Unless stated otherwise, all chemicals were purchased from Merck Eurolab and all antisera were from Sigma.

### Fermentation

A shake flask containing 100 mL of YPG medium (per liter: 10 g yeast extract, 10 g peptone, 10 g glycerol) was inoculated with one cryovial from the *P. pastoris* cell bank, and incubated at 28°C for approximately 24 hours and agitated at 180 rpm.

This culture was used to inoculate the starting volume in the bioreactor to a starting optical density (OD_{600}) of 1.0. Depending on the operation mode the starting volume was either 1.2 L for fed batch or 1.4 L for chemostat process.

Fermentations were carried out in a 2.0 L working volume bioreactor (MBR; Wetzikon, Switzerland) with a computer based process control (ISE; Vienna, Austria). Fermentation temperature was controlled at 25°C, pH was controlled at 5.0 with addition of 25% ammonium hydroxide and the dissolved oxygen concentration was maintained above 20% saturation by controlling the stirrer speed between 600 and 1200 rpm, whereas the airflow was kept constant at 100 L h^{-1}.

The media were as follows:

Batch medium contained per liter:

2.0 g citric acid, 12.4 g (NH_{4})_{2}HPO_{4}, 0.022 g CaCl_{2}·2H_{2}O, 0.9 g KCl, 0.5 g MgSO_{4}·7H_{2}O, 40 g glycerol, 4.6 ml PTM_{1} trace salts stock solution. The pH was set to 5.0 with 25% HCl.

Glucose fed batch solution contained per liter:

550 g glucose·1H_{2}O, 10 g KCl, 6.45 g MgSO_{4}·7H_{2}O, 0.35 g CaCl_{2}·2H_{2}O and 12 ml PTM_{1} trace salts stock solution.

Chemostat medium contained per liter:

55 g glucose·1H_{2}O, 2.5 g KCl, 1.0 g MgSO_{4}·7H_{2}O, 0.035 g CaCl_{2}·2H_{2}O, 21.8 g (NH_{4})_{2}HPO_{4} and 2.4 ml PTM_{1} trace salts stock solution, furthermore the pH was set to 5.0 with 25% HCl.

PTM_{1} trace salts stock solution contained per liter:

6.0 g CuSO_{4}·5H_{2}O, 0.08 g NaI, 3.0 g MnSO_{4}· H_{2}O, 0.2 g Na_{2}MoO_{4}·2H_{2}O, 0.02 g H_{3}BO_{3}, 0.5 g CoCl_{2}, 20.0 g ZnCl_{2}, 65.0 g FeSO_{4}·7H_{2}O, 0.2 g biotin and 5.0 ml H_{2}SO_{4} (95%–98%). All chemicals for PTM_{1} trace salts stock solution were from Riedel-de Haën (Seelze, Germany), except for biotin (Sigma, St. Louis, MO, USA), and H_{2}SO_{4} (Merck Eurolab).

After approximately 24 hours the batch was finished and – depending on the fermentation strategy – the feed and if required the harvest was started.

The continuous fermentation was initiated at a *D* = 0.15 h^{-1} and performed at least for 5 resident times *τ* to reach steady state conditions.

Then the dilution rate was decreased stepwise, always achieving steady state conditions before the next change of the dilution rate. At *D* = 0.0086 h^{-1} the procedure was reversed and the dilution rate was increased stepwise up to the critical dilution rate *D*
_{crit} = 0.2 h^{-1}. Samples were taken after 3 and 5 *τ* and analyzed as described below.

The standard fermentation strategy was a fed batch with a constant feed, this means that the batch phase was followed by the glucose fed batch with a feed rate *F* = 8.925 g h^{-1}. The fermentations were terminated at appr. t = 120 h. Samples were taken frequently and processed as described below.

The optimized fermentation strategy consists of different phases to perform the calculated growth kinetic. The batch phase was followed by an exponential feed phase with a growth rate of 0.2 for 3.6 hours, followed by a linearly increasing feed rate calculated by equation (16), where k = 0.0144 g h^{-2} and d = 36.8064 g h^{-1} for 16.0 hours.

*F*_{
L
}= *k*·*t*_{
L
}+ *d* (16)

### Method of calculation

#### 1. Setup of calculations

We divide the total feed period in equal intervals [*t*
_{
n
}, *t*
_{n+1}] (1 ≤ *n* ≤ *N*) of length *dt*. Therefore,

*t*_{n+1 }= *t*_{
n
}+ *dt* (17)

We start with an initial value *dt* = 1 [h]. The best value for *dt* is determined within the optimization process.

At every point of time *t*
_{
n
}we denote by *X*
_{
n
}= *X*(*t*
_{
n
}) the amount of biomass and by *P*
_{
n
}= *P*(*t*
_{
n
}) the amount of product in the bioreactor. At the beginning of the fed-batch process the initial values are *X*(0) = *X*
_{0} and *P*(0) = *P*
_{0}, as achieved at the end of the batch phase.

First we have to describe the growth of the biomass. We use the simplest model, the exponential growth model,

Since the specific growth rate *μ* of the biomass depends on time, we calculate (eq. 18) in discrete time steps

where *μ*
_{
n
}is the specific growth rate during the interval [*t*
_{
n
}, *t*
_{n+1}]. The initial values for *μ*
_{
n
}are chosen arbitrarily, for instance *μ*
_{
n
}≡ *μ*
_{max}. The optimal values for all of the *μ*
_{
n
}'s are determined within the optimization process.

Second we have to describe the accumulation of the product. We simply calculate the total product yield during the interval [*t*
_{
n
}, *t*
_{n+1}] by the following formula

*P*_{n+1 }= *P*_{
n
}+ *dP*_{
n
} (20)

with

*dP*_{
n
}= q_{
Pn
}· *X*_{
n
}·*dt* (21)

The relationship between the specific rate *q*
_{
P
}of product formation and the specific growth rate *μ* was experimentally determined in chemostat cultures. The dependence of *q*
_{
P
}on *μ* was described analogous to Monod equation:

The values for *q*
_{
Pmax
}and *k*
_{
q
}are derived from the experimental data by the method of least squares, i.e. the parameters *q*
_{Pmax }and *k*
_{
q
}are chosen that the sum of the deviations from the experimental data squared is minimal.

Next we have to calculate the amount of substrate *dS* which we must feed in the time interval [*t*
_{
n
}, *t*
_{n+1}]. To do this, let *S*
_{
n
}be the amount of substrate added to the bioreactor until the time point *t*
_{
n
}. Then the substrate consumption rate depends on the amount and on the increase of biomass, i.e.

where *m*
_{
S
}is the maintenance coefficient and *Y*
_{
XS
}is the true yield coefficient of biomass from substrate. Inserting formula (18) in (23) the amount of substrate feed in the interval [*t*
_{
n
}, *t*
_{n+1}] calculates as

To calculate the parameters *Y*
_{
XS
}and *m*
_{
S
}from experimental data of chemostat cultures by the method of least squares, we use the observed biomass yield coefficient *Y*'_{
XS
}depending on the specific growth rate *μ*. This is done by *dX* = -*Y*'_{
XS
}·*dS* and inserting formula (18) and the formula for the whole substrate consumption which implies

Formula (25) can be transformed to

From this double reciprocal plot *Y*
_{
XS
}and *m*
_{
S
}were determined by linear regression.

Last but not least we need the total volume for the calculation of the volumetric productivity. The model process starts with a batch volume of *V*
_{0} = 1 L. The total volume at each time interval is then

with the substrate concentration in the feed medium s_{
f
}and the density of the feed medium *ρ*
_{
f
}. Due to the high biomass concentrations achieved in *P. pastoris* fermentations, the cells occupy a significant fraction of the total volume, while the product is secreted to the liquid phase, the culture supernatant. In order to calculate the product concentration, the available liquid volume *V*
_{
l
}is calculated at each time interval with the specific volume of wet biomass, which is derived from dry biomass as the specific volume per dry biomass *ν*
_{YDM} = 0.0033 L g^{-1}.

*V*_{ln} = *V*_{
n
}- *X*_{
n
}·*ν*_{
YDM
} (28)

Finally, we calculate the biomass and the product concentrations. The product concentration *p* at the time point *t*
_{
n
}is calculated as

and the biomass concentration *x* at the same time point is

The medium feed rate *F*
_{n} at each time point is

These values are used to determine the feed rate profile of the optimized fed batch process.

#### 2. Optimization

The goal of our optimization problem is to find the best values for the specific growth rates *μ*
_{
n
}and the best value for *dt* (which implies that the total feed period undergoes the optimization process too) such that the volumetric productivity *Q*
_{
P
}calculated at the point of time *t*
_{N+1 }as

is maximized under the following constraints:

*μ*_{min} ≤ *μ*_{
n
}≤ *μ*_{max} for (1 ≤ *n* ≤ *N*) (33)

and

*X*(*t*_{N+1}) = *X*_{max} (34)

Here *μ*
_{max} = 0.2 h^{-1} is the maximum specific growth rate at just below washout in chemostat cultures. Since below *μ* = 0.02 *h*
^{-1} significant product degradation appeared, the lower boundary was set at *μ*
_{min} = 0.03 *h*
^{-1}. Also the biomass concentration needs to be limited. The upper limit is mainly defined by the cell separation step, which is practically limited with approximately 100 g*L*
^{-1} dry mass.

##### Remark

Additional constraints may be entered, e.g. the final product concentration may be set at a minimum level.

In the Excel sheet we set *N* = 150. The values *t*
_{
n
}, *μ*
_{
n
}, *X*
_{
n
}, ... are organized in columns, with each time point *t*
_{
n
}... a row. The values of *X*
_{
n
}, *P*
_{
n
}, *V*
_{
n
}, ... are calculated from the respective previous row using the equations provided above. The optimization process is performed by the Excel Solver as a black box. It maximizes the final *Q*
_{
P
}field by varying the *μ* fields within the boundaries and the *dt* field.

The Excel file used for this work is provided as an additional file.