FedBatchDesigner

Please provide some basic information about your process setup below. Parameters in the left column are shared by both stages. Parameters in the right column are specific to each stage. Any parameter not provided for the second stage specifically will be assumed to be the same as in the first stage.

Some of these parameters are determined by your experimental setup (e.g. \(V_\textrm{max}\)). Others can be found in the literature (e.g. \(\rho\) for your production organism) and some need to be estimated from experimental data (usually the specific productivities \(\pi_0\) and \(\pi_1\)). Tutorials on how to do this can be found on github.

Please see also the info panel for more details regarding the underlying assumptions etc.

\(V_\textrm{max}\) - \(V_\textrm{batch}\) is the volume of medium added during the feed phase. Three feed strategies are considered:

  • Constant feed: The feed rate doesn't change throuout the feed phase.
  • Linear feed: The feed rate increases linearly according to \(F = F_0 + t \cdot dF\).
  • Exponential feed: The feed rate increases exponentially according to \(F = F_0 \cdot e^{\mu t}\).

For linear and exponential feed, the initial feed rate \(F_0\) is chosen such that the total amount of biomass also increases linearly or exponentially (taking substrate requirements for product formation and maintenance into account).

For optimizing the exponential feed, specific growth rates up to \(\mu_\textrm{max}^{F,\textrm{exp}}\) are considered. The parameters \(\mu_\textrm{max}^\textrm{phys}\) and \(F_\textrm{max}\) are used to ensure that neither the maximum physiological growth rate of the organism nor the maximum feed rate of the reactor are exceeded by any feed strategy.

Maximum available volume in the reactor

Substrate concentration in the feed

Maximum specific growth rate to consider when optimizing exponential feed

Maximum feed rate of the reactor

The batch phase is assumed to have already been passed and won't be optimized.

Fluid volume in the reactor at the end of the batch phase

Biomass concentration at the end of the batch phase

These parameters (yield coefficients and specific ATP consumption and product formation rates) can change between the two stages and make up the specific substrate consumption rate according to

\( \sigma = \frac{\mu}{Y_{X/S}} + \frac{\pi_0 + \mu \pi_1}{Y_{P/S}} + \frac{\rho}{Y_{ATP/S}} \) .

As there is no growth in the second stage, some of these are only available for the first stage.

Maximum specific growth rate (physiological)
Yield coefficients

Yield coefficients of biomass, product, and ATP (in grams per gram substrate consumed).

Specific rates

Maintenance factor (specific ATP consumption rate)

Non-growth-associated specific product formation rate

Growth-associated specific product formation rate

These parameters (yield coefficients and specific ATP consumption and product formation rates) can change between the two stages and make up the specific substrate consumption rate according to

\( \sigma = \frac{\mu}{Y_{X/S}} + \frac{\pi_0 + \mu \pi_1}{Y_{P/S}} + \frac{\rho}{Y_{ATP/S}} \) .

As there is no growth in the second stage, some of these are only available for the first stage.

Yield coefficients

Yield coefficients of biomass, product, and ATP (in grams per gram substrate consumed).

Specific rates

Maintenance factor (specific ATP consumption rate)

Non-growth-associated specific product formation rate
Background

The aim of this web tool is to explore the design space of fed-batch fermentations with a growth-arrested second stage. The feed profile of the first stage is either constant, linear, or exponential. During the second, growth-arrested, stage, the feed rate is kept constant (at a value determined by the maintenance coefficient \(\rho\) and growth-independent production rate \(\pi_0\)).

The tunable variables for the process design are:

  • The parameter determining the feed profile during the first stage:
    • The feed rate \(F\) for constant feed.
    • The absolute growth rate \(dX\) for linear feed. This is the rate of change of the total amount of biomass during the first stage (e.g. 2 g/h). The initial feed rate \(F_0\) and its change over time \(dF\) are chosen such that \(dX\) is constant (for details see below).
    • The specific growth rate \(\mu\) for exponential feed. The initial feed rate \(F_0\) is chosen such that the feed rate \(F\) and the total amount of biomass \(X\) both grow exponentially and at the same rate (for details see below).
  • The point of switching between the first and second stage (\(t_{switch}\)) or, more practically, the fraction of the total feed volume that is fed during the first stage (\(V_{frac} = \frac{V_{s_1}}{V_{tot}}\)).

Outputs include visualizations of how the average volumetric productivity (also called space-time yield) and the final product titer relate to these parameters.

Thanks to analytical solutions for the calculation of biomass and product formation in the first stage, the whole design space can be evaluated with a brute force approach in little time. The assumptions that those analytical formulas rely upon are usually satisfied in microbial fed-batch settings with a growth-arrested second stage (for details on assumptions see below).

One crucial aspect to note is that we assume no growth in the second stage with the feed rate held constant at the exact value that satisfies the substrate requirements for maintenance and product formation (i.e. the best case scenario).

Underlying assumptions
General (both stages):
  • No substrate accumulation.\(S = 0\)
  • Product formation is proportional to biomass and growth.\(\dot P = X \cdot \pi_0 + \dot X \cdot \pi_1\)
  • There is a maintenance requirement.\(M = X \cdot \rho\)
Stage 1: Growth phase
  • Biomass growth is determined by the amount of substrate remaining after taking maintenance and product formation into account. \( \dot X = Y_{X/S} \cdot (F \cdot s_F - \frac{M}{Y_{ATP/S}} - \frac{\dot P}{Y_{P/S}}) \)
  • The feed rate is constant in case of the constant feed strategy. \(\dot V = F = const\)
  • For the linear feed strategy, the feed rate increases linearly with time. By choosing the appropriate initial feed rate \(F_0\) and rate of change \(dF\) (taking the substrate requirements for maintenance and product formation into account), total biomass growth is ensured to also be linear. In other words, when the user increases the value for the maintenance factor \(\rho\), \(F_0\) and \(dF\) will increase accordingly and biomass still grows in a linear fashion throughout the whole first stage. \( \begin{gather} F = F_0 + t \cdot dF\\ \mu_0 = \frac{dX}{X_0}\\ F_0 = \frac{X_0}{s_F} \left(\frac{\mu_0}{Y_{X/S}} + \frac{\rho}{Y_{ATP/S}} + \frac{\pi_0 + \mu_0 \pi_1}{Y_{P/S}}\right)\\ \alpha = \frac{1}{1 + \pi_1 \frac{Y_{X/S}}{Y_{P/S}}}\\ \beta = \pi_0\frac{Y_{X/S}}{Y_{P/S}} + \rho\frac{Y_{X/S}}{Y_{ATP/S}}\\ dF = \alpha \beta (F_0 - \frac{X_0 \cdot \beta}{s_F \cdot Y_{X/S}}) \end{gather} \)
  • In case of the exponential feed strategy, total biomass and the feed rate both grow exponentially and at the same rate (again by choosing the appropriate initial feed rate \(F_0\) similar to the linear feed). \( \begin{gather} F = F_0 \cdot e^{\mu t}\\ F_0 = \frac{X_0}{s_F} \left(\frac{\mu}{Y_{X/S}} + \frac{\rho}{Y_{ATP/S}} + \frac{\pi_0 + \mu \pi_1}{Y_{P/S}}\right) \end{gather} \)
Stage 2: Production phase
  • No growth\(\dot X = 0\)
  • The feed rate is constant at a value that exactly satisfies maintenance and product formation. \( F = \frac{1}{s_F} (\frac{M}{Y_{ATP/S}} + \frac{\dot P}{Y_{P/S}}) \)
Limitations
  • Productivity and volumetric productivity relate only to the feed phase (i.e. they don't take the duration of and amount of product produced during the batch phase into account).
  • Non-feed volume changes (e.g. evaporation, base addition for pH control) are not considered.