# Yield Maximisation(Bioprocess Optimisation)

(Version: February 2022)

The application presented demonstrates the benefit of model-based optimisation strategies for the development and optimisation of bioprocesses using the example of an UpStream process for the production of a recombinant bioproduct. In this example, the bioproduct is produced using a genetically modified organism.

## Application Example

The cultivation of the genetically modified organism $\color{#4daf4a}{X}$ (cf. E. coli) takes place in an ideally mixed classical stirred tank reactor. The formation of the recombinant bioproduct $\color{#377eb8}{P}$ is induced by the addition of an activator $a$ (cf. IPTG) at time $t_{\text{a}}$ [1]. Here, biomass growth and product formation compete for the same substrate $\color{#ff7f00}{S}$ (cf. glucose). At the end of the processing time $t_{\text{harvest}}$, the harvesting and downstream processing of the formed bioproduct takes place. The time series of the considered state variables of the non-optimised bioprocess are shown in the following figure:

As can be seen from the figure, high substrate concentrations $\color{#ff7f00}{S}$ inhibit biomass growth $\color{#984ea3}{\mu}$. Following the addition of the activator, much of the substrate is consumed in product formation. At this point, a noticeable inhibition of biomass growth is also visible due to the increasing product concentration. After depletion of the substrate, product formation stagnates. Likewise, the decay of biomass can be observed. The productivity of the bioprocess $Pr_{\text{P}}$ is calculated after termination of the bioprocess to be the quotient of the change in product concentration and the elapsed process time:

\begin{align} Pr_{\text{P}}=&\,\dfrac{P(t_{\text{harvest}})-P(t_{0})}{t_{\text{harvest}}-t_{0}} \label{eq:productivity} \end{align}

Furthermore, the model necessary for the model-based optimisation of the system is known and can be represented in the form of a differential equation system [2, 3]. In the present case, this would be given by the following equations:

\begin{align} \dot{X}=&\,\mu(S,P)\cdot X -k_{\text{d}}\cdot X \label{eq:ode_cX} \\[10pt] \dot{S}=&\,-\dfrac{\mu(S,P)\cdot X}{Y_{\text{X,S}}} -\dfrac{\rho(S,a)\cdot X}{Y_{\text{P,S}}} - m \cdot X \label{eq:ode_cS} \\[10pt] \dot{P}=&\,\rho(S,a)\cdot X \label{eq:ode_cP} \end{align}

Here, the parameters $k_{\text{d}}$ and $m$ correspond to the constant rates of biomass decay and the consumption of substrate to sustain the metabolism of the cell. The two yield coefficients $Y_{\text{X,S}}$ and $Y_{\text{P,S}}$ reflect the ratio between substrate consumed and biomass and product formed, respectively. The following equations Eq.  \eqref{eq:mu_kinetics} and Eq.  \eqref{eq:pi_kinetics} represent the model kinetics for the description of biomass growth or decay as well as product formation [2]:

\begin{eqnarray} \mu(S,P) & = & \frac{\mu_{\max}\cdot S}{K_{\text{S}}+S+\frac{S^2}{K_{\text{I,S}}}+\frac{S\cdot P}{K_{\text{I,P}}}}\label{eq:mu_kinetics}\\[10pt] \rho(S,a) & = & \begin{cases} 0 & \text{for }a=0\\ \frac{\rho_{\max}\cdot S}{K_{\text{P}}+S} & \text{for }a=1 \end{cases}\label{eq:pi_kinetics} \end{eqnarray}

The table below lists all model parameters used in this application example:

Symbol Description Value Unit
$\mu_{\max}$ maximum specific growth rate $0.32$ $\text{h}^{-1}$
$\rho_{\max}$ maximum specific product formation rate $0.020$ $\text{h}^{-1}$
$K_{\text{S}}$ half-saturation constant substrate $2.3$ $\text{g}_{\text{S}} \cdot \text{L}^{-1}$
$K_{\text{P}}$ half-saturation constant in product formation $1.6$ $\text{g}_{\text{S}} \cdot \text{L}^{-1}$
$K_{\text{I,S}}$ inhibition constant substrate $9.1$ $\text{g}_{\text{S}} \cdot \text{L}^{-1}$
$K_{\text{I,P}}$ inhibition constant product $0.10$ $\text{g}_{\text{P}} \cdot \text{L}^{-1}$
$k_{\text{d}}$ specific biomass decay rate $0.024$ $\text{h}^{-1}$
$m$ specific substrate maintenance rate $0.034$ $\text{g}_{\text{S}} \cdot \text{g}_{\text{X}}^{-1} \cdot \text{h}^{-1}$
$Y_{\text{X,S}}$ yield coefficient biomass/substrate 0.60 $\text{g}_{\text{X}} \cdot \text{g}_{\text{S}}^{-1}$
$Y_{\text{P,S}}$ yield coefficient product/substrate 0.053 $\text{g}_{\text{P}} \cdot \text{g}_{\text{S}}^{-1}$

## Methodical Background

In general, the optimisation of a bioprocess is considered to be the suitable modification of the operating mode of bioreactors. This modification is carried out with regard to one or more quantifiable objectives. The objectives may include the minimisation of process duration or energy, as well as expensive or toxic substrates [4]. In the present application example, the productivity $Pr_{\text{P}}$ of the bioprocess serves as the sole objective.

In addition, it must be possible to control the process to be optimised. These control variables are either known or to be determined by methods of data analysis or, if necessary, simulatively (e.g. sensitivity analysis). Furthermore, the optimisation is based on the prerequisite that a parameterised mathematical model of the bioprocess to be optimised is available, which is able to represent at least all major process characteristics. In the present application example, the substrate concentration $\color{#ff7f00}{S}$ at the beginning of cultivation $t_{0}$, the activation time of product formation $t_{\text{a}}$ and the harvest time  $t_{\text{harvest}}$ are designated as control variables.

Methodically, model-based optimisations can be segmented into two groups: One is the determination of an optimal control design for dynamic processes. This "dynamic" optimisation aims to find a control sequence for the control variable(s). Second, the determination of static values of the control variable(s), which is partly comparable to the parameter identification procedure. In both cases, control variables are determined to minimise or maximise the formulated objective variable. In the present application example, static optimisation approach is used:

As can be seen in the scheme above, the considered control variables $S(t_{0})$, $t_{\text{a}}$ and $t_{\text{harvest}}$ are vectorised as $\theta$. The optimum values of the control variables $\Omega$ are finally determined by maximising the above objective function Eq. \eqref{eq:productivity}:

\begin{align} \Omega = \underset{\theta}{\text{arg max}}\,Pr_{\text{P}}(\theta) \label{eq:optimisation} \end{align}

The optimisation is implemented as a mathematical minimisation or maximisation. A number of established methods are available for this purpose. These include deterministic (e.g. Gradient or Simplex methods [5]) or heuristic (e.g. Genetic Algorithms [6], Simulated Annealing) methods as well as their combination as hybrid methods.

Also, dynamic and static constraints of the control variables, which are described by systems of equalities as well as systems of inequalities, have to be taken into account with respect to the implementation. In the present case, this applies to: $S(t_{0}) \gt 0$, $t_{\text{a}} \gt 0$ and $t_{\text{harvest}} \gt 0$ (static) and $t_{\text{a}} \lt t_{\text{harvest}}$ (dynamic).

## Optimisation Results

By comparing the subfigures A "non-optimised bioprocess" and B "optimised bioprocess" in the following figure, it can be seen that the targeted productivity $Pr_{\text{P}}$ could be increased by $\approx70\,\%$.

This increase was achieved primarily by a suitably increased substrate concentration $\color{#ff7f00}{S}$ at the initiation of the cultivation as well as by a suitable prolongation of the product formation phase (earlier addition of the activator $a$). With regard to the harvest time $t_{\text{harvest}}$, only a minor correction was conducted.

Alongside the increased productivity $Pr_{\text{P}}$, the elevated absolute product concentration $\color{#377eb8}{P}$ as well as a reduced biomass accumulation $\color{#4daf4a}{X}$ are also to be noted. Both effects are very advantageous for the DownStream-process subsequent to the cultivation. The table below lists the control variables used in this application example as well as the resulting productivity. Values of optimised control variables are formatted in bold font:

Symbol Description Vaule
non-optimised
Value
optimised
Unit
$X_0$ initial biomass concentration $0.75$ $0.75$ $\text{g}_{\text{X}} \cdot \text{L}^{-1}$
$S_0$ initial substrate concentration $25$ $\mathbf{33}$ $\text{g}_{\text{S}} \cdot \text{L}^{-1}$
$P_0$ initial product concentration $0$ $0$ $\text{g}_{\text{P}} \cdot \text{L}^{-1}$
$t_{\text{a}}$ initiation time of product formation $24$ $\mathbf{14}$ $\text{h}$
$t_{\text{harvest}}$ harvest time $48$ $\mathbf{49}$ $\text{h}$
$Pr_{\text{P}}$ productivity $0.015$ $\mathbf{0.026}\text{ }(+73\,\%)$ $\text{g}_{\text{P}} \cdot \text{L}^{-1} \cdot \text{h}^{-1}$

In this application example, the initial concentrations of biomass $X_0$ and product $P_0$ have been excluded from the optimisation for the sake of simplicity. However, with respect to a specific application, all available control variables are to be investigated regarding their sensitivity to the objective and, if necessary, to be included in the optimisation strategy.

Likewise, a modification of the selected bioprocess regime may represent an aspect of the optimisation strategy, given its technological and economic feasibility. The present application example considers a simple discontinuous (batch) mode of operation. Possible modifications include the fed-batch or repeated-fed-batch mode of operation, as well as the conversion to a continuous (chemostat, turbidostat) mode of operation. Given the biotechnological feasibility, an immobilisation of the biomass can also be the subject of these modifications.

## References

1. [1] L. Gao, Y. Ren, Y. Ma, J. Lin, J. Lin: Modeling and simulation of production of metallothionein and red fluorescent fusion protein by recombinant Escherichia coli using graphical programming. In: Modeling, Programming and Simulations Using LabVIEW Software., IntechOpen, London, United Kingdom, 2011. doi: 10.5772/14091
2. [2] D. Voet, J. G. Voet: Biochemistry. John Wiley & Sons, New York, 1990. ISBN: 0-471-61769-5.
3. [3] A. Kremling: Grundlagen der mathematischen Modellierung. In: Kompendium Systembiologie, Vieweg+Teubner Verlag, 2012. doi: 10.1007/978-3-8348-8607-1_3.
4. [4] A. I. Bojarinow, W. W. Kafarow: Optimierungsmethoden in der chemischen Technologie. Verlag Chemie Weinheim, 1972. ISBN: 978-3527254682.
5. [5] J. A. Nelder, R. Mead: A simplex method for function minimization. In: The Computer Journal, 7(4):308–313, 1965. doi: 10.1093/comjnl/7.4.308
6. [6] D. E. Goldberg: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Boston, 1989. ISBN: 978-0-201-15767-3