C

_{AMOX}(t

_{f}) = 9.927 mM, allowing for an objective function, C

_{6-APA}(t

_{f}) = 10.067 mM (Point B). The limiting points (A and B) highlighted in Figure 12 resulted from the initial concentration conditions of the system.

_{AMOX}(t

_{f}) considerations) via the reaction mass efficiency (RME), calculated by Equation (19).

_{AMOX}) at the end of a batch run to the masses of starting materials (PHPGME (m

_{PHPGME}) and 6-APA (m

_{6-APA})) at the start (t

_{0}= 0). Values of RME for different product amoxicillin concentration constraints are shown in Figure 13. As the specified amoxicillin concentration constraint increases, the RME increases, which is expected. All values of RME were relatively midrange with respect to typical pharmaceutical manufacturing processes [33]. The effect of scale-up on material efficiencies, as well as plant-wide costs, is an important consideration during process design and optimisation studies such as this one [34].

## 4. Conclusions

- The first implementation of dynamic temperature profile optimisation in order to meet specific product quality constraints and minimise feedstock consumption in β-lactam antibiotic production.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Zaffiri, L.; Gardner, J.; Toledo-Pereyra, L.H. History of antibiotics. from salvarsan to cephalosporins. J. Investig. Surg.
**2012**, 25, 67–77. [Google Scholar] [CrossRef] [PubMed] - Balkhi, B.; Araujo-Lama, L.; Seoane-Vazquez, E.; Rodriguez-Monguio, R.; Szeinbach, S.L.; Fox, E.R. Shortages of systemic antibiotics in the USA: how long can we wait? J. Pharm. Heal. Serv. Res.
**2013**, 4, 13–17. [Google Scholar] [CrossRef] - Pulcini, C.; Beovic, B.; Béraud, G.; Carlet, J.; Cars, O.; Howard, P.; Levy-Hara, G.; Li, G.; Nathwani, D.; Roblot, F.; et al. Ensuring universal access to old antibiotics: a critical but neglected priority. Clin. Microbiol. Infect.
**2017**, 23, 590–592. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Russell, M.G.; Jamison, T.F. Seven-step continuous flow synthesis of linezolid without intermediate purification. Angew. Chemie Int. Ed.
**2019**. [Google Scholar] [CrossRef] - Lin, H.; Dai, C.; Jamison, T.F.; Jensen, K.F. A rapid total synthesis of ciprofloxacin hydrochloride in continuous flow. Angew. Chemie Int. Ed.
**2017**, 56, 8870–8873. [Google Scholar] [CrossRef] [PubMed] - Giordano, R.C.; Ribeiro, M.P.A.; Giordano, R.L.C. Kinetics of β-lactam antibiotics synthesis by penicillin G acylase (PGA) from the viewpoint of the industrial enzymatic reactor optimization. Biotechnol. Adv.
**2006**, 24, 27–41. [Google Scholar] [CrossRef] [PubMed] - Food and Drug Administration (FDA). Sales of antibacterial drugs in kilograms. 2010; 4–6. [Google Scholar]
- Food and Drug Administration (FDA). Sales of antibacterial drugs in kilograms. 2012; 5–8. [Google Scholar]
- Hamed, R.B.; Gomez-Castellanos, J.R.; Henry, L.; Ducho, C.; McDonough, M.A.; Schofield, C.J. The enzymes of β-lactam biosynthesis. Nat. Prod. Rep.
**2013**, 30, 21–107. [Google Scholar] [CrossRef] - Elander, R.P. Industrial production of β-lactam antibiotics. Appl. Microbiol. Biotechnol.
**2003**, 61, 385–392. [Google Scholar] [CrossRef] - Laxminarayan, R. The state of the world’s antibiotics in 2018. 3 July 2018. [Google Scholar]
- Gerogiorgis, D.I.; Jolliffe, H.G. Continuous pharmaceutical process engineering and economics investigating technical efficiency, environmental impact and economic viability. Chem. Today
**2015**, 33, 29–32. [Google Scholar] - Diab, S.; Gerogiorgis, D.I. Process modelling, simulation and technoeconomic evaluation of crystallisation antisolvents for the continuous pharmaceutical manufacturing of rufinamide. Comput. Chem. Eng.
**2018**, 111, 102–114. [Google Scholar] [CrossRef] [Green Version] - Rodman, A.D.; Gerogiorgis, D.I. An investigation of initialisation strategies for dynamic temperature optimisation in beer fermentation. Comput. Chem. Eng.
**2019**, 124, 43–61. [Google Scholar] [CrossRef] - Rodman, A.D.; Gerogiorgis, D.I. Dynamic optimization of beer fermentation: sensitivity analysis of attainable performance vs. product flavour constraints. Comput. Chem. Eng.
**2017**, 106, 582–595. [Google Scholar] [CrossRef] - McDonald, M.A.; Bommarius, A.S.; Rousseau, R.W.; Grover, M.A. Continuous reactive crystallization of β-lactam antibiotics catalyzed by penicillin G acylase. part I: model development. Comput. Chem. Eng.
**2019**, 123, 331–343. [Google Scholar] [CrossRef] - McDonald, M.A.; Bommarius, A.S.; Grover, M.A.; Rousseau, R.W. Continuous reactive crystallization of β-lactam antibiotics catalyzed by penicillin G acylase. part II: case study on ampicillin and product purity. Comput. Chem. Eng.
**2019**, 126, 332–341. [Google Scholar] [CrossRef] - Encarnación-Gómez, L.G.; Bommarius, A.S.; Rousseau, R.W. Crystallization kinetics of ampicillin using online monitoring tools and robust parameter estimation. Ind. Eng. Chem. Res.
**2016**, 55, 2153–2162. [Google Scholar] [CrossRef] - McDonald, M.A.; Bommarius, A.S.; Rousseau, R.W. Enzymatic reactive crystallization for improving ampicillin synthesis. Chem. Eng. Sci.
**2017**, 165, 81–88. [Google Scholar] [CrossRef] - Gonçalves, L.R.B.; Fernández-Lafuente, R.; Guisán, J.M.; Giordano, R.L.C. The role of 6-aminopenicillanic acid on the kinetics of amoxicillin enzymatic synthesis catalyzed by penicillin G acylase immobilized onto glyoxyl-agarose. Enzyme Microb. Technol.
**2002**, 31, 464–471. [Google Scholar] [CrossRef] - UK Department of Health. Antimicrobial resistance empirical and statistical evidence-base. 2016. [Google Scholar]
- Alemzadeh, I.; Borghei, G.; Va, L.; Roostaazad, R. Enzymatic synthesis of amoxicillin with immobilized penicillin G acylase. Trans. C Chem. Chem. Eng.
**2010**, 17, 106–113. [Google Scholar] - Schroën, C.G.P.H.; Van Roon, J.L.; Beefink, H.H.; Tramper, J.; Boom, R.M. Membrane applications for antibiotics production. Desalination
**2009**, 236, 78–84. [Google Scholar] [CrossRef] - Fodi, T.; Didaskalou, C.; Kupai, J.; Balogh, G.T.; Huszthy, P.; Szekely, G. Nanofiltration-enabled in situ solvent and reagent recycle for sustainable continuous-flow synthesis. ChemSusChem
**2017**, 10, 3435–3444. [Google Scholar] [CrossRef] [PubMed] - Pereira, S.C.; Castral, T.C.; Ribeiro, M.P.A.; Giordano, R.L.C.; Giordano, R.C. Green route for amoxicillin production through the integration with the recycle of the by-product (p-hydroxyphenylglycine). In Proceedings of the Brazilian Congress of Chemical Engineering, São Paulo, Spain, 7–12 June 2015; Blücher, E., Ed.; pp. 1823–1830. [Google Scholar]
- Marler, R.T.; Arora, J.S. Survey of multi-objective optimization methods for engineering. Struct. Multidiscip. Optim.
**2004**, 26, 369–395. [Google Scholar] [CrossRef] - Cuthrell, J.E.; Biegler, L.T. On the optimization of differential-algebraic process systems. AIChE J.
**1987**, 33, 1257–1270. [Google Scholar] [CrossRef] - Logsdon, J.S.; Biegler, L.T. Accurate solution of differential-algebraic optimization problems. Ind. Eng. Chem. Res.
**1989**, 28, 1628–1639. [Google Scholar] [CrossRef] - Wächter, A.; Biegler, L.T. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program.
**2006**, 106, 25–57. [Google Scholar] [CrossRef] - Čižniar, M.; Fikar, M.; Latifi, M.A. MATLAB dynamic optimisation code DynOpt. User’s guide, technical report. KIRP FCHPT STU Bratislava
**2005**. [Google Scholar] - Hicks, M.B.; Farrell, W.; Aurigemma, C.; Lehmann, L.; Weisel, L.; Nadeau, K.; Lee, H.; Moraff, C.; Wong, M.; Huang, Y.; et al. Making the move towards modernized greener separations: introduction of the analytical method greenness score (AMGS) calculator. Green Chem.
**2019**, 21, 1816–1826. [Google Scholar] [CrossRef] - Sheldon, R.A. Fundamentals of green chemistry: Efficiency in reaction design. Chem. Soc. Rev.
**2012**, 41, 1437–1451. [Google Scholar] [CrossRef] [PubMed] - Ribeiro, M.G.T.C.; Machado, A.A.S.C. Greenness of chemical reactions—limitations of mass metrics. Green Chem. Lett. Rev.
**2013**, 6, 1–18. [Google Scholar] [CrossRef] - Jolliffe, H.G.; Gerogiorgis, D.I. Plantwide Design and Economic Evaluation of Two Continuous Pharmaceutical Manufacturing (CPM) Cases: Ibuprofen and Artemisinin. Comput. Aided Chem. Eng.
**2015**, 37, 2213–2218. [Google Scholar] - Li, J.; Lai, T.C.; Trout, B.L.; Myerson, A.S. Continuous crystallization of cyclosporine: the effect of operating conditions on yield and purity. Cryst. Growth Des.
**2017**, 17, 1000–1007. [Google Scholar] [CrossRef]

**Figure 1.**Leading antibiotics used in the U.K. in 2016 [21].

**Figure 2.**Simplified reaction pathway presented for the semiempirical model [20].

**Figure 3.**Arrhenius parameter regression from experimental data [22].

**Figure 4.**Concentration profiles of amoxicillin: experimental [22] vs. Arrhenius modelled results.

**Figure 5.**Isothermal concentration response surfaces as a function of batch time and operating temperature for (

**a**) PHPGME, (

**b**) 6-APA, (

**c**) AMOX and (

**d**) PHPG.

**Figure 6.**Isothermal response surface of reaction rate ratios as a function of batch time and temperature.

**Figure 11.**Final concentrations (colour scheme congruent with Figure 9 trajectories).

Subclass | Antibiotic | Spectrum | Generation | Application | CAS # | MW | Sales [7,8] | Price |
---|---|---|---|---|---|---|---|---|

(g mol^{−1}) | (tonnes) | (10^{3} GBP kg^{−1}) | ||||||

Penicillin | Penicillin V | Narrow | 1st | Laryngitis, bronchitis, pneumonia, skin infections | 87-08-1 | 350.39 | 126.29 | 82.584 |

Oxacillin | Narrow | 2nd | Staphylococci infections | 7240-38-2 | 401.44 | 2.87 | 293.334 | |

Nafcillin | Narrow | 2nd | Staphylococci infections | 985-16-0 | 414.48 | 8.48 | 27.206 | |

Dicloxacillin | Narrow | 2nd | Bronchitis, pneumonia, staphylococci infections | 3116-76-5 | 470.33 | 7.35 | 71.100 | |

Ampicillin | Broad | 3rd | UTIs, pneumonia, gonorrhoea, meningitis, abdominal infections | 69-53-4 | 349.41 | 42.35 | 11.358 | |

Amoxicillin | Broad | 3rd | Tonsillitis, bronchitis, pneumonia, gonorrhoea, sinus infections, UTIs | 26787-78-0 | 365.40 | 1,122.41 | 37.057 | |

Ticarcillin | Broad | 4th | UTIs, bone + joint infections, stomach infections, skin infections | 34787-01-4 | 384.43 | 2.92 | 46.487 | |

Piperacillin | Broad | 4th | UTIs, bone + joint infections, stomach infections, skin infections | 66258-76-2 | 517.56 | 140.51 | 60.854 | |

Cephalosporin | Cephalexin | n/a | 1st | UTIs, upper respiratory tract infections, ear infections, skin infections | 15686-71-2 | 347.39 | 321.90 | 63.152 |

Cefadroxil | n/a | 1st | UTIs, staphylococci infections, skin infections | 66592-87-8 | 363.39 | 11.75 | 83.638 | |

Cefazolin | n/a | 1st | Respiratory tract infections, UTIs, skin infections, | 25953-19-9 | 454.51 | 39.39 | 5.930 | |

Cefdinir | n/a | 2nd | Bronchitis, pneumonia, skin infections, sinus infections | 91832-40-5 | 395.42 | 41.77 | 59.457 | |

Cefaclor | n/a | 2nd | UTIs, respiratory tract infections, sinus infections | 53994-73-3 | 367.81 | 2.95 | 33.487 | |

Cefprozil | n/a | 2nd | Ear infections, skin infections | 92665-29-7 | 389.43 | 11.12 | 33.813 | |

Cefoxitin | n/a | 2nd | UTIs, skin infections, pneumonia, bronchitis, tonsillitis, ear infections | 35607-66-0 | 427.45 | 4.30 | 156.519 | |

Cefixime | n/a | 3rd | Sinus infections, bronchitis, pneumonia | 79350-37-1 | 453.45 | 1.73 | 59.919 | |

Cefotaxime | n/a | 3rd | UTIs, pneumonia, abdominal infections, bone + joint infections | 63527-52-6 | 455.47 | 2.57 | 251.725 | |

Ceftriaxone | n/a | 3rd | Lower respiratory tract infections, UTIs, skin infections | 104376-79-6 | 554.58 | 29.90 | 212.500 | |

Monobactam | Aztreonam | n/a | n/a | Pneumonia | 1-5-7 | 435.43 | 3.72 | 39.492 |

**Table 2.**Properties of species in the reaction scheme for the batch enzymatic synthesis of amoxicillin.

Compound | Abbreviation | Type | CAS # | MW (g mol^{−1}) |
---|---|---|---|---|

p-hydroxyphenyl glycine methyl ester | PHPGME | Feed | 127369-30-6 | 180.18 |

6-aminopenicillanic acid | 6-APA | Feed/Side product | 551-16-6 | 216.26 |

amoxicillin | AMOX | Product | 26787-78-0 | 365.40 |

p-hydroxyphenyl glycine methyl ester | PHPG | Side product | 37784-25-1 | 167.16 |

Parameter | T = 5 °C | T = 25 °C | T = 35 °C | |
---|---|---|---|---|

T-dependent | k_{cat,1} (IU g^{−1} min^{−1}) | 0.57 | 0.59 | 0.64 |

k_{cat,2} (IU g^{−1} min^{−1}) | 9.16 | 3.07 | 1.77 | |

Fixed | K_{M1} (mM) | 0.20 | 0.20 | 0.20 |

K_{M2} (mM) | 27.47 | 27.47 | 27.47 | |

X_{max} (–) | 0.96 | 0.96 | 0.96 | |

k_{E} (mM) | 16.03 | 16.03 | 16.03 | |

k_{PHPGME} (mM) | 2672.04 | 2672.04 | 2672.04 | |

k_{AMOX} (mM) | 4.59 | 4.59 | 4.59 | |

k_{PHPG} (mM) | 4.51 | 4.51 | 4.51 | |

k_{6-APA} (mM) | 4550.28 | 4550.28 | 4550.28 |

Parameter | k_{cat,1} | k_{cat,2} |
---|---|---|

k_{0} (IU g^{−1} min^{−1}) | 1.89 | 4.74 × 10^{−7} |

E_{a} (J mol^{−1}) | 2.80 × 10^{3} | −3.88 × 10^{4} |

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## Share and Cite

**MDPI and ACS Style**

Cuthbertson, A.B.; Rodman, A.D.; Diab, S.; Gerogiorgis, D.I.
Dynamic Modelling and Optimisation of the Batch Enzymatic Synthesis of Amoxicillin. *Processes* **2019**, *7*, 318.
https://doi.org/10.3390/pr7060318

**AMA Style**

Cuthbertson AB, Rodman AD, Diab S, Gerogiorgis DI.
Dynamic Modelling and Optimisation of the Batch Enzymatic Synthesis of Amoxicillin. *Processes*. 2019; 7(6):318.
https://doi.org/10.3390/pr7060318

**Chicago/Turabian Style**

Cuthbertson, Andrew B., Alistair D. Rodman, Samir Diab, and Dimitrios I. Gerogiorgis.
2019. "Dynamic Modelling and Optimisation of the Batch Enzymatic Synthesis of Amoxicillin" *Processes* 7, no. 6: 318.
https://doi.org/10.3390/pr7060318