Abstract
Optimal feed rate strategy is studied for fedbatch culture of recombinant cells with plasmid instability and with different death rates for the plasmidfree cells (PFC) and plasmidbearing cells (PBC). Most of the fedbatch fermentation is known to have firstorder singularity and therefore a single singular arc. However, this study shows that a singular arc with secondorder singularity and therefore two distinct singular arcs are possible for a recombinant cell process if PFC and PBC are subjected to death, and their specific growth rates are proportional to each other. Two types of singular arcs are elucidated and analyzed. The optimal policies over the singular arcs are theoretically explored as these findings reveal qualitative information on the singular arc, which is critically important in providing the optimal initial conditions in numerical computation of optimal feed rate profile.
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Abbreviations
 F :

Feed rate (cm^{3}/s)
 F _{max} :

Maximum feed rate (cm^{3}/s)
 F _{min} :

Minimum feed rate (cm^{3}/s)
 F _{singular} :

Singular feed rate (cm^{3}/s)
 H :

Hamiltonian function
 PBC:

Plasmidbearing cells (g)
 PFC:

Plasmidfree cells (g)
 k _{d} :

Constant death rate (s)
 S :

Substrate concentration (g/cm^{3})
 S _{F} :

Feed substrate concentration (g/cm^{3})
 t :

Time (s)
 t _{f} :

Final time (s)
 V :

Bioreactor volume (cm^{3})
 V _{max} :

Maximum bioreactor volume (cm^{3})
 x _{1} :

Total plasmidcontaining cells (g)
 x _{2} :

Total plasmidfree cells (g)
 x _{3} :

Total substrate (g)
 x _{4} :

Bioreactor volume (cm^{3})
 x _{5} :

Time (t)
 X :

Cell mass concentration (g/cm^{3})
 Y _{ x/s } :

Cell mass yield (g/g)
 ϕ:

Switching function
 μ:

Specific growth rate (s)
 α:

Fraction of plasmidcontaining cells that revert
 β:

The ratio, μ^{−}(S) = βμ^{+}(S)
 η:

λ_{4}(t _{ f }) = η is a unknown adjoint variable corresponding to x _{4}(t _{ f }) = V _{max}
 +:

Plasmidbearing cells, PBC
 −:

Plasmidfree cells, PFC
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Acknowledgment
One (HSS) of the authors was supported in part by a fellowship from Korea Science and Engineering Foundation (KOSEF) and a grant from the University of California, Irvine.
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Appendices
Appendix 1
Rewriting the first ODE of Eq. (6),
where m = −(1−α)μ^{+} and \(n=(\lambda_2 \alpha {\lambda_3} \mathord{\left/ {\vphantom {{\lambda_3} {Y_{x/s}^}}} \right.}{Y_{x/s}^})\mu^+.\) On the singular arc m is always negative since α < 1 and μ^{+} > 0. If n = 0, λ_{1}(t) approaches zero asymptotically. However, n is always negative over the same region because λ_{2} > 0 and λ_{3} < 0. This means λ_{1}(t) is less than the negative value of λ_{1}(t) when n = 0 for all times over the singular arc. Therefore, we can say that λ_{1}(t) is always negative over a singular arc. Correspondingly, B is always positive over the singular arc.
Appendix 2
Equation (6) gives rise to the final adjoint variable vector, [−1, 0, 0, η, 0] and the Hamiltonian at the final time is −μ^{+} x _{1}(t _{f}). Therefore, the Hamiltonian is negative constant over all time because it must be constant. Over the singular arc where ϕ = 0 and since λ_{5}(t) = 0, we conclude that
Since λ_{2} is positive over the singular arc, λ_{2} k _{d} x _{2} > 0 and the sufficient condition for Eq. (A2) is
Substituting Eq. (9) into Eq. (A3) and recalling that B is positive over the singular arc, we obtain
Before going through a further analysis of the singular arc, let us restrict the form of specific growth rates μ^{+} and μ^{−}. As stated in the Introduction, the process uses the difference in death rates of the PBC and PFC. At very low substrate concentrations the PFC suffer death whereas the PBC do not. At low substrate concentrations, no inhibition takes place, and both specific growth rates μ^{+}and μ^{−} increase with the substrate concentration so that both μ _{ s } ^{+} and μ _{ s } ^{−} are positive over the affected ranges (very low substrate concentration). Then, \(sign({\mu^+} \mathord{\left/ {\vphantom {{\mu^+} {\mu^)_s}}} \right. } {\mu^)_s}\) is negative from Eq. (A4) because sign (μ _{ s } ^{+} ) is positive. This means that if substrate concentration increases with time \((\dot{S} > 0),\mu^+\) increases and \({\mu^+} \mathord{\left/ {\vphantom {{\mu^+} {\mu^}}} \right. } {\mu^}\) decreases correspondingly, but if \(\dot {S} < \,0, \mu^+\) decreases and \({\mu^+} \mathord{\left/ {\vphantom {{\mu ^+} {\mu^}}} \right. } {\mu^}\) increases. Although not simple to determine the definite sign of \(\dot{S},\) we can say that \(\dot {S} < \,0\) on a singular arc so that the substrate concentration moves in the direction of increasing the ratio of specific rates \(({\mu^+} \mathord{\left/ {\vphantom {{\mu^+} {\mu^)}}} \right. } {\mu^)}\) gradually at the expense of μ^{+}. It is in line with the fact that increasing \({\mu^+} \mathord{\left/ {\vphantom {{\mu^+} {\mu^}}} \right. } {\mu^}\) is favorable to PBC formation, even though low \(\mu^+\) and decreasing substrate concentration lead to taking more time for the process to reach the final reactor volume. This can be represented by the following
We can conclude that the substrate concentration decreases with time in the direction of gradually increasing the ratio of specific rates \(({\mu^+} \mathord{\left/ {\vphantom {{\mu^+} {\mu^)}}} \right. } {\mu^)}\) at the expense of μ^{+} and μ^{−} on a singular arc.
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Shin, H.S., Lim, H.C. Optimal fedbatch operation of recombinant cells subject to plasmid instability and death. Bioprocess Biosyst Eng 31, 655–665 (2008). https://doi.org/10.1007/s0044900802166
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Keywords
 Plasmidbearing cells
 Plasmid stability
 Different cell death rate
 Fedbatch
 Singular arc
 Optimal feed rate profile